STAR2 4 0CT161985

The Telecommunications and Data

Acquisition Progress Report 42-82

• ° .-

April-June 1985

E.C. Posner

Editor

(NASA-CR-176203) THE TELECOM_UNTC&TIONS AND N86-I_3£_

D_ ACQUISITION REPORT Proqress Peport, TH_n

apr. - Jun. 1985 (Jet Propulsion Lab.) N86-10,432

218 p HC AI_/MF AOl CSCL 17B Unclas

63/32 290]q

August 15, i985

National Aeronautics and

Space Administration

Jet Propulsion Laboratory

California Institute of Technology

Pasadena, California
The Telecommunications and Data

Acquisition Progress Report 42-82

April-June 1985

E.C. Posner

Editor

August 15, 1985

National Aeronautics and

Space Administration

Jet Propulsion Laboratory

California Institute of Technology

Pasadena, California
The research described in this publication was carried out by the Jet Propulsion

Laboratory, California Institute of Technology, under a contract with the National

Aeronautics and Space Administration.

Reference herein to any specific commercial product, process, or service by trade

name, trademark, manufacturer, or otherwise, does not constitute or imply its

endorsement by the United States Government or the Jet Propulsion Laboratory,

California Institute of Technology.
Preface

This quarterly publication provides archival reports on developments in programs

managed by JPL's Office of Telecommunications and Data Acquisition (TDA). In space

communications, radio navigation, radio science, and ground-based radio astronomy, it

reports on activities of the Deep Space Network (DSN) and its associated Ground Communications Facility (GCF) in planning, in supporting research and technology, in implementation, and in operations. Also included is TDA-funded activity at JPL on data and

information systems and reimbursable DSN work performed for other space agencies

through NASA. The preceding work is all performed for NASA's Office of Space Tracking and Data Systems (OSTDS).

In geodynamics, the publication reports on the application of radio interferometry

at microwave frequencies for geodynamic measurements. In the search for extraterrestrial

intelligence (SETI), it reports on implementation and operations for searching the microwave spectrum. The latter two programs are performed for NASA's Office of Space

Science and Applications (OSSA).

Finally, tasks funded under the JPL Director's Discretionary Fund and the Caltech

President's Fund which involve the TDA Office are included.

This and each succeeding issue of the TDA Progress Report will present material in

some, but not necessarily all, of the following categories:

OSTDS Tasks:

DSN Advanced Systems

Tracking and Ground-Based Navigation

Communications, Spacecraft-Ground

Station Control and System Technology

Network Data Processing and Productivity

DSN Systems Implementation

Capabilities for New Projects

Networks Consolidation Program

New Initiatives

Network Sustaining

DSN Operations

Network Operations and Operations Support

Mission Interface and Support

TDA Program Management and Analysis

GCF Implementation and Operations

Data and Information Systems

OSSA Tasks:

Search for Extraterrestrial Intelligence

Geodynamics

Geodetic Instrument Development

Geodynamic Science

Discretionary Funded Tasks iii
Contents

OSTDS TASKS

DSN Advanced Systems

TRACKING AND GROUND-BASED NAVIGATION

A VLBI Survey at 2.29 GHz ............................................................

D. D. Morabito, R. A. Preston, J. G. Williams,

J. Faulkner, D. L. Jauncey, and G. D. Nicolson

NASA Code 310-10-60-26-00

Comparison of the Calibration of Ionospheric

Delay in VLBI Data by the Methods of Dual

Frequency and Faraday Rotation ..................................................... 1 1

J. A. Scheid

NASA Code 310-10-60-60-00

COMMUNICATIONS, SPACECRAFT-GROUND

A New 70-Metar Antenna Quadripod With

Reduced RF Blockage ............................................................... 24

J. J. Cucchissi

NASA Code 310-20-65-64-00

Seismic Analysis of the Large 70-Meter Antenna,

Part I: Earthquake Response Spectra Versus

Full Transient Analysis .............................................................. 31

K. Kiedron and C. T. Chian

NASA Code 310-20-65-64-00

An Experimental TE12-TE11 Circular

Waveguide Mode Converter ............................... ........................... 43

D. Hoppe

NASA Code 310-20-64-22o00

Propagation and Radiation Characteristics

of a Multimode Corrugated Waveguide Feedhom ..................................... 57

D. Hoppe

NASA Code 310-20-64-22-00

Load-Deflection Tests and Computer Analyses

of a High-Precision Adhesive-Bonded

Antenna Reflector Panel ............................................................. 68

C. T. Chian and R. Levy

NASA Code 310-20-65-05-10

STATION CONTROL AND SYSTEM TECHNOLOGY

A Sequential Decoding Performance Analysis

for International Comet Explorer ..................................................... 82

D. Divsalar

NASA Code 310-30-71-83-04 •

The VLSI Design of Error-Trellis Syndrome

Decoding for Convolutional Codes ................................................... 92

I. S. Reed, J. M. Jensen, T. K. Truong,

and I. S. Hsu

NASA Code 310-30-70-84-08

['I{ECE,EdNG PAGE BLANK NOT FILMED
High-Rate Convolutional Code Construction

With the Minimum Required SNR Criterion ............................................ 108

P.J. Lee

NASA Code 310-30-71-83-02

DSN Systems Implementation

NETWORK SUSTAINING

D$.N. I:r.._..,=nry =n,,I Timing (:yi=t=m

Mark IV-85 .......................................................................... 1 13

B. W. Falin

NASA Code 314-40-41-81-13

DSN Operations

MISSION INTERFACE AND SUPPORT

JPL Emergency Support of TDRSS and

Compatible Satellites ......................................... : ...................... 120

N. A. Fanelli

NASA Code 314-40-22-34-01

TDA PROGRAM MANAGEMENT AND ANALYSIS

Planning for VLA/DSN Arrayed Support

,to the Voyager at Neptune ........................................................... 125

J. W. Layland and D. W. Brown

NASA Code 314-40-31-30-03

A VLA Experiment - Planning for

Voyager It Neptune ................................................................. 136

J. W. Layland, P. J. Napier, and A. R. Thompson

NASA Code 314-40-31-30-03

Periodic Binary Sequences With Very Good

Autocorreietion Properties ........................................................... 143

S. Tyler and J. Loftsson

NASA Code 055-47-11-10-01

A Binary Sequence of Period 60 With Better

Autocorreietion Properties Than the Barker

Sequence of Period 13 ................... ........................................... 159

J. Watkins, J. Loftsson, and S. Tyler

NASA Code 055-47-11-10-O1

GCF Implementation and Operations

The GCF Mark IV Implementation

end Beyond ......................................................................... 163

R. A. Crowe

NASA Code 314-20-40-00-08

vi
OSSA TASKS

Search for Extraterrestrial Intelligence

SETI Radio Spectrum Surveillance System

B. Crow, A. Lokshin, M. Marina,

and L. Ching

NASA Code 199-50-62-10-04

........................................... 173

Mobile VLBI and GPS Measurement of

Vortical Crustal Motion .............................................................. 185

P. M. Kroger, J. M. Davidson, and

E. C. Gardner

NASA Code 692-40-40-00.07

DISCRETIONARY FUNDED TASKS

Improved Mapping of Radio Sources From

VLBI Data by Leut-Squares Fit ...................................................... 199

E. R. Rodemich

NASA Code 404-00-73-02-55

Referees ............................................................................ 211

vii
TDA Progress Report42-82 April-June 1985

N86-10383

A VLBI Survey at 2.29 GHz

D. D. Morabito, R. A. Preston, J. G. Williams, and J. Faulkner

TrackingSystemsandApplicationsSection

D. L. Jauncey

CSIRO Divisionof RadioPhysics,Sydney,Australia

G. D. Nicolson

CSlR, Johannesburg,SouthAfrica

The Deep Space Network is establishing a high-accuracy VLBI celestial reference

frame. This article presents the results of a search for suitable radio sources to be used in

constructing this frame. The VLBI observations using DSN baselines at 2.29 GHz with

fringe spacings of about 3 milliarcseconds have been performed on 1398 radio sources

spread over the entire sky. Of those, 917 sources were detected including 93% of the

identified BL Lacertae ob/ects, 86% of Quasars and 36% of galaxies. The resulting catalog

of compact radio sources is also useful for various astrophysical studies and in the formation of VLBI celestial reference frames.

I. Introduction

This article presents the results of a systematic VLBI fullsky survey undertaken to establish a comprehensive catalog of

ultracompact celestial radio sources. The survey was conducted

by performing 2.29 GHz VLBI observations on known radio

sources to search for compact structure. Of 1398 radio sources

observed, 917 sources have been detected. Arcsecond positions for 787 of the detected sources have been previously

determined from the VLBI survey data (Refs. 1,2, 3) and are

being used to identify optical counterparts (Refs. 4, 5, 6).

The results of this survey are presently being utilized to

form a VLBI reference frame of 100-200 sources by determining precise relative positions (0701-0701) (see Ref. 7). Such

celestial reference frames (see also Ref. 8) will be at least an

order of magnitude more precise than previous stellar frames

and are nearly inertial since the extragalactic sources are

without measurable proper motions. They enable significant advances in various geodetic and astrometric studies

(e.g., crustal plate dynamics, earth rotational irregularities,

planetary dynamics, interplanetary spacecraft navigation).

A similar, but deeper, VLBI survey of the ecliptic zone has

been previously published (Ref. 9).

The VLBI survey is also useful for studying the characteristics of compact radio sources. The detected survey sources

coincide with the cores of quasars and galaxies. Understanding

the nature of these energetic cores is crucial in unraveling the

origin and evolution of the objects in which they reside. The

catalog will not only serve as a reference list for observers,

but it can be used in statistical studies of radio source properties and cosmological theories. Toward these ends, the catalog

has been supplemented with optical identifications, optical

magnitudes, redshifts and radio spectral indices derived from

the literature.

1
II. Sample Selection and Completeness

Candidate sources were selected primarily from the Parkas

survey (Ref. 10) and the NRAO-Bonn survey (Ref. 11)which

together span the entire sky (LbHL > 10 degrees). These surveys both provide total flux density measurements at 2.7 and

5.0 GHz for most sources. The sample observed with VLBI

covers the full sky and was chosen largely on the basis of

criteria placed on total flux density S($2.7) and spectral index

a(as_o ; S = S o f_), neglecting temporal variability. For exampla for those sources fnr whio_h tha Parkas and NRAO-Bonn

surveys give total flux densities at both 2.7 and 5.0 GHz,

100% of the sources were observed for which S _ 1.0 Jy and

a _ 0.0 (114 sources observed, 105 detected), and 89% for

which S >i 0.5 Jy and t_/> -0.5 (717 of 805 sources observed,

592 detected).

Also observed were 681 weaker or steeper spectrum sources

from the Parkes and NRAO.Bonn surveys, as well as from the

general literature. Our sample was intended to be purely

extragalactic, and identified galactic sources were eliminated

from the sample. However, some of the optically unidentified

sources that met our sample criteria could be galactic. Such

sources are highly unlikely to have been detected with VLBI

at our angular resolution and sensitivity.

Completeness of the observed sample is difficult to estimate, not only due to temporal variability but also because

the two finding surveys had different levels of completeness

for different sky regions, lacked two frequency information

for all sources, and had different primary survey frequencies.

Neglecting temporal variability, both finding surveys are nearly

complete for S ) 1.0 Jy and a ) -0.5, resulting in a combined

completeness of more than 97% for the sky area covered.

The spectral index criterion is necessary because the NRAO.

Bonn survey frequency was 5.0 GHz, not 2.7 GHz. Based on

these sample criteria, the VLBI survey is estimated to be 93%

complete, again neglecting temporal variability, with a total of

312 sources observed. Because the flux density limits of the

finding surveys varied depending on sky region, estimates of

completeness for sources with lower total flux densities do

not apply to the entire sky (see Table 1).

III. The Observations

The observations were performed at 2.29 GHz with pairs

of antennas on California-Spain, California-Australia, and

Australia-South Africa baselines (see Table 2) during 68 different observing sessions between 1974 and 1983 (see Table 3).

Right circular polarization was received and data were recorded

on the NRAO Mark II system (Ref. 12).

The fringe spacing sampled ranged from 2.5 to 4.1 milliarcseconds. For the mean fringe spacing of 3.3 milliarcseconds,

the normalized fringe visibility of a Gaussian source varies

from 0.9 to 0.1 as the half-intensity diameter increases from

0.5 to 2.2 milliarcseconds.

The 5o detection limit in correlated flux density was generally "0.1 Jy. The corresponding random uncertainty in

detected source strength is _0.02 Jy, but systematic errors

at about the 10% level dominate the random contribution for

most sources. To ensure that few compact radio components

would be missed due to a priori source position errors, the sky

was completely searched within 0.5 arcminutes of all nominal

source positions by cross-correlating over an appropriate range

of delay and delay rate.

Total flux densities at 2.29 GHz were also measured for

most sources at the time of VLBI measurement by means of

on-off measurements with a noise-adding radiometer. The

random uncertainties in total flux density measurements

typically range from 0.03 to 0.3 Jy, with systematic errors in

antenna sensitivity being ~3%.

IV. Results

Of 1398 sources observed, 917 (or 66%) were detected with

VLBI; 83% of the observed sources with S _ 0.5 Jy and

'a _-0.5 were detected. Figure 1 is an equal area sky distribution plot of the detected objects. Sparsity near the galactic

plane is evident. Figure 2 is a correlated flux density histogram

of the detected objects. There are 49 sources with correlated

flux densities greater than 1 Jansky, and 227 sources with

correlated flux densities greater than 0.5 Jansky.

Detection statistics as a function of optical identification

type appear in Table 4. Detection statistics as a function of

general optical class appear in Table 5: 93% of identified BL

Laeertae objects were detected, 86% of QSOs were detected

and 36% of galaxies were detected.

Figure 3 displays a sample page of the survey results.

Figure 4 displays a sample page of the reference table to the

supplementary data given in Fig. 3. Figures 3 and 4 will

appear in their entirety in a separate report. Descriptions of

the entries to Fig. 3 appear below.

Column Description

1

2/3

Source nar_le

1950.0 positions. Asterisked positions are determined from the VLBI survey data and have typical

uncertainties of 1 arcsecond (see Refs. 1-3). Other
Column

4

Description

positions are from the literature, and in most

cases, errors are < 30 arcseconds.

Spectral indices between 2700 to 5000 MHz

followed by corresponding reference number

(see Fig. 4). A few existing compilations of redshifts, optical identifications and optical magni.

tudes were useful aids in preparing our catalog

(Refs. 63, 84, 86, 111 and 232 in Fig. 4). How.

ever, in almost all cases we have drawn values for

these quantities from original references to enhance

accuracy. A star following the reference number

indicates a questionable or conflicting value, and is

explained in the notes to Fig. 3. For many Southern

Hemisphere source's, the optical characteristics

were obtained from an optical identification

program which utilized the radio source positions

determined by our survey (Refs. 4, 5, 6).

Redshifts followed by corresponding reference

number (see Fig. 4).

Column Description

6

8

9

10

II

12

Optical identifications followed by correponding

reference number (see Fig. 4). Optical identification codes are defined in Table 4.

Optical magnitudes followed by the corresponding

reference number (see Fig. 4). These values may be

visual, blue or red.

Experiment codes as defined in Table 3.

Measured 2.29 GHz total flux density (Jansky).

Measured 2.29 GHz correlated flux density

(Jansky). Values for 17 ecliptic sources marked

U_y CI_t._L_LO_D CU_ i,J. qrJl,lLJL L%_L..7.

Visibility is def'med as the correlated flux density

divided by the total flux density.

East-west (u) and north-south (v) spatial frequencies of the observations in units of lO6

wavelengths.

Computer readable versions of the catalog are available

upon request.
Acknowledgments

We are indebted to the instrumentation and observing assistance supplied by

L. J. Skjerve, D. J. Spitzmesser, and the operations personnel of the Deep Space Network,

the Australian National Radio Observatory at Parkes, Australia, and the Hartebeesthoek

Radio Astronomy Observatory, South Africa. We are also grateful to A.W. Harris,

E. Padua, M. A. Slade, G. B. Trager and J. Kidder for computational aid.

References

1. Morabito, D. D., Preston, R. A., Slade, M. A., and Jauncey, D. L. 1982, "Arcsecond

Positions for Milliarcsecond VLBI Nuclei of Extragalactic Radio Sources. I. 546

Sources," A.J., 87, 517-527.

2. Morabito, D. D., Preston, R.A., Slade, M.A., Jauncey, D. L., and Nicolson, G. D.

1983, "Arcsecond Positions for Milliarcsecond VLBI Nuclei of Extragalactic Radio

Sources. II. 207 Sources," A.J., 88, 1138-1145.

3. Morabito, D.D., Wehrle, A.E., Preston, R.A., Linfield, R.P., Slade, M. A.,

Faulkner, J., and Jauncey, D.L. "Arcsecond Positions for Milliarcsecond VLBI

Nuclei of Extragalactic Radio Sources: III. 74 Sources," April 1985, A.J., 90,

590-594.

4. Jauncey, D.L., Batty, M.J., Wright, A. E., Peterson, B.A., and Savage, A. 1984,

"Redshifts of Southern Radio Sources. VI," Ap. J., 286, 498-502.

5. Jauncey, D.L., Savage, A., Morabito, D.D., and Preston, R.A. 1985, "Optical

Identifications of Compact Radio Sources: Candidates for the Construction of an

Extragalactic Radio/Optical Reference Frame," in preparation.

6. Savage, A., Jauncey, D. L., Batty, M.J., Gulkis, S., Morabito, D. D., and Preston,

R. A. 1983, "Optical Identifications of Radio Sources with Accurate Positions Using

the UKST IIIa-J Plates," Proc. 1AU Colloquium No. 78, Astronomy with Schmidt

Type Telescopes, Asiago, Italy 481-487.

7. Fanselow, J. L., Sovers, O.J., Thomas, J. B., Purcell, G. H., Cohen, E. J., Rogstad,

D.H., Skjerve, L.J., and Spitzmesser, D.J. 1984, "Radio Interferometric Determination of Source Positions Utilizing Deep Space Network Antennas - 1971 to

1980," A. J. 89, 987-998.

8. Ma, C., Clark, T.A., and Shaffer, D.B. 1981, "Mark III VLBI: Astrometry and

Epoch J2OOO.O," Bull. AAS, 13, 899-899.

9. Wehrle, A.E., Morabito, D.D., and Preston, R.A. 1984, "Very Long Baseline

Interferometry Observations of 257 Extragalactic Radio Sources in the Ecliptic

Region," A. J. 89, 336-341.

10. Bolton, J.G. 1979, Magnetic Tape Version of Unpublished Master Parkes Catalog,

maintained by CSIRO, Division of Radio Physics, Sydney, Australia.

11. Kuhr, H., Nauber, U., Pauliny-Toth, I., and Witzel, A. 1979, "A Catalogue of Radio

Sources," Max-Planck-Institut fur Radioastronomie, Preprint No. 55, 1-337.

12. Clark, B. G. 1973, "The NRAO Tape Recorder Interferometer System," Proc. IEEE,

61, 1242-1248.
Table 1. Completeness estimates for various sample criteria

Spectral

Index Number Number

DecLination Flux Density Limit of of Completeness

Range, Limit (_ 5.0 Sources Sources of Sample,

deg ($2.7) , Jy 2.7 ) Observed* Detected* %

-90 to +90 > 1.0 • -0.5 312 258 93

-90 to +27 • 0.65 • -0.5 396 336 85

+70 to +90 • 0.5 • -0.5 30 29 97

*Includes small adjustment to account for estimated number of sources with total flux density given

at only one frequency which would have a > -0.5.

Table 2. Participating observatories

Location Designation Diameter,

m

Baseline Length

Kilometers

X 103

Wavelengths

X 106

Hattebeesthoek, S. Africa HT 26

Parkes, Australia PK 64

Tidbinbilla, Australia 43 64 --y --

42 26 --

Goldstone, California 14 64

13 26

Madrid, Spain 63 64

61 26 :_--

-10.6

m8.4

75

81

64
Table 3. Experiment summary

Experiment

Code

Date

YR MN DY

Observatories

1 74 07 31 14 42

2 74 08 13 14 42

3 75 06 17 14 42

4 75 06 18 14 42

S 75 08 23/24 14 42

6 75 08 23 14 62

7 75 09 15 14 42

S 75 09 21 13 63

9 75 10 26 i3 43

10 76 05 27 14 63

11 76 11 09/10 13 63

12 76 11 11 13 63

13 76 11 14 13 43

14 77 01 28 13 43

15 77 02 12 13 43

16 77 02 20 13 43

17 77 02 21 13 43

18 77 02 23 13 43

19 77 02 25 13 43

20 - 77 04 21 13 43

21 77 04 22 13 43

22 77 06 15 13 43

23 77 09 11 14 61

24 77 09 28/29 13 63

25 77 10 11 13 63

26 77 10 27/28 13 43

27 77 11 01 13 43

28 77 11 21 13 43

29 77 12 02 13 43

30 77 12 11 13 43

31 77 12 12 13 43

32 77 12 13/14 13 43

33 78 01 09/10 13 43

34 78 02 20/21 13 43

Experiment

Code

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

Date

YR MN

78 03

78 03

78 03

78 04

78 04

78 05

78 06

78 10

80 2

80 2

80 3

80 3

80 3

80 3

S0 3

80 3

80 3

80 4

80 4

80 6

81 1

81 1

81 1

81 3

81 4

81 5

81 10

81 10

81 11

82 2

82 2

82 2

82 4

83 6

DY

05

14/15

21

04

16

30/31

29/30

18

1

27/28

2

3

12/13

14/15

19

26

27

24-27

26

19

21

25

31

1

22

8

23

26

1

14

17

19/20

20

21

Observatories

13 63

13 63

13 63

13 63

13 43

13 43

13 43

13 43

14 43

14 43

13 43

13 63

13 63

13 63

13 43

13 43

13 63

PK HT

43 HT

14 43

13 43

13 43

13 43

14 43

14 43

14 43

13 43

13 43

13 43

43 HT

43 HT

43 HT

43HT

13 43
Table 4. Optical identification codes with VLBI detection statistics

Number Number

Percent

Optical Description of of Detected

ID Detections Nondetections

B Blue Stellar Object 34 6 85

CG Compact Galaxy 1 2 33

D Diffuse Galaxy 6 9 40

DB DB Galaxy 3 8 27

IM D4 Galaxy 0 2 0

E Elliptical Galaxy 10 28 26

E0 E0 Galaxy 3 4 43

E1 E1 Galaxy 1 2 33

E2 E2 Galaxy 2 2 50

E3 E3 Galaxy 1 1 50

E4 E4 Galaxy 3 2 60

E5 E5 Galaxy 0 1 0

EF Empty Field 84 65 56

G Galaxy 48 97 33

L BL Lacertae Object 56 4 93

N Neutral Stellar Object 14 3 82

NG N-Galaxy 11 5 69

N2 Type 2 N-Galaxy 0 2 0

PG Probable Galaxy 10 9 53

PQ Probable QSO 41 25 62

Q Quasi-Stellar Object 503 .67 88

R Red Stellar Object 7 1 88

S Spiral Galaxy 0 2 0

SB Sb Galaxy 0 1 0

SC Sc Galaxy- 0 1 0

SG Seyfert Galaxy 2 1 67

SO SO Galaxy I 1 50

U Unidentified Object 2 3 40

(Blank) No Information 74 127 37

Tot_s 917 481 66

Table 5. Optical class VLBI detection statistics

Number Number Percent

Class of of Detected

Detectiom Nondetections

Galaxies and Probable Galaxies 102 180 36

QSOs and Probable QSOs 544 92 86

BL Lacertae 56 4 93

Stellar Objects 55 10 85

Empty Fields 84 65 56

No Information and Unidentified 76 130 37

Totals 917 481 66
90° DECLINATION

_-"_" "-,a. i" • ' ,,/ _ •

/ • . _,. _#eo / • •

/"*" ".--"---'-/-- ' **-3-__ .,,""_.'¢. ,. • .;. "4. / _ -:',/ " ..-. ,.. "-....;

,"_".;.. " /-h " "". ,'th ;''" ".."

I ° ," o'. "" 20 _ .'6 .°. "., -l ..I

_.,.--'-T- - - T . ,--. ,

_, ".. . "\. _ . ...',.

'_: ,-, • ..\.. • . •_ -_..:. .

\\.. .... , • -_,. : ..-. • .

""." " • " ._---"",_ 'k • , Z*%

"_-'._,- " "t" \ "-:.. •

__ .".'.'.'.'.'.'.'." _ •...... 60 °

._. _....-:.-"---.=. ._..'j--__._

, g.. " ":_: :.'. _,. . :...j,u_30 °

: .,. .'\.. \.. : .. 2\,\

h':

" "" ; h'"-"5'

., _ . ..'-'-.8. _ "Z4 " . ".%b." I_h

•""'"". "l.._" " " ":)" _ "*

;::: : """ - / _7"

•" //

•

:.,

o• I"/_" M". .'.,•

"_S .'o "/" ... -30o

--,z.__ .. /_ .°

_j<oo

-90 °

Fig. 1. Sky distribution of sources detected with VLBI

RIGHT

ASCENSION

o

z

200

100-

_==--

50-

0

i ' ' _ ' I ' ' ' ' I

i

0.5 1.0 1.5 2.0

CORRELATED FLUX DENSITY, JY

Fig. 2. Histogram of correlated flux densities for detected sources
ORIGINAL F;,-_ i,.,

OF POOR QUALITY

III II II | III Illl |1 Ill I II ||ill II

o m_- m_m- _H-O_ _mo_ _o_ _o--_ _m _m_ -o_-_

IIIII IIIII I III IIII IIIII IIII IIII Illll IIIIt

_0 _0 _ _ _ 0 0

_ O0 _ _ _ OOO _ _ 0 O 0 0 0

- : . 0''"' 0 o00 oa ?• _? ? ? -?

--= _ ** _ * * ÷÷÷ * * * ÷ ;

0 oo oo0 0 0 o00 0 oo 0 Oo 0 oo 0 0

v v v v v

,,__ oo oo o- _ ooooo ooooo _ _ _ o_- o

_= 99 O0 Oo oo 0 ooo oo o _= .. ,, ,, ,, , ?,,,? ????? ? ? oo?,, ooIIII I

u_

00000 o0o00 00000 00000 00000 o0000 00000 00o0o Ooood

_V V V _ VV V VV V VV V_

mm

_ _ _0 _ 0 _®_ _ _ ® ®_ _ _ _

=u

_ _0_ _00 _00 _ ®_ 0 _ ® _ 0 _ _ _ O_ _

0

0

. o- - O- - O_ 0 0 0

_ 0_0_

m o00o_ 00o0o o_ 0,000 00000 0-000 00000 00000 -0-0 !IIII Ill I÷Iii t1÷1 II I I÷ I IIII Ill

_ .1+11 I I ""+_II_ ÷ ÷÷_÷I _"gg ÷II ,+,7 l++ll It+!+

Z_ ...........................................

00000 00000 00000 00000 00000 00000 00000 00000 00000

_ _d+_O +0+0 00_ _ _ 0 O_ 0_+0_ _0_

-- II_ II 0/_ I+l .II ÷0 BII /0

m00000 000 NO00_ 0_0_+ ÷+_ _ _+_ _0_ _

00000 0_000 0000 0000_ _000 _00_0 _0_00 00000 0000®

O0 O0 _ 000 _ 000 0 O_ _000 O0 0 0_00 O0 O0 0000

0

_.=

,,.,=

2

0

--I

0

0

IL
POOR QUALm

1. Burbidge, E.M. (1967). Astrophys. ,j. Letters, 149, L51.

2. Bolton, J.Q., Kinman, l.D., and Wall, d.V. (1968). Astrophys. d.

Letters 154, LJ05.

3. Searle, L., and Bolton,,j. g. (1968). Astrophys. d. Letters 154, L101.

4. Veron, M.P. (1971). Astron. A_trophgs. 110 1.

5. Wall,d.V., Shimmins, A.d., and Merheli,ln,,j.K. (1971). Aust. ,j.

Phgs. suppl. 19, 1.

6. Fanaro_,B. L. , and Blake, g.M. (1972). Mon. Not. R. astr. Soc. 157,41.

7. Baldwin, d.A., Bu_bidge, E. hl., Hazard, C., Murdoch,H.S., Robinson, L.B.,

and Wampler, E.d. (1973). Astroph.qs. J. 185,739.

8. Browne, I.W.A., C_owther, d.H., and AdgJe, R.L. (1973).

Nature 244, 146.

9. Peterson, B.A., Bolton,,j.B., and Shimmins, A.d. (1973).

Astrophgs. Letters 15, LI09.

10. Sandage, A (1973). A_..trophgs. J. 180,6_7.

11. Wills, B.J., Wills, D., and Douglas, d.N (1973). Ast_on. d. 78,521.

12. Hoskins, D , Murdoch,H.S., _dgie, R.L., C_owther,.J.H., and Gent, H.

(1974). Mon. Not. I_. astr. Soc. 166,23,5.

13. ,johnson, K.H. (1974). Ast_c_n. ,j. 79, 1006.

14. Strittmatter,P.A., Carswell,R.F., gilbert, g., and Burbidge, E.M.

(1974). Ast_ophgs. J. 190, 509.

15. Wills, D, and Wills, B.,j. (1974). Astrophgs. d. 190,271.

16. Bolton,,j.g., 5himmin_..,A ,j., and Nal],,j.V. (1975).

/_J._t. ,j. Phgs. Suppl. 34, J.

17. Colla, g., Fanti,C. , Fanti, R., g_.oia, I., Lari,C., Lequeux,,j.,

Lucas, R. , and Ulrich,M. H. (I_/'/5). Astron. Astrophgs. Suppl. 20, 1.

18. Condon, d. d. , Balonek,T. d. , and ,jaunce.q,D. L. (-1975). Astron. d. 80,887.

19. Edua_ds, T. , Kronbe_g,P. P. , aT=d Henard, g. (1975). Astron. ,j. 80, 1005.

20. Fanti,C., Fanti,R. , Fica1"ra, A. , F ormiggini,L. , giovannini,g. , Lari,C.

and Pad:ielli, L. (1975). Astron. Astrophgs. Suppl. 19, 143.

21. HcEuan, N. J. , 13_oune, I. W A. , C_outhe_,F. H. (1975).

Mer,_. R. astr. Soc. [¢0, 1.

22. Condon,,j. J. , lJalonek, T. d. , and ,Jaunceg, D.L. (1976). Astron. ,j. 81,913.

23. Kuh_,H. (1977). Ast]-on. A._,trophus. Suppl. 29, 139.

24. Pete_son, B. A. , Bolt¢,Ib, d. (_., and Savage, A. (1976).

Astrophgs. J. l-etter_- • 17,1oj37.

25. Peterson, B. A. , ,jaunreg,D L. , Wri_ht,A. E. , and Condon, J. ,j. "(1976).

A=.;troph_s. J. 207, L,',.

26. Savage, A. , _oune, I. W. A. , and Bolton,,J. g. (1976).

tdo;_. Not. R. ast_'. ,¢;oc. 177,77P.

27. Smith,H. E. , Spin_ad,H , and Smith, E.D. (1976). Publ. Astron.

Soc. Pacific 88,621.

28. Veron, M.P., Veron,_., Adg_e,R.L., and (_ent,H. (1976). Ast_on.

Ast_ophgs. 47, 401.

_9. Wills, B.J. (1976). A_tron. ,j. 81, 1031.

30. Wills, D., and Wills, B.d. 41976). Astrophgs. d. Suppl. 31, 143.

31. Boksenberg, A. , Briggs, S.A. , Carsuell, R.F. , Schmidt, M. and

Walsh, D. (1976). Mon. Not. R. astr. S¢_c. 177,43P.

32. Condon,,j. jj. , Hick, P.D. , and `jaunce.q,D.L. (1977). Astron. d. 82,692.

33. Me_kelijn, d.K., Shimmins, A.,j., and Bolton, J.g. (1968).

Aust. d. Phgs. 21, 5_3.

34. ghigo, F.D. (1977). A_t_ophgs. "j. Suppl. 35,359.

35. Schmidt, M. (1977). Ast_ophgs. J. 217,358.

36. Smith-Haenni,A. L. (1977). Astron. Astrophgs. Suppl. 27,205.

37. Spinrad,H. , Westphal,,j. , Kristian,,j. , and Sandage, A (1977).

Astrophgs. d. Letters _16, L87.

Fig. 4. Sample page of references to spectral indices, red shitts, optical identifications and optical magnitudes

10
TDA Progress Report 42-82 April-June 1985

N86-10384

Comparison of the Calibration of Ionospheric Delay in

VLBI Data by the Methods of Dual Frequency and

Faraday Rotation

J. A. Scheid

TrackingSystemsandApplicationsSection

When both S-band and X-band data are recorded for a signal which has passed through

the ionosphere, it is possible to calculate the ionospheric contribution to signal delay.

In Very Long Baseline lnterferometry (VLB1) this method is used to calibrate the ionosphere. In the absence of dual frequency data, the ionospheric content measured by

Faraday rotation, using a signal from a geostationary satellite, is mapped to the VLB1

observing direction. The purpose of this article is to compare the ionospheric delay

obtained by these two methods. The principal conclusions are: 1) the correlation between

delays obtained by these two methods is weak; 2) in mapping Faraday rotation measurements to the VLB! observing direction, a simple mapping algorithm which accounts

only for changes in hour angle and elevation angle is better than a more elaborate algorithm which includes solar and geomagnetic effects; 3) fluctuations in the difference in

total electron content as seen by two antennas defining a baseline limit the application

of Faraday rotation data to VLBI.

I. Introduction

In VLBI, two or more antennas track the same extragalactic source simultaneously, and at each station the received

signal is digitized and recorded on magnetic tape. The tapes

from a pair of stations are then correlated to determine the

time delay between the arrival of a wave front at one station

and the arrival of the same wave front at the other. The total

delay has geometric, tropospheric, ionospheric and instrumental components,

r=r 6 +r T +rib +rco, b=S,X (1)

The geometric delay r G is nondispersive. For the purposes of

this analysis, tropospheric delay rr is considered nondispersive.

Ionospheric delay _'t is dispersive. Instrumental delay r c

includes the station clock offsets as well as dispersive components due to differences in the antennas and electronics at

the two stations. Accordingly, the ionospheric and instrumental delays carry the subscript b to indicate the frequency

band of the observation.

The excess phase delay due to the propagation of radiation

through a dispersive medium is given for a single frequency by

r=l fds(n - 1) (2)

where n is the index of refraction, c is the velocity of light in

vacuum and the integration is over the ray path (see Ref. 1 for

11
basic ionospheric formulas). The major effect of the ionosphere is due to a plasma of free electrons. Under the assumption that damping due to electron collisions is negligible, the

index of refraction at a point in the ionosphere is given by

n = (1-r) 1/2

*:b:,,cos

(3)

where fb is the observing frequency in band b, fp is the plasma

frequency, fH is the frequency of precession of an electron

in the geomagnetic field, and ¢ is the angle between the wave

normal and the direction of the magnetic field. Plasma frequency in Hertz is related to electron density d in electrons

per cubic meter by

fp = 8.984dl/2 (4)

Expanding the expression for n, the quantity n - 1 to be

integrated along the ray path takes the form

1 I r2

n-l=- T r-_- ...

(5)

Normally for the plasma of the ionosphere, fn <fp < 15 MHz.

For frequencies of interest in VLBI experiments, fp << lb.

Consequently, a first order approximation to n - 1 is obtained

by neglecting the geomagnetic field and retaining only the

terms quadratic in r,

where TEC is the columnar total electron content at'zenith

for station i.

For VLBI observations, the quantity of interest is the

difference between the delays at the two stations with respect

to a common reference point,

= 40.31 [TECI TEC2 1

rlb c/2 LC"gg_S_l cos%

(9)

II. Dual Frequency Method

In reference to Eq. (9), the total observed delay may be

approximated by

N

% = ro + _ (10)

Thus a measurement of the delay at two frequencies makes

possible the determination of the constants ro and N so that

rb may be removed from the total delay at both frequencies.

In processing VLBI data, the ionospheric contribution to

delay is determined in the program CALIBRATE for both the

dual frequency and Faraday rotation methods. The frequencies used in CALIBRATE are weighted averages of the channel

frequencies in each band. In the dual frequency approach, the

quantities

n- 1=- 2 f_ (6)

In this approximation, the group delay at a single station

caused by the ionosphere is given by

40.31/" D(h)

| dh

_/bJ cos_

(7)

2

A=

Ox - _s)

B = 22

- )

(11)

where h is elevation in meters, D(h) is the electron density

profile in electrons per cubic meters, and a is the angle between

the local vertical and the tangent to the ray to the source.

Neglecting the bending of the ray, Eq. (7) is approximated by

40.31 TEC

_ (8)

c[_ cos

are computed (Ref. 2). If Eq. (10) were exact, the quantities

A and B would equal r ° and N respectively. Contributions to

dispersion other than those of Eq. (10), however, introduce

frequency dependent terms into A and B. Such contributions

are present but are known or assumed to be constant in time.

Terms known to be small were omitted by truncating the

expansion of the index of refraction and neglecting the geomagnetic field.

12
The data processed by CALIBRATE have been corrected

by phase calibration to remove instrumental effects. The

component of instrumental phase shift which is independent

of frequency appears in the delay observable as a term proportional to 1If. If this is not eliminated by phase calibration, it

will contaminate the determination of both r o and N. Any

instrumental delay inversely proportional to frequency

squared which is not removed by phase calibration contributes

directly to the value of N determined from the data.

III. Faraday Rotation Method

Faraday rotation refers to the rotation of the axis of the

polarization ellipse of an electromagnetic wave as it propagates through a magnetized plasma. The total Faraday rotation

in radians due to passage of the wave through the ionosphere

in the presence of the geomagnetic field is given by

- fdh r,, cos__2 COS O_

(12)

Faraday rotation is measured along the ray path of a

signal from a geostationary Applications Technology Satellite

(ATS) to a ground receiving station at Goldstone. Total

electron content for this slanted ray path is then computed

and mapped to the zenith at the zenith reference point. The

zenith reference point is defined to be the point along the

ray path between the ATS and the ground station at a reference altitude, typically 350 km. The result, the reference point

zenith electron content, is the form of the Faraday rotation

data which is ir_put to CALIBRATE.

In CALIBRATE these data are mapped to the ray path

through the ionosphere along the lines of sight from the VLBI

stations to the source. This mapping is designed to account

for several differences between the conditions of the Faraday

rotation measurement and those of the VLBI observation

(Ref. 3). These include factors to account for changes in hour

angle, solar-zenith angle, geomagnetic latitude and elevation

angle.

Solar-zenith angle is the angle between the observer's

zenith and the sun. The solar-zenith angle dependence of

vertical ionospheric electron profile has been modeled by

Chapman (Ref. 3) and by Yip, yon Roos and Escobal (Ref. 4).

Parameters in the SEASAT altimeter semi-empirical model of

S.C. Wu (private communication, 1977) applied in this

analysis are determined by least-squares fitting to the measured zenith electron content. Daytime zenith electron content is lower during summer than during winter while nighttime zenith electron content remains higher during summer.

Thus the model parameters vary during the year.

The geomagnetic adjustment is complicated by a "geomagnetic anomaly." During early morning hours, the concentration of ionospheric electrons is higher at the magnetic equator

and lower at the geomagnetic poles. As the ionosphere is illuminated by the sun, electrons drift to the north and south away

from the magnetic equator reducing the electron concentration at the magnetic equator by about 10 percent. This time

dependence is parameterized in the model of Wu.

The model used to compute ionospheric delay from Faraday

rotation data refers to the combination of mapping factors

used. In this analysis two models for treating the Faraday

rotation data are compared. Model 1 accounts for changes in

hour angle, elevation angle, solar zenith angle and geomagnetic latitude. Model 2 accounts for changes in hour angle and

elevation only. In either case, the difference in ionospheric

delay between VLBI stations is calculated according to Eq. (9)

from the mapped total electron content.

IV. Data

The VLBI data used in this analysis are for the 257.6.km

baseline between the Owens Valley Radio Observatory (OVRO)

near Big Pine, California and Deep Space Station 13 (DSS 13)

at the Goldstone Tracking Station near Barstow, California.

These data were collected using the Mark III VLBI data

acquisition system during Mobile VLBI experiments 81A,

81B and 81C conducted on February 15, 17 and 18 respectively in 1981. Correlation of these experiments was done at

the Haystack Observatory, Westford, Massachusetts. Phase

calibration was applied in all three experiments. The mean

frequencies for the S and X bands were 2289.901 MHz and

8437.9102 MHz, respectively. Typical system error for synthesized delay in these data is 0.05 nanoseconds.

Values of columnar total electron content at intervals of

one hour were extracted from Faraday rotation data covering

the time period of the VLBI experiments. These data, used in

CALIBRATE to interpolate to the mean time of the VLBI

observations, are plotted in Fig. 1 in units of 10 IT electrons

per square meter as a function of local time in hours.

For the purpose of comparing the ionospheric delay calculated by the dual frequency and Faraday rotation methods,

the delays were averaged over all observations in an experiment and the average was subtracted from the ionospheric

delays. Thus, the scatter of the ionospheric delay about the

experimental means are the quantities being compared. From

Eq. (13), the ratio of ionospheric delay at X-band to that at

S-band is roughly 0.08. Thus, S-band ionospheric delays are

dominant and they are the data used in this analysis.

13
In all plots there is one point for each observation. The

average time interval between observation mean times is about

twenty minutes. A few points in the data had anomalously

large delays. Observations resulting in delays outside of the

range -3.0 to +3.0 nanoseconds were excluded from the

analysis. Figures 2, 3 and 4 present the ionospheric delay

scatter obtained from both methods plotted against local

time for experiments 81A, 81B and 81C, respectively. In the

plots against local time, the experiments proceed from the

start time of the experiment to 24 hours on the right and are

then continued from 0 hours on the left.

V. Discussion

In Fig. 5, the Faraday rotation data for both mapping

models are plotted against the dual frequency data. In these

scatter plots the data of the three experiments are combined

and plotted in the form of deviations from the mean normalized with the standard deviations. Also shown in each figure

are the lines of regression, the slopes of which are the correlation coefficients. Since the data are presented in the form of

deviations from the mean, the line of regression passes through

the origin. The correlation coefficients are 0.23 and 0.47 for

models 1 and 2, respectively. Thus, while there is a positive

correlation between the dual frequency and Faraday rotation

methods of obtaining ionospheric delay, it is a weak correlation. Furthermore, the correlation is made worse rather than

better by the inclusion of mapping factors which incorporate

solar zenith angle and geomagnetic latitude effects.

In Fig. 6, the data from the three experiments are superimposed in the same plot against local time. In the three

experiments, the Faraday rotation data are seen to track very

closely, the dual frequency data less well. The same 24-hour

schedule was shifted and used for all three experiments. These

experiments were consecutive and were performed within a

four-day period so that the same sources were being observed

at the same local times during each experiment to within a

few minutes. The variation of the Faraday rotation data with

time of day was similar for the three experiments, so that the

kind of tracking seen in the Faraday rotation data of Fig. 6 is

expected.

Ionospheric variations may be classified in three categories:

large-scale spacial (>500 km), large-scale temporal (>I h)

and small-scale (<500 km or <1 h) variations (Ref. 5). Largescale effects are modeled in the mapping of the Faraday rotation data. Thus, to a first approximation the large-scale effects

may be removed from the dual frequency data by subtracting

from it the Faraday rotation data. This difference, plotted in

Fig. 7, should retain only the effect that small-scale ionospheric variations and irregularities produce in the dual frequency data by virtue of the fact that in this method ionospheric delay is measured independently at each station. We

see, however, that the data of model 1, in contrast to that of

model 2, contain a diurnal signature. This suggests that the

model 1 mapping function introduces a large.scale effect into

the data.

Based on the Faraday rotation data, day hours were taken

to be 0700 to 2100 local time. Standard deviations oD and

oN for the day and night periods, respectively, of the data in

Fig. 7 are given in Table 1. Let us assume that there is a

component of the fluctuation in the measured ionospheric

delay which is proportional to the total electron content. This

implies

a_ =R%_ (13)

where OlD and OIN are respectively the day and night standard

deviations of the ionospheric delay, and R is the ratio of the

average total electron content of day to that of night. In

Table 1, the ratio OD/ON is 1.48 and 1.35 for the data based

on mapping models 1 and 2, respectively. From the Faraday

rotation data, the average total electron content of the ionosphere, in units-of 1017 electrons per square meter, was found

to be 1.88 and 5.62 for night and day, respectively, so that

R = 2.99. The fact that R is different from OD/ON may be

interpreted to indicate that there is an additional component

oo of the ionospheric delay scatter which is independent

of changes due to sunlight. Thus,

2 2 U2

OD = tIID ÷ o

0"_/ = O_V +0'2 0

(]4)

From Eqs. (13) and (14) one finds the values of OZD, OZN

and oo listed in Table 1.

Two points can be made about the statistics of Table 1.

First, that the scatter in the data associated with model 1 is

significantly larger than that associated with model 2, reflecthag the weaker correlation of the model 1 Faraday rotation

data to the dual frequency data. Second, that Oo, the component of scatter in ionospheric delay which is constant, is

comparable in size to OlD, the component associated with

sunlight. System noise error for these data, based on the

number of bits correlated, is approximately 0.05 nanoseconds

and is therefore too small to account for oo. Possibly the

component which is independent of total electron content is

introduced during data acquisition and processing. On the

14
other hand, it may be that the assumption of Eq. (13)is

incorrect.

Rays from the ends of a baseline in the direction of an

extragalactic source penetrate the ionosphere along parallel

lines separated by the baseline distance. For the relatively

short baseline of these data, fluctuations in ionospheric delay

can be estimated according to

or =A(L/IO)°'Tsr (15)

where L is the distance in kilometers between points in the

ionosphere and A is a dimensionless constant (Ref. 5). Since

ionospheric delay is proportional to total electron content, the

estimate of the standard deviation given by Eq. (15) is proportional to total electron content and is therefore comparable

to the quantities of otD and Onv. Using the average values of

total electron content for day and night, the average ionosperic delay is computed from Eq. (8) to be 14.4 and 4.8 nanoseconds, respectively, for day and night observations. From

the values of OlD and OtN given in Table 1, we find the con.

stant A to be 0.0045 and 0.0027 for models 1 and 2 respectively. Callahan finds values of A in the range 0.0040 to

0.0031. This agreement supports the contention that the

delay scatter seen in Fig. 7 is due to small-scale variations in

the ionosphere.

Vl. Conclusions

The weakness of the correlation between ionospheric delay

computed by the dual frequency and Faraday rotation

methods supports the conclusion of J. M. Davidson (private

communication, 1981), obtained for the ARIES Project,

that the ionosphere has not been adequately modeled in the

mapping function applied to Faraday rotation data to calibrate VLBI data. Ionospheric delay computed from Faraday

rotation data depends on the measurement of total electron

content along a single ray path through the ionosphere.

Ionospheric delay determined by the dual frequency method

depends on the difference between the total electron content

along two rays through the ionosphere separated by the

baseline distance. The dual frequency method therefore

incorporates fluctuations due to differences in the ionosphere along the two rays. Since these fluctuations cannot be

modeled, the Faraday rotation method cannot be used to

compute the ionospheric delay calibration at the level required

for processing VLBI data.

In this analysis, one component of fluctuation irl ionospheric delay obtained by the dual frequency method is

associated with variations in the total electron content of the

ionosphere induced by sunlight. The other component is

found to be independent of day-night variations in the ionosphere. System noise error is too small to account for the

solar independent component.

Acknowledgments

The Faraday rotation data was supplied by H. N. Royden. The author wishes to thank

G. E. Lanyi, G. H. Purcell, J. L. Fanselow and H. N. Royden for helpful discussions.

References

1. Meaks, M. L.,Methods ofExperimentalPhysics, Vol. 12, part B. New York: Academic

Press, 1976.

2. Thomas, J.B., "An Analysis of Radio Interferometry with the Block O System,"

JPL Publication 81--49, Jet Propulsion Laboratory, Pasadena, California, December,

1981.

3. Chapman, S., "The Absorption and Dissociative or Ionizing Effect of Monochromatic

Radiation in an Atmosphere on a Rotating Earth," Proc. Phys. Soc. (London), Vol.

43, p. 25, 1931.

4. yon Roos, O.H., Yip, K. B. W. and Escobal, P.R. "A Global Model of the Earth's

Ionosphere for Use in Space Applications," AstronauticaActa, Vol. 18 (Supplement),

p. 215, Pergamon Press, 1974.

5. Callahan, P. S., "Ionospheric Variations Affecting Altimeter Measurements: A Brief

Synopsis," Marine Geodesy, Vol. 8, Numbers 1--4, 1984.

15
Table 1. Statistics for dual frequency daisy scatter minus Faraday

rotation delay scatter. The day and night data includes 107 and 68

data points, respectively.

Model 1 Model 2

o D 0.936 nsec 0.618 nsec

o N 0.631 nsec 0.457 nsec

OD/O N 1.48 1.35 _

aid 0.733 nsec 0.441 nsec

alN 0.244 nsec 0.147 nsec

a 0.592 nsec 0.433 nsec

o

A 0.0045 0.0027

16
"8i

m

l)

r_

o

z

N

z

o

U

z

@

U

2

10?L

8 m

7 --

6 --

5--

4 --

3

2 -

1

I I L I I I I I I I I I I I

_1_.,

81A 81A

81C

81B 81B

I I I I l I I I I I [ I i

2 4 6 8 10 12 14 16 18 20 22 24 2

LOCAL TIME, hrs

l ,

4

Fig. 1. Columnar total electron content of the ionosphere determined by

the Faraday rotation method in units of 10 electrons per square meter

versus local time, experiments 81A, 81B and 81C.

17
-2

r_

u 0

o

-2

-2

' I ' I ' I [ I ' I ' I ' I ' I ' i ' i ' I '

DUAl. FREQUENCY

i I , I , I a I , I l I L I , I t I , I , I ,

' I I J I 1 I I ' I ' I ' I ' I ' I ' I ' I '

FARADAY ROTATION, MODEL 1

I I , I i I i ] i I i I

' I _ i * I ' I ' I i I ' I i I ' I ' I i I i

FARADAY ROTATION, MODEL 2

l I l I i I i I L I I I i I 1 I L I i I l I I

2 4 6 8 I0 12 14 16 18 20 22

LOCAL TIME, hrs

Fig. 2. S-band ionospheric delay scatter versus local time, experiment 81A

24

18
DUALFREQUENCY

I v I v I ' I _ I ' I ' I ' t

I , I , I I t i L , I ,

' I ' I ; I ; I ' I ' I ' I ' I ' I I ' I ' i

eu 2 FARADAY ROTATION, MODEL 1

E

>.-

...,

o_ _

i I , ,, I i I , I z I J

-2

' I _ I _ I ' I ' I ' I ' I ' t ' I ' I I |

FARADAY ROTATION, MODEL 2

I I , t I I I I , I J I i I _ I I I _ 1 J i

0 2 4 6 8 10 12 14 16 18 20 22 24

LOCAL TIME, hrs

Fig. 3. S-band ionospheric delay scatter versus local time, experiment 81B

19
c

U

"I0

Z

0

-2

' I ' I ' I _ I ' I

DUAL FREQUENCY

' I ' I ' I ' I ' I ' i '

, I , I I I , I ; I L I , I , I , I L I , I L

' I ' I ' I ' I ' I ' i ' I ' I _ I ' l ' I '

FARADAY ROTATION. MODEL 1

°t b.-;v'¢---I-

-2 _ , I , l _ I , I , I i I ; I , I , I i I J I ,

' i ' i ' I ; 1 ' i ' I '_ I ' I ' I ' i ' I '

FARADAY ROTATION. MODEL 2

2-

,m A _JL _ __

, I , I , I _ I , I , I , I , I , I , I , I L

2 4 6 8 10 12 14 16 18 20 22

LOCAL TIME, hrs

Fig. 4. S-band ionospheric delay scatter vemu$ local time, experiment 81C

24

2O
I I

MODEL1

2 _

o °o •

• • _P ioM •

go o

0 •

'O • O •

O O O

• ,,°0

Do o °

--- -2- • ° @° °

• • •

o • •

i I ' ' I •

MODEL 2

__ 2 • --

°l o ° °

N • • _ :Q_

• • o_

• • • Q •

z _llt

r_t,_•o _j_ _ •

o • a. _ Ip_lLr"_aPeo _

"__ql_ "_ • - •

o '=o_0 •

_ • • •

-2

I

-2

•q

NORMALIZED DUAL FREQUENCY IONONSPHERIC DELAY, nsec

Fig. 5. Dual frequency delay scatter versus Faraday rotation delay scatter at

S-band, experiments 81A, 81B and 81C

21
2

u 0

=_

i ' I ' I ' I _ I ' I ' I ' I ' I _ I I ' I '

-2

t I I I L

FARADAY ROTATION, MODEL I

2

O'

-2

0 2 4 6 8 10 12 14 16 18 20 22 24

LOCAL TIME, hrs

Fig. 6. Dual frequency and Faraday rotation ionospheric delay scatter versus local time at S-band,

expedments 81A, 81B and 81C

22
T

MODEL !

-2

u

z

u

O

Z

o

o

MODEL 2

-2 I i i i l , , i i I I l ; L I i J _ , I l ,

0 5 I0 15 20

LOCAL TIME, hrs

Fig. 7. Dual frequency delay am.alter minus Faraday rotation clelily ak=atter versus io_il time st _aand,

experiments 81A, 81B and 81C

23
TDA Progress Report 42-82

NS(}-l 0385

April-June 1985

A New 70-Meter Antenna Quadripod With

Reduced RF Blockage

J. J. Cucchissi

Ground Antennas and Facilities Engineering Section

The new subreflector mount (quadripod) for the 64-meter to 70-meter antenna extension project was the result of many trial designs aimed at reducing RF spherical and plane

wave blockage and minimizing structural weight while satisfying strength and natural frequency requirements.. An optimum design emerged which has a gain improvement of

0.32 dB over the present 64-meter design. This article describes some of the trial designs

made and the final optimum configuration selected.

I. Introduction

The 64-Meter Antenna Rehabilitation and Performance

Upgrade Project at the Jet Propulsion Laboratory (JPL)

initially aimed at increasing the gain by about 1.9 dB at

X-band (8.45 GHz) for the upcoming Voyager-Neptune and

Galileo-Jupiter encounters. The effort entails the following:

(1) Increasing the antenna aperture from 64 to 70 meters

(+0.8 dB).

(2) Resurfacing the entire primary reflector with high

quality panels and aligning those panels with high precision optical theodolites and holographic methods

(+0.5dB).

(3) Shaping the subreflectorto an asymmetric surfaceand

themain reflectorto an axisymrnetricsurfaceinorder

to obtaina uniform RF radiationpattern(+0.3dB).

(4) Automating the axial(z-axis)and y-axisfocusingcontrollersof the subreflectorpositionerand addingstructuralstiffeningbracesto the main reflectorcentral

trussto reduce gainlossinduced by gravityloadswhen

the antenna is tilted at extreme elevation angles

(+0.3dB).

(5) Employing any modificationsthatmay add to thetotal

1.9 dB gainincreaseand add a levelof confidence"to

theupgrade project.

Since the subreflector mount was to be redesigned anyway, a

vigorous effort, funded jointly by the Advanced Systems Technology and the TDA Engineering offices at JPL, was undertaken to investigate different quadripod configurations that

showed a promise in reducing the RF blockage. When compared to the other structural and mechanical modifications,

the quadripod yielded the greatest potential gain improvement

for its cost. A thorough review of candidate designs was made,

and several were selected to be studied.

II. Design Requirements

For each new configuration, the following requirements

were in effect:

24
(1) The total RF blockage (planar and spherical wave) due

to the quadripod legs should be no greater than the

percentage aperture area blockage for the current 64-

meter antenna. If possible, the blockage should be

minimized.

(2) The natural frequency of the lowest mode of the

quadripod should be equal to or greater than that for

the present 64-meter antenna quadripod to be compatible with the existing serve system.

(3) The lateral (y-gravity) displacement of the subreflector should not be excessive due to the limited capability of the subreflector positioner to correct for it (9 in.

range).

(4) The quadripod should conform to the structural

strength criteria set forth in the ASCE Tower Code

(Ref. 1) under varying gravity loading, corresponding

to different antenna elevation angles. Furthermore, the

quadripod should accommodate a larger, heavier subreflector and subreflector positioner and be able to

withstand safely the lifting loads of the feedcone and

subreflector hoists.

(5) The quadripod should be an "optimum" design rather

than just a "working" design, i.e., it should satisfy the

above performance constraints with minimum truss

weight. Material cost savings are related to weight

reduction.

III. Definition of Blockage

The beam of transmitted or received energy includes all

rays parallel to the axis of the paraboloid that fall within the

aperture of the paraboloid. The beam intensity is assumed to

be uniform over its circular cross-section. The blockage area

is defined as that portion of the beam cross-section representing the qptically obstructed rays (Ref. 2). It can be considered

as the sum of two kinds of blockage: (1) the blockage that

occurs where the wavefront is spherical, and (2) the blockage

that occurs where the wavefront is plane. Spherical wave

blockage is the shadow of the quadripod legs projected on the

paraboloid when they are illuminated from the focus. In Fig. l,

this is the shaded area outside the dashed circle. Plane wave

blockage is the projection of the subreflector and quadripod

onto a plane and corresponds to the shaded area inside the

dashed circle.

Without an explicit function to relate optical blockage to

the RF antenna efficiency, the following empirical equation

valid for small B was used to relate the percent of blocked

area to aperture area, B, and aperture efficiency, r/,:

r/ = [1 - 1.2(B/100)] 2

For the purpose of this study, the quadripod legs were

assumed 100% opaque. Since the inner face of the legs would

be constructed from steel plates to simplify fabrication, this

assumption is not only conservative but accurate. Also, gain

loss due to the subreflector plane wave blockage is not included

in the gain loss values given in Tables 1 and 2, because its

effect is usually accounted for in the microwave efficiency

estimate. The "net blockage," as defined here, includes optical

plane wave and spherical wave blockage by the quadripod legs

only.

IV. Methodology

Three different options to reduce RF blockage by the quadripod are indicated in Fig. 2. In order of increasing reduction

potential, they are: (1) increase R 1 by attaching the legs to

the main refector surface at a point farther from the paraboloidal axis; (2) increase the pitch angle 3 to make the legs

closer to the vertical; and (3) change the cross-section of the

legs by reducing the widths of the inner and outer faces, Wi

and We. The first option would have required extensive modification of the existing main reflector rectangular girder structure to attach the quadripod. Excessive fabrication and erection costs caused this option to be eliminated. The second

option reduces spherical wave blockage by moving the inner

face of the legs farther from the primary focus of the Cassegrainian system. It also increases the size of the apex, Wa, but

this had to be done anyway to accommodate the larger subreflector. The magnitude of the pitch angle was restricted,

however, to that necessary for reasonable clearance of the subreflector. The third option became the key to the study and

proved to be quite effective.

Using the JPL-IDEAS structural optimization computer

program to perform analysis and design, a series of pin-jointed

truss finite element models of candidate designs was generated.

The performance of the various quadripod geometries was

determined for each of the required loading conditions. In

each case the member connectivity (or topology) was maintained while the dimensions or proportions of the quadripod

leg cross-sectional width profile (Fig. 2, Sec. A-A) were

changed. The widths of the parallel faces were selected so that

they were "balanced," that is,the outer face (I4/o) lay within

the spherical wave shadow generated by rays impinging on the

inner face (Wi). Minor changes in apex dimensions were made

as necessary to match the width changes. The selection of a

trapezoidal cross-section minimizes blockage while maximizing

quadripod leg torsional stiffness, which is a function of the

enclosed cross-sectional area. Also, due to the symmetry of the

structure, it was possible to use only half models; this reduced

computer computation costs.

25
V. Trial Designs

Table 1 compares the 64-meter antenna quadripod finite

element model with the four best 70-meter trial configurations. All were pin-jointed truss models. For each trial design

case, the objective was to meet or exceed the first mode (torsional) frequency of the 64-meter quadripod model while

minimizing the structure weight. The gravity displacement of

the subreflector was monitored but was not usually imposed as

a design constraint.

As the cross-section of the leg was reduced, so was the net

blockage. As shown in the table, this reduction is accompanied

by a decrease in natural frequency and an increase in gravity

displacement of the subreflector. The progressive drop in

weight in Models 1, 2, and 3 resulted from a reduction in

length of certain leg truss members as well as the relative ease

with which the structure met the design constraint.

Model 4 represents a different situation. The increase in

structure weight indicated a difficulty in meeting the constraint

with a very slender quadripod leg profde. Nevertheless, the

weight increase was within acceptable limits, and Model 4 did

meet the design constraint; it was selected as the final configuration because of its low blockage.

The "acceptable weight" limitation was not so much a

requirement as a guideline. Since these were stand.alone

quadripod models, it was not possible to determine directly

their effect on the reflector back up truss structure without

costly analyses of the larger tipping structure model. Also, the

quadripod would be counterweighted, and available space for

counterweight was limited. Nevertheless, subsequent analysis

of the composite back up structure with quadripod computer

models showed no significant penalties resulting from this

design.

VI. Final Design

Trusses are usually analyzed as pin-jointed structures. In

reality they are not pin-jointed because of bending and torsional stiffness at the rigid corners, but axial forces in the

truss members predominate so bending and torsional moments

are often neglected. An analysis of a rigid-joint 64-meter

quadripod model, however, showed a substantial increase in

torsional natural frequency (from 0.74 Hz to 1.22 Hz),

whereas a rigid-joint 70-meter model did not (from 0.78 Hz to

0.82 Hz). To achieve the higher 64-meter frequency, the final

configuration had to be redesigned.

Primarily a computer program redesign sequence consists

of resizing truss members without altering the overall propertions or dimensions of the model. Unfortunately, the topology

of Model4 made it unreasonable to attempt to meet this

higher natural frequency constraint by only resizing members;

the optimization algorithm in IDEAS, in achieving its best

solution, reached a point where the effect of additional

structural stiffness was offset by the associated increase in

structural mass. At this point, the natural frequency had been

maximized and could only be increased by reducing nonstrucrural weight, which was invariant.

Because of an unwillingness to abandon this configuration

(because of its blockage reduction potential), a new modification was proposed: outrigger braces, as sketched in Fig. 3,

were added to connect the lower portion of the quadripod legs

with additional points on the reflector back up truss. These

braces resist the rotation of the legs about their longitudinal

axis (which was evident from an examination of the first mode

shape) and thereby increase the frequency. The results are

hsted in Table 2. Compare the results to Model 4 in Table 1,

and note the reduced quadripod structural weight as an additional benefit. An insignificant increase in net blockage is due

to the outriggers.

VII. Supplementary Results

Additional observations, resulting from other" analyses performed during the course of the study, are briefly discussed

below: !

(1) For a given quadripod geometry, increasing the aperture area while keeping other parameters constant

causes an increase in spherical blockage area, and the

percentage increase in spherical blockage is much

greater than the percentage increase in aperture area.

(2) Varying the leg depth had no appreciable effect on

either blockage or the frequency of the first mode.

(3) The first mode shape indicated a rotation of the nearly

rigid apex about the focal axis; the legs showed weakaxis bending. An attempt to reduce this bending by

using K-braces to subdivide the bays of the outer

face produced improved but limited results due to the

extreme slenderness of the quadripod legs.

(4) Efforts to meet the torsional frequency requirement

included supplementing the apex structure by adding

members across the axis of symmetry. This was done

early in the study, and results indicated that the connectivity of the apex could be simplified without

affecting the first mode frequency. These supplemen.

tary members were then removed.

(5) The second mode of the stand-alone quadripod model

exhibited sideways bending at a frequency 0.5 I-Iz

26
(6)

higher than the first mode. Modes higher than two had

frequencies in excess of twice the first mode frequency.

For the composite back up structure with quadripod

model, the lowest mode was quadripod torsion at a

frequency slightly less than the first mode of the standalone quadripod model.

VIII. Conclusions

The following are conclusions from the quadripod blockage

reduction analysis:

(1) Varying the inner and outer face widths had a significant effect on the quadripod blockage and fundamental frequency.

(2) The torsional frequency pc_rformanee constraint con.

trolled the design. In all cases, the requirement on the

y-gravity displacement of the subreflector was easily

satisfied.

(3)

(4)

Experience with the 64-meter quadripod model indicates that a pin-jointed model yields a torsional natural

frequency much lower than measured on the actual

structure. A rigid-jointed model, which accounts for

bending and torsional stiffness of the truss members,

gives more realistic results. The 70-meter quadripod

model performance was insensitive to the joint continuity. Again, this is probably attributable to the leg

slenderness.

The quadripod legs are usually pin-connected to the

reflector truss back up structure. This type of connee.tion ........................... cannot resist bending nr twi_tino O mnrn_nt_ ............

and therefore allows the legs to rotate about their

longitudinal axis, which is evident from the mode

shape. Limiting this rotation greatly increased the

first mode frequency; it required the addition of

outrigger braces near the base of the quadripod legs

to provide torsional rigidity.

Acknowledgment

The author wishes to express his appreciation to Dr. Roy Levy, Chief Design Engineer

of the 70-Meter Extension Project, for his patience and guidance.

Many other individuals have contributed valuable information, time and assistance

throughout the course of this study: R.C. Clauss, F. W. Stoller, F. Lansing, H. Phillips,

M. S. Katow, S. Rocci, and D. Strain. To these, the author is indebted.

References

1. Guide for Design of Steel Transmission Towers, ASCE Manual on Engineering Practice,

No. 52, ASCE, New York, 1971.

2. Herndon, J., "Efficient Antenna Systems: A Program to Calculate the Optical Blockage by the Quadripod on Large Microwave Antennas,"JPL Space Programs Summary

37-48, Vol. II, pp. 58-63, Jet Propulsion Laboratory, Pasadena, CA.

" 27
Table 1. Trial designs to reduce RF blockage. All models are pin-jointed.

Parameter

64-m

Antenna

70-m Antenna

Model Model Model Model

I 2 3 4

Wi, in. (m)

Wo, in. (m)

H, in. (m)

Net Blockage, %

Loss, -dB

Structure Weight, lb (kg)

Gravity Displacement of Subreflector, in. (m)

Lowest TorsionalFrequency,Hz

18.0 (0.457) 18.0 (0.457) 18.0 (0.457) 15.0 (0.381) 11.0 (0.279)

36.0 (0.914) 36.0 (0.914) 32.0 (0.813) 28.0 (0.711) 20.0 (0.508)

96.0 (2.438) 96.0 (2.438) 96.0 (2.438) 96.0 (2.438) 96.0 (2.438)

6.34 5.67 5.01 4.44 3.32

0.68 0.61 0.54 0.48 0.35

41,600 54,258 54,174 53,174 59,146

(18,869) (24,611) (24,573) (24,119) (26,828)

0.89 1.25 1.26 1.30 1.48

(0.0226) (0.0318) (0.0320) (0.0330) (0.0376)

0.74 1.23 1.13 0.99 0.78

Table 2. Comparison of current 64-m quedripod with final 70-m design.

Both models are rigid-jointed.

Wi, in. (m)

Wo, in. (m)

H, in. (m)

Parameter 64-m Antenna 70-m Antenna

Net Blockage, %

Loss, -dB

Structure Weight, Ib (kg)

S/R and Positioner Weight, lb (kg)

Gravity Displacement of Subreflector, in. (m)

Lowest Torsional Frequency, Hz

Lowest Pitch Frequency, Hz

18.0 11.0

(0.457) 0.279)

36.0 20.0

(0.914) (0.508)

96.0 96.0

(2.438) (2.438)

6.34 3.42

0.68 0.36

41,600 53,250

(18,869) (24,154)

12,400 24,000

(5,625) (10,886)

0.89 1.16

(0.0226) (0.0295)

1.22 1.42

1.42 2.65

28
I

I I

\

\ ',,Mr, ,,;/

Fig. 1. Plane wave and spherical wave blockage

i Wo

SECTION A-A

Fig. 2. Parameters controlling RF blockage

29
OUTRIGGERS

Fig.3.Planviewof70-meter quadripodwithoutriggers

3O
TDA Progress Report 42-82 April-June 1985

N86-1038S

Seismic Analysis of the Large 70-Meter Antenna,

Part I: Earthquake Response Spectra Versus

Full Transient Analysis

K. Kiedron and C. T. Chian

Ground Antennas and Facilities Engineering Section

As a check on structure safety aspects, two approaches in seismic analysis for the large

70-m antennas are presented. The first approach, commonly used by civil engineers,

utilizes known recommended design response spectra. The second approach, which is the

full transient analysis, is versatile and applicable not only to earthquake loading but also

to other dynamic forcing functions. The results obtained at the fundamental structural

frequency show that the two approaches are in good agreement with each other and both

approaches show a safe _design. The results also confirm past 64-m antenna seismic studies

done by the Caltech Seismology Staff

I. Introduction

Since the trial structural-mechanical designs and the upgrading and rehabilitation effort of the present 64.m antenna to a

70-m aperture have been completed (Ref. 1), it is essential to

check the candidate design for severe environmental loading

conditions to satisfy safety requirements. The purpose of this

two part study is to investigate the earthquake response of the

large antenna structure, to present the modern methodology

and to check the structure safety aspects. The emphasis in

the first part is placed on the mathematical description of the

method of analysis with detailed design safety features to be

presented in the second part.

During the early design phase of the 64-m antenna network,

the safety against earthquake loading was one of the important

tasks that was taken into consideration. The conclusion of

that early seismic analysis, conducted by Prof. G. W. Housner

of the California Institute of Technology (Ref. 2), was that the

structural design should withstand the horizontal acceleration

of about 0.25G where G is the acceleration of gravity.

Today, with a wider access to modern digital computers

and new numerical techniques in structural mechanics, more

subtle dynamic analysis of structures is available. This first

part of the study will compare the results of two approaches:

the response spectra approach versus the full transient analysis.

Cross verification of the results should add to a better confidence in understanding the behavior of the complex antenna

structure in response to random earthquake excitation.

II. Methodology Description

The analysis of earthquake-excited structures needs to take

into account the nonperiodic form of the external forcing

term. Such a problem requires the use of special analytical

procedures, as presented in Refs. 3, 4, and 5, which are classified into two broad possibilities:

(1) The frequency response procedure

"(2) The modal analysis procedure

31
Procedure (1) simply determines the natural undamped frequencies of the structure which the designer uses in comparison against the frequencies of the external forces in an effort

to avoid resonance. Procedure (2) is more important and

widely used in practice and will be described in this study

in detail. The differential equation of the structure under

dynamic loading is written in the form:

[M] y+ [Cl )+ [rl y+ 95 =0 (1)

where y is generalized displacement, [M], [C], [K] are mass,

damping, and stiffness matrices, respectively, and {f} is an

external force vector where f(t) is an arbitrary function

of time. Equation (1) is the basic equation describing the

dynamic behavior of the structure.

The first step in solving Eq. (1) is the determination of the

free undamped response where no damping or forcing terms

exist. The special dynamic form of Eq. (1) is reduced to

[M] ._+ [K] y=0 (2)

The general solution of free responses represented by Eq. (2)

is written in the form:

Yo

?i

= Eyoi exp qcoit) (3)

i= 1

where _i are the eigenvalues (natural frequencies) of the

system, Yoi are the eigenvectors, n is the degree of freedom

and j =

For forced responses represented by Eq. (1) the solution

can be written also in a linear combination of modes as

n

Y = E Yoi zl(t) = [Yo] {z) (4)

i = 1

where the matrix lye] lists all the modes (assumed to be

normalized) and za(t) are scalar mode participation factors,

i = 1,..., n, zi(t) is a function of time and represents the

proportion of motion in each mode.

An important advantage of the linear mode superposition

in Eq. (4) is that an approximate solution can be obtained by

truncating modes and including only part of the total modal

contributions. In general, lower modes make the principal

contribution to the dynamic response, and good approximation is usually obtained by considering only the first few

modes in the analysis.

If Eq. (4) is substituted into Eq. (1) and the result is

premultiplied by Eve] T, then

[Yolr[M] LVo]{z)+ Lvo] r [Cl LVo](_)

+ Lye]r [K] lye] (z}+ Lvo] r 95 = 0 (5)

By the orthogonality property,

(Yoi) T [M] {yoi ) = IM.* i :_= jiJ

(6)

where i, j = 1..... n and M_ is a generalized mass.

Also by definition of the eigenvalue problem,

then

(

0 i_ej

lyon)r [K] f.Voj)=

M." i --/

(7)

The matrix [C] is of such a form that

l 0 i_j

_Yo_)[cl _Yoj)= (8)

2 w i _i Mff i = j

where _i is the damping ratio. The natural frequency of a

structure affects its response to an earthquake. Most of the

energy content of an earthquake is in the 1 to 20 Hz frequency range. The duration of "violent" shaking may last

20 or more seconds. If a structure has a fundamental natural

frequency in the 1 to 20 Hz range, then it will have time to

build up a resonant response. The level of resonance built

up depends on the structural damping. For antenna structures

considered here, a damping ratio of 7% for the concrete and

4% for the steel is reasonable. Table 1 lists recommended

damping ratio values (Ref. 6). For parameterization, we took

damping ratios as 2, 5 and 10% as described later in Section III.

Then, the system of Eq. (5) contains only diagonal terms

and forms simply a set of ordinary differential equations

which, after the modes are normalized, are written as

32
£1 +2/;1 c°l zl +_°21zl = - {Yol }r {f_/M_

z2 = - _o2} r _/m;

z_ +2_. %_. +,o_z = - _Vo._r _/M"

(9)

Each ordinary differential equation in Eq. (9) can be solved

in an elementary manner, and the complete solution is obtained by superposition as in Eq. (4). In the earthquake case,

forces {f(t)}, at each node, vary in the same manner with

th-ne. ""--- Will,It ,t._ LIL_antenna is ...t.:..+._ +.... +t..... _.... +;^,, _tl.lOJ_L_ldL LO _Ol LIIK,IU_O.III_ ' lllKatLI_llO 9

the foundation is subjected to a certain "forcing" acceleration a(t), which tends to move with the ground. Since the

motion is relatively rapid, it causes severe stresses and deformation throughout the antenna structure. If a mechanicalstructural component of the antenna is rigid, it will move with

the same acceleration motion of its base, and the dynamic

forces acting on it will be very nearly equal to those associated

with the base acceleration. By superposing and opposing

motions on the whole foundation-to-top structure, we can

consider the moving foundation as equivalent to a fixed

foundation with forces (.-Ma {r}) acting on the nodes as shown

in Fig. 1. The vector (r} is an influence vector (ReL 3) which

geometrically connects the acceleration at nodes. The {r)

consists of ones and zeros only.

A typical modal differential equation from Eq. (9) (for i =

1..... n) is written as

z i + 2 _, 6o, z i + w_ z'i = a(t) (lO)

with

Z. = ¢_.Z _.

I I J

and

_t i = {Yoi }T[M] {ff_./M/_ (11)

The solution of Eq. (10) can be written simply as the Duharnel

integral

t

zi, = _1 f a(r) exp [- _i _°i ( t - r)] sin 6oi(t - r) dr 60

t O

(12)

In practice, two approaches to solve Eqs. (10) through (12)

are followed:

Approach 1: Full transient analysis. The complete transient

response analysis is obtained from integration of Eq. (12),

carried out numerically. In principle, superpositions in Eq. (4)

will result in the full transient responses required. Often a

simple calculation is carried out for each mode to determine

maximum responses followed by a suitable "addition" of these

responses. More details are given in Appendices A and B.

Approach 2: Response spectra. Direct earthquake response

spectra are obtained without the necessity of carrying out

complete transient analysis. For various input earthquake

motions, the responses of a single degree of freedom (typical

of Eq. [10] ) have been evaluated to determine the "envelope"

response spectra. These are available in the literature and are

known as the earthquake response spectra, as in Fig_ 2 and 3.

The response spectra are useful to design engineers because

they embrace the spectra of many observed earthquakes. A

structure designer can safely select these design response

spectra as inputs that describe the statistically justified excitation of the ground at a given site in the United States. They

included three x, y, z motions (two horizontal and one

vertical).

III. Computational Results

Twenty values of natural frequencies and the participation

factors from Eq. (11) are made available from the JPL IDEAS

program for the 70-m antenna. The two approaches are compared for the natural frequency at the first mode, with three

different damping ratios 2, 5 and 10%. Table 2 represents the

value of permissible displacement Sa and acceleration Sa

taken from spectrum curves proposed by Housner in Refs. 7

and 8. The first row in Table 2 concerns an earthquake of the

intensity expected at Goldstone, the second row an earthquake of the 1940 E1 Centro intensity. The third row represents values from the full transient analysis. The computer

program TRANST (see also Appendix A) was used to solve

Eq. (10) representing the motion of a single degree of freedom

of the system. The program solves optionally the eigenvalue

problem before proceeding to the transient response solution.

The fundamental frequermy f = 1.59 Hz was used first for

computation. A "simplified" 1940 E1 Centro earthquake was

selected as the acceleration input function a(t) in Eq. (10)

(Fig. 4). Transient displacement responses are shown in Fig. 5

for damping ratios 2, 5 and 10%, respectively.

Another numerical method was also used to evaluate the

Duhamel integral Eq. (12) representing the transient response

solution of Eq. (10). This numerical method uses trigonometric

identity and converts the original Duhamel integral into a

summation of closed form solutions. A description of this

method, as well as a flow chart of the computer program are

33
given in the Appendix B. The transient response for the fundamental frequency f = 1.59 Hz, with damping ratios 2, 5 and

10% are given in Fig. 6. The results obtained from both

numerical methods are in excellent agreement.

IV. Conclusions

The development of methods and practices suitable for

structural design of the DSN large antennas should include

the analysis of the structure for earthquake resistance. In this

study two approaches were presented, one which utilizes

known recommended design response spectra, and the second

which is the full transient analysis, applicable also for a different type of dynamic loading. The comparative results obtained

at the fundamental frequency show that the two approaches

are in good agreement. The preliminary results agree with past

64-m antenna study done by Caltech Seismology Staff that

shows that the center of the mass of the structure should not

exceed about 0.25 to 0.35G. In this first part of the study the

emphasis was on the mathematical tools to solve the responses

and on the cross verification of different approaches. The

second part of this study will compute the forces developed in

antenna structure components due to seismic excitation and

will compare these with seismic design requirements according

to building codes.

Acknowledgement

The authors acknowledge the assistance given by Dr. F.L. Lansing, Dr. R. Levy,

W. D. Merrick, B. Saldua, D. Strain, J. Cucchissi of JPL and Prof. G. W. Housner of the

California Institute of Technology during the various execution steps of this work.

References

1. McClure, D.H. and McLaughlin, F.D., "64-Meter to 70-Meter Antenna Extension,"

TDA Progress Report 42- 79, Jet Propulsion Laboratory, Pasadena, Calif., pp. 160-164.

2. TDA Technical Staff, The NASA/JPL 64-Meter-Diameter Antenna at Goldstone

California: Project Report, JPL Technical Memorandum 33-6 71, Jet Propulsion Laboratory, Pasadena, California, July 15, 1974.

3. Clough, R.W. and Penzien, L., Dynamics of Structures. New York: McGraw-Hill,

1975.

4. Wiegel, R. L., Earthquake Engineering. New York: Prentice-Hall, Inc., 1970.

5. Zienkiewicz, O.C., The Finite Element Method in Engineering Science. New York:

McGraw-Hill, 1977.

6. Newmark, N. M., and Hall, W. J., Procedures and Criteria for Earthquake Resistance

Design in Building Practices for Disaster Mitigation, Building Science Series 46, U.S.

Dept. of Commerce, National Bureau of Standards, pp. 209-236, Feb. 1973.

7. Housner, G. W., "Behavior of Structures During Earthquakes," Proc. ASCE 85 (EM4),

1959, pp. 109-129, Proceedings Paper 2220.

8. U.S. Atomic Energy Commission (1963). Nuclear Reactors and Earthquakes, TID7024, Washington, D.C., Office of Technical Services.

34
Table 1. Recommended damping ratios

Type and condition Percentage of

Stress level of structure critical damping

Working srress, no more than

about I/2 yield point

At or just below yield point

Vital piping

Welded steel, prestressed concrete,

well reinforced concrete

(only slight cracking)

Reinforced concrete with considerable cracking

Bolted and/or riveted steel, wood

structures with nailed or bolted

joints

Vital piping

Welded steel, prestressed concrete

(without complete loss in prestress)

Prestressed concrete with no prestress left

Reinforced concrete

Bolted and/or riveted steel, wood

structures, with bolted joints

Wood structures with nailed joints

0.5tol.0

2

3 to5

_to7

2

5

7

7 to 10

10 to 15

15 to 20

Table 2. Displacement and acceleration for fundamental frequency

o_ = 9.99 rad/s

/" = 1.59 Hz

T = 0.629 s

Damping ratio, % Displacement Sd, in. (cm) Acceleration, Sa/G

2 1.4 (3.556) 0.362

2.4 (6.096) 0.620

3.05 (7.747) 0.826

5. 0.9 (2.286) 0.233

1.56 (3.963) 0.403

2.76 (7.01) 0.747

10 0.65 (1.651) 0.168

0.96 (2.438) 0.248

2.31 (5.867) 0.260

35
_---. o (t)

- (t)

Fig. 1. Foundation motion as equivalent force

<.

_>

o.

_n

00

,o_'x'/i/'k/<_ ",4x 7<'/://,.J' ,.><rx _._7-'.,2 \ \x ,',o

_.-//3< ,/_ 3_\\4.,__//_'_% //X /\ x\r'o--"

6 , cP 6

0

, \_+" .X'/

• ., 0,4

" ' _ 0,2

o._ , _ _ X o.1

0.01 0.02 0.04 0.060.080.1 0.2 0.4 0.6 0.8 1 2 4 6 8 10

PERIOD OF VIBRATION, T, s

Fig. 2. Combined earthquake response spectra (adapted from Fief.3)

36
1.4

1.2

1.0

i 0.8

0.6

0.4

0.2!

0

_ =0%_

1 2

PERIOD OF VIBRATION, Ts s

Fig. 3. Average displacement response spectrum of 1940

El Centro intensity (adapted from Fief. 3)

3.00

- 2.25

-2.25 -

•-3.OO --__._..1_ _

0 1

m_

3 4 5 6 7 8 9 10

TIME, s

Fig. 4. Earthquake acceleration inp_ function

|

I-.

z

8Ol i IlL I I I I I I i

6°t-<°> /I "

4,0 " -- _.i- .II^^A _ ..

oF ^ll II II I_ II A /\ A A A ^ _,, 2

vv v -sol I I I I I I i I [

!

VVVV vvvv- _!

-eo/] I I I I I I I ]

eo r_ _ T T T

__'2° f V v " " v " .... i-

-8011 1 I I 1 I

0 ! 2 3 4 5 6 7 8 9 10

TIME, s

Fig. 5. Transient displacement response damping ratio:

(a) 2%, (b) S%, and (c) 10%

37
EE

2

..m

8o  ̧

6o

40

20

o

-2o

-40

-60

.8o

8o

60

4o

20

o

-20

..40

-60

-8o

80

6o

4o

20

o

-20

-4o

-60:

-80__

I I I I I I I I I

_--(_)_^ _ AAA ^ AAAA_ ^ :;

 VVvv vvvv,, :_

I I I I I I I ] I

I 2 • 3 4 5 6 7 8 9 10

TIME, s

Fig. 6. Numerical evaluation of Duhamel integral for damping ratio:

(,,)2%,(b)S%,(©)10%

38
Appendix A

Adams-Moulton Method

The first numerical solution of the antenna transient

response follows the Adams-Moulton method in solving a

system of linear differential equation of motion, written in the

matrix form as:

[M] {Z} + [C] (Z)+ [K] {Z} = (a(O} (A-l)

and

wh@r_ th_ matrie_ [M1 [(7.1 and [K1 ar_. of the. order At" X N.

..................... L-'-J ) L--J • _--- t--J ...................

Equation (A-l) can be written as

{2}+ [M]-I [C] (i}+ [M]-_ [K] (Z}= [M]-_ (a(0}"

(A-2)

assuming [M] -l exists.

Equation (A-2) can be written as

{Z)=- [M] -l [C] (Z}- [M]-l [K] (Z} + [M] -l (a(t)}

(A-3)

Equation (A-3) can be written in the form of

if we let

{l)}= [A] {Y} + {B} (A-4)

(A-S)

(A-6)

where [A] is the coefficient matrix of the order 2N X 2N,

{B} is a vector of dimension 2N, and [O] and [I] are the

N X N null and identity matrices, respectively.

Equation (A-4) is a set of 2N simultaneous first-order

differential equations which are solved by using the AdamsMoulton numerical technique (JPL computer library subroutine SVDQ). The technique uses linear multistep predictorcorrecter formulas. Such a technique has the advantage that

from successive approximations to each value, an estimate of

the truncation error is made.

The fourth-order Runge-Kutta method is used to generate

the approximate values of the first four points (n - 3, n - 2,

n - 1, n), since the local truncation error is of order h 5. Values

at these previous four points are needed to" predict or correct

the value at the point (n + 1). The integration order is selected

in such a way as to maximize the step size and to reduce the

computation time, consistent with meeting the requested user

accuracy.

In the first-order equation

(A-9)

integrating between x n and Xn+ l :

/n+ 1

n+l --dY dx =

x dx

n x

n

f (x, y) (A-1 O)

_ [Ol [I1 ]]

[A] = (A-7)

[M]- _ [r] - [MI-' [C

The Adams-Moulton method, like all predictor-corrector

methods• starts by predicting Yn+l from an initial value of

Yn, and then provides successive improvements of Yn+l, or

else corrects Yn+t before calculating the next step.

I 39
The Adams-Moulton method uses the following predictor:

Yn+l = Yn + "_4 (55 fn- S9 fn -, + 37fn-2 - 9 fn-3)

and the following corrector

R = O(h 5) (A-11)

h

Yn+I = Yn + "_" (9fn+l + 19 fn - 5 fn-I +fn-2)

R = O(h s) (A-12)

where h is the step size and R is the truncation error.

By using the Eq. (A-11) as a predictor and Eq. (A-12) as

a corrector, the function y = y(x) is obtained. Further details

on the method can be found in Ref. A-1.

Reference

A-1. Carnahan, B., Luther, H., and Wilkes, J., Applied Numerical Methods. New York:

John Wiley, 1969.

INPUT DATA: IM], [KI, ICl, 1. (t)})

t

 SO.RQ' TINE .,.VD,

USE RUNOE- UTT" I

METHOD TO GENERATE THE FIRST

4 POINTS OF SOLUTION

t

SOLVE NO N-HO MOG EN EOUS

EQUATION BY PREDICTORCORREC TOR METHOD

_,=Ay +B

(SUBROUTINE SVDQ)

t

PROVIDE (SUBROUTINE TRANSIENT FOVT) RESPONSE J

t

_sUBROUTINEOUTPUt:'(" DMOUT, VOUT, MOU_

OUTPUT:

EIGENVALUES,

EIGENVECTORS,

HOMOGENEOUS

SOLUTIONS

Fig. A-1. Flow chart of TRANST program

(using Adams-MouIton method)

4O
Appendix B

Numerical Evaluation of the Duhamel Integral

The Duhamel integral

0

a(T) exp [-_iwi(t - T)] Sill Wi(t - T) dT

(B-l)

representing the transient response solution of the equation of

motion (Eq. [i0]) can be evaiuated numerically. Since the

acceleration input a(r) is given in a tabular form, the inte.

grand can be divided into a number of band functions added

together in such a manner as to form the original aO') function.

Equation (B-l) is converted into the sum of exact solutions;

each resembles Eq. (B-l) but with a unit forcing function a(t).

The sum solution can be used when the actual input is approxi.

mated by band segments.

The approach is explained as follows: Using the trigono.

metric relation

sin%(t - r) = sincottcos co:-- cosw: sin_i r

(S-2)

substitute Eq. (B-2) into Eq. (B-l). The Duhamel Integral

becomes

Zi(t) = exp(-_twit ) Sin wtt f,

o

a(r) exp(_i%r) cos wit dr

exp(-_i%t ) cos wit (t

wl 1o

Using the relationships

a(r)exp(_twtr ) sin wit dr

(s-3)

_z) cos _ dx = exp(_x) (a cos_ + #sinax)

O_2 + f12

(B-4)

and

fexp(ax) sin dx = exp(ax) (a sin _ - fl cos _)

_2 + 1_2

(]3.5)

let

a = _

fl=%

(B-6)

and assume that the acceleration a(¢) is constant (_) between

r = nat and r = (n + 1.)At, where Aris the step of summation.

Equation (B-3) then becomes

z',(t) -

exp(-_._iOsin ¢a/t

X . E _ Iexp(_tm:) (_ica_cos %r + %.sin _:)T :(n÷ ' ) ar

all A¢ Jr=n At

exp(-/_twl t) cos %t

(B-7)

for integration step between r = nat and r = (n + OAr where

n is an integer.

The integral in Eq. (B-l)is then transformed into the summation of the individual contributions from r = 0 to r = t. A

flow chart of the FORTRAN program used to evaluate Eq.

(B-7) is shown in Fig. B-1.

41
I NII_JT

NT = TOTAL TIME

DT = TIME INCREMENT

OMEGA = FREQUENCY (w|)

CI = DAMPING RATIO (_i)

t

READ

T (I) = TiME O')

ACCEL (I) = INPUT FORCING FUNCTION (a(z'))

t

EVALUATE

AI = i(o_- ¢iw;t),i. (,,it]/[ w_ (I + ¢2)I

h =[("_"_;')') _' w;,)/[w_o+e_)]

A2= a(T) (exp _i ¢u|'r) (_[| WiCm_iT + wisin wiT)

i B2= el(T) (*xp _iw|'r) (_[iwisln wiT- w;coswiT)

SUM

N N

i=1 i=l

Fig. B-1. Flow chart of numerical evaluation of

Duhamel integral

42
TDAProgressReport 42-82

N86-I0387

April-June 1985

An Experimental TEl2 - TEll Circular

Waveguide Mode Converter

D. Hoppe

RadioFrequencyandMicrowaveSubsystemsSection

This article describes theoretical and experimental results for a prototype TEt2 -

TE 1] circular waveguide mode converter. The system which requires such a device, the

high power Ka-Band transmitter is described briefly. A short review of coupled mode

theory is given, and the theoretical performance of the final converter design is given.

Experimental results for the fabricated converter are presented and compared with

theory. A method of identifying the various circular waveguide modes in a multimode

device is described. Given the close agreement between the theoretical predictions and

experimental results, the computer code may be used with confidence in the design of

future multimode tapers and mode converters.

I. Introduction

Previous reports have described the conceptual design of a

high power Ka-Band transmitter (Ref. 1), theoretical calculations of mode purity effects on the performance of the system

(Ref. 2), and the feed which will be used with the Ka-Band

transmitter and transmission line (Ref. 3). This article focuses

on another component in the system, the TEl2 - TE11 mode

converter.

Conventional klystrons are not capable of producing

400 kW CW at 34.5 GHz, which is the required power level

and frequency for the Ka-Band transmitter. The problem

arises since, as the klystron output cavity is scaled to higher

frequencies, its dimensions are decreased. It soon becomes

impossible to extract 400 kW CW from the beam in the

reduced interaction volume without exceeding a power density of 1 kW/cm 2 on the cavity walls. The power density of

1 kW/cm 2 represents an approximate upper limit which is

set by the present state of the art in cooling technology.

In order to increase the interaction volume, an unconventional microwave tube, the gyroklystron, will be used. Two

possible configurations for such a device are shown in Fig. 1.

In both cases several circular waveguide cavities operating in

the dominant TE11 circular waveguide mode prebunch the

beam. The energy is extracted in the output cavity, which is

an open-ended resonator that resonates in the higher order

TEl2 circular waveguide mode. This allows the output cavity

dimensions to be increased thus allowing 400 kW to be

extracted from the beam without exceeding 1 kW/cm 2 on the

cavity walls. The microwave energy then exits the output

cavity in a rotating TEl2 circular waveguide mode.

Unfortunately the TEll mode, not the TEl2 mode, is the

most suitable mode to use in the rest of the system. The

TE_2 mode has a radiation pattern which is totally unsuitable

for illuminating the subreflector. The pattern is multilobed

with the main radiation appearing off the waveguide axis. The

TEl i mode on the other hand has the conventional dominantmode pattern which is easily modified by the feed to give the

43
optimum Gaussian pattern for illuminating the shaped subreflector. For these reasons it is necessary to develop a device

which will convert the TEl2 mode to the TEl1 mode. Two

different locations for the converter are possible, one inside

the vacuum envelope of the tube (Fig. 1[a]), and the other

in the 1.75-in. transmission line (Fig. 1 [b]). This article

describes experimental and theoretical results for a small

diameter device suitable for the in-tube configuration depicted

in Fig. 1 [a].

II. Mode Converter Theory

When the diameter of a circular waveguide exceeds 0.766

_o at the frequency of operation, the microwave signal may

propagate in more than one circular waveguide mode. For a

perfectly straight circular waveguide, these modes are orthogonal and no energy is exchanged between them. When the

guide deviates from perfection, either by design or by accident,

mode conversion occurs, and the modes become coupled.

The general deformed waveguide may be specified by writing the radius as a function of z, and @as follows (Ref. 4, and

J. Doane, "Propagation and mode coupling in corrugated

and smooth wall circular waveguide," Plasma Physics Laboratory (internal document), Princeton, New Jersey):

When the _ = Im - il perturbation is chosen the incident

TEl mode is coupled to all the TE(i+_)p and TM(i±_)p modes.

In order to enhance the coupling to only the desired TE(i+Q),

mode the perturbation is repeated at a specific interval in z,

which is approximately given by the beat wavelengt h between

the two modes of interest. The beat wavelength between

modes 1 and 2, _1,2 is given by

_'1 h2

Xl, 2 = IX1 _X21 (2)

In summary the rough design of the TEi/- TE,nn converter

consists of a £ = li-ml radial perturbation repeated longitudinally at an interval given by Eq. (2) where mode 1 is the

TEi/mode, and mode 2 is the TEmn mode.

In order to accurately determine the number of perturbations required, their magnitude, and their exact placement, a

detailed analysis of the coupled mode problem must be undertaken. The propagation in an arbitrarily deformed circular

waveguide can be described by the following matrix equation:

aA(z)

dz - --/ [/3(z)] A(z)+ [C(z)] A(z) (3)

r(z,@) = ro+ E ct_(z) cos£@+ E a_(z) sink@ (1)

k

In general, the azimuthal order of the perturbation (£ and k)

determines which modes will be coupled. For example, a pure

£ = 1 perturbation causes coupling between the TEtj , TMi]

mode group and the TE(i±l)r a and TM(/_I)m mode groups. The

= 1 perturbation corresponds to curvature. Therefore an

incident TEll mode is coupled to the TEora, TMora, TE2ra and

TM2m modes through the curvature. Similarly a radial perturbation with no azimuthal variation, £ = 0, and only a longitudinal variation, couples modes with the same azimuthal

index. This is the type of coupling that occurs in circular waveguide tapers and horns.

The first step in a mode converter design is to determine

what order of azimuthal variation is required. For example, a

TE.. - TE converter requires a perturbation of order £

t! mn

where £ = Im - il. For modes of the same first index a circularly symmetric (£ = 0) perturbation is required, for modes

differing by 1 in first index a curvature (£ = 1) perturbation is

needed, modes differing by 2 require an elliptical deformation,

and so on.

Here

A(z) = a vector containing the mode amplitudes

[/3(z)] = a diagonal matrix containing the propagation

coefficients

[C(z)] = a matrix containing the coupling coefficients for

the local waveguide perturbation.

Each term of [C(z)], C.., is determined by the azimuthal tl

order of the local perturbation, its magnitude, and the specific

modes i and / (Ref. 4, and J. Doane, "Propagation and mode

coupling in corrugated and smooth wall circular waveguide,"

Plasma Physics Laboratory (internal document), Princeton,

New Jersey). The propagation of coefficient l_ii is determined

for each mode by using the local radius.

A computer program was developed to solve Eq. (3) for the

special ease of arbitrary radial perturbations with no azimuthal

variation r(z, ¢)= r(z), and modes with first index I. This is

sufficient for the specific mode converter design required,.

TEl2 - TEl 1' i.e., £ = 0.

44
F.oUowing the method outlined by Moeller (Ref. 5) the

radial perturbation was taken to be sinusoidal with respect

to Z,

[ 27rz

r(z) = ro + Ar cos _-'_-B-B] (4)

The average radius was chosen to be 0.423 inches which is

slightly larger than the radius of the tube's output cavity. The

number of ripples was chosen to be seven, and in order to

obtain maximum conversion efficiency at the design frequency

of 34.5 GHz optimum values of 0.039 inches and 1.364 inches

were found for Ar and Xa respectively. These values were

op '" " ' by SUIVIII_I ___ r_Ll'r'- _._)t'_" tOlZ" ...... : ....... _OlllLIllldl.lOllbk:---': .... Ulg AF tlmlzeu vat to u,,

and Xa. Seven ripples were chosen since the theoretical ripple

magnitude required for the seven ripple device was small

enough that no modes become trapped in the ripples at

the design frequency of 34.5 GHz. This condition is required

for accurate modeling of the interaction with the existing

computer code. A cross sectional view of the final device is

shown in Fig. 2.

The mode converter is a reciprocal device. That is, an input

TEl2 mode will be converted to an output TEI_ mode with

exactly the same efficiency as an input TE l _ mode is converted into a TEl2 mode. Although the final device will be

used as a TEl2 - TE 11 converter it is simplest to test it reciprocally as a TEll - TEl2 mode converter. Figure 3 shows the

mode composition as a function of z when a TE 11 mode is

incident on th.e device. In this plot, and throughout the

remainder of this paper, dBc denotes the amount of power

carried by any waveguide mode with respect to the input

TEll mode power. The theoretical final output mode composition is summarized in the last column of Table 1. The

efficiency of the conversion is found to be 99.77%. The device

efficiency when it is operating in the reciprocal mode will

also be 99.77% but the spurious output power will be contained in other modes. The conversion efficiency is plotted as

a function of frequency in Fig. 4. The computer program was

also used to determine the allowable tolerances on re, At, and

Xa to be given to the machine shop in order to ensure high

conversion efficiency for the fabricated device. The most

sensitive of the parameters was found to be the average radius,

re, which must be held to -+13.001 inches to maintain 99%

efficiency. An error of about -+0.002 inches was found to be

acceptable for the ripple magnitude, At, while significant

errors in the ripple period, Xa, were permissible.

III. Experimental Results

The device depicted in Fig. 2 was fabricated in 3 sections,

one section of 3 ripples, and two containing two ripples each.

In addition to making the fabrication easier, dividing the

device up in this manner also allows experimental measurement of the mode content after 2, 3, 4 and 5, as well as the

total number of ripples, 7. Two additional tapers were fabricated, one from the existing rectangular to circular waveguide

transition output radius of 0.184 inches to the converter

input radius of 0.462 inches, and one from the converter output radius of 0.462 inches to the proposed transmission line

radius of 0.875 inches.

As we discussed earlier the mode converter is tested reciprocally as a TEl1 - TEl2 mode converter. A block diagram of

the experimental set-up used on the antenna range is shown in

Fig ¢ A _;.... i............ ;a= ,.......... ,o )h .... ,angul°, ,,_

circular transition output diameter to the input diameter of

the converter. The TE 11 mode undergoes some mode conversion in this taper, and the TEl1 plus the spurious TEIn and

TMln modes then enter the rippled sections. The mode most

strongly coupled in this taper, the TM II mode, was measured

to be at level of approximately 22.0 dB below the TE 11 power

at the first taper output. The method of determining mode

content via pattern measurement is discussed in Appendix A.

This sl!ghtly impure signal then passes through the mode converter, and through an up taper (which also causes a slight

amount of additional mode conversion) to the final diameter

of 1.75 inches. By inserting a variable number of rippled

sections between the two tapers, and measuring the far field

pattern of the taper-ripple-taper chain, the mode content after

0, 2, 3, 4, 5 and 7 ripples was determined.

The patterns measured when no sections are inserted

between the two tapers are shown in Fig. 6, along with the

ideal TEl I patterns which would be measured if there were no

mode conversion in the two tapers. Figure 7 shows the measured patterns when the entire converter (7 sections) is insorted between the tapers. The theoretical TEl2 patterns,

which would be measured if the tapers caused no mode

conversion and the mode converter were perfect, are also

plotted in Fig. 7. The intermediate patterns measured when 3

sections were inserted between the tapers are shown in Fig. 8.

Measurements were also taken for 2, 4 and 5 ripple configurations, at the design frequency of 34.5 GHz.

By using the methods described in Appendix A, the mode

composition at the output of the final taper was estimated for

each of the configurations mentioned above. In particular,

estimates for the O, 3, and 7 ripple cases are summarized in

Table 1. The estimated mode content from the measurements

is also plotted on Fig. 3 for comparison with theory.

A f'mal set of measurements was made to experimentally

determine the conversion efficiency vs. frequency characteristics of the mode converter. For these measurements the

45
total converter was inserted between the tapers, and pattern

measurements were taken for frequencies of 34.0, 34.2, 34.4,

34.5, 34.6, and 34.8 GHz. The mode composition was then

determined and the results for the measured efficiency vs.

frequency are plotted in Fig. 4 for comparison with theory.

IV. Discussion

In comparing the experimental and theoretical results for

the patterns of the tapers connected together with no rippled

sections between them, we see that excellent agreement is

found between the theoretical TE l_ H plane pattern and the

measured H-plane pattern. The agreement in the E plane is not

as good. This is expected since the taper system produces

spurious TM11 and TM12 modes whose effects are only seen in

the E plane. Figure 8 again shows good agreement in both

planes, with more error in the E plane. This can be explained

by considering the taper effects again. The short 0.368-

0.924 inch taper generates some spurious TM 11 power which

enters the mode converter. Computer simulations show that

spurious TM 11 power entering the device will be converted primarily into TM modes at the output of the device. Once again,

the effects of these modes are only visible in the E plane. The

slight assymetries which are detectable in the measured patterns are probably due to slight misalignments between the

taper and mode converter sections.

When the theoretical mode content along the converter and

that derived by pattern measurement after 0, 2, 3, 4, 5 and 7

ripples are compared in Fig. 3, excellent agreement is found

for the TE_I and TEl2 modes. However, poor agreement is

found for the TM11 and TMI: modes. As was mentioned

earlier a spurious TMll signal at a level of approximately

-22 dBc was found at the output of the first taper. This signal then enters the rippled sections where it is converted primarily into TM modes. These spurious effects overwhelm the

calculated TM 11 and TM 12 conversion effects for a pure TE 11

mode input, but have little effect on the dominant TEll -

TEl2 interaction. This explains the good agreement seen for

the TEl1 and TEl2 modes and poor agreement for the TM

modes. Despite the spurious effects an overall efficiency over

99.5% was measured for the device (See Table 1), which is

also in good agreement with the calculated value of 99.77%.

For the bandwidth results shown in Fig. 4 the best agreement between theory and measurement is found for the points

within 100 MHz of the design frequency. For frequencies further removed from 34.5 GHz some of the disagreement may

be attributed to errors in the theoretical calculations. For all

frequencies, four forward traveling modes were used to model

the interaction in the ripples. For the higher frequencies

shown in the figure this is probably not sufficient since the

TEl3 mode may also propagate. For the lower frequencies the

TM12 mode becomes trapped in between the ripples, and

reflected waves which are not included in the computer model

become important.

V. Conclusions

In conclusion, theoretical and experimental results for a

prototype TEl2-- TE 11 mode converter have been presented.

Good agreement between theory and experiment was found in

most cases, and reasonable explanations have been given for

the instances where some disagreement has been found. The

computer code used to generate the theoretical results presented in this report may now be used with confidence in the

design of multimode tapers, mocle converters and other

devices. The pattern measurement technique for mode identification has also been proved, and may be used to characterize

other components. Future work will include an upgrade of the

computer code allowing the inclusion of reflected modes. This

places confidence in the design of a possible future mode converter which may be placed in the 1.75 inch diameter transmission line for use with a system configured as shown in

Fig. 1[b].

46
Acknowledgments

The author would like to thank H. Reilly who supervised the fabrication of the tapers

and mode converter sections, and S. Stride who assisted in the far field pattern measurements.

References

1. Bhanji, A., Hoppe, D., Hartop, R., Stone, E., Imbriale, W., Stone, D., and Caplan, M.,

High power Ka-band transmitter for planetary radar and spacecraft uplink, TDA Pro-

_r_._.¢Report d2-7_ (Anril-June | 9R4_. Jet Propulsion Laboratory. Pasadena. CA. vv.

24-48.

2. Hoppe, D., ImbrUe, W., and Bhanji, A. The effects of mode _purity on Ka.ba,d systern performance, TDA Progress Report 42-80 (Oct.-Dec. 1984), Jet-Propulsion

Laboratory, Pasadena, CA, pp. 12-23.

3. Hoppe, D., Propagation and radiation characteristics of a multi.mode corrugated waveguide feedhorn, TDA Progress Report 42-82, Jet Propulsion Laboratory, Pasadena, CA.

4. Levine, J., Rippled wall mode converters for circular waveguide, International Journal

of Infrared and Millimeter lqaves, Vol. 5, No. 7, pp. 937-952, 1984,

5. Moeller, C., Mode converters used in doublet III ECH microwave system, Int. J. Elect.,

Vol. 53, No. 6, pp. 573-585, 1982.

47
Table 1. Measured TEll-TEl2 mode converter perlormance. Mode composition after N mode

converter sections.

Mode

Measured Theory

N=0 N=3 N=7 N=7

TEl 1' % 98.23 59.09 * 0.005

TM11, % 1.66 2.45 0.19 0.17

TEl2, % * 37.99 99.57 99.77

TM12, %. 0.0025 0.48 0.21 0.05

Others, % * * 0.025 "0

* No power measured down to system sensitivity.

48
(Q)

_- TUNING /_ OUTPUT • /--- UP TAPER /_ DOWN TAPER

\DIAPHRAM / CAVITY / /

PIERCE _ BUNCHER / (TE121) /COLLECTOR /

..... _ CAVITIES / _ _ _dF

L .....

i ELECTRON BEAM

/DRIVE I MODE CONVERTER WINDOW

/-- WIGGLER (TEl2 "-_TE II) (TEl I) _ HELICAL

MUU_..

FILTER

(b)

MAGN ETRON

I NJ ECTIO N

SPLIT TUNABLE

BUNCHER

CAVITIES (TEl 11)

LIPTAPER

CAVITY

(TE121)

COLLECTOR

ANTENNA

INPUT

DRIVE mmm

SCRAMBLER

MAGNETS

(TEl2)

MODE CONVERTER

(TEl 2"-_"TEl I )

Fig. 1. Two possible gyroklystron configurations: (a) pha_ 1, (b) phase 2

_e-!. 364---_

TEl2

R(Z) = R0 +Z_R _ (2_ Z/X B)

TEl2

Fig. 2. TEI2-TE_l mode converter

TE11-.-..D,-

@TEl 1

49
I I I I I I I I I

TEl2

-10

"1o

_.1

-40

0

0

0

/ \

I TMI 2

-_ / I

I I / I

I I / I

I I I / I

I I i// _/ _..

I I o

| I EXPERIMENT V _ THEORY

I I O TEll, TM11, AND TEl2 --_

II

I o TMI2

I I I I

1 2 3 4 5 6 7 8

Z, in.

Fig. 3. Mode composition as a function of longitudinal coordinate for the

TE_l-TE12 mode converter

I0

SO
z

U

=.

Z

o

==

>

Z

O

U

100/ I I I I I I I I

0

96

94

9O I I I I I I I

34.0 " 34.2 34.4 34.6

f, GHz

THEORY

O EXPERIMENT

I

I I

34.8 35.0

Fig. 4. TF.._l-TE12 mode ¢onverldon efficiency Vl frlqllll_

WR 28

DETECTOR

(MIXER)

I

TO

RECEIVER

mWR 28 TO

0.368" ID

TRAN SI TIO N H.,O,,L.O _3"'"-°-'-"1 ,,--.. I s,c,,o.s °F'''E°pt"°F"'° o.,.,,.,.,s,. ,o I H

Fig. 5. Antenna range test set up

ILLUMINATING

ANTENNA

51
-5

-10

-15

-2O

-25

-3O

-35

-- (a)

I .I I I I I I I

I I I i

I I I I

I

-120

M

I I

I I t i

I I

I I I I

I

-100

I i Ill

I I, U

I |!

I |!

I II

40 -60 -40

I t I

-2O 0 20

POLAR ANGLE, d_

I/'_ i \

i1 / \

rl l , ',

IW\',,

I I 1 t

Fig. 6. Taper radiation patterns: (a) H plane pattems, (b) E planepatterns

-- THEORY

------ EXP_IMENT

I I I I

I I I I

I

100

I I I

120 140 160 100

52
-5

-I0

-15

-2O

-25

-3O

-35

Q

2

-40

< 0

-I0

-15

-2O

I

I I I I I I 1 I

n

r

i I i

THEORY

---- EXPERIMENT

-25

-3O

-35

I I I I I I I I I I I I I I

Cb)

•-40 I I

-180 -160 -140 -120 -100 -80 -60

_. , i\

I

/ i I i

"\ I"\

l I \

40 6O 8o

I I

•-40 -20 0 20 100 120

POLAR ANGLE, dag

I

140

Fig. 7. Total mode converter radiation patlerns: (a) H plane paltema, (b) E plane patterns

160 180

$3
,.u"

2

.<

-5

-10

-15

-2O

-25 -

-30--

-35

-40 I

-180 -160

I

-140

I

-120

I I I t I I I I

------ E PLANE

-- H PLANE

I I I

100 120 140 160 180

Fig. 8. Three dpple mode converter. Measured radiation patterns.

54
Appendix A

Mode Identification Using Far Field Radiation Pattern Measurements

In this appendix a method of mode identification in a

multimode circular waveguide is explained briefly. I The far

field radiation pattern for an arbitrary combination of circular

waveguide modes assuming small reflection from the aperture

is given by (Refs. A-l, A-2):

E(R, 0, 0) = k exp(-/RR)(l + cos 0) a2 2R

m,n

- sin0

_(ka sin0)

kasin0

Jm(kasm0) \

7- - _ -- -|sinm Ca.

x_,. - (h .n 0)7 "

],_(ka sin 0) 1

x C,,,,,(e,..)'t2 exp(/¢,.,,) , • costao%

(x_,.)-(kasin0)" J

(A-l)

where

k = 27r/Xo

ko = Wavelength

R = Far field radius

a = Waveguide radius

Pmn = Power carried by the T_mn mode

_mn = Power carried by the _mn mode

Cmn = Phase of the TEmn mode

¢mn = Phase of the TMmn mode

Cmn = Normalization constant for the TEmn mode

Cmn = Normalization constant for the TMmn mode

Xmn = Zero OfJm(X )

z A more detailed discussion can be found in Z. Zhang, M. Thumm, and

R. Wilhelm, "Far field radiation patterns from open-ended oversized

circular waveguides and identification of muRimode outputs of gyrotrons," Institute fur Plasma fur schung, Universtat Stuttgart (internal

document).

_n = Zero OfJm(X )

M, N = Indices chosen large enough to include all propagating modes in the waveguide.

Equation (A-l) assumes that only one of the two orthogonally

polarized sets of modes, the set where E, _ sin _, exists in the

waveguide. If we further assume only modes with one azimuthal

variation exist, m = 1, as in the TEl2 - TE 11 mode converter,

or tapers, we may write the foUowing equations for the form

of the radiation in the E and H planes, respectively, as

E plane

/

E = L__,_ [C'n(Pln) 112 exp(/C_ln) '11(ka sin O)

n=z \-i ka sin 0

ka sin 0 Jl (ka sin O)1

- _n(_n)'/2 exp(j_n) X2----£-(--kasi'_0-_ _ /

H plane

N

E cz E Cln(Pln )1/2 exP(/_In)

n=l

l[ (ka sin 0)

(X'ln): - (ka sin 0) 2

(A-2)

(A-3)

From Eq. (A-3) we see that only TE modes contribute to the

radiation in the H plane. Furthermore, if we examine the

radiation in this plane at the point ka sin 0 = Xtlp , where

J' (X')= 0, it is found that all TEln modes have a null at 1 Iv

tlais point, except the mode TElt , where the denominator and

numerator in Eq. (A-3) vanish. Thus, by examining the H

plane pattern at the points ka sin 0 =X'ln n = 1,... ,N the

relative levels of all the TEln modes may be determined, in

terms of power, since the normalization constants Czn are

known.

Similarly the relative levels of the TMln modes may be

determined by examining the E plane pattern at the points

where ka sin 0 Xln. In this manner all of the TEln , and TMIn

modes may be identified. The method may be extended to

include the orthogonal set of m -- 1 modes by including polarization considerations, and the modes with other azimuthal

variation, m = 0, 2, 3 .... may be identified by taking more

pattern cuts)

2Z. Zhang, M. Thumm, and R. Wilhelm, op. cir.

55
A-1.

A-2.

References

Silver, S., Microwave Antenna Theory and Design, Rad. Lab. Ser., Vol. 12. New

york: McGraw-Hill, pp. 336-338, 1949.

Ludwig, A.C., Radiation pattern synthesis for circular aperature horn antennas,

IEEE Trans. Antennas and Propagation, Vol. AP-14,pp. 434-440., July 1966.

56
TDA Progress Report 42-82

N86-10388

April-June 1985

Propagation and Radiation Characteristics

of a Multimode Corrugated

Waveguide Feedhorn'

D. Hoppe

RadioFrequencyandMicrowaveSubsystemsSection

This report describe= a prototype of the multimode corrugated feedhorn which will

be used in the 400 kW CW Ka-band radar system. A rough design is done using coupled

mode theory and standard comtgated waveguide modes. A more exact analysis using

mode matching techniques is then used which takes into account the effect of a finite

number of" corrugations per wavelength and determines the modes which are reflected

from the device. A prototype feedhorn has been constructed and measured. These experimental results are then compared to the theoretical predictions which agree satisfactorily

closely.

I. Introduction

A previous report described a conceptual design for a

400 kW CW Ka-band radar transmitter (Ref. 1). In order to

transport this large amount of CW miUimeter wave power, a

multimode transmission line must be used. By using optimally

designed waveguide tapers, mode converters, and monitoring

devices nearly all of the microwave power at the end of the

transmission line will be contained in a rotating TE II circular

waveguide mode. The chosen diameter of the multimode

waveguide system is essentially the correct size for optimally

illuminating the subreflector, but the beamwidth in the

E and H planes should be equalized for best antenna efficiency

(Ref. 2). Equal E and H plane beamwidths may be obtained

by transforming the TE 1] mode into the balanced HE 11 mode

in multimode corrugated waveguide. This report describes

the theoretical and experimental results for a multimode

corrugated waveguide section that accomplishes the above

transformation.

II. Theory

Two types of horn antenna possessing equal E and H plane

patterns are commonly used. The first is the dual mode horn

(Ref. 3), and the second is the corrugated horn (Ref. 4).

In the dual mode horn a dominant mode circular waveguide is connected to another guide of slightly larger diameter,

where modes up to TM_ 1 may propagate, via a step transition.

57
The step size is chosen to generate the precise amount of

TMI l mode from the TE 11 mode so that when the two modes

travel through the flared horn section which follows, the E

and H plane patterns are equalized. The bandwidth of this

horn is limited since the two modes must arrive at the horn

aperture in phase, and the two modes have phases velocities

which vary differently with frequency.

In the corrugated horn the single mode smooth wall waveguide is connected to a corrugated waveguide which supports

only the HE 11 mode. Some matching between the waveguides

is provided by gradually changing from ),/2 slot depth to

>`/4 slot depth in a short transition region. Throughout the

transition region only the HE11 corrugated waveguide mode

may propagate, and the E and H plane radiation patterns of

this mode become nearly equal when the balanced condition

is reached (slot depth _- >,/4). The bandwidth of this horn is

larger than that of the dual mode horn since the transverse

electric field pattern and hence the radiation pattern of the

HE11 mode are relatively insensitive to small changes in slot

depth around the balanced condition (slot depth _ >`/4). After

the HE11 mode is established in the single mode corrugated

waveguide, the guide is gradually flared, without changing the

slot depth, to the required aperture size.

Special problems arise when multimode smooth and corrugated waveguides are used. In the specific system considered in Ref. 1 a 1.75" I.D. circular waveguide is used at 34.0

GHz allowing over 100 waveguide modes to propagate. The

radiation pattern of the TEll mode from the open ended

multimode waveguide may be improved by either abruptly

converting some of the energy to the TM11 mode, as in the

dual mode horn, or gradually changing the TE 11 mode into the

balanced HEll mode as in the corrugated horn.

Any attempts to mimic the performance of the step in the

dual mode horn in a multimode waveguide are unsuccessful,

since a single step transition, slot, or iris generates a large

number of modes other than TMll. These modes ruin the

pattern symmetry of the TE 11- TM11 combination.

The second alternative is to gradually change the TEI I

mode into the HEt l mode using a multimode corrugated

waveguide in which the slot depths change from _ X[2 to

X[4. A cut-away view of such a device is shown in Fig. 1.

Under multimode conditions new problems arise in making

this transition. A 1.75" diameter corrugated waveguide

operated at 34.0 GHz also supports many propagating modes,

even when the slot depth is constant. When the depth is

changed along the guide, coupling occurs between the various

corrugated waveguide modes (Ref. 5), in much the same way

as radius changes in smooth multimode waveguide cause

mode conversion.

The smooth walled circular waveguide may be considered

to be a special case of corrugated waveguide for which the slot

depth is zero or _/2 at the operating frequency. The propagation in a corrugated waveguide of varying or constant slot

depth, D, may be written in terms of coupled wave equations (Ref. 6):

a_.A_(z) = 4_(o) Ai(z) dz

N

- E l,

]= 1

O)

In this equation Ai(z ) is the amplitude of the ith corrugated

waveguide mode and _z(D) is the phase constant of the ith

mode, which is a function of the slot depth at the particular value of z. If the slot depth is changing along the guide

(dD/dz _: 0), then coupling occurs between the possible corrugated waveguide modes. The magnitude of the coupling

between modes i and j is represented by Cii(D), and it varies

approximately as 1/[/3z(D)-/31(D)], becoming larger for modes

with more equal phase velocities. The quantity/3(D) is found

by solving a transcendental equation whose solutions depend

on the guide slot depth. For slot depth = 0 or >`/2, the solutions reduce to those corresponding to TE and TM modes in

the smooth walled waveguide. For depths of approximately

_[4 and 3X/4 the solutions are those corresponding to the

balanced HE and EH modes of corrugated waveguide. The root

for the TEll mode (1.841), D = k/2, changes to that of the

balanced HEll mode (_2.401) as D decreases to ),/4. If the

coupling term in Eq. (1) is neglected, then we find that the

TE H to balanced HEll mode conversion is perfect (100%).

This is the case when only one mode propagates in the guide,

as in the transition section of the standard corrugated horn.

In the case of a multimode corrugated waveguide where

D changes from X/2 to ),[4, the TEll- HE11 branch is only

coupled significantly to the TMII to EH12 branch, which has

the closest phase constant. In this case Eq. (1) may be

rewritten:

d

--azA_(z) = 4_(D)A_(z)-Q2(D) -_za° a_(z)

dD

a A2(z) = 4_2(D)A 2 (z) - Q_(D) --_ a_(z)

(2)

Here,

A1(0 ) = TEll input power

58
A2(0) = TMll input power = 0

AI(_) = HEll output power

A2(£) = EH12 output power

= transition section length

The design objective is to determine a slot depth profile where

D varies from ),/2 to approximately ),/4 which maximizes

AI(_ ), and minimizes A2(_ ) when Eqs. (2) are solved numer.

ically. Several profiles for the slot depth variation were con.

sidered, including a linear profile, and one which provides

constant coupling along the transition section, making

cl:(z_ ) _- =4 (3)

Here a1 contains the input mode amplitudes, b t the reflected

modes, and b2 the output modes. The matrices [811 ] and

[821 ] are scattering matrices which are determined by the

computer code. Their entries are a function of frequency

and device geometry. This computer program was used extensively in another development (Ref. 8), which calculated the

theoretical radiation patterns for the feed under different

excitation conditions.

A device of the type described above was designed using

the simple coupled mode theory. The more exact scattering

matrix theory was then used to predict the radiation patterns

and reflection characteristics of the corrugated waveguide

section. The part was then built and measured on the antenna

range. These theoretical and experimental results are described

in the next section.

where K is a constant (Ref. 6). Of the profiles examined, this

profile gave maximum conversion efficiency in a minimum

length £. For this profile the slot depth changes most gradually near depths of ),/2, where the phase constants of the two

branches are closest, and more rapidly as D approaches _/4.

This profile was chosen for the actual device which was built

and tested.

Certain assumptions are made in the above analysis. In

particular, an ideal corrugated waveguide is modeled where an

infinite number of slots per wavelength are assumed. Reflections are ignored, and only the main coupling interaction is

considered. A more exact analysis of the-problem which

eliminates all of these faults can be undertaken. In this method

the corrugated wavegnide is modeled as a large number of

straight waveguide sections of different radii which are connetted in series. Electric fields are described as sums of smooth

walled circular wavegnide modes, and are matched at each of

the discontinuities. The result is a scattering matrix which

relates the reflected circular waveguide modes at the input to

the device and the output modes, to the incident modes at

the input end. The theory of this approach is given in detail

by James (Ref. 7), and a computer code to carry out the

computations has been written by the author I (see also

Ref. 8, section B). The scattering matrix approach may be

summarized by two matrix equations:

(4)

I See D. J. Hoppe, "Scattering Matrix Program for Circular Waveguide

Junctions," Interoffice Memorandum No. 3335-84-071 (internal

document), Dec. 5, 1984, Jet Propulsion Laboratory, Pasadena, CA.

III. Calculated and Experimental Results

The experimental model of the corrugated feed consisted

of 92 slots cut in the wall of a 1.75 inch I.D. circular waveguide. The width of the slots and lands was chosen to be

0.035 inch. These parameters were' chosen for the test device

for several reasons. First, Eq. (2) was solved for a device of

various lengths using the constant coupling slot depth profile.

From this rough analysis it was found that feed lengths in

excess of 15), gave an HEll mode purity level above 95%.

Next a very conservative number of slots per wavelength, 5,

was chosen for the test device. The final consideration was the

machining complexity. It was decided to keep the number of

slots under 100, and 92 were chosen which corresponded to an

HEI 1 mode purity of 96.5% from the simple analysis. The slot

depth profile was determined by the methods described in the

previous section, and each slot depth was specified individually. The final design parameters were then used as input

data for the scattering matrix program which produced the

final theoretical radiation patterns. The detailed calculation

of the scattering matrix for the device involved the cascading

of 370 40-by-40 scattering matrices representing the straight

sections and waveguide junctions making up the device.

The scattering matrix method calculates the matrix for

each subsection of the total device (i.e., after 1 slot, 2 slots,

.... up to 92 slots). From this we may determine the aperture

modes that would be present if the device were terminated

after'any number of slots. The results of this calculation for a

frequency of 34 GHz are shown in Fig. 2. The figure shows the

increasing level of TM 11 and decreasing level of TE 11 as more

slots are added, as well as a few of the other modes which are

present. The final aperture modes and reflected modes are

shown for the 92 slot device in Table 1. The aperture modes

were then used to determine the E and H plane far field

patterns of the feed. These results are shown in Figs. 3 and 4.

59
The phase center of the feed was calculated as being exactly in

the aperture plane. The efficiency of the 64-meter antenna in

its present hyperboloid-paraboloid configuration, neglecting

surface tolerance errors, was calculated using the theoretical

patterns of Figs. 3 and 4. This efficiency was calculated to be

80.8% using the equivalent paraboloid method, and 79.5%

using the GTD program (Ref. 8). Some theoretical results for

the bandwidth were also determined. The calculated feed

patterns for frequencies of 33.5 and 34.5 GHz are shown in

Figs. 5 and 6, respectively. The theoretical antenna efficiency

at 33.5 and 34.5 GHz was determined to be 81.0% and 80.4%

respectively, both numbers being calculated using the equivalent paraboloid method.

A block diagram of the experimental test set up used on the

mesa is shown in Fig. 7. The transition from single mode

(0.368 I.D.) to multimode (1.75 I.D.) waveguide is a profiled

circular waveguide taper. An ideal taper would produce a

perfectly pure TEI I mode at the 1.75 I.D. end. For this

particular taper the TMI1 mode, which is the mode most

strongly coupled to the TE II mode in the taper, was measured

to be about 29 dB below the TE 11 mode at the taper output.

The measured patterns for the device are plotted in Figs. 3

and 4 for comparison. Excellent agreement is found in both

planes with the best agreement in the H-plane. The slight

differences in the E-plane may be accounted for by recalling

that the calculated pattern is for a pure TE 11 mode input, but

in the experiment some spurious TM 11 power is also present.

The effects of this slight mode inpurity appear primarily in the

E plane (see Ref. 8, Figs. 4 and 6). The experimental phase

center of the feed was also determined to be in the aperture

plane, which is also in agreement with theory.

IV. Conclusions

In conclusion, excellent agreement between the theoretical

predictions of the scattering matrix program and experiment

were obtained for a multimode corrugated feed section at

34 GHz. Since the program is also capable of analyzing corrugated horns of arbitrary profile and various other circular

waveguide devices, it is expected to prove to be a valuable

tool for analysis in the future.

6O
Acknowledgments

The author would like to thank H. Reilly who supervised the fabrication of the various

waveguide components and S. Stride who assisted in the pattern measurements.

References

1. Bhanji, A., Hoppe, D., Hartop, R., Stone, E., Imbriale, W., Stone, D., and Caplan, M.,

High power Ka-band transmitter for planetary radar and spacecraft uplink, TDA

Progress Report 42-78 (April-June 1984), Jet Propulsion Laboratory, Pasadena, CA,

pp. 24--48.

2. Potter, P.D., and Ludwig, A., Antennas for space communications, JPL Space

Program Summary No. 37-26, Vol. IV, Jet Propulsion Laboratory, Pasadena, CA,

April 30, 1984, pp. 200-208.

3. Potter, P. D., A new horn antenna with suppressed sidelobes and equal beamwidths,

Microwave Journal, June 1963, pp. 71-78.

4. Clarricoats, P.J., and Saha, P.K., Propagation and radiation behavior of corrugated

feeds, Part 1, Prec. IEE, September 1971, pp. 1167-1176.

5. Dragone, C., Reflection, transmission and mode co.nversion in a corrugated feed,

Bell System Tech. J., July-August 1977, pp. 835-867.

6. Doane, J.L., Mode converters for generating the HE 11 (Gaussian-like mode) from

TEol in circular waveguide, Int. J. Elect., Vol. 53, December 1982, pp. 573-585.

7. James, G. L., Analysis and design of TE11 and HE 11, corrugated cylindrical waveguide

mode converters, IEEE Trans. Microwave Theory Tech., Vol. MTT-29, October 1981,

pp. 1059-1066,

8. Hoppe, D., Imbriale, W., and Bhanji, A., The effects of mode impurity on Ka-band

system performance, TDA Progress Report 42-80 (October-December 1984), Jet

Propulsion Laboratory, Pasadena, CA, pp. 12-23.

61
Table 1. Aperture and reflected modes for a pure TEl1 mode incident on the feed

Percent of dB Below

Modes Incident TE 11 Power, Incident TE l I Power,

% dBc

Reflected Modes

TEl I " 0.0002 -58

TM11 0.0012 --49

TEl2 0.0001 -62

TMI 2 0.0010 -50

TEl3 < 0.0001 -65

TM] 3 0.0015 -48

TEl4 < 0.000l -65

TMI4 0.0054 --43

TEl5 0.0001 -62

Aperture Modes

TEl I 81.95 -0.86

TM 11 17.76 -7.5

TE 12 0.054 -33

TMI2 0.010 --40

TEl3 0.0013 -49

TMI 3 0.2003 -27

TEl4 0.0003 -55

TM14 0.0090 -40

TE 15 0.0008 -51

62
OTEll INPUT MODE

L

o.,,4._r l_X'/9"/"/"/'/"__/'_/'_-/"__o.,o,.

CORRUGATED SECTION WITH

VARYING SLOT DEPTH

Fig. 1. The antenna feed

roll

BALANCED HE11 OUTPUT MODE

63
10

T_

TEll

TM11

1

2

u

z

15

20

25

30

35

TM13

TEl2

40

45

50

0 16 32 48 64 80

CORRUGATION NUMBER

96

Fig. 2. Mode content vs corrugation number

112

64
-I0

<

-25

-3O

-35

-40

-5-

I

-180 -160

I I I I I T I

I

-140

I

-120

I

-I00

- A

-80 -60 -40

I A I I I

f _

I

P_

il IA_

II

-20 0 20 40 60 80 100

POLAR ANGLE, deu

I I I I I

_l THEORY

EXPERIMENT

I I I

120 140 160

Fig. 3. Theoretical and experimental E-plane patterns at 34.0 GHz

uJ

2

<

0 I

-5-

-10 -

-15 -

-20 --

-25-

--30

-35-

-4o I

-180 -16O

I I I I

I

-140

I

-129

I I

/ \

f I

I

-100 -80 -.60 -40 -20 0 20 40 60 80

POLAR ANGLE, deg

Fig. 4. Theoretical and experimental H-plane patterns at 34.0 GHz

I I I I I

lJl THEORY

EXPERIMENT

I

1GO 120 140 160 180

65
2

-5

-10

-15

-20

-25

-20

-35 -

-4O

-180

I I I I I I t J

I I I I I I

-160 -140 -120 -100 -80 -60

M I I I I I I I I

0 20 40

POLAR ANGLE, deg

Fig. 5. Theoretical feed pattems at 33.5 GHz

------ E-PLANE

-- H-PLANE

I I I I I I

_ I_ 120 I_ I_ 18U

I I I I I

I I 1 I

-140 -120 -100 -80

I I t /_ I I

-60 -40 -20 0 20 40

POLAR ANGLE, deg

Fig. 6. Theoretical feed patterns at 34.5 GHz

i I I I I I

------ E-PLANE

-- H-PLANE

I I I I I I

60 80 100 120 140 160 1gO

66
WR 28

DETECTOR

(MIXER)

TO

RECEIVER

WR 28 TO

0.368" ID

TRANSITIO N H PROFILED H 1.75" ID

TAPER CORRUGATED

0.368" - 1.75" SECTION

Fig. 7. Antenna range test set-up

ILLUMINATING

ANTENNA

67
N86-10389

TDA Progress Report42-82 April-June 1985

Load-Deflection Tests and Computer Analyses of a

High-Precision Adhesive-Bonded

Antenna Reflector Panel

C. T. Chian and R. Levy

GroundAntennasandFacilitiesEngineeringSection

New adhesive-bonded panels are being investigated as a part of an effort to extend and

upgrade the 64-m to a 70-m antenna network. Load-deflection tests were conducted on a

sample high-precision adhesive-bonded panel for comparison with design criieria. Two

computerized structural models were developed in order to predict the deformation patterns under different types of distributed and concentrated loadings. The main purpose

was to obtain empirical stiffness factors for the slit beams and girders in the panel structure. With determination and use of the empirical stiffness factors, there is a good agreement between the theoretically predicted deflections and the test measurements. It was

also found that the new bonded panels satisfy the stringent design specij_wations and

surface tolerance bounds.

I. Introduction

New high-precision adhesive.bonded panels are being investigated under the Advanced System Program's research and

development effort in order to replace the traditional rivetted

panels, reduce fabrication costs, and improve the surface tolerance characteristics.

The new adhesive-bonded panels, if successful, would be

used in the current 64-m antenna network rehabilitation and

extension project at X-band. The new panels are expected to

reduce about 0.5 dB of gain loss at X-band.

II. Panel Description

The test panel is adhesive-bonded and has a solid 0.18-cm

(0.070-inch) skin thickness with no skin perforations. It has

circumferential beams at 35.6-cm (14-inch) centers and a

radial girder at each of two sides to support the beams as

shown in Fig. 1. Beams and girders are zee sections, slit and

reinforced (Figs. 2 and 3). The test panel is fabricated under

a JPL contract to Toronto Iron Works (TIW) of Sunnyvale,

California and a subcontract to COSPAL in Bergamo, Italy.

The purpose of this test was to evaluate the response of

the bonded panels for distributed loadings that are representative of environmental wind and gravity loadings, and to

determine empirical stiffness and residual deflection factors

due to the slitting of girders and beams.

A series of tests were conducted to measure panel deflections due to stimulated wind loadings and specified, concentrated shoe loads up to 135 kg (300 lb). The geometry of the

test panels is similar to panel number 8 of the new 34-m

68
AZ/EL antennas, recently built by TIW for Deep Space Stations 15 and 45. In order to provide a higher structural stiffness, the new test panels were designed with ten back-up circumferential beams, instead of seven, as originally employed

for the 34-m antennas.

III. Theoretical Models

Two computerized structural models were developed for

simulation of the deflections of the test panel. The first model

for theoretical panel deflection analysis is the FORTRAN program PANELDEFL. PANELDEFL was designed to provide

analysis for the full set of panel configurations used in a complete antenna. The analysis model is automated using a minimum description of the essential geometry and construction

features of each panel of the set. The PANELDEFL program

provides a microwave pathlength analysis and summary for

the set of all of the p.anels, as well as for each individual panel.

The method of deflection analysis is to integrate the EulerBernoulli differential equations for the beams and girders. A

more detailed description of the PANELDEFL program can

be found in Appendix A.

The second panel structural model was developed by using

the NASTRAN finite element program for the tested panel as

a check. Since the PANELDEFL program does not use finite

element approximations, it is expected to be more accurate

than NASTRAN for the present problem. A description of

the NASTRAN panel deflection model is given in Appendix B.

Comparison of the outputs of the two programs showed the

results to be almost identical.

The results of theoretically predicted panel deflections

from PANELDEFL were compared with the measured data.

This procedure has provided good validation and correlation

of the computer models and good estimates of the material

elastic properties.

IV. Test Loading Configurations

Five series of tests were conducted on the adhesive-bonded

panel, which comprised both the distributed loading and the

concentrated loading cases (Table 1). The test configurations

were of two kinds:

A. Panel without center supports

B. Panelwith center supports.

The test bed consists of a steel surface plate and I-beam

fixture that provided rigid panel support and measurement

points. The whole test bed, together with the sample panel,

is shown in Fig. 4. A schematic diagram of the test panel is

given in Fig. 5. A total of 19 dial indicators were used to

measure the panel deflections in various locations, which are

shown in Fig. 6. The test configurations are described as

follows.

A. Panel Without Center Supports

The maximum distributed panel loading was 39 kg/m 2

(8.0 PSF). This distributed load for the sample panel corresponds to a uniform thickness of about 2.54 cm (1.0 inch) of

sand. Up to 56 sandbags with predetermined weights were

used to represent the distributed loading as shown in Figs. 7

and 8.

-"lne gird er ....................... oenumg stze_s wax _itllllitl._tl to I.. U¢ Ullly ^_,.. ^1. at/utlt ....

210 kg/cm 2 (3000 psi), which is less than one-tenth of the yield

stress. After the distributed load test was completed, the concentrated load test was conducted. A concentrated load of

85.5 kg (190 Ib) was used. Steel plates 30 × 30 × 2.54 cm

(12 X 12 X 1 inch) were placed on top of a smaller steel plate

10 X 30 × 2.54 cm (4 X 12 × 1 inch) to simulate the concentrated shoe load as shown in Fig. 9.

B. Panel With Center Supports

Both the distributed load test and the concentrated load

test were also conducted for this panel test configuration. The

distributed load tests were performed first. The maximum

distributed loading, with center supports in place, was about

78 kg/m (16.0 PSF) on the gross area, which corresponds to

about 5.08 cm (2.0 inches) of sand.

This is a typical estimated wind load that the panel is

required to support, allowing for 40% perforation area. This

loading occurs when the antenna is tilted to the zenith position

under a 160 km/hr (I00 mph) wind. Up to 84 sandbags with

predetermined weights were evenly distributed on the panel.

After the distributed load test was completed, the concentrated load test was then conducted. Concentrated loads of

99, 126, and 135 kg (220, 280, and 300 lb) were used. Again,

steel plates of 30 X 30 X 2.54 cm (12 × 12 × 1 inch) were

placed on top of a smaller I0 X 30 × 2.54 cm (4 X 12 × 1

inch) to simulate the concentrated shoe load. Each of the

30 × 30 X 2.54 cm steel plates weigh about 18.5 kg (41 lb),

while the smaller plate (10 × 30 × 2.54 cm) weighs approximately 6.3 kg (14 lb). Each load increment was intended to

have a minimum of 20 minutes duration time. The recorded

data reflected this time effect.

V. Comparison Between Theoretical and

Test Results

Verification of the two panel structural models was made

against the measured data to compare the two different

69
numerical approaches. Measured deflection data taken from

the 19 dial indicators were recorded and compared with the

theoretically predicted values.

The results are tabulated in Tables 2 and 3. Table 2 shows

comparisons between the two numerical approaches for the

following two cases: (1)Panel without center supports, with

distributed load at 39 .kg/m 2 (8 PSF); (2) Panel with center

supports, with distributed load at 78 kg/m 2 (16 PSF). The

results from the PANELDEFL and NASTRAN programs

differed only by 3%.

Empirical stiffness factors for the panel back-up beams and

girders were used to account for the slits. These stiffness

factors were determined in an iterative process to achieve a

good agreement with the measured deflection data. For the

sample panel tested, it was found that for Girder Stiffness

Factor (GSF) of 1.00 and Beam Stiffness Factor (BSF) of

0.80, the test and theory agrees to within 12%, for most of

the data (Table 3). Therefore, moments of inertia for the slit

beams, in the current configuration, are computed as 80% of

the moments of inertia for the beams without slitting. The

rivetted panels had been load-tested previously.

Comparisons of stiffness factors obtained from different

tests indicated that the empirical stiffness factors are strongly

dependent on the panel configurations.

Table 4 shows the comparison of empirical stiffness factors

for panel girders and beams for three tests. A 3-day duration

test was also conducted to study the hysteresis of the panel.

A concentrated load of 135 kg (300 lb) was applied at the center of the panel and maintained for three days. Daily readings

of the dial indicators were made. However, after the load was

removed, no permanent deformation was observed.

Vl. Summary and Conclusions

At the specified level of loads, the tested adhesive-bonded

panel was found to deflect linearly with the load, for both the

distributed and concentrated loading cases as shown in Figs. 10

and I 1. Deflections showed insignificant hysteresis when the

load is removed, even for the 3-day-long duration test. The

adhesive-bonded panel withstood the specified level of wind

loads, as well as the concentrated shoe load, without any

apparent degradation.

The empirical stiffness factors for the slit beams and girders

used in the panel structural models were found to be strongly

dependent on the panel configurations. The variation of the

panel configurations include whether the panel is rivetted or

bonded, and whether the panel skin is solid or perforated. The

spacing of beams and the overall geometry of the panel also

influence the determination of the empirical stiffness factors.

Because of such a wide variation of the panel configurations,

and because there is no simple theoretical way to compute or

predict the stiffness factors, it is concluded that a test is

needed for each type of panel configuration in an antenna set.

Acknowledgments

The authors acknowledge the assistance given by F. Lansing, S. Rocci, and the Ground

Antennas and Facilities Engineering Section field engineering technicians during the

various steps of this work.

7o
Table 1. Panel load-deflection tests

Test Center Loading Load level

series supports type

1 No Distributed

2 Yes Distributed

3 Yes Concentrated

4 Yes Concentrated

5 No Concentrated

19.5 kg/m 2 (4 PSF); 39 kg/m 2 (8 PSF)

39 kg/m 2 (8 PSF); 78 kg/m 2 (16 PSF)

99 kg (220 lb); 135 kg (300 lb)

126 kg (280 lb)

85.5 kg (190 lb)

Table 2. Comparison of theoretically predicted deflections (Unit: cm (inch),

distributed loading)

(a) Panel Without Center Supports, at 39 kg/m 2 (8 PSF)

Indicatot

bet 8 9 10 16 17 21

0.106 0.102 0.092 0.076 0.096 0.066

NASTRAN Model(A) (0.0419) (0.0404) (0.0362) (0.0297) (0.0379) (0.0264)

0.104 0.I00 0.089 0.075 0.095 0.065

PANELDEFL Model (B)

(0.0410) (0.0395) (0.0352) (0.0294) (0.0372) (0.0259)

Ratio A/B 1.02 1.02 1.03 1.01 1.01 1.01

(b) Panel With Center Supports, at 78 kg/m 2 (16 PSF)

1 Indicator

bet 8 9 10 t6 17 21

0.0625 0.0544 0.0318 0.0706 0.0653 0.0564

NASTRAN Model (A)

(0.0246) (0.0214) (0.0125) (0.0278) (0.0257) (0.0222)

PANELDEFL Model (B)

0.0605 0.0526 0.0307 0.0683 0.0638 0.0577

(0.0238) (0.0207) (0.0121) (0.0269) (0.0251) (0.0227)

Ratio A/B 1.03 1.03 1.03 1.03 1.02 0.98

71
Table 3. Comparison of theory and test data (unit: cm}

(a) Panel Without Center Supports, Distributed Load at 39 kg/m 2 (8 PSF)

l Indicator

ber 6 7 8 16 17 20 18 21

Test (A) 0.086 0.099 0.104 0.074 0.104 0.094 0.089 0.061

Theory (B) 0.086 0.097 0.099 0.074 0.092 0.099 0.086 0.064

Ratio A/B 1.00 1.02 1.05 1.00 1.13 0.95 1.03 0.95

(b) Panel With Center Supports, Distributed Load at 78 kg/m 2 (16 PSF)

Indicator

ber 6

Test (A) 0.038

Theory (B) 0.030

Ratio A/B 1.27

7 8 16 17 20 18 21

0.056 0.061 0.038 0.076_ 0.066 0.074 0.058

0.053 0.061 0.043 0.064 0.061 0.061 0.056

1.06 1.00 0.88 1.19 1.08 1.21 1.03

Table 4. Effective stiffness factor for girders and beams

(7/84, Panel Numbers (I0/84, Modff. (4/85, Modff.

3 and 9) Panel Number 8) Panel Number 8)

Goldstone test on COSPAL test on Goldstone test on

Test rivetted panel bonded panel bonded panel

Center support No No Yes No No No Yes

Panel skin Solid Perf. Perf. Solid Solid Solid Solid

No. of beams 10 7 10 10

GFACT a 0.60 0.50 0.48 0.87 0.74 1.00 1.00

BFACT b 0.75 0.75 0.68 0.90 0.95 0.80 0.80

aGFACT : Effective stiffness factor for girders.

bBFACT : Effective stiffness factor for beams.

72
POOR QUALITy

169.67T(66.8) _--

I

2l

(8.5)

213.36

(84.o)

m_

A

193.29

m (76.1)

(9.25) ]

-21.s; -2,.I_ -24.13-

(8.5) (9.5) (9.5)

Fig. 1. Panel dimensions and back-up beam spacings [unit in cm (inch)]

i

Fig. 2. Test panel showing silt girder

. 73
OF POOR QUALITY

Fig. 3. Test panel with slit beams end girders

I

Fig. 4. Sample" panel with dial indicators

74
Fig. S. panel IoadinQ conttgurStions: (a}

cen_ supportS; (b) with center supports
Fig, 7. Distributed load test using sandbags

Fig. 8. Up to 84 sandbags were used to represent the distributed load

76
0'_I_..:_'_! _ _,,--.,,.,, ,_._

OF POOR _'" ....",

Fig. 9. Steel plates were placed on top of the panel to simulate the concentrated load

0.204

(o.o8o)

o.lo2

_Q (0.040)

19.5 39.0

(4.0) (8.o)

LOAD kg/m 2 (PSF)

Fig. 10. Panel deflection at dial indicator number 8 as a function of

the distributed load (panel without center supports)

0.122

(o.o,m)

o.o61

O_ (0.024)

k43.0 86.0

(95.0) (190.0)

LOAD, kg (Ib)

Fig. 11. Panel deflection at dial indicator number 17 as a function

of the concentrated load (panel without center supports)

77
Appendix A

Computer Program for Panel Deflection Analysis (PANELDEFL)

I. Program Function

This computer program determines the deflections for one

or a number of paraboloidal microwave antenna-reflector

surface panels. In addition to performing the deflection

analysis, each panel is best fit to minimize the mean square

deflection errors from its ideal surface. Three fitting parameters

are determined by a least squares analysis. These parameters

consist of a shift in the coordinate normal to the plane that

approximates the panel surface, and the two independent rotations about the panel's X- and Y-coordinate axes, which are

as shown on Fig. A-1. Independently best-fitting individual

panels is not always a valid procedure, so that the mean, the

root-mean square (rms), and the standard deviation of the

panel errors are supplied both with and without the fit. These

are furnished both for the direction approximately normal to

the panels' surface and also for the microwave pathlength

direction. Weighting factors are applied to the analysis to

approximate the area associated with each point of deflection

calculation.

A comprehensive analysis of the entire set of antenna

panels can be obtained by modelling one panel from each of

the annular rings of panels that form the antenna surface.

When this is done, the deflection analysis for each ring is synthesized to provide the rms microwave half-pathlength error

for the entire antenna's panels. If the external loadings on the

panels are specified to be the gravity load (weights)of each

panel, then the synthesized deflections for each ring are assembled to provide the rms error for the entire surface at different

antenna elevation angles of 0, 30, 60, and 90 degrees. These

results are also recomputed with a common shift in the

antenna focal axis direction of the entire set of panels to minimize the assembled surface error. When entire antenna surface

calculations are made, each panel ring is weighted by its area

relative to the entire aperture area.

calculations will be made. The first and the last of these points

will actually fall on the girders. A trapezoidal configuration

for each panel is determined by the inner (RI) and outer (RO)

radii of the panel location within the antenna and by the

central angle (TH). For RF pathlength analysis, it is necessary

to specify the focal length (F) of the paraboloidal surface, or

an approximating focal length of a quasi-paraboloidal surface.

Elastic properties of the structure are determined by

specifying the bending moment of inertia of the girders

(GIRDI) and beams (BEAMI), the modulus of elasticity (E)

and bending efficiency factors for the girders (GFACT) and

beams (BFACT). These factors are usually less than unity

because of the effects of slitting the girders and beams in the

fabrication process or because of other effects, such as web

shear deflections. The factors are best determined by test.

They conceivably could be greater than unity because of integral combining action of the surface skin sheet with the girder

and beam cross sections. The surface skin is assumed to be

parasitic and not to contribute to the strength of the panel.

The typical panel is assumed to be supported at two points

on each girder at a distance (ENDG) in from the end of the

panel. The axes of the girders are inward a distance (ENDB)

from the radial edges of the panel. Thus, each beam span is the

width-of "the trapezoid at the beam axis less twice the girder

edge distance. The girders can also be specified to have an

interior (ICENT) redundant support, which can be located

by several methods (RINT).

Although not used in deflection calculations, the weight

of each panel is computed from the cross-sectional areas of

the beam (AB), girders (AG), panel skin thickness (THICK),

and the density (DENS). A weight breakdown is given to show

the contributions of each of these three types of components

as well as the weight per unit of planform area.

II. Panel Configuration and Model

For deflection analysis each panel comprises a pair or

radial girders that support a number (NRING) 1 of approximately circumferential (actually chordal) beams as shown

in Fig. A-1. A number (NRIB) of equally spaced points along

the axis of each beam can be specified at which deflection

1Capitalized terms within parenthesis are FORTRAN namelist input

parameters supplied by the user. See the definitions following the

$INPUT in the user instruction section.

III. Deflection Analysis

The loading for deflection analysis is a uniform surface

load (PSF) to simulate either a wind pressure loading or a

gravity loading. If the gravity loading is not known a priori,

it is reasonable to run all the other data through the program

just for the purpose of obtaining panel weight to establish the

gravity loading.

The deflection analysis is made for beams and girders by

integrating the usual differential equation that equates the

78
bending moment to the second derivative of displacement with

respect to the longitudinal coordinate. The axes of beams and

girders are assumed to be straight and contained in the secant

plane. The secant plane is the plane that contains all four

corner (or near-corner) girder supports. The deflections are

• computed normal to this plane. The loads on the beams are

the distributed line loads determined from the surface loading

and the beam spacing and are applied normal to the secant

plane. The girder loads are concentrated reactions of the

beams. The case of the redundant interior girder support is

solved by the method of consistent displacements; the girder

deflection is determined for a unit load at the redundant

support and the unit load is scaled and applied to annihilate

the girder deflection that is computed from the beam reactions

when ignoring the interior support.

User Instructions

Runstream for JPL UNIVAC 1100/80 E computer:

@XQT

$$INPUT

$$END

52219*RIL.PANELDEFL/MAP.

One set of namelist data per representative

panel for a maximum of 24 rings.

$$INPUT Last panel.

$$END

@EOF

The NAMELIST input data are supplied for each panel

ring as follows (see Fig. A-I for sketch showing geometric

variables):

F Focal length of parent parabola.

RI Inner radius of the panel ring projected on the

aperture plane.

RO Outer radius of the panel ring projected as

above.

NRING Number of rings within each panel. This is also

the number of circumferential beams, including the beams at RO and RI. Maximum = 30.

NRIB Number of radial lines within the panel. This

sets the number of points on each ring beam for

deflection calculations. The first and last lines

are on the panel radial edge girders.

Maximum = 30.

TH

AB

AG

GIRDI

BEAMI

GFACT

BFACT

ICENT

RINT

DENS

E

_F

THICK

SLOPE

IBUG

ENDG

ENDB

$$END

Central angle of each panel in the ring, degrees.

Cross-sectional area of the beams.

Cross-sectional area of the girders.

The bending moment of inertia of the girders.

The bending moment of inertia of the beams.

Effective stiffness factor for girders.

Default = 0.75.

Effective stiffness factor for beams.

Default = 0.75.

If not zero the girder has an interior support in

addition to the end supports.

Distance to the interior girder support (for

ICENT not zero).

If RINT = 0.0: Support will be at the girder's

center.

If RINT = positive: It is the radial distance

from the center of the antenna, measured

perpendicular to the focal axis (same coordinate system as for RO, RI).

If RINT = negative: It is the slope distance

along the girder starting from RI.

Material density.

Young's Modulus.

Superimposed loading normal to the surface of

the panel in pounds per square foot. This is the

only loading for which deflection analysis is

made and this could be different for each panel

ring.

Average thickness of panel skins (allow for perforation) used for weight calculations.

The secant slope of the panel in the parent

parabola. This is computed and echoed by the

program. The loading is applied normal to this

slope.

0 = minimum printout.

3 = maximum printout.

Distance from the end of the girder to the

corner support point.

Distance from the center line of the girder to

the side edge of the panel.

79
NOTES:

1. Theminimumprintoutconsistsofthe deflection matrix

of the panel, panel weight, and summary rms analysis for each

panel. For rms analysis, interior points are weighted by unity,

edge points by one-half, and edge corner points by one-quarter.

2. Although the panel weight (beams plus girders plus skin)

is computed, deflection analysis is made only for the uniform

loads defined by PSF.

3. The load for deflection analysis is applied normal to

the approximating plane of the panel skin. If PSF is made

equal to the gravity weight of the panel then the associated

deflections can be considered as the built-in bias at manufacturing. At specific antenna elevations (0 °, 30 °, 60 ° and 90 °)

the composite pathlength error is computed for all panel rings

and for the variable spring-back deflection for each of the

panels in a full 360 ° panel ring. For the composite analysis of

the entire surface each antenna ring is weighted by its area.

4. Abest fit rms pathlength analysis is also determined for

the composite antenna on the basis of a shift in the antenna

Z (axial) coordinate.

5. The girder effective stiffness is computed as GFACT*

GIRDI*E. Similarly the beam stiffness is BFACT*BEAMI*E.

6. Unless noted above, units are customary English inches

and pounds.

7. The namelist data for successive panels does not need to

repeat any information for prior panels that continue to be

applicable.

NOTE: THE PRINTOUT

PANEL DEFLECTION MATRIX

IS TOPOLOGICALLY SIMILAR

TO THE FIGURE:

I. ROWS REFERTO BEAMS

I-NRING

2. COLUMNS REFERTO

THE NRIB POINTS

SUPPOI_T

INTERIOR

SUPPORT

IF ICENT = 0

BEAM NRING

NRIB LINES OF DISPLACEMENT

CALCULATION POINTS

(NRIB = 8 IS SHOWN)

Fig. A-1. Panel configuration

8O
Appendix B

Panel Structural Model by NASTRAN Program

A panel structural model was developed by using the NASTRAN (NASA Structural

Analysis) finite element program for the tested panel.

The essential input data and procedure are summarized in Fig. B-1.

EXECUTIVE CONTROL DECK

APPLICATION DISPLACEMENT

SOLUTION I (STATIC ANALYSIS)

CASE CONTROL DECK

SELECT OUTPUT POINTS DESIRED, SPECIFY

LOADINGS AND CONSTRAINTS, REQUEST

DISPLACEMENTt FORCE AND STRESSAS

OUTPUTS

BULK DATA INPUT

GEOMETRY OF TESTPANELt PROPERTIES OF

BACK-UP GIRDERS AND BEAMSS MATERIAL

ELASTIC MODULUS AND POISSON RATIOt

BOUNDARY CONDITIONS, LOADINGS

_ 1

PANEL DEFLECTIONS

Fig. B-1. Rowchart of the esmmtild input data

and procedure

81
TDA Progress Report 42-82

N86-10390

April-June 1985

A Sequential Decoding Performance Analysis for

International Comet Explorer

D. Divsalar

Communications Systems Research Section

This article analyzes the effect of a noisy reference carrier on the performance of

International Comet Explorer (1CE) Sequential decoder. Ideal reference models for

frame deletion probability are given for the Telemetry Processing Assembly (TPA) and

the Linkabit (LS4815) sequential decoders. Based on these Meal reference models the

deletion probability in the presence of noisy reference carrier is computed for both the

high and the low data rate cases. A medium data rate performance model is then derived

using an interpolation method. The derived medium rate performance model depends on

the phase locked loop bandwidth-integration time product. Results are obtained for

frame length of 1024 bits, data bit rates of 1024 bps and 512 bps, modulation index of

60 °, and threshold loop bandwidths of 10 Hz and 3 Hz. The medium rate model agrees

with experimental data to within measurement uncertainty. For the 10 Hz loop band°

width and modulation index of 60 ° at frame deletion rate of 10 -4 , the required total

received power to noise ratio is 36 dB.

The analysis given in this article is general and applicable to any sequential decoder,

provided that the ideal reference model of the decoder is available.

I. Introduction

Sequential decoding is a useful technique for communicating at low undetected error rates from deep space probes.

But a failure mode known as computational overflow or

erasure limits the operation of the decoder at very high data

rates. The erasure (deletion) of a data frame occurs when the

decoder has not finished decoding that frame at the time that

it must be output. Increasing the speed of the decoding computation and the size of the decoder buffer provides some

improvement in performance, but only linearly. On the other

hand, if the buffer size is increased too much, erasures occur

in bursts rather than singly. Such bursts decrease the expected

improvement. Also if the speed of decoding is increased too

much with respect to data rate, then the decoder idle time

waiting for incoming data will be increased. Therefore much of

the increased capability is wasted.

Based on the received symbol sequence and past decoded

bits, the sequential decoder makes local best estimates of the

current data sequence. If the received symbols are relatively

noiseless, decoding proceeds rapidly with no searching. On the

other hand, if the received symbols are noisy, some of the

local estimates of current data will be wrong. The decoder

must eventually recognize that an error has been made and

search systematically backward through the local estimates

to correct those in error. The amount of searching depends on

the amount of noise in the received symbol sequence.

82
In this article, the effect of receiver phase-locked loop

(PLL) phase jitter on the frame deletion probability Pa of the

sequential decoder is modeled for the International Cornet

Explorer (ICE) link. The approach.is first to obtain an idealreference (baseline) model for the deletion probability of the

sequential decoder. This has been obtained using models and

simulation restilts in Ref. 1. These simulation and measurement results do not completely match our problem. Therefore

approximation was used for the ideal-reference models. For

the Linkabit (LS4815) sequential decoder, there are only

two measurement points available. So the ideal-reference

model used for LS4815 is also an approximation. Using these

baseline models, first the expected deletion probability is

calculated by averaging over the deletion probability conditioned on the PLL phase error, with the assumption that the

PLL phase error is varying sufficiently slowly over a frame.

The expected deletion probability, under this assumption,

serves as an upper bound on the actual deletion probability

and will be called the high data rate model for frame deletion

probability. Then a lower bound on the deletion probability

is obtained, which corresponds to the situation when the PLL

phase error is varying rapidly over one-bit time. For this case

the effective SNR has been computed and is inserted in the

ideal reference model. The result is called the low data rate

model. With these extreme models, an interpolation method

similar to that in Refs. 2 and 3 has been used to interpolate a

more realistic performance approximation between these

bounds which is called the medium data rate model.

II. System Model

The ICE telemetry system is shown in Fig. 1. The convolutional code is rate 1/2, the constraint length K = 24, and the

data frame length 1024 bits. The output symbol sequence of

the encoder is Manchester coded before it phase modulates

the carrier. At the receiver, the channel noise with two sided

power spectral density No2 is added to the received signal.

Let Pr be the total received power. Then Pr/No represents the

total received power to noise spectral density ratio. Throughout the deletion probability calculation, we assume nominal

0.5 dB degradation loss due to signal conditioning, symbol

synchronization and Manchester decoding assembly.

III. Ideal Reference Frame Deletion

Probability Models

Based on simulation results in Ref. 1, the frame deletion

probability has been approximated by a function for Helios

frame size of 1152 bits (Refs. 1 and 4) as

= exp Y]AiiR- (_V] -2 (1)

i = 1 j= 1

where matrix A with coefficients Ai! is given as

[2.397 8.824 -0.9887]

A = I-0.5331 -6.788 1.569 /

L 0.02303 0.8848 -0.8543..]

In Eq. (1), R = EffN o is the bit signal-to-noise ratio and N

is the average number of computations per bit. If the computational speed of the decoder is C computations/s, the

decoder buffer size is B bits, the frame size is F and the

information rate is rn bps, then

r.n C.B._

N - F- rb F (2)

where Tb is the bit time.

In order to find noisy reference deletion probability, we

should first find the ideal reference (perfect carrier reference)

frame deletion probability Pa (R). Using as a model Eq. (I)

and noting that ICE frame length size is close to Helios frame

• length of 1152, with some approximation the following

models for frame deletion for TPA and LS4815 decoders

are proposed:

Pa (R) = exp (% + n_1 R + tv 2 R 2 ) (3)

where '_0' 'el and '_2 are given in Table 1 for various cases. To

f'md the noisy reference medium data rate model for deletion

probability, we first find two extreme cases, namely the high

data rate model and the low data rate model.

IV. The High Data Rate Model

For the high data rate model we assume that the carrier

phase error is constant during one frame computation. Therefore the expected frame deletion probability can be defined

by

f ff

l'ah = /'a( R c°s2 ¢1¢)p(¢)d_ (4)

where Pa(R cos 2 01g}) is the conditional deletion probability,

for a given phase error ¢, which can be found by replacing R

with R cos 2 ¢ in Eq. (3), the ideal reference deletion probability model. In Eq. (4), the P(0) is the PLL phase error probability density, which is given (Ref. 5) approximately by

p(0) = exp Lo cos0) I¢,l<rr (5)

2. I o (p)

83
where p is the PLL signal-to-noise ratio, given by (Ref. 5)

PT COS20

(6)

p -- NoS,.re

and lo(" ) is the modified Bessel function of order zero. In

Eq. (6) 0 is the modulation index; BL is the PLL bandwidth

given by (Ref. 5)

1+ roa/% )

sL = SLo 1+,.0

(7)

where BLo is the loop bandwidth at threshold, ro is the

damping parameter at threshold, and t_ is the loop suppression factor given by (Ref. 5)

......0.7854 Pin + 0.4768 p_- .

_t = _/ 1 + 1.024 Pin + 0.4768 02

(8)

and Pin is the input signal.to-noise ratio to the bandpass

limiter and an IF filter having bandwidth B1y. Defining the

Carrier Margin (CM) by

PT COS2

CM='

NO2 (s,.o) (9)

then Pin can be expressed as

Pin = CM X PO (10)

with

a 2BLo

(1])

P O = BI F

In Eq. (7) a 0 is loop suppression factor evaluated by Eq. (8)

with Pin = Po" Finally F c in Eq. (6) is the limiter performance

factor given by (Ref. 5)

1 +Pin

Fc = 0.862 + Pin (12)

V. The Low Data Rate Model

For the low data rate model, we assume that the carrier

phase error is varying rapidly during the bit time. This is true

if loop bandwidth-bit time product is much larger than 1, i.e.,

BLr_>> 1 (13)

To compute degraded deletion probability for this case we

should replace the ideal reference bit signal-to-noise ratio R

in Eq. (3) by the effective bit signal-to-noise ratio R 72, where

is given by

11(p)

_ = E {cos_,)= lo(p) (14)

In Eq. (14), 11 (-) is the modified Bessel function of first

order. Then we can get the low data rate model for frame

deletion probability as

p,_ = P,_(R_ 2) (15)

where Pa (') is given by Eq. (3).

Vl. The Medium Data Rate Model

In sequential decoding, when the data rates are roughly

between 2BI.o and 2BI.oF bias, they should be considered as

medium data rates. Therefore the performance of the sequential decoder under the noisy reference for the medium data

rates should lie somewhere between the high data rate and

the low data rate performance models. In order to have an

accurate medium data rate model, we should predict the

effective memory duration or the effective integration time

for the decoder to make a decision on each decoded bit. Let

Tm denote the average integration time (time required on

average to test different braches through the tree diagram).

It is true that

Tb < T <F Tb (16)

But if a long search occurs, the search pattern is more

likely to look like a full tree search. In Ref. 1 the effective

integration time T,n has been found to be well approximated

by

1 N

Tm = 2T b (1-_-(log 2 (1+-_--))) (17)

The following computation has been used in Ref. 1 to

derive Eq. (17). If ¢n denotes the number of tests required in

a full binary tree with branch depth of n, then ¢n can be

found from recursion

Cn = 1 + 2 ¢n-1 (18)

84
The solution to Eq. (18) is

en = 2n-1 (19)

Similarly if we consider both forward and backward moves

contributing to a search of N steps then the branch depth n

is the solution to the equation

or

N

¢. = _ (20)

.... °z_" 2/ "-'"

On the other hand the accumulated length of tested

branches On can be found from recursion

On = n + 2 0 n-I (22)

The solution to Eq. (22) is

On = 2 n+l-n-2 (23)

Then

0

Tm = _n Tb " - (24)

where n satisfies Eq. (20) or (21). Using Eqs. (19), (21) and

(23) in Eq. (24) results in Eq. (17). For large N from Eq. (17)

T _' 2 Tt, (25)

As follows, we represent the medium data rate deletion

probability model as an interpolation between the high and

the low data rate models. We will find the irgerpolation

parameter in terms of loop bandwidth-integration time product Bz. T. This approach is similar to a technique used for

uncoded telemetry and block codes (Ref. 2). Let's define the

conditional effective bit signal-to-noise ratio as

Ref r = R x 2 (26)

where

T

x

/,= _m1 fom cos¢(t) dt

(27)

Note that for B/. T >> I the phase error ¢(t) changes rapidly

over a bit time; thus

x • _" = E (cos O]" (28)

BLTm "-*=

where we have assumed ¢(t) is a stationary ergodic process.

When Bt. T << 1, ¢(t) remains essentially constant over

the integration time T and thus

x • cos ¢_ (29)

B L Tin"0

Clearly these two extreme cases result in the low data and

the high data rate models for deletion probabilities, respectively. Now for anything between these two extremes we can

write

Pare = f Pa (Rx2 Ix) p(x) dx (30)

where x is defined by Eq. (27) and p(x) is the probability

density function of the random variable x. The approximate

probability density function of x is given in Ref. 1, which

enables us to evaluate Pare from Eq. (30). But rather than

doing so, we prefer to fred the high. and low data rate models

by using more accurate phase error distributions and then

finding the interpolation parameter approximately. Using

the ideal reference frame deletion probability models in

Eq. (3), we have

Pa(RX2 Ix) = exp (ao+alRx2+a_R2x4) (31)

Using the Taylor's series expansion of Eq. (31) around

_" = E (cos ¢), we obtain

Pa(Rx 2 Ix) = Pa (R_Z) (1 +131(x-x--) +/32 (x- x--)2 + .... )

(32)

where

and

/_1 = 2 a I R_'+ 4 a 2 R 2 _3 (33)

02 = a 1 R + a I R_'+ 6 a 2 R 2 _-2 + 2 0t 2 R 2 _-3 (34)

85
Noting that Pa_ = Pa (/_2) and using Eq. (32) in Eq. (30)

we get

Pal,. = ed_ (1 +t_: a:x + .... ) (35)

Also if we use Eq. (32) in Eq. (30) with assumption that

BL T << 1, then we get

Pah = Pa_ (1 + flu 02 + .) (36)

cos_ " " "

From Eqs. (35) and (36), we conclude that if 02 and a2 . are

x COS_

small and if other central moment terms are ignored, we

obtain the required interpolation formula as

0'2 ) 02

x

Pd_ + -Z--- Pdh e.. = I- O so (37)

Since we modeled ¢(t) as a Gaussian process with the autocorrelation given in Eq. (38), we can easily find the joint

density function of ¢(tl) and ¢(t2). Having this joint density

function, we can derive the following:

and

4+ 2

E{¢Z(t 1)¢2(t:)) = o_ 2R_(t l-t:) (43)

R___-(q -t:)_- -_1 R,2 (t, - t2) (44)

Note that

ocos0 2 = Rcos¢.. 7 (0) _ T1 R 2¢ (0)

Using Eq. (44) in Eq. (39) with Eq. (45) we get

At this point we model O(t) as a Gaussian process having the T T

samevarianceandbandwidthastheactualO(t) with an autocorrelation function derived from aP rocessand first-order °2x 2 =, _'-1 fmf"

PLL (Ref. 4): °cos_ ra

R¢ (z) = o: exp {--4Bz t r 1) (38)

1 1

Hence, 4 BL TM 32 BZLT:m

RcosO__(tl - t 2 ) dt I dt 2

(39)

05 = E(x-_)2

x

where

RcosO_x(tl - t2) = E {(cos ¢(tl) -x-') (cos ¢(t2) -_))

(40)

Using the approximation

cos _(t) _- 1 - ¢2(t)/2 (41)

we get

exp {-gB L [ t] - t21) dt 1 dt2

(45)

[E {¢2 (t I ) ¢2 (h)) - o:] (42)

Rc°s0-x (tl - t2) _ 4

(1 - exp (-8 BLT)) (46)/

Substituting Eq. (46) in Eq. (37), we have the medium

data rate frame deletion probability Pare"

VII. Conclusion and Numerical Results

for ICE

In this article, we have obtained models for ideal reference

frame deletion probabilities, based on results in Ref. 1, for

TPA and LS4815 sequential decoders. Then we have derived

the high and low data rate models for noisy reference frame

deletion probabilities. Finally, using an interpolation method,

we drived the medium data rate model for noisy reference

frame deletion probability. The results are shown in Figs. 3

through 8. In Fig. 2, the PLL bandwidth vs PT/No is depicted.

In Figs. 3 through 6 Pavs PT/No for various models is shown

for two-sided threshold loop bandwidth of 10 Hz and 3 Hz,

and for bit rates 1024 and 512 bps, for TPA decoder. In

Figs. 7 and 8, the results are given for the LS4815 sequential

decoder with 1024 bps. These theoretical models are in good

agreement with measurement results (J. W. Layland, private

communication).

86
Acknowledgment

-The author wishes to thank J. H. Yuen of Sec. 331 and J. W. Layland of the Telecommunications and Data Acquisition Office for their useful comments and suggestions.

References

1. Layland, J.W., "A sequential decoding medium rate performance model," in The

Deep Space Network Progress Report, Jet Propulsion Laboratory, Pasadena, CA,

Dec. 15, 1973, Tech. Rep. 32-1526, vol. XVIII, pp. 29--40.

2. Tausworthe, R.C., "Communications Systems Development: Efficiency of Noisy

Reference Detection," in The Deep Space Network Progress Report, Jet Propulsion

Laboratory, Pasadena, CA, Dec. 15, 1968, Space Programs Summary 37-54, vol. III,

pp. 195-201.

3. Lesh, J.R., "Seqllential Decoding in the Presence of a Noisy Carrier Reference,"

IEEE Trans. Communications, Vol. COM-23, Nov. 1975, pp. 1292-1297.

4. Yuen, J.H., Deep Space Telecommunications Systems Engineering, Plenum Press,

New York, Oct. 1983.

5. l.,indsey, W. C. and Simon, M. K., Telecommunications Systems Engineering, PrenticeHall, Inc., Englewood Cliffs, N.J., 1973.

87
Table 1. Model coefficients for frame-deletion rate

C, B, rb,

Computational Buffer size, Bit rate,

speed, bits bps

comp/sec

o_o (xI _2

TPA

DLS4815

92,500 3 X 1024 1024

92,500 3 × 1024 512

106 4 X 1024 1024

106 4 x 1024 512

3.71

2.98

0.936

0.23

1.9

3

6.12

7.2

-3.89

--4A9

-6.177 o

-6.77

88
lilT

STREAM

DIGITAL

CAJT_UT i

o,j

DECODER

J FRAME SYI_

F :. I(_4

- CODE STREAM k_'_lCH ESTEl

RF CARRIER

SYMBOL SYNC

AND MANCHESTER

DECODING

I I I

SDA U (NOTO,E_TIONAL)__

J-'-JONLY FOR SIGNAL _ - i II C°NDITI°NINGI [.._ ,LL

RF

DEMODULATION

Fig. 1. Telemetry link configuration

s(t)

I

No

R0

60

40

20

34.0

l I I I I _ ' _ I I ' ' ' ' I J I '

2BI.O = I0 Hz

, , , J I , , , , I , _ _ I I I I I I

34.5 35.0 35.5 36.0

PT/I'40, dB

Fig. 2. PLL bandwidth vs PT/N 0 for 2 BLO = 10 HZ

i0 °I

10-2

10-3

10"4

34.0

.... I .... I .... I ....

_'_ _ _ ..... FRAME = 1024 bits /

_ \ ,_,. _ MI: 60_.g -I

\ "x_ _"Lo:l°'z

r_ _ _J_ MEASUREMENT

IDEAL \ _ _ RESULTS "1

REFERENCE \ _ "i',- /

34.5 35.0 35.5 36.0

PT/N0, dB

Fig. 3. Frame deletion probability models vs PT/No for TPA,

bit rate : 1024 bps and 2 BLo = IOHz

89
10"1

10-2

10-3

1o-4

34.0

.... I ' ' ' "' I .... I

TPA

RATE = 1024

_ FRAME = 1024 bits

_ COMP SPEEO = 92.5 kc/s

_ %it= 3.F_JUaE

%%% MI: 6od.g

_ 2BLo = 3 Hz

IDEAL _

_, REFERENCE _\\ H H RATE MODEL ,o

- \

34.5 35.0 35.5 36.0

PT/N0. dB

Fig. 4. Frame deletion probability models vs PT/No for TPA,

bit rate : 1024 bps and 2 BLo : 3 Hz

10-I

10-2

io-3

10-4

31.0

'\' ' I"J,_ ' ' I ' ' ' ' I .... I ' ' ' ' I ' ' ' '

__ TPA

_ RATE: 512_,

_ _ FRAME = 1024 bits

_ _ COMP SPEED= 92.5 k¢/,

\ \ ,,<_.,o..

X MEDIUM _

RATE _

\ _OE_ \ \X_

\ LOW+ \ \

\ RATE \ \

\ ,_A_ \_OOE, \ -

REFERENCE \

\_o0,_ \ \

31.5 32.0 32.5 33.0 33.5 34.0

PT/NO, dB

Fig. 5. Frame deletion probability modes ve PT/No for TPA,

bit rate = 512 bps and 2 BL0 = 10 Hz

i0-I .... I ' ' ' ' I .... I ' ' 'T'PAI. .... I ....

N RA'rE: 512_,

\\ FRAME = 1024 bits

\_ COMP SPEED= 92.5 kc/s

\\\ B.,:3-f_E

,^-2 . \ \\ MO: 6o_o

\\\

\ \_..--_ow RATE _OE_

| \ WEAL. \ _ .

I \ REFERENCE\ ./N N "

I \_o+,, g \\

10"41 , , ,_, I , , , , _, , , ,N ,\, , , I , , , , I , , , ,

31.0 31.5 32.0 32.5 33.0 33.5 34.0

PT/No, dB

Fig. 6. Frame deletion probability models vs PT/N o for TPA,

bit rate = 512 bps end 2 BL0 = 3 Hz

90
10-1j. .... I I I

t.$4815

RATE = 1024

_,_ HIGH FRAME = 1024 bits

_ RATE COMP SPEED= 1000 kc/s

I_ _MODEL E'elf = 4. FRAME

.,-2 J.- _ _ MI = 60 deg -

,n.3t- ....... \ _

'DtR%CE ,ATE \ \

34.0 34.5 35.0 35.5 36.0

PT/N0, dB

Fig. 7. Frame deletion probability models vs PTINo for LS 4815,

bit rate : 1024 bl_ and 2 BLo = 10 Hz

I- Ls,,e15 -I

I- RA=: ;m4_ 4

l FRAME = 1024 bits |

L - COMP SPEED= 1000 kc/s_

/ Beff : 4*FRAME /

,^.2l M_: 6Od,e -I

,v _ 2BLO = 3 Hz "1

10-3 .

REFERENCE \ _

4 ' MODEL _ %% . .

10- _, , , , I ,\_ _,,%_ I _ _ _ i I , L _ t

34.0 34.5 35.0 35.5 36.0

PT/N0, dB

Fig. 8. Frame deletion probability models vs PT/No for LS 4815,

bit rate = 1024 bps and 2 BLO = 3 I-Iz

91
TDA Progress Report42-82

N86-10391

April-June 1985

The VLSI Design of Error-Trellis Syndrome Decoding for

Convolutional Codes

I. S. Reed and J. M. Jensen

Universityof SouthernCalifornia,LosAngeles

T. K. Truong and I.S. Hsu

CommunicationsSystemResearchSection

In this article, a recursive algorithm using the error-trellis decoding technique is developed to decode convolutional codes (CCs). A n example, illustrating the VLSl architecture

of such a decoder, is given for a dual-K CC. It is demonstrated that such a decoder can be

realized readily on a single chip with NMOS technology.

I. Introduction

Recently, the authors (Refs. 1, 2) developed a new errortrellis syndrome decoding scheme for convolutional codes

(CCs). This new method involves finding minimum-error paths

in an error-trellis. It was shown (Ref. 1) that the computation

of the error-trellis is accomplished by finding the solution of

the syndrome equations explicitly in terms of the actual error

sequence. This syndrome decoding scheme was then applied to

a rate 3/4, one-error-correcting systematic Wyner-Ash code

(Refs. 4, 5).

In this article, the error-trellis decoding is applied to decode

a rate 1/n, dual-K nonsystematic CC. The special example bf a

rate 1/2, dual-3 nonsystematic CC is treated in this article.

It is demonstrated in Ref. 6 and this article that the real

advantage of error-trellis decoding over Viterbi decoding of

CCs is the reduction of the number of states and transitions

between any two frames..A recursive algorithm for finding the

path of minimum error in the error-trellis is found which

realizes a rate 1/2, dual-3 nonsystematic CC. This recursive

algorithm eliminates all paths in certain fixed frames except

the path with minimum error. A VLSI chip architecture is

developed to realize this new recursive algorithm for decoding

the dual-K CC. The designs developed for this decoder are

regular, simple, expandable and, therefore, naturally suitable

for VLSI implementation.

II. The Properties of Convolutional Codes

In this section a brief review is presented of properties of

CCs needed in the following sections.

The input X to a k.dimensional CC encoder is represented

as an infinite sequence of vectors, X/, of form,

_x= [x0, x,, x2.... ] (1)

where X i = [xl/, x2/ ..... xk/] is a k-component vector of

elements from the Galois field GF(q), where q = pn with

p a prime integer. Each vector X/of k symbols for (/= 0, 1,

2.... ) is sometimes called an information or input frame (see

Ref. 4, Sec. 12.1). Similarly, the output Y of a CC of length n

is an infinite sequence of vectors, Yi, of form

92
-Y = [Yo,r,, .... ] (2) ,,,

where Yi = [Yli, Y21 ..... Yni] is an n-component vector of

elements from GF(q). Here vector Yi is called a codeword

frame or more simply, a code frame (Ref. 4, Sec. 12.1).

In a CC encoder the input X in Eq. (1) and output in

Eq. (2) are linearly related; hence there exists what is called an

infinite generator matrix G such that

Y = X. G (3)

For the CC to have finite memory, G has the form

G

... G

m

where the submatrices_G/are k X n matrices of form

(4a)

-- [G,.,iI (4b)

and the elements Grs/ belong to GF(q) for 1 < r< k, 1 <s< n

and 0 </< m. Multiplying the subvectors of x in Eq. (1) by

the matrix G in Eq. (4a) yields, by Eq. (3), the fundamental

identity,

min(j,m)

Y/ = E Xj_ i-G i (5)

i=0

which is the convolution of sequence {Xo, X 1.... } of information frames with the sequence {Go,G1,... ,G,n} of

matrix operators. The integer m in Eq. (4a) is the memory of

the convolution Eq. (5). The value of m is the maximum

number of past input frames Xj needed to compute Eq. (5),

recursively.

The convolution property (Eq. [5]) of finitely generated

CCs can be realized conveniently for some applications by the

operational calculus over a finite field GF(q). To accomplish

this one defines first the generating functions or, what are

sometimes called, the D-transform of the sequences {At/},

and the {(7/} and {Yj} matrices, as follows:

j=O

(6a)

g(D) = _ G/D j (6b)

/=0

and

Qtl

j=O

(6c)

where D is an operator variable. It is not difficult to verify that

by equating the coefficients in the matrix relationship

Y(D) = X(D). G(D) (7)

the fundamental convolution property (Eq. [5] )of a convolutional code of memory m is derived. Hence identity (Eq. [7] )

is precisely equivalent to _he defining relationship (Eq. [3] ) of

a convolutional code. Finally if D is identified with a unit

delay circuit element, it is not difficult to show that G(D)

maps directly onto an encoder circuit diagram.

By Eq. (5) the /th output of an n-vector or eodeword

frame, Yj is dependent on at most the m + 1 present and past

input k-vectors or information frames. Hence it is natural (as

suggested by Blahut [Ref. 4, Section 12.1]) to define

k I = (m+l)k (8a)

to be the word length of the CC. Then the word length k, is

extended by the encoding process of Eq. (5) to, what is called,

the block length nI of the CC. The block length of CC is

k 1

nl = (m+ 1). n =-R-- (Sb)

where R = k/n is the rate of the code. By Eq. (5) the block

length n I = (m + I) n is the length of the subsequence of Y

which, during encoding, can be influenced by a single information frame.

The minimum distance of the code is interpreted to be the

Hamming weight of the smallest weight code word segment of

= m + 1 which is nonzero in the first frame. Suppose for

some CC that.at most t errors occur during transmission in

the first £ code word frames, and that

2t+l<_d

is satisfied I_y the code. Then those errors which occur in the

first block length of CC can be corrected using feedback

decoding. Such a CC is called a t-error-per-block-lengthcorrecting CC or more simply a t-error-correcting CC (Ref. 4,

Sec. 12.3).

93
Another distance between code words of a CC which is

commonly used is the free distance dfree :

dfr _ = min WH (X(D) " G(D))

X(D)*O

Since clearly, d <_ dfree (Ref. 4), designing a CC with minimum

distance d guarantees that the code has a free distance ofd or

greater.

To find the minimum distance d of a CC, either the following k 1 × n1 submatrix is used

"Go G1 G2

6 o Cl

a

wherek 1 =(m+l)kandn I

• ° . IJ

m

• .. Gin_ 1

GO_

(9)

= (m + 1) n, or its corresponding

parity check matrix H. The techniques used to find the minimum distance for block codes apply also for finding d using

matrix G in Eq. (9) or the associated parity-check matrix H.

Sometimes (see Ref. 4, Sec. 3.3) matrix G in Eq. (9) is called

the basic generator matrix of the CC.

III. Error-Trellis Decoding

In this _ection error-trellis decoding as developed in Refs. 3

and 6 is reviewed briefly. First in order to avoid catastrophic

error propagation the D-transform G(D) in Eq. (6b) is

restricted to have the Smith normal form

G(D) = A(D) [Ik,O] B(D) (1o)

where A =A(D) and B =B(D) are, respectively, k X k and

n × n invertible matrices with elements in F[D], the ring of

polynomials in D over GF(g). The elements of the inverses

A -1 and B -1 of matrices A and B, respectively, are also in

F[D] or are polynomials in D. For descriptive brevity the

D-transform G(D) is called the generator matrix.

IfB = B(D) in Eq. (10), let

B = [B1,B2]TandB -' = [B,,B2] (11)

where the first k rows of B constitute submatrix B_ and the

remaining n - k rows are B2_Similarly the first k columns of

B -1 constitute submatrix B_ and the other n-k columns

are B2 . It was shown (Refs. 1,2, 6) that

m

G.HT(D) = G.B 2 = 0

where H(D) is a parity-check matrix for G(D).

(12)

Let Z(D) = [Z l (D) ..... Zn(D)] be the vector D-transform

of received sequence_Z. Then the D-transforms of the transmitted and received sequences are related by

Z(D) = Y(D) + e(D) (13)

where e(D) = [e,(D) ..... en(D)] is D-transform of error

sequence, henceforth called, simply, the error sequence•

From Eq. (12) the syndrome of the received code word is

S = Z'H T = e''ff22 (14)

This is a nonhomogeneous system of linear equations for the

unknown error sequence e(D). The problem of syndrome

decoding CCs is to solve this system of equations for e(D).

It was shown (Refs. 1,6) that the general solution of Eq. (14)

is given by

e = uG +ZR (15a)

where

R = B2 "B 2

with B2 and B2 defined in Eq. (11).

(15b)

Let ea replace e in Eq. (13) as the actual error sequence;

then a substitution of Eq. (13) into Eq. (15a) gives by Eq. (12)

e = uG+ZR = uG+(Y+ea)B2B 2 = uG+eR

which is independent of the transmitted codeword Y.

Using the solution (Eq. (15a)) of the syndrome equation an

estimate of the error sequence can be obtained from

I1_11= rain IluG + ZRII (16)

where II IIdenotes Hamming distance norm or weight and the

minimization is taken over all k.vectors u over F[D]. By

Eq. (16) the minimum weight error sequence is

; = &G+ZR = _G+eaR (17)

where u is thek-vector with elements in F[D] for which the

minimum weight in Eq. (16) is obtained. It was shown (Ref. 1)

94
that _ in Eq. (17) is a correction factor such that the original

message is estimated by

J" = z. c-' -u (is)

Substituting Z = Y + ea into Eq. (17) yields

= (y +e,)a = X +e (19)

Let E be the set of all error sequences which can be decoded

correctly. Then, if ea e E, the most likely error sequence

found by the minimization in Eq. (16) is equal to ea, and

therefore, by Eq. (19), u = ea G -1 . Thus, the minimization

in Eq. (16) has only to be taken over those sequences u which

belongs to E(-l) = {_ = e G -1 : • e E}. Hence

Ilitl = rain IluG + ZR II (20)

ueE(-D

Note that if ea e E, the most likely error sequence found by

either Eq. (16) or (20) is identical and equal to ea.

In order to actually perform the minimization in Eq. (20)

over E(-D, the sets E and E(-D must be identical. This is

generally impossible. However for systematic CCs, it was

shown (Ref. 6) that E(-_) can be approximated by the set

E_ -l) = {u:WH( _ ..... ui+m)_t, forall/_O)

(21)

where t = [(dfree - 1)/2] and m is the length ofmernory. For

a more detailed discussion of the relation between E(-D and

E_I), see Ref. 6. Thus, for systematic CCs, Eq. (20) becomes

I1511= min IluG +ZRII (22)

ueE(l-l)

In order to take the minimization in Eq. (22)over E_-I), a

specific procedure was found to "prune" the error-trellis

(Ref. 6). Also it was shown (Ref. 6) that the number of states

S and transitions T needed for error-trellis decoding of an arbitrary systematic CC is

and

rain{ t,mk}

S= E (mik) (q- 1)i (23)

i=0

min(t.(m+l)k)

T= E ((m + l )k) (q -1)ii (24)

i=0

respectively. Note that the standard Viterbi decoding (Ref. 3,

See. 4.17) requires qmk states and q(m+l)k transitions within

a frame time.

In the next section error-trellis decoding is developed for an

important class of nonsystematic CCs, called dual-K CCs. The

dual-K convolutional codes were invented and developed by

Viterbi and Odenwalder. These COs are nonbinary codes over

the field GF(2 x) and are used in practice in channels which

experience fading such as UHF tropospheric scatter channels,

etc.

IV. Error-Trellis Decoding of DuaI-K CCs

Dual-K (n, l) convolutional codes are of rate 1/n, of

memory m = 1, and with symbols in the f'mite or Galois field

GF(2 K) (see Ref. 7). The generating matrix G is a special case

of Eq. (4a), namely,

where"

G(]) =

-GO G 1

GO

Go G 1

(25a)

GO = [1,1,I ..... 1]

(25b)

GI = [gll'gl2 ..... gin]

with gl/_ 0 and gl/e GF(2 _:) and the gl/'s are all distinct,

for 1 _/_n.

From the above definition of a dual-g CC, it is readily verified that the minimum distance of the code is d = (2n - 1) and

the free distance is

dfr _ = 2n (25c)

Hence if no more than t symbol errors occur in the first 2 code

word frames and 2t+l_d=2n-1 or t_n-1, then

those errors which occur in the first frame can be corrected. In

other words, the dual-K CC is a t-error-per-block-lengthcorrecting CC where

(dr 1

t= ree._ =n-1

and [x ] denotes the greatest integer less than x.

95
If error-trellis decoding is applied to the dual-K CCs, then

from Eq. (20) the most likely error sequence _ is found as

I1_11= min IluG+ZRII (25d)

ueE (-1)

In Appendix A it is shown that E(-D can be approximated by

the set

_7(-1) = {u: WH(u.,ul+l)<t, foralli)O}

(250

Thus error-trellis decoding of dual.K CCs is performed by

taking the minimum in Eq. (25d) over the set _'(-l) in Eq.

(25e). But by Eq. (21)this set is equal to E_ -1) which is used

in Eq. (6) also for systematic CCs. Therefore, the trellis can be

"pruned" using the procedure in Eq. (6), and also the number

of states and transitions within a frame time is as given in

Eqs. (23) and (24). Consider the example by Odenwalder

(Ref. 7, Fig. 1).

Example 1. Let the Galois field GF(23) be generated by the

polynomial x 3 + x 2 + 1, irreducible over GF(23). If a is a root

of this polynomial, then

Ot,a2,693,_ 4 = 1 +t_+t_2,0t s = 1 +OqO 6

= l+a,a 7 = 1,and0

are the eight elements of GF(23). The generating matrix of

type Eq. (6b) for a rate 1/2, dual-3 CC is

G = [1 +D, 1 +aD] (26a)

The output of encoder in terms of input is

V = [YI,Y2] = X[I+D,I+aD]. (26b)

Ifone applieselementarycolumn operationsto G in Eq.

(25b),itisnot difficultto show that

G = [1,0]

is the Smith normal from Eq. (10). Hence

(27a)

B-I =[{Ot 3'0t2 +_3D 1

2,0/2 + Or2 DJ

are the matrices needed in Eqs. (1 la) and (15b).

(27b)

m

R =

ot + c¢3D 1

= [l,_]

a2 + a2DJ

o_2 + o_2D , o_3 + oL3D J

(28)

is the matrix R needed in the error-trellis solution, Eq. (15b),

of the syndrome Eq. (14). From Eq. (25c), dfree= 4 and

hence, the present dual-3 code will correct at most 1 symbol

per block length of 2. From Eq. (24), the number of transitions in one frame time needed in the error-trellis is

T =E '(,)(m-+l 1) (q-l) i = "(8-1) i =15

/=0 i=0

For the standard decoding trellis q(m+l)k _ 8 2 = 64

transitions are required. This yields a fractional reduction of

15/64 - 1/4 in the number of transitions needed for errortrellis decoding between that required for standard Viterbi

hard decoding. Also from Eq. (23), the number of states is

i--O i=0

which is equal to the number of states of Viterbi decoding,

i.e., qrak = 8.

The "pruned" error-trellis is shown in Fig. 1. And the construetion of the trellis is described in the following. The labels

on the pruned error-trellis shown in Fig. 1 correspond to the

solution, Eq. (15a), of the syndrome equation for the actual

error equal to the all-zero sequence. That is,

e = [el,e2] = uG = u (l+D,l+otD}

= {u+Du,u+aDu) (30)

is the output of the trellis. For example, at frame time ] and

state 0 if u = ,_4, then e = [,_4 + 0, a4 + a • 0] = [a 4, a 4] is

96
the label on transition from state 0 to state 0"4. Such a transition represents an attempt to "cancel" a simple error in the

error-trellis equation, Eq. (15a). If such an error does, in fact,

occur at frame/, then no further errors are allowed to occur at

frame / + 1. Thus a transition to other than state zero must be

followed by a transition back to state 0 in the next frame as

shown in Fig. 1.

Next suppose a transition to state 0"4 occurs, i.e., Du =

0"4. Then since u = 0, the transition from state Du = 0"4 back

to 0 is given, using Eq. (30) by e = [0 + t,4, 0 + 0" • 0"4] =

[0"4, o's]. The remaining labels to the "pruned" error-trellis

are obtained in a similar manner.

To illustrate error-trellis decoding of the dual-3 CC let the

generating sequence be

x(D) = 1+ ,,D (31)

Then the code word sequence is by Eqs. (7) and (26b)

Y(D) = X(D) G(D) = [1 + _,SD + aD z, 1 + 0"2D2 ]

Next, let the actual error sequence be ea (D) = [D 2 , 0"] so that

Z(D) = Y(D) + e (D) = [1 + 0"SD + 0"SD2, 0"s +0"2D2]

02)

Hence by Eq. (28),

ZR = [1 + _,SD + 0"SD2 , 0"SD2 ] i

+ 0"3D, 0"3 + or4

02 + 0"2D, 0"3 + 0"3D.j

= [0"3,0"4] + [0"3,0"4] D + [0"2,0"3] D + [ n3' 0"4] D 3

(33)

The finding of the minimum weight error-path _(D) in

terms of u(D) as given by Eq. (17) is equivalent by Eq. (15a)

to finding the code word u(D) G(D) which is closest to Z(D) •

R(D) as given in Eq. (33). Hence the minimum-weight errorpath can be found by applying the Viterbi decoding algorithm

(Ref. 3) to the pruned error-trellis in Fig. 1. To accomplish

this, the frames of ZR in Eq. (33) are added to the output uG

in the pruned error-trellis in Fig. 1 as shown in Fig. 2.

In order to illustrate the Viterbi algorithm as applied to the

pruned error-trellis suppose the decoder has reached frame 4.

The output of the transition from state 0"3 to 0 is

Coef [u(D) • G(D) + Z(D) R(D)] = [ n'3, t_4] + [0"3, a,4]

o 3 = [0, 0]

with Hamming weight 0. A similar calculation for the other 7

possible transitions shows that the transition from 0"3 to 0 is

the only one with Hamming weight 0. The path segment from

0"3to 0 is chosen since it has minimum weight.

At frame 5, Fig. 2, the minimum weight estimate of the

D-transform of the error sequenceis _(D) = [0, 0"] + [1,0] D 2.

Hence the estimate u(D) of the message correction factor

which achieves e(D) is

= a 3 + (_3D2 (34)

Finally, using Eqs. (27), O2), (34) in Eq. (18) yields, by

Table 1,

oz 0"3,0"2 + 0"3D1 [1]

+0"2o3

--tl+0"so+D',0"s+,:o2]I:]

- [0"3 +0"3D2 ] = 1 +0"D

the original encoded message in Eq. (31).

V. Recursive Algorithm for Error-Trellis

Syndrome Decoding of Convolutional

Codes

For the dual-3 CC described in See. W, error-trellis decoding of CC is used to correct one error in every £ frames. For

this example a recursive algorithm is developed to determine

the path with minimum Hamming weight for every _ frames.

This recursive algorithm for the error-trellis syndrome decoding of dual-3, rate 1/2, one-error-correcting nonsystematic

convolutional codes is described with a flowchart as shown

in Fig. 3.

The recursive algorithm is illustrated in example 1 of the

last section. In this example, u = 1 and £ = 2. By Eq. (33)

ZR = ([Or3, 0"4], [Or3,_4], [0"2,0"3], [0"3,0"4], [0, 0])

(35)

97
In Eq. (35), the first value of ZR is not equal to zero. Thus it

is assumed that one error occurs in the first two frames. To

find this error, the pruned error-trellis with a no-error output

u(D) G(D) in _ = 2 frames is computed. This partial trellis

is shown in Fig. 4. An error-trellis over these two frames is

created by adding the vectors [_3, a4] and [a3, a4], successively, to all labels, in the pruned trellis in Fig. 4. This resulting error-trellis for these two frames is shown in Fig. 5.

In Fig. 5 one needs only to find the path with minimum

weight which ends up in state o = 0 at the end of £ = 2 frames.

Using Viterbi decoding the path with minimum weight is

([_, 0], [0, 0] ).

Next, the input value following [a3, a4] and [a 3, a 4]

again is not equal to zero. Again it is assumed that only one

error occurs in the next two consecutive values of [a2, a_]

and [a s, a4]. These values are added again to the pruned

trellis of two frames given in Fig. 4 for generating an errortrellis. Again using Viterbi decoding one finds the path with

minimum weight to be ([1, 0], [0, 0]). Finally, the input

value following [a 2, a 3] and [c_3, c_] is [0, 0]. Hence the

estimated error for these two frame times is [0, 0]. Thus the

overall path with minimum weight is

$ = ([0,a], [0,0], [1,0l, [0,0], I0,0])

As a consequence, the correction factor _ (D) is _ (D) _ a 3 +

c_ZD2. Hence, from Eq. (18), the estimated message is X = 1 +

aD. The same procedure applies similarly to a systematic oneerror-correcting Wyner-Ash CC presented in Ref. 5. In this

code,£ =m + 1 =2+ 1 =3.

VI. A VLSI Design for Error-Trellis

Syndrome Decoding of Convolutional

Codes

In this section, a VLSI architecture is developed for the

recursive algorithm for error-trellis decoding of convolutional

codes presented in Fig. 3. This VLSI processor for selecting

the path with minimum Hamming weight is composed first of

d basic cells, where d is the number of paths in the error-trellis.

These d cells are followed then by a weight comparison circuit. A basic cell computes the path weight incrementally.

That is, if symbol A is not equal to symbol B, then the weight

of that particular path increases by one; otherwise, the weight

remains unchanged. The VLSI architecture of the error-trellis

syndrome decoder is illustrated in the following example.

The calculations used in the present example were given in

the last two sections. The VLSI architecture for this convohitional code is illustrated in Fig. 6. In this figure there are 8

basic cells corresponding to the eight possible paths in the

error-trellis. The function of each basic cell is described as

follows.

The ith basic cell corresponds to the ith path in 2 frame

times in the error-trellis. Thus if A is not equal to B, then the

weight of the ith path increases by one (otherwise it remains

the same) where A is the input value and B is the precalculated

value stored in the ith basic cell. In this example,A is (ZR) k,

for (k = 1,2, 3,4), and B is (u • G)j, for (] = 1,2, 3, 4), where

Z is the received code sequence, R is defined in Eq. (28), u is

an arbitrary k-vector of elements in F[D], and G is the

generator matrix.

First the received code sequence Z is multiplied by matrix

R and G -1 ,as shown in Fig. 7,to form both the input sequence

to the basic cells and the inverse of the received message, i.e.,

Z • G -1 . The input sequence ZR is then sent to all the 8 basic

cells as well as a zero detector simultaneously. The inverse of

the received message ZG -] is sent then to a delay line to wait

for the completion of the set of operations needed to estimate

the correction factor _. Then Z • G -1 is added to _ to obtain

the estimated message _ by Eq. (18). The purpose of this zerodetector is to check if the input vector ZR is zero or not. If

the two components are zero, then all the outputs of the

weight comparator, which are described in the following, are

also equal to zero. This indicates that the estimated correction factor u is zero, i.e., no error has occurred in the received

code sequence Z. If ZR is not equal to zero, then the two

components of ZR, i.e., [ZR 1, ZR2], in the first time frame

are sent to the equality check circuit sequentially as shown in

Fig. 8. The TGt's (for i = 1, 2, 3, 4) shown in Fig. 8 are 3-bit

registers. They are used to store the precalculated ui • G values

'for the/th path. Since it requires only 2 frame times to choose

a minimum Hamming weight path, four _egisters are needed to

store uj • G. The reason for the use of 4 registers instead of

one in this design is to avoid a more complex sequential

computation of ui • G.

The loading of the TGi's into the equality check circuit is

controlled by a 2-bit counter which is capable of generating

the required 4 different states. Because only 4 pairs of values

need to be checked in every 2 frame times, four clock cycles

are needed to finish the loading operation.

At the first clock cycle, ZR 1 and TG 1 are loaded into the

equality cheek circuit. At the next clock cycle, ZR 2 and

TG 2 are fed into the same circuit in sequence. The equality

check circuit is implemented by the XOR arrays and an OR

gate as shown in Fig. 9. For example, if ZR 1 is equal to TG l ,

98
then the output of the equality check circuit has logic level

zero; otherwise it is one.

The output of the equality check circuit is sent to a 3-bit

counter which accumulates the weight of a path in the errortrellis. After 4 clock-time, all the 4 pairs of the input sequence

Z • R and u/• G of each path on the error-trellis are compared.

The output of the 3-bit counter which is the calculated weight

of each path on the error-trellis is then sent to a weight com.

parator circuit. The weight comparator consists of Programmable Logic Array (PLA array). This is denoted by BLA1 in

Fig. 10.

Also shown in Fig. 10 is an array of control gates and a

table-lookup PLA. The inputs to each PLA1 in Fig. 10 are

two 3-bit registers, I¢/ and _, which denote the weights of

ith and /th path, respectively, in the error-trellis. The outputs of PLA1 are I4//or _ depending on which is smaller, and

a 1-bit signal, LR, to indicate if I¢t is smaller than W/. If t¢/

is smaller than I¢/, then LR is zero; otherwise LR is one. The

PLAI's are configured in a tree structure. The outputs of the

first level PLAI's are sent to the second level PLAI's as their

inputs and so forth.

For example, if I¢l is the smallest of all the weights, as

indicated in Fig. 10 at point A, it is logic zero. This will turn

on gate T1 and turn off gate T2 . Then the value of C which is

zero will be transferred through gate Tl to point B. Since I_1

is the smallest value, the value at point D is zero. This turns

on gates Ta and T,t and turns off gates Ts and T6. Therefore

the values at points A and B, which are zero, together with

the value at D which is zero as well, are transferred to the

inputs of another type of PLA, labeled by PLA2 in Fig. 10.

The function of PLA2 is to form a mapping between the

path and correction factor _. Since there are eight different

paths in the error-trellis, there are, correspondingly eight

different _'s. Finally, the correction factor _ is added back

to ZG -1 , by Eq. (18), as the estimated information ._. The

estimated information ,_ is then shifted out of this circuit

sequentially.

References

1. Reed, I. S., and Truong, T. K., "Error.Trellis Decoding of Convolutional Code" to be

published in PtoceedingslEE.

2. Reed, I.S., and Truong, T.K., "New Syndrome Decoding for (n, 1) Convolutional

Codes," Electronic Letters, Vol. 19, No. 9, April 1983, pp. 344-346.

3. Viterbi, A.J., and Omura, J.K., Principles of Digital Communication and Coding.

New York: McGraw-Hill, 1979.

4. Blahut, R. E., Theory o/Practice o/Error Control Codes. New York: Addison-Wesley,

1983.

5. Wyner, A.D., and Ash, R.B., "Analysis of Recurrent Codes," IEEE Trans. Info.

Theor. IT-9, pp. 143-156, 1963.

6. Jensen, J. M., and Reed, I. S., "Error-Trellis Decoding of Convolutional Codes," submitted to IEEE Trans. on Information Theory.

7. Odenwalder, O.P., "Dual-K Convolutional Codes for Noncoherent Demodulated

Channels," Proceedings of International Telemetry Conference {ITC), Vol. 12, pp.

165-174, 1978.

99
STATE

Du

[o,o] [o,o]

1⁄2

1⁄2

D

FRAMES: 2 3

Fig. 1. Pruned error-trellis with no error outputs u(D) G(D)

100
Du

ZR___:[a3,a4 ]

FRAMES:

^ )

U'C[

i

+ dD 2

a , a4]

[i,a6]

[e 2, a 3]

(I , e]

a , a6]

[a 3, a2]

a , a4]

L._

1. a 5]

0, a]

[0, O]

[a 6, a 2]

1, (16]

2 3 4 5

Fig. 2. Minimum-error path _ in pruned error trellis

[0, O]

a , a 3]

[a 5, a 5]

1, 1]

lOl
1 I CALCULATE Z.R I

ICALCULATE I

ZG-I I

DELAY

LINE

Z.R

, NO

I CALCULATE WEIGHTOFEACHPATH

WEIGHTCOMPARISON,

PATHWITHSMALLEST

WEIGHTIS CHOSEN

YES

D

NO

I _-o

Z'G-1 = SUMMATION q

ESTIMATEDINFORMATION'- Z.G-1 +'_

Fig. 3. Flow chart of error-trellis syndrome decoding of

dual-3, rate 1/2, convolutional code

STATE

Du

[o,o] [o.o]

1⁄2

1⁄2

1⁄2

D

FRAMES 1 2

Fig. 4. A basic cell of a pruned error-trellis with no

error outputs u(D) G(D)

102
1⁄2

FRAMES I 2

Fig. 5. Mimmum-error path • m pruned error trellis

in__

103
Z

Z'G -1 AND Z.R CALCULATING CIRCUIT

.'3

DETi

7

Fig. 6. Block diagram of a dual-3, rate 1/2, convolutional code decoder

104
ZRI'Z I ( a2+ a3D)+Z2( a2+ aZD)

ZR2"ZI( a3+ a4D)+Z2( a3+ a3D) 3

Z1

Z2

rT_

,f ZG -I

-_ ZR I

t2-BITCOUNTER CONTROLLEDBY CLOCK i

=R=ZR_. =R,=R_ ¢-- ¢' ¢' ¢'

)-BI_T EQUALITYCHECK

1-BIT

I )-BITCOUNTER I

;)-BIT

WEIGHT

Fig. 8. Block dlagram of each c_mfor _uhmng w_ht

Fig. 7. Block diagram of circuit for calculating Z- R and Z- G-1

Fig. 9. Logic diagram of equality check circuit

105
PATH

NUMBER 000 001 010 011 100 101 110 111

r8

LEVEL1

LR W1 orW 2 LR LR W5 or W6

PLA1

LR

LEVEL

T1

S(W1, W2,

S(W5, W6, W7, W8

LEVEL3 PLA1

T4

PATHNUMBER

PLA2

Fig. 10. Block diagram of weight comparatorcircuiL The inputsto this circuit are

weights of each path. The outputs are estimated corrector factors.

106
Appendix A

Approximation of the Set E(-1) for DuaI-K CCs

Let G -1 (D) be a right inverse to the generator matrix (25b)

of a dual-K CC of length n. Then E(-O = {eG -1 : eeE}. Def'me

_-1) : (u: I¢H (ul,_+l)<t, foraU/;_O}

(A-l)

In the following it will be shown that _-1) is a suitable

approximation to E (-1). Since for t _ 2, E(-1) *gE'I-'), the

UIUy .v.,_.v,., _a.._ is _u_--t = I, .ra. is ,ul n = 2. It 1-_ ....... I canny'-

verified that

G-1

L(_,2+_u)' _12 +_.)j = [g1,s2]T

(A-2)

is a right inverse matrix to the generator matrix of a dual-K CC

of length 2. Note'that gt and g2 are nonzero elements in

GF(2 x ).

Since the dual-K CCs of length 2 are 1-error-per.block.

length-correcting CCs, the set of sequences

E = {v: W_ (v/,v]+l)< 1,forall]->0}

(A-3)

is a desirable approximation to E. Thus

"_(-t) = (v = vG -l :veE} (A-4)

is an approximation to E(-1). Also from the following lemma,

one has that _'(-1) = _'l-l).

Lemma. Let n = 2, then _(-D = _'t-1).

Proof: Let u(D) egg-l). Then

u(D) : E _ " D] : v(D). G -1

]=0

where v(D) e/_ And from (A-2)

u(D) = _ (v_j-%)Z_ [s_,t2] T

1=o

= _ (vl/gl +v2]g2)D/

/=0

Thus, •

WH (Up 14]÷1) = WH (1_1/ gl 4" 132]g2, F1]÷I gl "I"V2/÷I g2 )

< wH (vl/gl, vi/+l gl' v2/g2' v2/+1 g:)

= wH (vl/, vl/+1, v2], v2]+l)

= wu(vl,v2)<l

From (A-3), since v(D) e E 1, this implies that E (-I) C_E_-1).

To show that _-') C_/_(-'), let u(D) e E_-'), construct

the sequence v_D) = (v _(D), v2(D)) = (g;_ u(D), 0). Sinc..e

g-_ • u(D) e E_ -_), it follows from (A-3) that _D) is in E.

Hence v(D) - G -_ = v(D) • [gr g2 ]r = u(D) is in E (-D.

107
TDA ProgressReport 42-82

N86-I0392

April-June 1985

High-Rate Convolutional Code Construction

With the Minimum Required SNR Criterion

P. J. Lee

Communications Systems Research Section

New short constraint length, high-rate convolutional codes which minimize the

required SNR are found and tabulated for rates 2/3, 3/4, and 4/5, and for constraint

length K up to 10. When compared with previously reported codes, most of the new

codes reduce the required SNR only slightly. However, there are some pairs of K and

code rate for which the new codes require considerably less SNR. The most significant

one is the new K = 8, rate 4/5 code which requires 1.25 dB less SNR than the known

code with the same parameters, for a desired bit error rate of lO -6. •

I. Introduction

For a convolutional coding system employing a Viterbi

decoder, the decoded bit error rate (BER) is well upperbounded by the transfer function bound (Refs. 1; and 2,

Chap. 4)

DO= exp(-_/No)

co = O(_/U_/No)exp(dfZ,/_o)

where NO is the one-sided noise power spectral density, Es

is the received signal energy per channel symbol, and

BER < c o •

oo

T(D,Z) = co

D=Do,Z=I i=df

(1)

where the coefficient c o and transfer function T(D, Z) depend

on the code and type of channel used. The quantity D Ois the

Bhattacharyya bound (Ref. 2, p. 63) which depends on the

channel only, df is the free distance of the code, and aI is the

number of bit errors in all incorrect coded symbol sequences

with Hamming distance i. For an additive white Gaussian noise

channel with binary PSK signaling (BPSK/AWGN channel)

without quantization, we have (Refs. 1 ;and 2, p. 248)

Q(w) = / exp (-t2/2) dt/Vt-_

W

Many researchers have used the maximum d[ criterion, or

the criterion of maximum df together with minimizing the

first few a_s in Eq. (1) for determining the goodness of a code

in their code search procedures. However, we have shown in

Refs. 3 and 4 that, for low rate codes, these criteria do not

necessarily lead to codes which minimize required signal-tonoise ratio (SNR) for a certain desired BER. Direct use of

Eq. (1) for BER evaluation in the search procedure provides

much better results.

108
This new minimum required SNR criterion is applied here

to the searches for good high-rate codes, which are useful for

systems with limited bandwidth. In the next sections, our

notation is introduced and the code search procedure is

briefly explained. Search results are then listed and discussed.

II. Preliminaries

Let mo, ko, and no be the number of binary memory cells,

inputs, and outputs of an (m o, ko/no) convolutional encoder,

where ko/n o is the code rate r. A typical nonsystematic, time

invariant encoder structure is shown in Fig. 1. A group of

I" ;nfnrrn2tlnn hlte ;e ehift_d intn a _hift register of length

K (= m o + ko), and outputs of no modulo-2 adders are sampled

and sequentially transmitted. The parameter K is called the

constraint length of the code, while m o is called its memory

length. Notice that the number of states in the Viterbi decoder

trellis is 2m°. The low-rate codes considered in Refs. 3 and 4

are special cases with k o = 1.

Besides these key parameters, the code performance is

determined by the connections from K shift registers to no

modulO-2 adders. These connections are often represented

by an no X K binary matrix G, called the code generator

matrix, where "l" stands for connection and "0" for nonconnection. As an illustration, a (2, 3/4) encoder and a

(3, 2/3) encoder are shown in Fig. 2, whose code generator

matrices are given, respectively, by

and

G

l 1 1 1 I 1

1 0 0 0

0 0 1 0

0 0 1 0

G

 :100:] 0 0 1

I I I

For short hand notation, G is often represented by

(g(1) ..... g0) ..... g("o))

The transfer function bounding technique on the BER at

the Viterbi decoder output will not be discussed here; it can

be found in many references, including Refs. 5, 6, and 7.

III. Code Searching Procedure

In this study, we restricted our searches to high-rate

codes with no = k o + 1. Notice that the number of (too, ko/

(k o + 1)) codes in the whole code space is 2(mo÷xo)x (_o+13.

For example, there are over 4 billion (5, 3/4) codes. Further.

more, according to our criterion, to test a code we have to

evaluate the transfer function bound, which requires a matrix

inversion. Therefore an exhaustive search is prohibitively

difficult except for very small m o and k o. Oniy partial

searches are possible to obtain results in a reasonable length

of time.

In the previous searches for low-rate codes, we developed

several effective techniques for reducing the code search space.

Many of these techniques are applied to the high-rate code

searches, with appropriate modifications.

First, for a given pair of m o and k o, we made a list of some

r = 1 codes (actually these are not codes since there is no

redundancy) which are to be used for the generation of

r = ko/(k o + 1) codes. In this list, by using the simple fact that

exchanges of g(j)'s do not affect the performance of the code,

identical codes are discarded. Also, codes with too small free

distance (less than d x - 2, where d x is the maximum known

free distance of (m o, ko/(k o + 1)) codes, or its bound if not

known) are deleted. This procedure is based on the observation (Ref. 3) that the Hamming distance (from the all-zero

output) on each branch (in the state diagram) of a lower rate

code is always larger than or equal to that of the higher rate

code, used as a seed for its generation. Catastrophic codes are

not discarded at this time, as good r = ko/(k o + 1) codes are

often found from catastrophic • = ko/k o codes.

Each code in the list is used for generation of lower rate

codes. Among the generated codes, identical codes and catastrophic codes are deleted. Codes with free distance smaller

than d x - 1 are also discarded. For each remaining code, the

BER performance is found by the transfer function bound

with a SNR at which the best code is expected to achieve BER

of 10 -6 .

where g(j) is the/th row of G, in octal. For the codes in

Fig. 2, G = (37,21,5,4) and G = (31,23,16), respectively. By

"code search" we imply the search for a code generator G

which provides good performance among the codes with

same key parameters.

IV. Search Results

The code search results are summarized in Table 1, where

the code generators of best codes are shown with their free

distance and the upper bound on the required bit SNR

lO9
(Eb/No,Eb = Es/r= Es • (k o + 1)/ko) value for desired BER

of 10-6 . These new codes are compared with the codes

reported in Refs. 8 and 9. If the best code in the sense of

minimum required SNR does not have maximum free distance, then the best code among maximum free distance

codes is also listed.

Notice that some of new codes have parameters never

considered before. For all codes, we were able to find better

codes than the previously reported codes. But the amount of

SNR saving is usually very small except for a few cases. For

the ease of (4, 4/5) code (or equivalently K = 8, r= 4/5

code), we found a code which not only requires 1.25 dB less

SNR but also has larger free distance than the previously

reported code. Since we could not exhaust the code search

space for most cases, there might be some better codes.

However, we expect that better codes, if they exist, would

improve the performance very slightly.

References

1. Viterbi, A.J., "Convolutional codes and their performance in communications systems," IEEE Trans. Commun. Tech. COM-19, pp. 751-772, Oct. 1971.

2. Viterbi, A. J., and Omura, J. K., Principles of Digital Communication and Coding, New

York: McGraw-Hill Book Co., 1979.

3. Lee, P.J., "New short constraint, rate 1/N convolutional c-odes which minimize the

required SNR for given desired bit error rate," IEEE Tram: Commun. COM-33,

pp. 171-177, Feb. 1985.

4. Lee, P.J., "Further results on rate I[N convolutional code construction with minimum required SNR criterion," TDA Progress Report 42-80, pp. 97-102, Jet Propulsion Laboratory, Pasadena, CA., Feb. 1985.

5. Omura, J. K., and Simon, M.K., Modulation/Demodulation Techniques for Satellite

Communications, Part IV: Appendices, JPL Publication 81-73, Jet Propulsion Laboratory, Pasadena, CA., Nov. 1981.

6. Lee, P.J., '_Fransfer function bound for partial-unit-memory convolutional codes

based on reduced state diagram," TDA Progress Report 42-76, pp. 70-76, Jet Propulsion Laboratory, Pasadena, CA., Feb. 1984.

7. Lee, P.J., "A very efficient transfer function bounding technique on bit error rate

for Viterbi decoder, rate I[N convolutional codes," TDA Progress Report 42-79,

pp. 114-123, Jet Propulsion Laboratory, Pasadena, CA., Nov. 1984.

8. Paaske, E., "Short binary convolutional codes with maximal free distance for rate 2/3

and 3/4," 1EEE Trans. Info. Theory IT-20, pp. 683-689, Sept. 1974.

9. Daut, D. G., Modestino, J. W., and Wismer, L. D., "New short constraint length convolutional code construction with selected rationale code rates," 1EEE Trans. lnfo.

Theory 17"-28, pp. 794-800, Sept. 1982.

11o
Table 1. Best (mo, ko/no) binary convolutional codes with no : k0 + 1which minimize the

required SNR for BER -< 10 -s and performance comparison to previously reported codes

(toO, ko/no) df Eb/No, dB Note Code generator G, in octal

(1, 2/3) 2 9.023 A 5 3 2

2 9.032 D 6 3 2

(2, 2/3) 3 7.360 A 15 13 12

3 7.570 D 17 15 6

3 7.798 P 16 13 11

(3, 2/3) 4 6.292 A 31 23 16

4 6.320 D 33 22 15

4 6.341 P 37 22 11

(4, 2/3) 5 5.870 A 61 46 37

5 5.888 P 61 56 27

5 6.169 " D 75 72 27

(5 2/3) 6 5.531 A 171 112 73

6 5.580 P 177 112 55

(6, 2/3) 6 5.171 A 366 241 163

7 5.211 P 337 236 155

(7, 2/3) 7 4.846 A 751 522 343

8 4.853 A 673 465 262

8 4.883 P 751 532 367

(8, 2/3) 8 4.632 A 1671 1322 423

(1, 3/4) 2 8.633 A 15 12 4 2

2 8.639 D 15 14 13 2

(2, 3/4) 3 7.527 A 37 21 5 4

3 7.634 D 36 32 14 7

(3, 3/4) 4 6.629 A 67 51 43 25

4 6.652 D 61 47 25 13

(4, 3/4) 4 6.042 A 157 122 41 24

4 6.336 D 172 127 106 45

(5, 3/4) 5 5.735 A 255 236 164 127

5 5.776 D 357 216 124 45

5 5.797 P 367 244 141 72

(6, 3/4) 6 5.449 A 723 657 "-" 345 261

6 5.452 P 512 467 - 311 274

(7, 3/4) 6 5.185 A 1752 1233 756 377

(1, 4/5) 2 8.578 A 34 23 10 4

2 8.820 D 36 26 13 11

(2, 4/5) 2 8.003 A 71 53 34 10

2 8.507 D 67 57 52 26

(3, 4/5) 3 6.760 A 153 137 51 25

3 6.838 D 174 132 56 23

(4, 4/5) 4 6.316 A 373 254 225 215

3 7.561 D 337 274 255 237

(5,4/5) 4 5.993 A 765 613 571 537

(6, 4/5) 5 5.710 A 1537 1351 1145 1053

2

4

4

15

15

13

112

156

II0

730

NOTES: A Found by the author

P Found by Paaske (Ref. 8)

D Found by Daut, et al. (Ref. 9)

111
k0 m0

no bits

Fig. 1. A lyplCll (m o,ko/no) encoder structure

O

(o) (b)

Fig. 2. Examples: (a) A (2, 314) code and (b) a (3, 2/3) code

112
TDA ProgressReport 42-82

N86-10393

April-June 1985

DSN Frequency and Timing System, Mark IV-85

B. W. Falin

TDA Engineering Section

As part of the DSN Mark lira Implementation Program, tht DSN Frequency and

T_ming System is currently undergoing changes. With the implementation of Signal

Processing Centers (SIC) at each Deep Space Communications Complex (DSCC), major

changes to the frequency and timing distribution equipment were necessary. This article

provides a functional description of the Mark IVA Frequency and Timing System (FTS)

as it exists today and planned capabilities through 1988.

I. System Definition

A. Background

The necessity to upgrade the Mark I11-77 FTS was driven

by two factors. First, the consolidation of the DSN stations at

a central complex required an increased distribution capability

for time and reference frequencies. It also required centralized

control and monitor of the frequency and timing equipment.

Since a complex would be supporting concurrent operational

activities, a need for the generation and distribution of simulated time was necessary. The second factor was the growing

need to replace aging equipment that had been in service for

twenty years.

B. Functional Definition

The DSN Frequency and Timing System provides precision frequency and timing data required at the Deep Space

Stations throughout the DSN.

The DSN Frequency and Timing System performs two

main functions, providing frequency and timing data and

validation. Frequency and timing data consists of sinusoidal

reference frequencies, timing pulses, and epoch time codes.

Frequency and timing validation assures that the primary

frequency standards throughout the entire DSN are maintained within prescribed tolerances relative to the National

Standard-United States Naval Observatory/National Bureau of

Standards (USNO/NBS).

C. Functional Description

The Frequency and Timing System functions are conceptualized as follows:

(1) Generation of precision frequency signals to provide

references for the generators of sinusoidal signals,

timing pulses, and time codes within DSCC's.

(2) Generation and distribution of sinusoidal reference

frequency signals, timing pulses, time codes, and provision for time displays within DSCC's.

(3) Control and monitoring of configuration status and

operating mode of the FTS at DSCC's.

(4) Network-wide synchronization of all Frequency and

Timing System signals to NBS, validation of frequency

113
standard calibration, and traceability of time to Universal Time, Coordinated (UTC).

Figure 1 depicts the Mark IVA Frequency and Timing System configuration.

D. Key Characteristics

The key characteristics of the Frequency and Timing System are:

Knowledge of time synchronization (to -+10 /as) and

frequency offset (-+3 parts in 10-13) between DSCC's

Knowledge of time synchronization (to -+5 gs) between

the DSN and the National Standard (USNO/NBS)

Knowledge of time synchronization (to -+10 /as) and

frequency offset (to -+I part in i0 -11) within each

DSCC

Performance validation of frequency and timing

functions for each Front End Area (FEA) and SPC

at a DSCC

Maintenance of a permanent record of frequency

and time parameters including configuration and

synchronization

Simulation time sdectable from the DSCC Monitor

and Control Subsystem (DMC).

II. System Elements

A. Functional Elements

The major elements of the Frequency and Timing System

consist of a Frequency and Timing Subsystem located at each

DSCC and a Network FTS Data Base subsystem located at

DSN Network Operations Control Center. The Frequency and

Timing Subsystem consists of six major subsets of equipment.

(The relationships between the subsets are depicted in

Fig. 2.)

Frequency reference standards

Frequency synthesis and distribution

Reference frequency cable stabilization

Time and timing generation and distribution

Time synchronization

Monitor and control.

B. DSS Frequency and Timing Subsystem

The master frequency standard consists of two Hydrogen

Masers backed by two Cesium Beam Frequency Standards.

These standards, which are located in an environmentally

controlled area, are provided with an uninterruptible power

source for emergencies. Any of the four standards can be

selected by the complex operator as the prime standard.

However, if the prime standard fails, another standard is

automatically selected.

The frequency synthesis and distribution subset consists of

a Coherent Reference Generator (CRG), which includes discrete synthesis, distribution amplifiers, and an interface panel.

The CRG receives 0.1 MHz, 1 MHz, 5 MHz, and 100 MHz from

the on-time frequency standard. It generates 10.1 MI-h,

45 MHz, 50 MHz, and 55 MHz and provides distribution ports

for user subsystems.

Sinusoidal reference signals are also distributed to users at

the antenna areas. This is accomplished by the use of active

stabilized cables that compensate for small phase changes in

the distribution path. These actively stabilized cables are used

for VLBI to provide the same stability as that of the Hydrogen

Maser reference frequency standard. The stability requirements for the standard reference frequency are defined in

Table 1.

The time and timing subset consists of a triple redundant

master clock, simulation time generator, time insertion and

distribution, and time code translators. The master clock

equipment provides the following new capabilities for the

DSN:

,Year End-Automatic reset

Leap Year-Automatic extra day addition

Leap Second-Automatic subtraction or addition of

leap second

Resetability-Simple clock adjustments

Time codes and timing pulses are provided to each user

via a time code translator. Two types of translators are provided for user application, and the time code is available in

binary milliseconds or binary-coded decimal (BCD) seconds.

Time codes are also distributed to users at the antenna areas

via fiber optic cables.

Time synchronization of the three complexes to the National

Bureau of Standards is accomplished via the Global Positioning Satellite (GPS) System by utilizing a GPS receiver at

each complex. Also, other sources for time synchronization

include: WWV, LORAN C, "IV line 10, and traveling clocks.

Time.sync data are also available from the VLBI system.

The functional capabilities of the Frequency and Timing

Subsystem can be controlled locally at the equipment level or

114
.remotely from the complex operator position. The complex

operator is provided with various displays that contain both

status and performance parameters. Such parameters will be

forwarded to NOCC for application in the Network FTS

Data Base.

monitoring the FTS system, and coordinating changes to complex time and frequency standards, as weU as generating

monthly reports containing both time and frequency measurement data.

C. NOCC Subsystem

The Network FTS Data-Base Subsystem, to be implemented

in 1988 at NOCC, will receive configuration data, status data,

and performance data from the FTS subsystem at each DSCC.

Also it will receive GPS time-sync data from the Tracking

System and VLBI time-sync data from the VLBI System.

The data will be analyzed as to time and frequency offsets

between complexes, clock behavior, and Hydrogen Maser

performance. The results will be available to complexes and to

Network Operations.

Presently, the analysis function is being performed by the

Network Operations and Analysis Section (NOA) as a DSN

supporting element. NOA is responsible for generating reports,

III. FTS System Schedule

The fotlowing additional implementations will complete

the Mark IVA FTS:

In support of the X-hand uplink implementation,

stabilized cables and distribution equipment are

required for distributing 100-MHz reference signals

on the antennas at DSS 45,65, and 15.

Time analyzer assemblies and frequency standard

monitor equipment will be implemented as part of

the FTS monitor and control in FY 86.

FTS Data.Base Subsystem at Network Operations

Control Center will be implemented in FY 88.

115
Table 1. _ Frequency Stability

Allen Variance 0")

T

Signal Processing VLBI Antenna-Mounted

Center Equipment

1 _c i x 10-12 I x 10--12

104 sec I x 10-14 1 x 10-14

12 hours I x 10-14 1 x 10-14

10 days 1 x 10 -13 1 x 10 -13

116
"" I""

%.I ,_

;'i (3

_7 a.

Ld 4..I

"="-0

_0

i_ [_.,

o

.!_

3--

4

1

L

I

I

I

I

F

Ell __.1

I

I

I

I

o2

I o

z

oz\xx _

_ _----_-

i

i

c_J

i

I °"

zO

o

o_- z

i

.,,.._{ _ I

8 I

I

J

1

I

._ I

._ _ I r.-.m

I,..,

..,,, _

o

t_J

.L.
EXTERNAL CALIBRATION

SIGNALS

1. SINE WAVES

a. REFERENCESTANDARDS

b. STABLE FREQUENCIES

c. CONSTANT PHASE

d. ISOLATION AND DISTRIBUTION

e. ADJUSTABLE FREQUENCIES

OFF_T

4. CALIBRATION

OFFSET

v

a. OFFSET AND STABILITY

MEASUREMENTS

b. INDEPENDENT PERFORMANCE

ESTIMATES

c. PRECISE SETTABILITY

d. TRANSMIT AND RECEIVE

CALIBRATION SIGNALS

2. TIMING PULSES

a. ISOLATION AND DISTRIBUTION

b. CONSTANT PHA3E W!TH S!NE

WAVES

c. ADJUSTABLE PHASE

REFERENCEFREQUENCY

TIME PULSE

TIME CODE

9

3. EPOCH TIME

a. iSOLATION AND DISTRIBUTION

b. DISPLAYS

c. LOCATES A TIME REFERENCE

IN DSS/DSCC

d. ADJUSTABLE TIME

_ REFERENCE FREQUENCIES

_ TIMING DI n CEzc

j_ , vbe_

TIME DISPLAY

STATION

TIME

REFERENCE

TIME CO DES

_ CALIBRATION SIGNALS

CO NFIGURATIO N_L_J%

COMMAND, AND L_J

CALIBRATION DATA

5. FTS CONTROLLER

o. MONITOR AND CONTROL OF

SELECTED FTS ELEMENTS

b. SUPPLY PERFORMANCE DATA

TO DMC

¢. STAND ALONE OPERATION

OF FTS CAPABILITY

_ PERFORMANCE STATUS DATA

STATION OPERATION

POWER

6. UNINTERRUPTABLE OPERATION

a. AUTOMATIC FAILURE DETECTION

AND RECOVERY

b. MAINTENANCE WITHOUT

DISRUPTING USER

c. POWER OUTAGE iMMUNE

_ POWER FTS ELEMENTS TO ALL

Fig. 2. FTS functional elements and relationships

I,RECEDING p/_GE BL_kNK NOT FILMED

119
TDA Progress Report42-82

N86-10394

April-June 1985

JPL Emergency Support of TDRSS and

Compatible Satellites

N. A. Fanelli

TDA Mission Support and DSN Operations Office

The Tracking and Data.Relay Satellite System (TDRSS) will consist of three identical

satellites in geosynchronous orbits and a dedicated ground receiving station. The first two

satellites (TDRS East and TDRS West) will form the operational TDRS service network

providing near-global real-time user satellite cove_'age. The third TDRS satellite will act

as an in-orbit spare. Since the TDRSS satellites are supported by a single ground station,

a method of providing emergency support for TDRS and user satellites is needed. This

article describes the support to be provided by JPL 's Deep Space Network.

I. Background

With the concept of an orbiting Tracking and Data Relay

Satellite System (TDRSS), NASA's Office of Space Tracking

and Data Systems (OSTDS) adopted a policy of discontinuing

the support of low earth orbiting satellites with a ground based

network of tracking stations.

Goddard Space Flight Centers (GSFC), Flight Projects

Directorate, requested a reconsideration of this policy in 1983.

The Jet Propulsion Laboratory (JPL) was requested to assess

the impact of providing support for a limited number of

projects under emergency conditions.

JPL's study showed that for a nominal cost, the 26-meter

subnetwork of antennas could provide the requested support.

OSTDS then approved the concept and assigned JPL the

responsibility of providing this support.

II. Support Conditions

Low earth orbiting satellites will normally be supported

by the TDRSS and the White Sands, New Mexico, receiving

station. In the event of a failure, either ground Station or

satellite, which would prevent or limit the normal communications functions, JPL may be requested to provide emergency

support. GSFC is responsible for the JPL notification that an

emergency exists.

III. Supported Projects

The following satellites will be supported.

A. Solar Maximum Mission (SMM)

This satellite studies solar activity occurring in active

regions of the sun, sunspots, and solar flares. Observations are

120
conducted in the ultraviolet, X-Ray, and Gamma-Ray regions

of the spectrum with earth based instruments providing

coverage at radio and optical wavelengths.

B. Land Satellite 4 and 5 (Landsat 4 and 5)

Landsat provides data continuity of earth resources information for worldwide users. Each satellite contains a thematic

mapper and a multispectral scanner imaging device plus

mission unique hardware.

C. Earth Radiation Budget Satellite (ERBS)

ERBS studies the earth's interaction with solar energy, a

critical factor in predicting the earth's climate.

D. Space Telescope (ST)

The Space Telescope is planned as a national facility. It

will consist of a 2.4-meter aperture Ritchery-Chretein cassegrainian telescope weighing approximately 9525 kilograms

with various energy detectors designed for observations of

infrared, visible, and ultraviolet wavelengths.

E. Gamma Ray Observatory (GRO)

GRO is an earth orbiting satellite that will study sources

of localized, galactic, and extragalactic gamma radiation.

F. Upper Atmosphere Research Sateilite(UARS)

UARS is designed to study the radiation, chemistry and

dynamics of the upper atmosphere at low, middle, and moderately high altitudes and the coupling between these properties in order to determine the seasonal correlations.

G. Space Transportation System (STS)

The Space Transportation System is the manned reusable

launch vehicle (Shuttle) used to carry into space nearly all of

the nation's payloads for military, private industry, universities, research organizations, and foreign governments and

organizations.

H. Tracking and Data Relay Satellite (TDRS)

The payload of each TDRS provides a Telecommunications service system which relays communications signals

between low earth orbiting user spacecraft and the TDRS

ground terminal. The service is provided by two types of

links: (1) a multi-access system, with one 30-element S-Band

phased array antenna system; and (2) a single-access system,

either S-Band or K-Band, using two 4.8-meter parabolic

antennas.

IV. Support Requirements

A. Initial Action

Upon initial notification of a declared satellite emergency

requiring JPL support, JPL will take the steps necessary to

provide station support. The DSN stations will acquire realtime engineering telemetry so that the project may assess the

satellite health. They will also acquire high rate telemetry

data recorded on the satellite's magnetic tape recorder. The

recorder playback will be activated by ground command. The

supporting DSN stations will generate and transmit radio

metric data. All data will be provided to GSFC in real time or

near real time.

B. Subsequent Passes

For subsequent passes, JPL will acquire real time engineering telemetry data, transmit ground commands to reload the

spacecraft memory, if requested, acquire spacecraft memory

dumps of telemetry data, acquire radio metric data, and provide data to GSFC in real time.

C. Prolonged Support

For prolonged emergency situations, reduced mission

objectives will be pursued. Telemetry data will be acqu.ired in

real time, science data will be acquired via spacecraft recorder

playbacks lasting from six to eight minutes, commands will

be transmitted on a scheduled basis, radio metric data will be

acquired, data rates exceeding the line capacity will be recorded on-station and played back during the post-pass period,

and spacecraft coverage will be provided as defined in a

negotiated schedule.

D. Support Configuration

TDRSS spacecraft emergency support will be provided by

JPL's Deep Space Network (DSN) 26-meter subnetwork of

stations located at Goldstone, California, Deep Space Station

16 (DSS 16); Canberra, Australia, (DSS 46); and Madrid, Spain,

(DSS 66). The configuration delivered by the networks consolidation program will be used and will have the capability of

supporting the data rates listed in Table 1. Real time transmissions will normally be limited to data rates which can be

accommodated by a 56-kilobit transmission line rate. Under

certain emergencies this capability will be increased to a

transmission rate of 112 kilobits. Figure 1 shows a typical

support configuration. Documentation will consist of a network operation[ plan (NOP) for each mission to be supported.

This plan will document the plans, procedures, and configurations to be followed by the DSN stations, the DSN operations

control team, and the DSN ground communications facility

following the declaration of an emergency. The data interface

for each mission to be supported will be documented in the

121
interface control document (ICD) for GSFC and JPL. The

ICD will describe the operational, technical and communications aspects of the data transfer between centers and projects using the DSN.

E. Support Scenario

When an emergency situation develops, GSFC will notify

the DSN operations chief using the coordination voice network. This voice network is active 24 hours per day, seven

days per week and connects the GSFC network control center

to the DSN network control center. GSFC will inform the

DSN of the circumstances for declaring the emergency, the

parameters needed to acquire the spacecraft such as downlink

frequency or channel, telemetry data rate and coding, uplink

frequency or channel, last received uplink and downlink

signal levels, etc. Arrangements will also be made for the

transmission of vectors needed to point the 26-meter antenna

at the spacecraft.

The operations chief will assess the request and negotiate

and schedule station support. A network controller will be

assigned to monitor the station to project interface; when

necessary, communications lines will be activated between the

assigned station and the project's operations control center at

GSFC. The station will be notified by the DSN controller of

the declared spacecraft emergency and will be asked to terminate the activity in progress and prepare for the emergency

support situation. The controller will brief the station on the

nature of the emergency and provide the parameters needed

for spacecraft acquisition.

The GSFC Flight Dynamics Facility will provide state

vectors to JPL or transmit an improved interrange vector

(IIRV) directly to the station. These vectors will be used for

antenna pointing. The station will use the NOP to configure

the station for support. Following the completion of the

preacquisition checklist the station will point the antenna at

the spacecraft and acquire the downlink signal. At this point

the station will notify the DSN controller of the acquisition

of signal (AOS).

The project will instruct the DSN Controller on requirements for recordings, ranging, and command. During the pass,

the DSN controller will monitor the net and provide assistance

to the project or station. Upon completion of the pass, the

station will be released for the next scheduled activity. The

project will determine if continued emergency support is

required and l_rovide the DSN scheduling office with the

requirements.

122
Tlbll 1. EIn_llnq s_tellite date

Mission Initial R/T Contact Rates: Subsequent R/T Contact Rates:

Telemetry, kbps Recorder Dumps, kbps Telemetry, kbps Memory Dump, kbps

LANDSAT 8 256 8 32

SMM 16 512 16 32

ERBS 1.6 32 1.6 -

ST 0.5 1024 4 -

GRO 32 512 I 32

UARS 32 512 1 32

STS 128 - 128 t28 128

TDRS 1 - 1 -

123
ANTENNA J

DSS 16 I

DSS46 I I I I

DSS 66 J J LOW

NOISE

AMPLIFIERS

EXCITER

TRACKING

DATA

PROCESSOR

T

RECEIVERS

SPACECRAFT J

COMMAND 9

ENCODER

_i - PROGRAMMABLE DATA

FORMATTER

"l _]_

J BIT SYNCHRONIZERS

/ FRAME

J SYNCHRONIZERS

_J COMMUNICATIONS L

INTERFACE |

Fig. 1. A typical support configuration

TO/FROM

GSFC

124
TDA Progress Report 42-82

N86-10395

April-June 1985

Planning for VLA/DSN Arrayed Support to the

Voyager at Neptune

J. W. Layland

TDA PlanningOffice

D. W. Brown

TDA EngineeringOffice

Preplanning for the use of the National Radio Astronomy Observatory's Very Large

Array (VLA) in support of Voyager at Neptune has been underway since early ] 982.

When arrayed with the DSN antennas at Goldstone, CA, the VLA more than doubles

the potential data return over the American longitude for the 1989 Voyager encounter.

This report summarizes the background, rationale, and current status of planning for

VLA-DSN Arrayed Support to the Voyager at Neptune.

I. Introduction

The Very Large Array (VIA) is an array of twenty-seven

25-meter antennas in a triradial configuration in the high New

Mexico desert. The primary role of this array is developing

maps of radio-bright objects in the sky, and it incorporates a

large mapping processor which is capable of cross-correlating

the 351 (=27"26/2) baselines of the array in real time (Ref. 1).

One of the optional products of this mapping processor is a

combined output which represents the coherent sum of the

signals being received at each of the antennas. This combined

output of the mapping processor will represent about two and

a half of the DSN's 64-meter apertures when all of the VIA

antennas are outfitted with X-band low noise amplifiers.

A study of the possible use of the VIA for the support of

the Voyager encounter of Neptune in August of 1989 was

commissioned by an exchange of letters in early 1982 between

Dr. B.C. Murray, then Director of the JPL, and Dr. M.S.

Roberts, then Director of NRAO. The study was handled at

JPL concurrent with the somewhat broader Interagency

Array Study which considered the total complement of

possible non-DSN receiving apertures that might be appropriate for support of the Voyager (Ref. 2). That study reported

its conclusions in early 1983, with positive recommendations

for the VIA and a few other observatories. Engineering

studies have continued since that time, and are reflected in

this article.

At the present time, Voyager is virtually unique in the

benefits that can be derived by arraying. The reason becomes

clear when one realizes that the Voyager was fundamentally

designed to give the superb coverage it did at the Jupiter

and Saturn encounters, but with a flight path which would

much later take it past the far outer planets Uranus and

Neptune (Ref. 3). Compensating for the increased distance

is being done partially by changes to the on-board control

software, but to provide outer planet coverage akin to that at

125
Saturn will require a substantial increment to ground receiving capability. Equipping a new mission for Voyager-style

return at these far outer planets would take many years,

and many hundreds of millions of dollars. The flight of the

Voyager presents a once-in-a-lifetime opportunity for today's

scientists and scientific institutions.

The main message from the study was this: Considering

all aspects, including the timeliness and costs of building

antenna facilities, and the scientific potential and uniqueness

of the Voyager, it is appropriate that most of the additional

aperture needed to support the Uranus and Neptune encounters be obtained by enlisting the help of the world's

large receiving facilities, providing that such help could be

mutually arranged. For the particular case of the VLA, configuring it to provide the equivalent of 2.5 times today's

DSN 64-meter antenna capability would boost the receiving

aperture over the American }ongitude so that coverage is

available for both the best compressed imaging data rate, and

for several of the alternative (higher) data rates without

image data compression.

This article updates the VLA material from the Interagency Array Study Report which appeared in TDA Progress

Report 42-74 (Ref. 2).

II. Voyager Mission Characteristics

The objectives of the Voyager mission to Neptune are,

generany, to extend the comparative studies of the outer

planets to include the environment, atmosphere, surface and

body characteristics of the planet Neptune; one or more of

its satellites, with emphasis on Triton, and a search for rings.

Typical specific scientific objectives to be addressed at Neptune include measurements of the gross morphological structures of the planet and satellites;" determination of the Neptune atmospheric composition, structure, and dynamics;

determination of the Neptune rotation period; detailed magnetospheric and plasma studies; a study of the satellite surface

features, temperatures, and probable Triton atmosphere; and

a study of the Neptune rings, if they exist.

The Voyager trajectories, from launch in 1977 onward,

are shown in Fig. 1. As can be seen, the decline in Voyager

signal strength from Jupiter to Uranus is some 10 dB due to

the increase in distance, and the decline from Uranus to

Neptune will be another 3.5 dB, most of which must be

accommodated on the ground. Figure 2 shows the configuration of the DSN as we expect it to be in 1989, together with

the more prominent of the world's.apertures which were

considered by the interagency array study. It should be

noted that the DSN as pictured in this figure is not identical

to the network as it will exist in 1986, but includes a performance upgrade of the present 64-meter antennas, and a

new high-efficiency 34-meter antenna in Spain.

The telecommunications link performance of the Voyager

spacecraft with the 1989 DSN and selected other apertures

arrayed is shown in Fig. 3. The horizontal axis of this figure

is indexed by GMT hours (Earth-received time) for the day of

encounter in August, 1989. The time of closest approach to

Neptune is at 08:06, and the time of closest approach to

Triton, the largest satellite, is at 13:16 hours. The VLA is

well positioned to return the final several hours of Neptune

bright-side imaging, as well as to off-load the tape recorder

to make room for Triton images. The left axis of this figure is

labeled in terms dB of total received signal-to-noise ratio for

a reference data rate of 1 bit/s, and the arcs of the figure

indicate the effective (normalized) received signal strength.

The cruves represent 90% confidence levels, and include the

nominal weather statistics and availability of individual antennas. The right-hand axis is labeled to indicate the threshold

levels applicable to various Voyager data rates. Communication is feasible at a given rate whenever the received signal

strength exceeds the associated threshold level. The information content of the various rates is shown in Table 1. The

higher of the two compressed imaging data rates is made

feasible by the arraying with two apertures: Parkes Radio

Telescope in Australia, and the VLA (2.5 equivalent 64-

meter antennas) in the American longitude.

Ill. Arraying Configuration and

Requirements

The functional block diagram for arraying between the

DSN Goldstone site and the VLA is shown in Fig. 4. The

X-band signal of the Voyager is received concurrently at the

DSN facilities and by the observatory. The X-band receiver

-provides. phase-coherent detection of the spacecraft signal,

and emits a baseband version of it which contains the same

data as the comparable signal at the DSN station. Functional

requirements include the provision of a real-time link between

the observatory and the DSN site, as well as local recorders

at both sites. These would always record the signals, and could

enable near.real-time arraying, but would primarily provide

backup in case of difficulty with the real.time link. The combiner must be capable of operating with either the link or

with the recorders.

With the configuration as shown, the choice is available to

operate either using the link as the primary pathway, or with

the recorded signal as the primary pathway. Using the realtime link is preferred from an operational standpoint because

it provides immediate validation of the array operation,

gives the flight project personnel immediate visibility into

126
their data, and avoids the effort and cost of transporting and

processing tapes, unless link outages occur.

Figure 5 shows the currently planned VIA configuration

for Voyager support. The items to be added to the VLA

include X-band feedhorns, low-noise amplifiers and downconverters at the front ends, and the phase-locked receiver

and coherent detector at the combined output of the VIA

processor. Use of the VLA's intrinsic signal transmission

facility has the advantage that the spacecraft signal reception

capability can be tested, and perhaps operated, almost independent of the location of the antennas within the array.

It has the disadvantage that the Voyager signal is subjected

to a 1.6 ms gap in signal reception per 52 ms control cycle,

and to a 3-level quantization in the VIA processor. Analysis

has shown the gap to be tolerable when the VLA is arrayed

with Goldstone (Refs. 4, 5).

Two channels of the VIA signal transmission and processing equipment may be used: one with a 6 to 12 MI-lz IF bandwidth to carry the spacecraft data signal at roughly full precision, and a second (not shown in Fig. 5) with narrower bandwidth to be used to self-calibrate the VIA: i.e., to control

the differential phase and delay in the system. Careful calibration or control of the differences between these two would

be needed. Tests in 1983 had shown that it was feasible to

self-calibrate the VLA on a natural signal source at a strength

analogous to that of the Voyager spacecraft's signal in the

narrow bandwidth (Ref. 6). More recent testing with the

Voyager signal itself holds promise that the self-calibration

can be achieved with the full bandwidth data channel (Ref. 7).

The baseline design X-band low-noise amplifier is a cryogenicaUy cooled FET amplifier, similar to those in place at

the VLA for its other frequency bands. The first two preprototype X-band LNAs are now in place on antennas at the

VLA, mated with JPL-provided feedhorns, to enable testing

of the X-band receive characteristics. A modern alternative

to the FET, the so-called HEMT (high electron-mobility transistor), with the potential for a much reduced system temperature, will be evaluated for use in the LNAs (cf. NRAO Voyager

Front-End Construction Plan, Rev. A, NRAO internal document, Feb. 15, 1985).

IV. Expected Level of VLA Support

The day of closest approach of the Voyager to the planet

Neptune is August 24, 1989 (PDT). At that time, Voyager

is visible daily from the VIA for an approximately eight

hour "pass." That day, and the several days immediately

surrounding it, are the times of greatest importance for the

data to be gathered from the encounter. The time intervals

for which arrayed support would be requested by the Voyager

project include these critical passes plus several passes for

testing the correctness of both configuration and operational

procedures plus a number of additional passes both preceding

and following the days of close encounter. Times not specifically requested, such as when Voyager is not visible, are presumed to be available for normal VLA astonomy purposes.

It is planned that the VLA will be made available for

Voyager-directed support or operational testing for up to

40 observing intervals of up to 8-1/2 hours each. Operational verification tests will begin in April 1989, and continue into May. Actual spacecraft support could begin in

mid July, about six weeks prior to encounter, and continue

through four weeks after encounter. In addition to the spacecraft operations and operational testing, subsystem testing

and verification would be required.

V. Organization Roles

The four organizations involved, NASA, NSF, JPL, and

NRAO, will each play a part in preparing the VLA to support

the Voyager at Neptune. As the principal beneficiary of this

effort, NASA is providing the funding needed to SUpport all

of those changes to the VLA which are needed for Voyager.

This includes work at both NRAO and JPL. The DSN organization will bear the.primary responsibility for system design,

planning, and management of the project. Implementation

responsibility will be shared between NRAO and JPL. The

mechanism implementing these roles is outlined in the "Management Plan for the VLA-GDSCC Telemetry Array Project"

(D.W. Brown, JPL Document 1220-1, March 15, 1985

[internal document], Jet Propulsion Laboratory, Pasadena,

CA).

With reference to Fig. 5, all equipment which is implemented into the VIA as an integral part of it, such as the

X-band LNAs and down-converters, will be instrumented in

a way that is suitable for general use by NRAO, and will be

retained by NRAO-VLA upon completion of the Voyager

support. All equipment which is adjoined to the VLA specfically for the Voyager support, and which is endemic to spacecraft data handling, such as the phase-locked receiver backend,

specialized recording capability, satellite link to Goldstone,

etc., will be de-implemented by JPL upon completion of

Voyager support.

Again with reference to Fig. 5, NRAO will be responsible

for the design and implementation of the X-band receiving

equipment. This equipment is at least similar to if not identical

to comparable elements already a part of the VLA for other

frequency bands. The feeds have been designed by JPL for

NRAO.

127
The equipment which is Specific for spacecraft support,

such as the phase-locked receiver and coherent detector,

specialized recording, and communications to Goldstone,

will be entirely the responsibility of the DSN, as will the

overall operational coordination of the array. The interface

between this equipment and the VLA instrumentation will be

the single signal combined output of the VLA processor.

Vl. Implementation Planning

Instrumentation of the VLA for Voyager support is a long

and tedious task because of the large number of antennas

involved. Rapid startup is essential if we are not to run afoul

of manufacturing, installation, or checkout problems later.

The current VLA implementation schedule is shown in Fig. 6.

Target date for completion of all installation work is January 1989. The installation of the front-end assemblies

throughout the 28 VLA antennas will be allotted two and a

half years to provide an orderly process which does not conflict with other necessary VLA activities. Some of the mechanical work for mounting feeds, etc., will take four years because it must take place within the maintenance hangar,

and was started at the beginning of 1985. The phase-locked

receiver and other back-end elements are planned to be

direct extensions of the implementation of the Parkes Radio

Telescope for the Voyager encounter with Uranus (Ref. 2).

At a system level, there are tests scheduled in late 1984 and

through 1985 which are essential to assuring that the desired

capability is achievable. The initial single antenna tests, using a

prototype LNA which was developed at the NRAO Charlottesville facility, together with a JPL "spare-parts" feed, have confirmed overall expectations of front end performance as well

as provided the first opportunity to view the Voyager spacecraft signal through a part of the VLA's electronics. The dual

antenna tests extend the signal path through the VLA correlator/combiner to the output port which would be in use in

1989, albeit with many fewer antennas involved in them. The

early lab tests and analysis should support and help the interpretation of the testing at VLA.

A number of review decisions are embedded in the schedule. HEMTs are an improved technology for the FET amplitiers which show promise for significantly reducing their

effective temperature, but require evaluation of reliability as

well as performance. Back-up power, if deemed necessary, and

real-time link equipment would be leased commercial equipment, each requiring appropriate lead time for ordering, contracting, and installation.

VII. Other Users

Other missions and other ground-based radio science users

were explored for possible joint interest in a VLA configured

so it was capable of 8.4 GHz signal reception. One existing

mission, ICE, the International Cometary Explorer, is an Sband mission, and its critical need for large aperture support

is occurring in September 1985, long before such capabilities

could be available. Of the future mission options examined

during the study, most are at modest data rates, and only one,

a Titan Probe (1996), showed any benefit from the type of

ground aperture increase available through the VLA. Even

this benefit seemed minor, and it was concluded that the

arrangements with the VLA for Voyager support should be

considered as unique to the Voyager, over the next dozen

years or so.

There is however additional ground-based science which

becomes possible with the X-band capability added to the

VLA. The extra frequency, for example, is of some interest

in monitoring the spectrum of variable stars and novae, and

possibly for Faraday rotation effects in polarization studies.

However, there are no spectral lines of more than modest

interest within the band, and the planned capability provides

no increased sefisitivity for detection of continuum sources

(A. R. Thompson, NRAO-VLA, private communication,

April 23, 1984).

There is some significant interest in the VLA X-band capability for planetary radar, with the DSN Goldstone 64-meter

site as transmitter and the VIA as receiver. One instance of

this is topographic mapping of the nearer planets such as

Venus and Mars. With existing radars such mapping is done by

a delay-doppler technique which depends upon some assumptions concerning the general form of the target. The resolution

of the VLA would allow these assumptions to be relaxed, thus

providing a more direct measurement of the surface shape.

Radar observations of more distant targets such as the outer

planet satellites and rings would capitalize upon the greater

sensitivity of the VLA. Observing proposals for planetary radar

observations will be input to the regular VLA proposal evaluation process.

Observations for VLBI at the DSN frequencies for astronomy or other applications could also be of interest. The X-band

on the VLA will provide another band in common with the

planned Very Long Baseline Array (Ref. 8), which would be

useful, though not sufficient by itself to cause it to be

implemented.

128
Acknowledgment

The concept described here for VLA support to the Voyager at Neptune owes a great

deal to the support received from the staff at NRAO, and from many members of the

JPL Telecommunications Engineering and Voyager Project organizations. Noteworthy

here are Dr. A. R. Thompson, Dr. P. Napier, and Dr. S. Weinreb of NRAO, and

C. E. Kohlhase, D. A. Bathker, R. C. Clauss, Dr. S. J. Kerridge and B. D. Madsen of JPL.

References

1. Napier, P. J., Thompson, A. R., and Ekers, R. D., "The Very Large Array: Design and

Performance of a Modem Synthesis Radio Telescope," IEEE Proceedings, pp. 1295-

1320, Nov. 1983.

2. Layland, J. W., et al., "Interagency Array Study Report," TDA Progress Report

42- 74, pp. 117-148, Jet Propulsion Laboratory, Pasadena, CA, August 15, 1983.

3. Stone, E. C., "The Voyager Mission: Encounters with Saturn," Journal o/Geophysical

Research, Vol. 88, No. A11, pp. 8639-8642, November, 19_3.

4. Deutsch, L. J. "The Performance of VLA as a Telemetry Receiver for Voyager Planetary Encounters," TDA Progress Report 42-71, pp. 27-39, Jet Propulsion Laboratory,

Pasadena, CA, November 15, 1982.

5. Deutsch, L. J. "An Update on the Use of the VLA for Telemetry Reception," TDA

Progress Report 42-72, pp. 51-60, Jet Propulsion Laboratory, Pasadena, CA, February 15, 1983.

6. Layland, J. W., Napier, P. J., and Thompson, A. R., "A VLA Experiment -Planning

for Voyager at Neptune," TDA Progress Report, 42-82, Jet Propulsion Laboratory,

Pasadena, CA, this issue.

7. Resch, G. M., TDA Progress Report, in preparation.

8. KeUerman, K. I., "The VLBA, Scientific, Technical, and Planning Overview," in

Multidisciplinary Use of the Very Long Baseline Array, National Academy Press,

Washington, D. C., 1983.

129
Table 1. Voyager Uranus and Neptune data rates

Equivalent

Data Rate, kb/s Data Type Full lmages/hr

29.9 Full frame imaging 13

21.6 Compressed imaging 13 + 6

and playback

19.2 Half frame imaging 6

14.4 Compressed imaging - 13

11.2 Compressed imaging 9

8.4 Compressed imaging 5

7.2 General science and None

engineering

130
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131
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135
TDA Progress Report 42-82

N86-10396

April-June 1985

A VLA Experiment - Planning for Voyager at Neptune

J. W. Layland

TDA Planning

P. J. Napier and A. R. Thompson

National Radio Astronomy Observatory

A VLA engineering experiment was conducted on the night of July 22. 1983 to

explore one aspect of the potential for the VLA to support Voyager at its Neptune

encounter in August of 1989. Specifically, the experiment tested the ability of the VLA

to self-calibrate on a natural radio source whose effective signal strength is the same as

Voyager's will be at its Neptune enGounter. The experiment was successful and supported

the belief that the VLA would be able to be self-calibrated with Voyager's signal.

I. Introduction

The Very Large Array (VLA) is an array of twenty.seven

25-meter antennas in a triradial configuration in the high New

Mexico desert. Reference 1 provides a comprehensive overview

of the VLA. Details on the VLA, its hardware and software,

operation, and the theory behind it may be found in Ref. 2.

The primary role of this array is developing maps of radiobright objects in the sky, and it incorporates a large mapping processor which is capable of cross-correlating the 351

(= 27*26/2) baselines of the array in real-time. One of the

optional products of this mapping processor is a combined

output which represents the coherent sum of the signals being

received at each of the antennas. This combined output of

the mapping processor would represent about 2-1/2 of the

DSN's 64-meter apertures if all of the VIA antennas were

outfitted with X-band cryogenic field effect transistors

(FET's). As noted in the Interagency Array Study Report

(Ref. 3), this capability can be of significant benefit to

Voyager at its encounter with Neptune in 1989.

Configuring the VLA to be able to receive signals from

Voyager will require significant investments of time and

money. One of the many questions asked before seeking to

use the VLA was whether the VIA could self-calibrate on the

signal from Voyager, and thus be able to easily combine the

signals from all 27 antennas without recourse to a stronger

nearby natural radio source as a calibrator. As the VLA

could not receive at X-band, obtaining a preliminary answer

to this question entailed observing some radio source which

provided a signal in one of the existing VLA receive bands

which was a suitable analogy to the signal expected from

Voyager at Neptune in 1989.

Voyager's total signal from Neptune is predicted to be

3.6 X 10-21 W/m 2 (B. D. Madsen, Voyager Telecom, private

communication, May 2, 1983). If we filter it into a 60 kHz

bandwidth, selecting one sideband and maximizing the bandpass SNR, the signal is cut by 5dB to 1.1 × 10-21 W/m 2.

Since Voyager's signal is matched to the feed polarization and

will appear in only one of the two polarizations (right-hand

136
circular, but not left-hand circular), while the total flux of a

natural radio source is measured in both polarizations, the

total cataloged flux of the comparable natural radio source is

3.6 Jy. The VIA processor develops integrated cross products

each 10 seconds (or a multiple thereof), which are used to

estimate the differential phases between antennas. These cross

products do not differ in character whether the input bandwidth is 60 kHz or 50 MHz. A natural source of 130 mJy

strength in the full VLA bandwidth of 50 MI-Iz should provide

the same SNR in the integrated cross-products as a 3.6 Jy

source in 60 kHz and, hence, the same as Voyager's signal.

This is significantly below the strength at which the VLA is

normally calibrated.

In the experiment itself, radio sources at both 0.5 Jy and

0.14Jy were included in the observation catalog to help

expose any sensitivity to source strength at this level. The

observing control "deck" was set up using the time sharing

PDP=10, and then called by the array operator into the Modcomp real-time system when needed that evening.

The observation interval occupied 3 hours and contained

the observations listed in Table 1. All observations were taken

in the 6-cm band, with a nominal system temperature of

60 K. Except where noted, the integration time is 10 seconds

and the closed-loop tracking of the differential phases was

performed with a loop gain of 1/4. All observations appeared

to lock-in solidly, and suggest that self-calibration on still

lower source strengths would be quite possible. The estimated

performance of the phase tracking operation was extracted

by post-processing the recorded data in the VLA computer.

II. Experiment Description

For most of the experiment the VLA was operated with

its fundamental integration interval set at ten seconds. Each

ten seconds, the 351 complex integrated cross products of

the signals received at each of the 27 antennas were recorded

for final processing. Also each ten seconds, the 26 cross

products for each of the antennas paired with the selected

reference antenna were sampled and used to estimate a differential phase which can represent atmospheric shifts, uncom.

pensated equipment drifts, source position offset, etc. When

the array is being calibrated, this differential phase is used

as the error signal input for a very narrow-band first-order

sampled-data phase-locked loop. In normal operation with a

10-sec integration time, the loop gain is set to 1/4, which

results in an effective memory time of about 40 sec, or a

bandwidth of 0.025 Hz. For the purposes of the engineering

experiment being described, the final processing of the

recorded data is devoted to estimating how well this sampleddata phase-locked loop was able to track the actual phase

variations.

The final non-real-time processing of the data is performed

by the VLA program ANTSOL (Ref. 2) which uses the stored

integrated cross products to perform a global least squares

estimate of the apparent gain and phase of each of the 27

antennas. This is the relatively well-known multi-parameter

form of the simple least-squares estimator, which will be used

but not developed here. The program uses all 351 of the

antenna pairs to provide a global solution for all 26 of the

phases measured relative to the reference antenna, which is

better by a factor of 2/27 smaller variance than the phase

estimates based upon a single integrated cross-correlation.

The global solution can be obtained using the elementary

(e.g., 10-seconds) integration intervals which were sampled in

the rea!-ti_me processing, or the elementary integrals can be

accumulated to increase their SNR before the global solution

is done. Assuming there is no significant external perturbation of the relative phases, their sample variance is a measure

of the performance of the real-time phase tracking process.

All of the data from the July 23rd engineering experiment

were non-real-time processed with an integration time of 120

seconds. A small part of the data was also processed at 30 seconds to improve visibility into the phase-estimator behavior.

The approximate relation between the observed phase variance

in the final processing, and the variance of the phase attributable to real-time processing is calculated in the Appendix.

III. Observations

As listed in Table 1, the experiment consisted of 10 observations of 6 distinct sources, with distinguishing characteristics

to be noted in the following. Sources are identified by their

standard cataloged positions, with the first source, 1803+784

being at right ascension of 18 hrs, 03 rain, and at declination

+78.4 degrees. This source is one of the VLA calibrators,

and its recorded flux at 2.52 Jy was used as the reference

against which the strengths of other, weaker sources were

compared.

The time of this first observation was 04:12 to 04:21

(IAT), and the elevation angle at which it was observed was

approximately 45 degrees. Final-processing of this source with

a 2-rain integration time resulted in an observed phase jitter of

2.5 ° rms. Converting this into an estimate of the jitter in the

real-time control loop by the methods of the Appendix yields

a value of 3.7 ° rms for the phase-tracking process with a 10-sec

T! and loop gain of 1/4. The data from this source was not

processed with 30-sec intervals.

For each source, the very beginning of the recorded data is

ignored until the acquisition transients have subsided. Then,

as many 120-sec (30-sec) segments as will fit into the allotted

time are accumulated and the phases and effective gains are

137
estimated for each of the 27 antennas, using the 351 cross

correlations between antenna pairs. The phase parameter of

the selected "reference" antenna is arbitrarily fixed at zero.

For each segment, the rms of the 26 non-reference phases

was calculated. The average, and one-sigma, uncertainty of

this rms value was then calculated over the (10 to 20) segments and appears in Table 1 in the columns identified as

R_MS PHASE. The column identified as SIG-PH is the estimated rms phase variation in the real-time phase tracking

process as derived from 120-sec segments by the methods

of the preceding section.

The second, third, and fourth observations were all processed with both 120-sec and 30-sec segments. The rms phase

at 30-sec integration time is approximately the same as the

rms jitter in the real-time control loop. In each case this estimate is larger than that developed on the basis of the 120-sec

segments.

These three observations (numbers 2 through 4) were

planned to compare the behavior at high (78 °) versus low

(16 °) elevation, and at about 0.5 versus 0.14 Jansky source

strengths. It is clear from this set that phase jitter is much

more dependent upon elevation angle than upon source

strength - at least for the levels at which we need to be

concerned. There are presumably variations of {he atmospheric path lengths that dominate the observed phase jitter.

ing time beyond 10 to 20 seconds would filter out some of

the real variations. Another way of lengthening the effective

measurement for the real-time control operation is to decrease

the control loop gain, as was done for observation number 10,

where g = 1/16. This gain change was effected by changing

a Right Arithmetic Shift instruction in the MODCOMP realtime software system. This gain change, by a factor of four,

could have effected a factor of two decrease in phase variations if they behaved as independent variables, but only .a 20%

decrease was observed.

Observations numbers 5 and 6 were "drift" tests, where

the VLA was calibrated on the selected radio source at the

beginning of the time interval, after which phase control was

turned off - the loop gain set to zero - and the phase variations observed as they drifted free over a 15 to 20 minute

interval. Starting and ending rms phases listed in Table 1

indicate considerable growth.

Observation number 9 was inserted to test the scaling of

source-strength with bandwidth, which was used to set the

inferred source strength believed to be Voyager-comparable.

The 2.6 Jy source via 3-MHz passband should result in approximately the same phase behavior as 0.5 Jy in 50 MHz. This

it did, but the observation was inconclusive because the

non-SNR.related effects appear to dominate the phase

variations.

Later observations number 7, 8, and 10 used the same

0.14-Jy radio source with varying real-time processing strategies. For number 7, the unit integration time was doubled, to

20 sec, which resulted in an apparent 20% decrease in phase

variation, as compared to a 40% decrease which could theoretically have resulted if the source of the phase variation was

independent between the 10-sec segments. Further, lengthening the segment time to 60 sec (and correspondingly increasing the loop gain to 1/2) actually resulted in an apparent 10%

increase in phase variations. The expected response if the

sources of the phase variations were independent between

the 10-sec segments would have been about a 25% decrease.

While these changes are not significant in a statistical sense

because of the wide uncertainty of the observed phase variations, they strongly suggest that lengthening the unit observIV. Conclusion ,,

Taken together, the several observations in the experiment

demonstrate that one should be able to self-calibrate the VLA

with a signal comparable in strength to that expected of

Voyager at Neptune, without suffering serious SNR loss to

the combined signal from the jitter in the real-time phase

tracking process. Application of that result to the Voyager

signal involved analogies in both the received signal frequency

and the VLA processing bandwidths, so further testing of

self-calibration with the Voyager spacecraft signal itself

will be needed as the X-band capability begins to be assembled on the VLA. The phase variations induced at low elevations are a concern which needs further exploration, as Voyager

will be at quite a low elevation throughout its encounter

pass.

138
References

1. Napier, P. J., Thompson, A. R., and Ekers, R. D., "The Very Large Array: Design and

Performance of a Modern Synthesis Radio Telescope," IEEE Proceedings, pp. 1295-

1320, Nov. 1983.

2. HjeUming, R. M., Ed., An Introduction to theNRAO Very Large Array, 1983 edition,

document available from NRAO to users and potential users of the VLA, (National

Radio Astronomy Observatory, Socorro, NM 87801).

3. Layland, J. W. et al. "Interagency Array Study Report," TDA Progress Report 42-74,

pp. 117-148, JPL, August 15, 1983.

139
Table 1. Summary of observations

RMS Phase RMS Phase

OBS Source Flux IAT EL Angle Sig-Ph. Notes

No. @ 2 rain @30 sec

1 1803+784 2.52 4:12 - 4:21 45.1 - 45.3 2.5 3.7 - Calibration

2 1714+219 0.53 4:25 -4:37 77.8-77.5 3.2*- 0.9 4.8*- 1.4 6.5 *- 1.0 Ref@ Hi Elevation

3 2037-253 0.57 4:42 - 4:51 15.4 - 17.5 6.0 ± 2.0 9.0 ± 3.0 10.0 *- 2.6 Ref_ Lo Elevation

4 1831-126 0.14 4:57 - 5:09 41.8 - 42.6 3.8 -+ 0.9 5.7 ± 1.4 6.3 *- 1.0 Nom. Vgr. Flux

5 1748-253 0.50 5:10 - 5:29 30.5 - 30.1 2 --, 5 ,3 -- 7 - Drift Test No. 1

6 1748-253 0.50 5:30 - 5:47 30.1 - 29.5 4 -- 10 6 _ 15 - Drift Test No. 2

7 1831-126 0.14 5:50 - 6:04 43.3 - 43.0 3.4 -* 0.3 4.2 *- 0.36 - 20 sec T1

8 1831-126 0.14 6:05 - 6:21 43.0 - 42.4 4.2 *- 1.2 4.7 *_ 1.3 - 60 sec TI

9 1908-202 2.6 6:22 - 6:38 35.8 - 35.6 2.7 *- 0.3 4.0 ± 0.45 - 3 MHz Scale

10 1831-126 0.14 6:39-6:55 41.4-40.2 3.9*-0.9 4.1.-0.9 - 10 sec@g = 1/16

140
Appendix

Extraction of Phase Jitter

Let the variance of the elementary phase estimates be

denoted as Vlo, and assume that this is due entirely to thermal

noise or other independent noise sources. Assume further that

Vlo is small enough that nonlinearities can be ignored. These

estimates are fed in real-time into the first-order phase-locking

process with gain g = 1/4. The result of this process is that the

individual local oscillators of the VLA antennas contain a

refined estimate of the differential phase between the reference antenna and each of the others. Let ntt denote the noise

at time t on the phase-difference measured between antenna

number i and the reference antenna, and let err be the accumulated phase (error) in the antenna number i local oscillator relative to the reference antenna phase. Then assuming

the nominal phase to be zero gives

Ct,e = ¢i,e-t +g(ni, e-i - ¢i,t-l)

=g.

t-I

(1 _g)t-l-s. ni, s

$=--¢m

for the phase of the ith local oscillator. The variance of the

filtered local oscillator phases denoted V_o/, is

VIot.= VlO*_9 2(1-g)2i

i=0

VIoy = Vlo *g/(2-g)

Vlo/ = Vlo/7 forg = I/4

If the global solution is performed in final processing with

the 10-second integrated cross products, the variance of those

resultant phase measurements, denoted Vra 1o, is a combination of the residual phase offsets held by the local oscillators,

and the additive noise during the measurement interval. In

this case the additive noise of the current measurement interval is independent of the actual phase errors which depend

only upon past measurement intervals. Let V_0g denote the

noise variance on the global solution phases based upon a

10-second interval. Then by this independence,

Vm 10 = VIOl + FlOg

Although the noises on the 351 cross products are 'not

independent, the individual noise components in the 27 correlated signal streams are. Hence, it is easy to show that the

cross products are pair-wise uncorrelated, and therefore the

covariance matrix of the inputs to the global least-squares

process is diagonal with all non-zero entries equal to Vlo. The

formal covariance matrix of the 26 globally solved phases

has all diagonal entries equal to 2=V10/27 and all non-diagonal

entries equal to I:1o/27.

Thus:

2

Vlos' 27 Vlo

Then, solving for the actual real-time phase errors, we find

V1o/ = 0.79 Vm 1o

Actual measurement intervals used are 30 sec and 120 sec,

in order to reduce the amount of data to be handled to a more

manageable level and to reduce the additive noise. As noise

samples are independent between each 10-see sub-interval,

adding '£' of these together (£ = 3 or 12) will reduce the noise

variance by a factor of 1/£ on all cross products (phasedifference measurements) except those involving the reference

antenna. When phase-difference measurements involving the

reference antenna are extended, the filtered local oscillator

phase is anticorrelated with respect to the noise in the previous 10-second intervals.

Let Xi, t denote the £-interval average of the phase-difference measurement with respect to the reference antenna

2-1

Xi,t = "_1E D (i - r, t-])

/=0

= "_ /=0 i,t-/- g s=-** (I _g)t-/-l-s . n_ s

Where D(i - r, .) denotes the 10-second phase-difference

measurement, and nts as before denotes the additive noise on

141
that measurement. By collecting like terms, and simplifying,

the expression for Xtt becomes

1 (1 - g)J

/=0

I_-I t-£

-g" T (1 -g)J (1 _g)t-s. ni,s

= $=--oo

The variance of the Xtt is simple to compute as long as

the n_s is independent and becomes

2 - 2 (1 -g)_

V = VlO (2g-g2) £2

For £ = 1, this expression degenerates to the obvious Vx =

Vlo + Vlof. For £ > 1, the Vx can be considerably less than

Vlo/£, which is the contribution of the noise alone.

The _-sample average of the local oscillator phases can be

extracted from the above expression for the Xi, r by removing

the additive noise terms from the first summation. Denote the

variance of this Q-sample averaged phase as V{@_}. This

variance is found to be

Vlo I£_ 2(1-g)[1-(1-g)_] I

For £ = 1, this expression degenerates (as it should) to the

expression for Vlof. Inserting numbers,

V{@_} = 0.060 • Vlo for _ = 12, g= 1/4

= 0.113 • Vlo for £ = 3, g=1/4

Global least-squares estimates of these @_ are extracted, and

their statistics computed to derive Vlof. Of the 351 baselines

used in the global solution, 325 contain additive noise terms

which are uncorrelated from each other and from the @_. The

26 baselines which are used in the real-time phase tracking

process contain noise which is uncorrelated with the noise on

the other baselines, but anticorrelated with the associated ¢_.

Since this apparent suppression of the added noise affects

less than 8% of the baselines, and those only partially, we will

assume in the following that this correlation between the

real-time filtered phase estimates can be neglected, and solve

for I,'1of as determined from the accumulated measurements at

30 and 120 seconds. In retrospect, it is clear that the need for

this assumption could have been avoided by discarding the

reference antenna and the affected baselines from the global

solution process. Using the assumption, we find:

Vlof = 1.14 V 3o using 30 second intervals

Vlof = 2.27 I"m 120 using 120 second intervals

A few other cases are also of interest, and can be extracted

from the foregoing calculations: A) loop gain = 1/16, 10 sec

real-time and 120 sec global integrals.:

V{¢_} = 0.786 * Vlo f, for 120 seconds, g = 1/16.

Inverting as before,

Vlo f, = 1.13"V12o for 12= 12, g = 1/16

B) loop gain = 1/4, 20 sec real-time and 120 sec, global integral

V{@_} = 0.619 * II2of for 120 sec, g = 1/4

and

V2of = 1.51*V m12o for£ =6,g =1/4

C) loop gain = 1/2, 60 sec real.time and 120 sec global integral

V{¢_} = 0.75 * V6o f for 120 sec, g = 1/2

V6o f = 1.24 * Vm 1so for I_= 2, g = 1/2

These conversion factors will be used to convert from measured phase variations to an estimate of phase variations

embedded within the real-time phase-lock process.

142
N86-10397

TDA Progress Report 42-82 April-June 1985

Periodic Binary Sequences With Very Good

Autocorrelation Properties

S. Tyler and J. Loftsson

ReliabilityEngineeringSection

Computer searches were performed using both an 8086 microprocessor and a Cyber

750 mainframe to find repeated binary phase coded waveforms with very good ma'tched

and mismatched autocorrelation properties. The best results for every period up to 64

are given. Sequences with optimal peak sidelobes were discovered for each of these

periods. These sequences have extensive applications in radar and communications,

particularly in situations when there are very unfavorable signal-to-noise ratios. The best

sequence of period 64 when processed using a mismatched filter giving no sidelobes

has a reduction in the main lobe of less than 0.23 dB.

I. Introduction

This article discusses binary sequences of lengths up to 64

with very good periodic autocorrelations.

For short periods, optimal sequences are well known. For

sequence periods which equal 3 rood 4 and are prime or are

of the form 2n - 1, direct methods for finding a sequence

with, in some respects, an optimal autocorrelation are also

known.

However, for most other periods, there is no known practical algorithm for deriving optimal sequences (Ref. 1). Since

sequences of lengths of about 40 or more are too long to

be subjected to an exhaustive computer search (Ref. 2),

smaller searches must be made and practical algorithms developed which may lead to sequences with good periodic autocorrelations.

Periodic sequences such as these have great practical value

in radar (Ref. 3) and communications (Ref. 4), especially in

situations with extremely adverse signal-to-noise ratios. These

sequences also have value in artificial intelligence, since analog

solutions and pattern recognition methods may apply and be

far superior to digital methods for finding excellent (but not

the best) long sequences. In addition, such sequences can be

used in cryptography to provide derivable "code books" in

situations where "two-key" encryption is not desired. Finally,

optimal sequences are ideal for use in searches for extraterrestial intelligence; not only are they easy to detect, they

also advertise current levels of technology.

One example of the use of such a sequence is the Venus

ranging experiment by MIT's Lincoln Laboratory in 1959

and 1961. A binary "pseudorandom" shift register sequence

of period 213 - 1 = 8191 was used to determine whether to

143
transmit "pulse" or "no pulse" in consecutive time intervals.

This sequence was very easy to synthesize and had the property that its autocorrelation was recoverable despite a noiseto-signal excess of many decibels (Ref. 5). Due to the difficulty of analyzing an 8191-sequence, planetary ranging sequence periods are generally shorter (28 - 1 = 255 is common)

or at least factorable. Even a sequence of period 8192 would

be relatively easy to analyze using Fast Fourier Transform

techniques. However, the .computers of 1959 did not have

sufficient capability to permit the synthesis of an adequate

8192-sequence. The ranging systems used at the Jet Propulsion Laboratory have tended to use Boolean combinations of

several shorter sequences to facilitate rapid acquisition. The

combination is used to specify phase modulation on a continuous wave carrier; this technique requires less maximum

power output than does amplitude modulation (Ref. 5).

This article will discuss the following:

Matched Periodic Binary Sequences: A description

and example of how to evaluate the autocorrelation

of a sequence.

Calculation of Mismatched Values: A description

and example of how to calculate the main-lobe loss

when a sequence is analyzed by a mismatched filter.

Sequence Generation Techniques: A description of

the techniques used to generate "good sequences.

Results: Tables of the best values found, both for

matched values and those analyzed by a mismatched

filter.

Some of the values listed are "optimal"; others are merely

the best the authors have been able to obtain to date. The

main emphasis has been on finding the best value for a sequence of period 64 - the best previous value was improved

by over 27%.

II. Matched Periodic Binary Sequences

A binary sequence (or binary code) is a string of bits. It

can be thought of as a vector, e, where each ci is a plus one or

a minus one. A periodic sequence is one which is continuously

repeated; for a binary sequence of period L _÷j = c_ for all £.

The "autocorrelation," a, of a sequence c of period j is:

k----1

When the autocorrelation is normalized by dividing it by

/, it is called the "autocorrelation function."

Here a., a_., a^. and so forth are "main lobes," the remaining a_ are "sldelobes;" a is considered to have j "elements,"

one main lobe and j - 1 sidelobes.

For a sequence to have "good matched autocorrelation

properties," it must satisfy at least one of the following

criteria:

(1) The peak sidelobe in the autocorrelation is small.

(2) The sum of the squares of the sidelobes in the autocorrelation is small.

These concepts are illustrated by means of an example.

Consider a sequence of period 7:

--++-++

To get the elements of the autocorrelation, suppose the

following:

- - + + - + + is the original sequence. Then

+ - - + + - + is the sequence shifted one position.

-+-+--+ is the arithmetic product for each

position. The number -1 is the sum of

these products; it is the first sidelobe

element of the autocorrelation.

Shifting by 2,

..... dt-_

The number -5 is the sum; it is the second sidelobe element of

the autocorrelation.

Shifting by 3,

+-+-+++

The number +3 is the sum and the next element in the autocorrelation.

Shifts by 4, 5, and 6 positions are equivalent to those of

3, 2, and 1. The last element of the autocorrelation is the main

lobe. It corresponds to the original unshifted sequence. The

other elements are the sidelobes (the main lobe is not a sidelobe). Thus, the autocorrelation of- - + + - + + is -1, -5,

144
3, 3, -5, -1,7. Here P = peak sidelobe magnitude = 5;M = sum

of squares of sidelobes = 70.

An "optimal" sequence for period 7 is + + +- - + - and

has the autocorrelation =1, -1, -1, -1, -1, -1, 7, where P = 1,

M=6.

For period 8, + + + + - + - - is an optimal sequence. It

has the autocorrelation, 0, O, 0, --4, 0, 0, 0, 8. Here P = 4,

M= 16.

The remainder of this section shows the connection between

the autocorrelation and the Fourier transform.

As a preliminary, it should be noted that a_ = a/+_ = a/__ for

all £:

a]+_

i i 2÷i

-- = E c _ ck

k=l k=l k=2+l

i i

k=l k=l

It is useful to have a matrix Z which satisfies a = Zc, that is:

/

k=l

However, the "circular convolution matrix" R of the

sequence e is actually a/× / matrix satisfying:

Since

i

a__t = _ Rk_ ck

k=l

a__ 1 = a_+i_ I

i

= E ck Ck+_+/-I

k=l

Then Rk_ = ck+ 1-2

/

= _:_ Ck Ck+l-2

/¢=1

The "Fourier transform" 3. of the sequence e, a vector

X = De, satisfies:

/

X_ = EDk_ck

k=l

where

= l._l__(k-l)(Q-1)

vT

co = exp(2ni//)

i = ,v/:-'f

The sequence e can be. restored from k by means of the

"inverse Fourier transform":

i

ck = E D*km _*m

m=l

where

_m*= the complex conjugate of Xm

De_ _ 1 _/-(k-l)(m-l)

vT

The circular convolution matrix can therefore be expanded

in terms of an inverse Fourier transform:

/

Rk£= Ck+l__ - N__/m_=l ¢oj-(k+l-£-l)(rn-1)_k* m

i

=%_-m_=l OJ]-(k-l)(rn-l) _k* Go(rn-l)(_-l)m

/

= ,4- >,*D.,,

m=]

Thus

R = ,v_-D* A*D

145
where

A = 8z7 I

8.. = 1 i=j

= o i--/:j

The criterion of optimality that average sidelobe response

be minimized with respect to mainlobe response means minimizing

j-1

2

_=1

a?

!

Since a_ always equals 1_ , this is equivalent to minimizing

the sum of the squares of the sidelobes:

j-I

2

_=1

which is in turn the same as minimizing

/ J

2

E a_ = E* al_ _ al__ =c*R*Rc

_=1 _=1

= jc*D*IAI 2Dc =]X*IAI 2 3,

This is equivalent to minimizing

J

Ihil4

i-- 1

If evaluations of sequences are to be performed on a computer in a language which includes no bit manipulation instructions, calculating the hi is more efficient than evaluating the

autocorrelation. It requires ]2 multiplications to calculate

the autocorrelation unless individual bits are used. To calculate the h i using a Fast Fourier Transform (FFT) requires

fewer than j (log 2 j) operations. However, in an assembly

language, a maximum of j + (j/word length)logical operations

are needed to replace the j2 multiplications in calculating the

autocorrelation.

When calculating the hi, it is helpful to check the normalization• By Parsev.al's theorem:

!

Ih112 = /

i= 1

III. Calculation of Mismatched Values

When sequences are used for ranging, they are phase coded

rather than amplitude modulated• Since the signal-to-noise

ratio (SNR) is expected to be very low, it is generally favorable

to use maximum amplitude throughout; amplitude modulation would be inconsistent with this requirement. Similarly,

periodic rather than aperiodic waveforms are generally used

to increase the redundancy of the information• A single

(aperiodic) sequence uses 2n bits to transmit only n bits of

information (the information transmitted being the displacement of the starting point of the sequence)• However, a

repeating (periodic) sequenc.e uses m × 2" bits to transmit

the same information, where m is the number of repetitions

which are processed (whether this is truly the best way to use

m X 2 n bits to transmit n bits of information in a noisy environment is not the issue).

The detection procedure is equivalent to comparing the

incoming signal io a template consisting of the original sequence and "moving the template around" until it matches

the signal• In this situation the autocorrelation is the "output"

of a "matched receiver•"

Although the signal is not amplitude modulated, the

template may have some amplitude modulation• The incoming

sequence is then no longer correlated with itself but with a

similar "weighted" sequence. By varying the amplitude of each

bit in this template sequence, the sidelobes can be reduced or

even eliminated.

• •

The cross-correlation, x, of two sequences, b and c, each of

period / is:

J

X_ = E bk Ck+_

k=l

Let c be a binary sequence which is to be cross-correlated

with b, a sequence composed of real numbers. Sequences c

and b are related by the "weighting function" t.

b_ = t_ c_

In this case, b can be considered a "mismatched filter" to

c. To normalize this mismatched filter:

]

2

E b_=J

t_=1

146
For a given sequence, there usually exists a mismatched

filter which can be used to mathematically operate on the

sequence so as to reduce all the sidelobe values to zero. For

/ > 4, however, the main-lobe value will also be reduced somewhat. The sequence with the best mismatched (cross-correlation) properties is the one which has the smallest decrease in

main-lobe value and therefore has the smallest ratio of

/2 - f2

x. /

! E bkck

k=l

The circular convolution matrix R for mismatched

sequences is:

]

xQ-1 =E R_b k

k=l

The average-to-peak cross-correlation response is then:

j-1

i= 1

x.2

/

Minimizing the above is equivalent to minimizing

b* RR* b

b* cc* b

which occurs when

b = (RR*) -1 c

=ID* IA 1-2 De

1

This gives b k proportional to

J

This choice of b zeros the cross-correlation (Ref. 5). The only

constraint necessary for RR* to be nonsingular is that IXil2 > 0

for all i, i.e., that RR* is positive definite

The best sequence is the one which minimizes

] i=t IXil z

In practice, smaller weights may be chosen to reduce the

SNR loss. In this case, the sidelobes will be reduced rather

than eliminated.

The following example illustrates the derivation of the

appropriate mismatched filter for a given sequence.

Consider the sequence of length 8 discussed in the previous section. The sequence + + + +- +-- has autocorreo

lation 0, 0, 0, --4, 0, 0, 0, 8, where P = peak sidelobe magnitude = 4 and where M = sum of squares of sidelobes = 16.

It will be shown that when this sequence is analyzed with

the appropriate mismatched filter, b, the cross-correlation

becomes (0, 0, 0, 0, 0, 0, 0, :9)"

The elements of the mismatched filter, bk, must be normalized so that

8

1 2ffi

-_- _'_ bk 1

k=l

in order to obtain the correct value of x.

The loss in SNR for the mismatched filter is then L =

(8/x/) 2. To actually calculate L:

1i ,

L=T k=l

where the h_ are elements of the Fourier transform of the

original sequence:

-)_1 "

X2

_3

_'4 =

?'s

_'6

where

i

co

B

1

x/8

"1 1 1 1 1

1 60 i -co* -1

1 i -1 -i 1

1 --co*-i 60 -1

1 -1 1 -1 I

1 --60 i co*-1

1 -i -1 i 1

1 co* -i -co -I

1 1 1

-co -i co*

i -I -i

co* i -co

-1 1 -I

co -i -co*

-i -1 i

-co* i co

= x/ST = exp (i7r/2)

= _ = exp (in/4) = (1 + i)/x/r2

1

1

1

1

-1

1

-1

-1

147
Thus

1

2

(1 +i) - V/2"(1 -i)

2/

2(1-i) + _(l+i)

-2

2(1+i) + X/2-(loi)

-2i

2(1-0 V_" (1 + i)

1

IXl 2 = -_-(4, 12, 4, 12, 4, 12, 4, 12)

1

IXlz

- (2, 2/3, 2, 2/3, 2, 2/3, 2, 2/3)

L =_- k=l

= 1 (8 + 8/3)

8

= 4/3

= 1.3333

The loss in dB is I0 loglo (1.3333) = 1.250 dB.

For this particular case, the mismatched filter elements

can be found by inspection. They also can be calculated as

follows:

/ Xk

bi= E Da

k = 1 IXk 12

where D_.k is the inverse Fourier transform.

There is no need to normalize the bi at this point; it can

always be done later since

/

=J

i= 1

The filter b is proportional to

-1 1 1 1 1 1 1 l

1 co* -i -w -1 -4.o* i co

1 -i -1 i 1 -i -1 i

1 ..¢o i co*-1 ¢o -i -¢o_

1 -1 1 -1 1 -1 1 -1

1 --co* _ co -1 _* i -w

1 i °1 -i 1 i -1 -i

1 w i -to* -1 .-_ -4 w*

6

!(1 + i) - x/2 (1 - i)

6i

2(1-i) + v_ (1 +i)

.-6

2(1 +i) + v_ (I-i)

-6i

2(1 -i) - ,,/5 (1 +0

=8

1

1

1

-1

3

-1

-1

With b properly normalized:

b = ! (1,3, 1,1,-I, 3,-1,-1)

It is easily verified that the cross-correlation of

(1,1,1, I,-I, 1,-1,-1)

and

is

1V_(1,3, 1, 1,-1,3,-1,-1)

(0,0,0,0,0,0,0,4 V_

IV. Sequence Generation Techniques

A. Iterative Improvement

The basic concept in deriving a good sequence is to start

with an arbitrary sequence of period / and perturb one bit of

the sequence to create a new sequence. Then the criteria of

optimality developed in the previous section are used to

decide whether the new sequence is superior to the old one.

If the new sequence is inferior, a new bit of the old sequence

is perturbed; if the new sequence is superior, it replaces the

old sequence and is itself then modified by one bit. Eventually

a "locally optimal" sequence is thus obtained. A new initial

sequence is chosen, and the procedure is repeated as often as

desired.

Perturbing a sequence by 2 bits or 3 bits was also tried;

it yielded significantly inferior results to 1-bit modifications.

It may appear that one should look at, say, all /C 4 sequences which vary by four bits from the best sequence

148
obtained by iterative improvement once one-bit, two-bit, and

three-bit modifications have failed to improve it rather than

abandon the sequence and start over. Certainly, it can not

hurt to apply this technique to the best sequence ever found.

However, in general, the time spent in an often futile attempt

(via 4-bit modifications) to improve one already "locally

optimized" sequence could be better spent by locally opti.

mizing a thousand new sequences from scratch.

The iterative procedure also produces good sequences

faster than does an exhaustive search. For ] = 43, at least one

out of every 107 sequences examined had M = 42; for an

exhaustive search, only one out of every 101° sequences

would have that value. Even if the exha-stive search examined sequences only when

/

_"_.c_= +-I

i= I

only one of 109 sequences would have M = 42. An actual

exhaustive search might be restricted to sequences beginning

with two or more +l's followed by a -1; this would not

increase the rate of finding M = 42 sequences.

The best results were obtained by minimizing M, but good

results should also be obtainable by minimizing L. When

iteratively improving a sequence by calculating L, one need

not continually recalculate the sum in

k=l

If element c of the original sequence is to be perturbed,

r

the new Fourier transform elements, 2,_, are always simply:

P

h_ = h_ -+2Dn_ = 2,_ -+(2/V7) exp (2rri(n - 1) (£ - i)/])

which is noticeably more efficient, especially when ] is prime

so that an FFT does not help.

It is also possible to modify a sequence iteratively by noting

which elements of the Fourier transform are farthest from

unity and then perturbing the appropriate bit or bits in the

sequence to improve these worst values of )'c

B. Choosing an Initial Sequence

Since numerous sequences were to be chosen, an important

criterion was to avoid accidentally repeating initial sequences.

This was done by making each 4-bit "nibble" of the first

initial sequence different and then systematically changing

the sequence of nibbles. Rotations of initial sequences are

unlikely to give identical results after iteration (but ones

complementation will). Thus for/ = 64, this procedure can

supply 16!/2 > 1013 initial sequences while for] = 48 it gives

16!/(2 × 4!)> 4 × lO II initial sequences.

Attempts to improve on the choice of initial sequences by

modifying sequences of periods j +--1, j -+ 4, 2/', ]/2,1/4,

]1 + ]2 = 1' and so forth (where the unmodified sequence had

good autocorrelation properties) did not yield better results.

It is also difficult to synthesize a large number of such initial

sequences.

It may seem that a minor modification of a sequence with,

e_,r f = lfY')'_ _nd it4" = IN')') will give _ gnnd ¢_nll_nc_ with

j = 1024; this is simply not true. An even worse idea would

be to create a j = 1023 sequence with a shift register and then

pretend it has j = 1024 and analyze it with a Fast Fourier

transform.

C. Methods of Finding Good Sequences

by Inspection

1. Quadratic residue sequences. For odd prime periods,/,

one forms a quadratic residue sequence by setting to -1 all

elements ci for which i = n2 mod / for some integer n </'/2.

The remaining elements are set to +1. These sequences give

• M =/' - 1 (always optimal) for/' = 3 mod 4. For / = 1 rood 4,

half of the as equal +1 and the remaining ai are -3, thus M =

5(/" - 1) which presumably is never optimal, merely good.

For example let/= 11. Elements 1, 4, 9, 5 = 16 mod 11,

and 3 = 25 mod 11 are set to -1. The remaining elements are

+1. This gives a sequence withM = 10.

An integer ! is a quadratic residue modulo n if m z = I

(modulo n) has a solution for some integer m and (I, n) = 1.

When p is an odd prime, the Legendre symbol (lip) is defined

as:

I 1 if/is a quadratic residue modulo p

I - 1 otherwise

For this reason, quadratic residue sequences are also referred

to as Legendre sequences.

Quadratic residue sequences always have sums

_-_c/ = -+1

i= l

As will be shown later, such sequences do not have good

mismatched autocorrelation properties. It would be more

useful to discover an algorithm (if one exists) to produce the

149
P = 3 sequences with sums of -+7 that dominate Table 2 for

/" = 1 mod 4 rather than use quadratic residue sequences with

P = 3 and sums of -+1.

2. Double prime sequences. For periods, j, which are the

products of two odd primes: j = JJ2 where J2 > Jl' one can

synthesize "double prime" sequences. These sequences are

called "twin prime" when ]'2 = Jl + 2. Twin prime sequences

are always optimal, with M = j - 1. Double prime sequences

are based on the Jacobi symbol [l/j] where j = jlj2 and

For double primes, one sets to -1 all elements ci for which

(i, ]) = 1 and [i/j] = -1 as well as those for which (i,/) :_ 1

and i = 0 mod/'2" The remaining elements are set to +1. For

example, let j = 35. For (i, j) 4= 1 and i = 0 mod 7, elements

7, 14, 21, 28, and 35 are set to -1. For [i/j] = -1 elements 2,

6, 8, 18, 19, 22, 23, 24, 26, 31, 32, and 34 are set to -1. The

rest of the elements are +1. This gives M = 34.

For 1"2= ]1 + 4, this method gives sequences with autocorrelation element values exclusively of 1 and -3, which are

presumably good but hot optimal. For j2 =]1 + 6, the method

gives autocorrelation element values exclusively of-I, +3,

and -5, which are not necessarily even good.

Triple prime sequences can also be generated using the

Jacobi symbol. The first one with three odd primes has j =

105, a period which is not investigated in this paper. In addition, it is not manifest that such a sequence should be good,

let alone optimal.

3. Shift register sequences. Forj = 2n - 1, a "shift register"

sequence can always be created with M = j - 1. For example

let j = 15 = 24 - 1. Then one must find an irreducible polynomial of order 4 such as:

r4+r3+ 1 = 0

with a-recursion of

rk = rk_3+rk_4

where addition is mod 2. Applying this recursion to 0011

• gives 001101011110001 which, when one replaces the O's

with -l's, hasM = 14.

These sequences are produced by "shift registers" which

are devices of n consecutive binary storage positions which

shift the contents of each position to the next position down

the line at regular intervals. To compute the new first position, a mod 2 sum of the contents of some of the previous n

positions (n = 4 in the above example) is used.

For j _< 127, a recursion relation for j = 2n - 1 can always

be found of the form:

rk = rk__ +rk_"

where 0 < It< n, which gives a sequence with M =/" - 1.

4. Multiplication of sequences. When one multiplies two

sequences of relatively prime periods bit by bit, their autocorrelations are also multiplied, element by element. Thus if

sequence c, with autocorrelation a, is formed by multiplying

the sequences d and e, with autocorrelations u and v respectively, then

c.=d.e.

I 1 I

and

a. = u. 12.

I i 1

If e is optimal (P = 1) with some odd prime period/'e = 3

mod 4 and d is optimal (P = O) of period 4, then c has period

]c = 4]e and M = 16 (/'e - 1) = 4]'c - 16. For example if d is

+ + +-,and e is + + +- - +-,then write d as

(+++- +++-+++-+++- +++-+++- +++-)

andeas

(+++--+-+++--+-+++--+-+++--+-)

_encis

(++++_+__++_++-+-+ .... ++___++)

To get the autocorrelation:

u=(0004000400040004000400040004)

v=(-1-1-1-1-1-1 7-I-1-1-1-1-17-1-1-1-1-1

-17-1-1-1-1-1-1 7)

So

a = (000-4000-4000-4000-4

0 00-4000-4000 28)

150
An M = 192 sequence for j = 52 can be obtained by multiplying the 4 and 13 sequences. The 4-sequence, d, is: dI =

1, d2 =-l,d 3 = l,d 4 ----1, and d i =d._ a.Inhexnotation,

replacing the -1 by a 0, this is E. Repeating this thirteen times,

d = EEEEEEEEEEEEE. The 13-sequence, y, is, in hex notation: 1F35. Repeating this four times, y = F9AFCD7E6BF35.

Then z/= d_i gives z = E8BEDC6F7AE24 which hasM = 192

as well as an L equal to that of the 13-sequence, namely 1.040.

The "exclusive nor" (the ones complement of the "exclusive

or") operation is used to perform this multiplication in hex

notation.

5. Golay sequences. There is another method to obtain an

M = 1Q'2 _etauenee for i = 52.

...... at .... , -

Let y = F9AFCD7E6BF35, as before. Then let

d. = 1

!

!

di= 1

In hex, d = FFFFFFC001FFF

1 _i_26

27 _i_39

40_i_52

z. = d.rvi now gives z = F9AFCIM195F35

which is a sequence of the form S S S S where each S is a

sequence of length 13. The sequence z has L = 1.050. Sequences of the more general form S 1$2S 1S2- are called Golay

sequences. Sequences of the form S S S S do not generally

have the property of producing good autocorrelations. However, they do have the property that, when S is of length _,

the first _ = j/4 sidelobes on either side of the main lobe are 0.

" D. Demonstration of Equivalence

Two sequences which are equivalent always have the

same M value (the converse is not true). Two sequences,

e and d, can be shown to be equivalent, e _ d, as follows:

(1) c_c.

(2) If c _ e and e _ d, then c "" d.

(3) If c"- d, d" c.

(4) If ci = -di for all i then e "" d.

(5) Ifc. = d.+k for all i and some k, then e "" d.

(6) If c. = d/x k for all i and some k where (i, k) = 1, then

cud.

As a special case, when k = j - 1, ci = d]_ i, a "mirror image"

sequence.

When ah exhaustive search is made, only one sequence

from each equivalence class need be examined: an efficient

algorithm for performing such a search is not known.

One method for restricting a search to a small number of

members of each equivalence class is to examine only sequences beginning with several +l's followed by a -1; it is

not evident that any significant advantage can be obtained in

this manner, however.

E. Proof of Optimality

All of the best values of P discovered for each ] as well as

a number of M values are given as optimal. Most of the proofs

of optimality are trwmi and based only on the fact that when

j = N mad 4, each element of the autocorrelation is also N

mad 4. For example, when M = j - 1 for odd j, one may be

sure that no smaller value of M can be obtained. Similarly,

M = 40" - 1) must be optimal for/= 2 mad 4.

Other useful facts for proving optimality are:

(1) No sequence of/> 4 hasP = 0 (Ref. 7).

(2) No sequence of/> 13 and/= 1 mad 4 hasP = 1.

(3) _nai = c

i= 1 i= 1

This latter equation is easy" to derive. Suppose ci has h l's

and £ -l's where h + _ =/. Then

!

_"_c/= h-_

i = 1

To find the autocorrelation sum, it is sufficient to realize that

every element of the sequence will be multiplied by h l's and

£ -l's. thus,

/

a_ = (h - 2) 2

$=1

F. Examples of Optimality Proofs

To illustrate methods of optimality proofs, two examples

are given

1. Proof for j = 36. The period ] = 0 mad 4, so each a i must

be 0 mad 4

i_l ai = Ci

"= \ i= 1

151
The right hand side of the above equation is an even square

(0, 4, 16, 36, 64,100 .... ). The left hand side is

Thus either

or

If

then M >/80. If

35

36 +E ai

i= 1

_a_l =0

i= 1

then either every ai = 0, which is impossible, or some lai[ >1 4.

In this case _'-i = ai gives two equal deviations from zero

(unless j - i = i) which can be negated only by two other

elements of the autocorrelation. With 4 nonzero elements,

M ) 64. If j - i = i, then this deviation from zero must be

negated by two or more elements of the autocorrelation; thus

M'_ 82 + 42 + 42 "96. So/I/= 64 is optimal.

2. Proof for j =. 41. The period / = 1 mod4, so each ai

must be 1 mod 4.

-'lai = ci

i= 1 i= 1

The right hand side is (I, 9, 25, 49, 81 .... ); therefore

4O

41 +Eaz.

i= 1

equals the right hand side (RHS). If RHS = 81,40 a,_ = 1 give

M = 40 and P = 1, which is impossible. If 36 a: = 1,2 a: = -3,

and 2 a: = 5, thenM-- 36 + 18 +50= 104. IfRHS=49,

32 a: = 1 and 8 as =-3 giveM = 32 + 72 = 104. If RHS = 121

30 a: = 1 and 10 a: = 5 give M = 30 + 250 = 280. These are

the minimum deviations from all a i = 1 for the RHSs closest

to 81. ThusM = 104 is optimal.

G. Further Evidence of Optimality

Evidence of optimality can also be obtained even when an

exhaustive search has not been performed and a proof attempt

indicates that lower values of M may be possible. When/= 44,

for example, 244 sequences are possible. However, as our

discussion of equivalent sequences has shown, if one sequence

has a given M value, so do a number of others:

44 rotations

× 2 +/- interchanges

× 20 modifications by taking every nth element where

(44, n) = 1.

This appears to give 1760 sequences with the same autocorrelation properties. However, some of these sequences are

identical, so the number of equivalent sequences is less than

1760. Nevertheless, for some values of M, more than 21°

equivalent sequences with a given M value must-exist if any

do. So roughly every 10 billionth sequence would have that

value. If anywhere near that many sequences were examined

iteratively without finding a given M value, it would strongly

suggest that either no sequence giving that value existed or

that such a sequence could not easily be derived by iterative

one-bit modifications of better and better sequences.

When the M-value is underlined in Table 2, the authors feel

that no better value exists. When no proof of optimality

exists, the best evidence for this is the accumulation of a large

number of sequences, many of which are equivalent, of that

value. When one hundred sequences ofM = 144 forj = 44 are

found but none ofM < 144 are discovered, there is considerable circumstantial evidence that M = 144 is optimal for/= 44.

On the other hand, M = 112 for j = 60 may seem very surprising. Prior to the discovery of such a sequence, one might be

excused for believing that no such sequence will be found.

Yet, when one or two sequences with M = 112 are discovered,

the evidence against a sequence with j = 60 and M = 80 or 96

is not overwhelming. Thus, the authors have decided not to

underline a value for M in the table unless at least 30 sequences

with that M have been discovered independently.

Of course, this criterion is no guarantee of optimality. For

example, for j = 43, thirty sequences with M = 138 were discovered prior to the appearence of a lower value (M = 42).

The generation of P = 1 sequences for / = 43 and / = 47 by the

iterative method in less than 30 minutes of Cyber processing

152
time also indicates that the M-values listed for/< 47 are likely

to be optimal.

H. Ratio of Ones to Minus Ones

As can be seen from the preceding section, the best

sequences have

(the DC Fourier element)

j.

i= 1

for best results. Thus one expects that the best codes of

/ _ 64 will have

]

_2c/ = -+8

i= 1

For a period of 64, this gives 36 l's and 28 -1 's (or vice versa).

Robert Keston has pointed out that if one splits'such a

/ = 0 mod 4 sequence, e, into two sequences, f and g, where

Jei = C2 i- 1

gi = c2 i

then either f or g should have an equal number of l's and -l's

(R. Keston, personal communication, May 1984).

This can provide assistance in selecting initial sequences or

discarding unwanted sequences.

If one is looking for a particular value of M for a given ],

it may be helpful to look at the properties of the ai and c.

that must be satisfied. For example, at one time the best

known M for / = 48 was 112. It was hoped that anM of 96

could be obtained. To accomplish this, one must have:

2

E48ai = C.

i= 1 i = 1

= 16,36,64,100 ....

= RHS

But IRHS - 481 > 24 would mean M > 96 so only RHS = 36

or 64 are possible. For RHS to equal 36, there must be an

odd number of fours in the autocorrelation; this can not give

M = 96. Also, RHS = 64 cannot work with 6 fours, since

5 of them, including the middle element would be positive and

one (the middle element again, which is impossible) would be

negative. So .the only sequence which works must have an

autocorrelation with +8 in the middle and a +4 on either side

so that 48 + 8 + 4 + 4 = 64. The sequence itself must have

either 20 or 28 l's. If one takes every other element of the

sequence, one will get 8, 12, or 16 l's. These restrictions could

make it easier to hunt for such a sequence. Luckily in this

case, even without using them, there was ample time to find a

e_nn_nr_ ,ulth _ ----OA

I. Sequences With Good Matched but Poor

Mismatched Autocorrelation Properties

Sequences with excellent mismatched autocorrelation

properties generally have very good matched autocorrelation

properties. The converse is not true and is most typically

false for quadratic residue, double-prime or shift register

sequences. The reason is that for such sequences,

i= 1

So the first element of the Fourier transform,

1

_1 "-",/7-

Suppose/= 63. Then

1

,/g3

1

- 63

2

X1

Using Parseval's theorem:

63

E X_ :63

i = 1

Thus

63 1

E Xff : 63- 6--_

i= 2

153
At best,

these

all the k i for 2 _< i _< 63 are equal. Then each of

-- 1-646 

So

6_ 1 - 63+ 62X63 _ 632

-= IXil2 64 32

Therefore the L value is, at best, 63/32 = 1.96875, which is

very poor. The L value of the shift register sequence with

j = 63 and M = 62 in Table 2 actually is 63/32; the remaining

Fourier components are equal.

By the above argument the best possible L value for a shift

register sequence of] = 2n - 1 is:

2 n - 1

L2r/--1

Unless n is small, L _ 2, or about 3 dB.

The DC component of the Fourier transform, greatly

elevated due to the near equality of l's and -l's in the sequence, always produces an L value which represents roughly

a 3 dB loss in signal; this compares very unfavorably with the

0.15 dB to 0.25 dB losses corresponding to some of the

sequences with better ratios of l's to -l's. Sequences of

period 4j formed by multiplying an L = 1 sequence of period

4 by a shift register sequence are no better, as the L-value of

the sequence with period 4j equals that of the sequence with

period j.

This provides another incentive for not investigating shift

register sequences exclusively. Not only is it a nuisance to

analyze such sequences; in addition their autocorrelation

properties are, in some respects, not very good.

John Bailey has offered a solution to this problem; eliminate the DC component. One method would be to have

+1 and -1 be out of phase by other than 180 ° (Ref. 8).

For a two phase sequence the modification is:

Element of

Shift-Register Sequence

Element of

Modified Sequence

+1

-1

+1

- exp i/3

where

/3 = tan -1 ( /-"_-2V']")

For example, if the unmodified sequence is

+++--+-

then the modified sequence is

c = (1, 1, 1, - exp i/3, - exp i/3, 1, - exp i/3)

For complex elements, the autocorrelation is:

/

at = E c_c_+_

k=l

In our example, a7 = 7

at = 3 - 2 (exp i/3) - 2 (exp - i/3).

= 3 -4 cos/3

forl_<£_<6where

/3 = tan-I 3

9

tan 2/_+1 = 1 =_ cos 213 = 1"6

COS 2

Putting _ in the first quadrant, cos/3 = 3/4, so at = 0 for 1 _ It

_6.

In general, when ] = 2N - 1

_]-I ]'+I

at 2 2 cos/3

for I _<It <]. Thus

and

tan 2 13 -

COS 2 /3 -

4/'

q - 1)2

O' - I)2

(/+ 1) 2

154
which, with _ in the first quadrant, gives

a_ = 0 (_0mod/)

One could also derive a 3-phase sequence as a product of two

2-phase sequences:

Elements of Element of

Unmodified Sequences Modified Product

1 1 1

-1 -1 1

1 -1 - exp (-i_)

-I 1 - exp i_

Once again,

If one wishes to zero all sidelobes without a decrease in

SNR, one can also let / -- 221v and create a sequence with

phases; a complete discussion of this would be too far afield

of the topic of binary sequences.

V. Results

Table 2 shows the best sequences for periods 28 to 64 for

both matched and mismatched cases. Table 1, showing the

results for periods 3 to 27 (Ref. 9) is incl_uded for

completeness.

In Table 2, the heading / gives the period (length) of the

sequence;P gives the lowest value of the peak sidelobe;M gives

the lowest sum of the s.quares of the sidelobes discovered for

any sequence of period/. When the sequence with the optimal

peak sidelobe has a higher M, both values are given. When two

references are given on the same line, the first one refers to

the matched sequence and the second one to the mismatched

sequence. A [t_lel't_ll_¢ ....... of "X ....LUL_L_* _...... tO tiltht......attl_lU.:^1^

When the value for P, M or L is in parentheses, the authors

feel that a better, but as yet undiscovered, sequence may

exist. When the value is underlined, it is unlikely that a better

value exists. In all other cases, the value can be proved to be

optimal. All values for P are optimal unless two values are

given for a specific/, in which case the lower one is optimal.

The sequences are written in hex notation. The first bit is

always a plus sign. For example, the sequence for 29 is given

in hex as 14A7C111. In binary this would be 0001 0100 1010

0111 1100 0001 0001 0001. By replacing 0 with a minus

sign, and 1 with a plus sign, and removing the leading zeros,

we get the sequence:

+_+__÷_+__+++++ ..... +___+___+

155
References

1. Boehmer, A., Binary Pulse Compression Codes, 1EEE Transactions on Information

Theory, Volume 13, No. 2,.p. 156, April 1967.

2. Lindner, J., Binary Sequences up to Length 40 with Best Possible Autocorrelation

Functions, Electronic Letters, Vol. II, No. 21, p. 507, 16 October 1975.

3. MacMullen, A., Radar Antennas, Transmitters, and Receivers, pp. PC-1-56, Technology Service Corporation, April 1977.

4. Hoffman de Visme, G., Binary Sequences, English Universities Press, 1971.

5. Golomb, S., Baumert, L., Easterling, M., Stiffler, J., and Viterbi, A., Digital Communications With Space Applications, Prentice-Hall, 1964.

6. Brennan, L. E.. and Reed. I. S.. Theory of Adaptive Radars, IEEE Transactions on

A erospace and Electronic Systems, Vol. AES-9, March 1973.

7. Turyn, R., Optimum Codes Study, Sylvania Electronic Systems Final Report AF19

(604)-5473, 29 January 1960.

8. Bailey, J., Modified PAr Codes With Optimum Autocorrelations, Technology Service Corporation, August 1978.

9. Tyler, S., and Keston, R., Optimal Periodic Binary Codes of Lengths 28 to 64, TDA

Progress Report 42-57, March and April 1980, Jet Propulsion Laboratory, Pasadena,

CA.

10. Bailey, J., and Tyler, S., Periodic Bina_ Waveforrns with Optimum Autocorrelation

Functions, Technology Service Corporation, September 1978.

11. Tausworthe, R., Correlation Properties of Cyclic Sequences, JPL Technical Report

No. 32-388, 1 July 1963, Jet Propulsion Laboratory, Pasadena, CA.

12. Watkins, J., Lof,[sson, J., and Tyler, S., A Binary Sequence of Period 60 with Better

Autocorrelation Properties than the Barker Sequence of Periodic 13. TDA Progress

Report 42-82, April-June 1985, Jet Propulsion Laboratory, Pasadena, CA.

13. Barlog, M., Horn, T., Ulrich, D., and Variot, M., Binary String Manipulation, CSUN

Report CS480, May 1985.

156
Table 1. Best matched and mismatched sequences for periods 3 to 27

Matched Sequence Mismatched Sequence

] P M Sequence, hex L Sequencel hex

3 1 2 4 1.5000 4

4 0 0 E 1.0000 E

5 1 4 1E 1.1111 1E

6 2 20 25 1.3125 28

7 I 6 4B 1.5400 40

8 4 16 CB 1.3333 E5

9 3 24 IF4 1.6650 104

I0 2 36 350 1.6761 25D

II I i0 716 1.2909 67A

12 4 16 941 1.1250 941

13 1 12 1F35 1.0400 1FAB

14 2 52 36A3 1.2153 27F5

15 1 14 647A 1.1520 698F

16 4 -48 FAC4 1.2589 EED8

17 3 64 19A3D 1.2165 1128E

18 2 68 31EDD 1.2843 21419

19 1 18 7A86C 1.1119 465D0

20 4 64 F6ESE 1.1111 C5640

21 3 52 117BCE 1.1097 170848

22 2 84 3D1231 1.2178 28312B

23 1 22 6650FA 1.1114 7CEA2D

24 4 32 DC20D4 1.0607 C3DEA6

25 3 72 18B082E 1.1195 128COBC

26 2 100 2C 1AEB 1 1.1240 34AFBC9

27 3 74 5A3C444 1.0965 7D3472B

157
Table2.Bestmatchedandmismatched sequences forperiods28to64

Matched Seqeunce _',. Mismatched Sequence

/ P M Sequence, hex L Sequence, hex Reference a

28 4 80 B30FDD4 1.1305 F3DDD21 9, 10

29 3 92 14A7Clll 1.1384 14A7Clll 9

30 2 116 3FAD938A 1.1260 3FAD938A X

31 1 30 4B3E3750 1.0898 45C2D660 11, 10

32 4 80 89445BC1 1.0950 89445BC1 10

33 3 64 18A5C240D 1.0656 18A5C240D 9

34 2 132 29D3BB82D 1.1337 29D3BB82D 10

35 1 34 71F721592 (1.0732) 722B92F3E 11, X

36 4 64 F397A6517 1.0055 F397A6517 9

37 3 84 1BD623E3B6 1.0771 1BD623E3B6 9

38 2 148 302162B8B6 1.1085 302162B8B6 10

39 3 86 60CD4F47BE 1.0528 60CD4F47BE X

40 4 80 DBgE._E05BC 1.0575 DB9EAE05BC 9

41 3 100 1079731045A 1.0723 1079731005A 10

42 2 164 2CF51397B7C (1.0523) 2CF51397BTC X

43 I 42 653BE2E08D6 (1.0786) 5189822FC34 11, X

44 4 144 AFDESAF8665 1.1022 AFDE8AF8665 9

45 3 124 17473C9BFAD0 1.0667 17473C9BFAD0 9

"46 2 180 2A2818CDBCI6 ' (1.0842) 2A2818CDBC16 X

47 1 46 421A8D93A9EF (1.0727) 795220A780EC ll,,Y

48 (8) 96 99803C312AB6 1.0375 99803C312AB6 X

48 4 (112) CBF089223A51 9

49 3 (144) 1COB504676CB0 (1.0799) 1COB504676CB0 X

50 2 196 236D4FF70651E (1.0984) 236D4FF70651E X

51 3 (146) 6DSDECF8433E8 (1.0700) 6D5DECF8433E8 X

52 4 (128) FDEE871D85B44 1.0000 E8BEDC6FTAE24 X

53 3 (164) 11CAA3E46F7B65 (1.0706) 11CAA3E46F7B65 X

54 2 212 3917B588A2C302 1.0826 3917B588A2C302 X

55 3 (214) 7BCFB32717DOA5 (1.0837) 7BCFB32717DOA5 X

55 (5) (182) 7FOAA13316DC34 X

56 4 (208) 852659EBA181B8 (1.0993) 852659EBA181B8 X

57 3 (184) 16A38C8BC7FD1AD (1.0637) 16A38C8BC7FDIAD X

58 2 228 3B64AAF8FDCE520 (1.0896) 3B64AAF8FDCE520 X

59 1 58 5D49DE7C1846D44 (1.1003) 6CF43BE8A12CF9D ll, Z

60 4 (112) EC757781362D6F9 (1.0352) EC757781362D6F9 12

61 3 (204) 1481F734DC7EEA74 (1.0624) 1481F734DC7EEA74 X

62 2 244 225746DC62583D20 1.0638 225746DC62583D20 9

63 i 62 4314F4725BB357E0 (1.0830) 408AB703D6597390 11, 13

64 4 (240) B24FEAE7E4529CF0 (1.0538) CDgBFFOE16D2AB98 ZZ, X

a X Refers to this article.

Y T. Safer, personal communication, April 1985.

Z N. Lee, personal communication, October 1984.

ZZ J. Watkins, personal communication, June 1985.

158
TDAProgressReport 42-82

N86-10398

April-June 1985

A Binary Sequence of Period 60 With Better Autocorrelation

Properties Than the Barker Sequence of Period 13

J. Watkins

TRW, Redondo Beach, California

J. Loftsson and S. Tyler

Reliability Engineering Section

A binary sequence of period 60 has been discovered which in some respects has better

autocorrelation properties than the Barker sequence of period 13. When both sequences

are processed using appropriate sidelobe-eliminating mismatched filters, the Barker

sequence's main lobe is reduced by a factor o[1.040 or 0.17dB, while the new sequence's

main lobe is reduced by a factor of only 1.035 or 0.15 dB. This sequence is the first

counterexample known to the authors of the hypothesis that the autocorrelation properties of all sequences of periods greater than 13 are inferior to those of the Barker

period-13 sequence. Sequences of this type are very useful in radar and deep space communications, especially in situations where there is an adverse signal to noise ratio.

I. Introduction

A binary sequence is a string of bits. It can be thought of as

a vector C where each C/is a plus one or a minus one. A periodic sequence is one which is continuously repeated; for a

binary sequence of period j:

Ci+ i = C i

for all i. The autocorrelation,a, of a sequence, C, of period ] is:

i

ai = E CK CK+ i

k=l

The element a/ is called the main lobe; the remaining a i are

sidelobes. For example, if C = (1, I, 1, -1), a = (0, 0, 0, 4). In

this example, the sidelobes are zero. For a sequence to have

good autocorrelation properties the sidelobes should be small.

The example used is perfect, but for sequend_ periods other

than 4, the autocorrelation sidelobes are never all zeros.

The cross-correlation, X, of two sequences, B and C, each

of period j is

J

•, =Z; cK+,

K=I

Let C be a binary sequence which is to be cross-correlated

with B, a sequence composed of real numbers. The sequences

C and B are related by the weighting function, T.

B i = T.C.

In this case, B can be considered a "mismatched filter" to C.

159
For a mismatched filter B to be normalized,

J

i=1

Starting with a binary sequence, for example;

C l = (1,-1, 1,-1,-1,-1,-1,-1, 1, 1, 1,-1,-1, 1,-1,-1)

and the (normalized) weighting function,

Ti = _(1,4,1,2,1,4, 1,2, 1,4,1,2,1,4, 1,2)

then the normalized mismatched filter

Bi =

× (1, -4, 1, -2, -1, -4, -1, -2, 1,4, 1, -2, -1,4, -1, -2)

and the cross-correlation

xi=(0,0,0,0,0,0,o,0,0,0,0,0,0,0,0,_)

Examples of periodic binary sequences with good autocorrelation properties are available from a number of sources

as shown in the companion paper to this article (Ref. 1). Such

sequences are utilized extensively in satellite and planetary

probe ranging and comrnunications (Ref. 2). Mismatched

weighting functions can often suppress the cross-correlation

sidelobes to arbitrarily low levels, but only at the expense of

a loss in signal to noise ratio (SNR) (Ref. 3).

This loss equals the square of the sequence period divided

by the square of the main lobe. In the above example of a

sequence of period 16 analyzed with a normalized mismatched

filter to give no sidelobes, the loss is:

11 11

loss = 256 × 2048 8 1.375

The loss in dB is 10 loglo (1.375)= 1.38 dB. A method for calculating sidelobe-eliminating weighting functions is given in

Ref. 1.

A periodic sequence is a Barker sequence if its sidelobes are

all +1. A Barker sequence can have two periods, 5:

++++-

and 13:

+++++ -++-+-+

The Barker sequence of period 13 has a reduction in main

lobe of a factor of 1.040 when processed by a mismatched

filter which completely suppresses all cross-correlation sidelobes, corresponding to a loss in signal to noise ratio of

0.17 dB (Ref. 4). Until now, the authors had not been able to

discover a counterexample to the hypothesis that no sequence

of period greater than 13 will have better autocorrelation

properties than those of this Barker sequence. For example,

the best codes listed for periods 36, 39, and 64 all have somewhat greater SNR losses (0.19, 0.22, and 0.23 dB, respectively)

than does the Barker code of period 13.

II. A New Sequence of Period 60

A search of binary sequences of period 60 was performed

on a Cyber 750 computer. The Cyber was chosen due to its

availability, speed, and the inclusion in its assembly language

of an instruction which counts the number of ones in a word.

This latter property greatly facilitates a search for sequences

which have good matched-filter autocorrelation properties

(Ref. 5). The period was chosen to correspond to the Cyber's

word length of 60. Good sequences were found iteratively by

looking at all one-bit modifications of the previous best

sequence.

The sequence evaluator, written in COMPASS, was decisive

in permitting an adequate number of sequences to be evaluated

in a reasonable amount of time. The program segment in Fig. 1

was used for this task.

The Cyber processed approximately 10s sequences per

hour. Within 50 computer-hours, two sequences were discovered each having an excellent matched-f'flter autocorrelation.

One of them is:

++r +-++--+-+---++-++-+ ........ +-

---.-,.--....---.-.,.--.---.

Represented in hexadecimal notation, the sequence is:

D651B4021678B91

The other sequence, in hex notation, is:

EC757781362D6F9

By starting with bit 52 of the second sequence and taking

every 53rd bit modulo 60, one obtains the ones complement

of the first sequence. Both sequences have SNR losses of only

0.15 dB when analyzed by the appropriate mismatched filter.

160
IlL Implications for Aperiodic Sequences

An aperiodic sequence does not repeat. For it, the autocorrelation, a, of a sequence, C, of length _ is

a,: eK+,

K=]

The main lobe is ao; the remaining a i are sidelobes.

An aperiodic sequence is a Barker sequence if the absolute

values of each of its sidelobes never exceeds 1. Aperiodic

Barker sequences exist for all sequence lengths less than 6

as well as for lengths 7, 11, and 13.

The question of whether the aperiodic Barker sequence of

length 13 is superior to aperiodic sequences of all other lengths

has not yet been completely resolved. M. Golay has defined

the "merit factor" for aperiodic sequences to be the square of

the sequence length divided by twice the sum of the squares

of the sidelobes (Ref. 6). His original upper bound for the

merit factor was 2e 2 = 14.778, which is approached only by

the Barker-13 sequence's merit factor of 14.08. Golay states

that 12.325 is a reasonable upper bound on the merit factor

of all other aperiodic sequences. He has performed an exhaustive search of skew symmetric sequences of lengths up to 59

to support this. Whether or not the discovery of the present

periodic sequence of length 60 stimulates further tests of

Golay's bound, it should help to dispel the notion that the

autocorrelation properties of the best periodic as well as

aperiodic sequences degrade with increasing length.

References

1. Tyler, S., and Loftsson, J., Periodic binary sequences with very good autocorrelation

properties, TDA Progress Report 42-82, Jet Propulsion Laboratory, Pasadena, CA,

August 1985.

2. Golomb, S., Baumert_ L., Easterling, M., Stiffler, J., and Viterbi, A., Digital Communications With Space Applications. New York: Prentice-Hall, 1964.

3. Ackroyd, M., and Ghani, F., Optimum mismatched filters tbr sidelobe suppression,

IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-9, pp. 214-217,

March 1973.

4. Rihaczek, A.W., and Golden, R.M., Range sidelobe suppression for Barker codes,

IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-7, pp. 1087-

1092, Nov. 1971.

5. Corning, B., Heraper, D., Lee, R., Rothman, M., and Watkins, J., Binary sequences

of periods 60, 64, and 72, CSUN Report CS380, California State University, Northridge, CA, May 1984.

6. Golay, M., The merit factor of long low autocorrelation binary sequences, IEEE

Transactions on Information Theory, Vol. IT-28, pp. 543-549, May 1982.

161
MVAL

LOOP

IDENT

ENTRY

EQ

SB2

SBI

SX5

SX6

SA2

BX3

LX3

BX4

CX4

IX4

IX4

IX4

IX6

SB2

NE

IX6

IX6

EQ

END

MVAL

MVAL

$+1S17

30

1

30

B0

Xl

X2

1B

X3-X2

X4

X4-X5

X4+X4

X4*X4

X6+X4

B2-B1

B2,LOOP

X6+X6

X6 -X4

MVAL

Call assembler with ID

Define entry point

Put number of shifts into B2

Used for decrementing

For (number of l's - 30)*2

Sum of squares of sidelobe

Put parameter value in X2

Put sequence to be shifted in X3

Left rotate the sequence by one bit

XOR the shifted and original sequence

Count the ones in the result

Subtract 30 from the number of l's

Double the difference

Square the difference

Add this to the total

Decrement loop counter

Loop if not equal to 0

Double the sum

Avoid doubling middle term

This returns

Fig. 1. Sequence evaluator

162
N86-10399

TDA ProgressReport 42-82 April-June 1985

The GCF Mark IV Implementation and Beyond

R. A. Crowe

TDA Engineering Section

This article presents a brief overview of the major subsystems that comprise the

Mark IVA GCF along with key information on basic functionality that may be inadequately or imperfectly understood. Concluding paragraphs describe the evolving GCF

System as it is currently being designed.

I. Introduction

The Ground Communication Facility (GcF) was modified

in order to meet the requirements of the DSN Mark IVA

implementation now nearing completion. The key characteristics of the GCF did not change markedly during this time

frame (see Figs. 1-3). The underlying architecture of the GCF

was left intact while modifications were initiated in order to

expand the capacities of its constituent subsystems. Communications support at Goldstone, Spain and Australia was unified and consolidated under the aegis of the incipient Deep

Space Communications Complex (DSCC). Economics of

shared functionality have evolved whereby the collective communications capacity at each DSCC is shared by the competing

demands of its DSS components. The DSN was expanded in

this time frame to begin assuming responsibility for the scheduling and operation of the 26-meter network. One 26-meter

antenna is located at each of the three DSCC sites. Voice,

data and teletype circuits required for Spain andAustralia

continue to be ordered from the NASA Communications

(NASCOM) network. NASCOM Engineering has delegated to

the GCF the overall responsibility for communications support

between Goldstone and JPL. This has resulted in expansion of

the intersite Goldstone microwave facilities that link DSCC 10

• with the Communications Center known as GCF 10, and the

installation of a fiber-optic communication path between the

DSCC and the outlying 26-meter station.

Current plans call for continued GCF expansion in the area

of capacity and augur significant changes in the area of key

characteristics. High-rate missions and Very Long Baseline

Interferometry (VLBI) tracks late this decade and into the

1990s provide the principal motivations toward added capability. Efficiency, generality, automation and simplification

militate toward a restructuring of the communication architecture to embody layering techniques, variable size data units,

and interfaces to external networks.

II. Digital Communications Subsystem

The Digital Communications Subsystem performs the

exchanges of digital data blocks between the DSCC's and the

Central Communication Terminal (CCT) at JPL (Fig. 4).

Block multiplexing on the available communication circuits

permits shared use of the lines among projects. Communication capability at each DSCC has been implemented in a

modular fashion in order to save on circuit lease costs. Four

configurations are possible at each DSCC:

(1) One 56 kilobits/sec (kb/s) duplex circuit

163
(2)

•(3)

(4)

One 56 kb/s duplex circuit and one 56 kb/s simplex

circuit

One 56 kb/s duplex circuit and two 56 kb/s simplex

circuits

One 56 kb/s duplex circuit and one 224 kb/s simplex

circuit.

All simplex circuits carry data in the direction of the DSCC

at JPL only. Configuration changes are possible by scheduling/

descheduling of circuits. No software changes are required.

The Digital Communications Subsystem is composed of the

following principal assemblies:

A. Station Digital Communication (SDC) and Central

Digital Communication (CDC) Subsystems

The SDC and CDC Subsystems consist of all data transmission equipment including line interfaces, data sets, Network Encoders Decoders (NED), Communication Buffers

(CB), digital and analog test equipment, and patch facilities

for trouble isolation as well as a front-end line interface

between the actual data sets and NED appearances (see Figs. 5

and 6).

The data transmission equipment is provided by NASCOM

including the line interfaces/converters, data sets and common

carrier interfaces associated with the overseas DSCCs.

The CDC implementation includes a 40-by-40 digital line

switch that provides the connection from the data set to a

particular NED/CB combination that plugs in to a specific

port on one of the five Error Correction and Switching (ECS)

front.end computers at the CCT. This switch expedites normal

activation of data circuits and simplifies the switchover to the

backup ECS in the event of hardware failure.

B. Area Routing Assembly (ARA)

The ARA is the DSN computer component that performs

the GCF functions at the DSCC. Two identical Mod Comp

II/25 computers are so configured: one for prime and one for

backup. The ARA receives data blocks from all DSCC computers and multiplexes them to JPL on the available circuitry.

In the reverse direction, data received from JPL is distributed

to DSCC computers according to destination code and data

type.

The ARA in cooperation with the ECS will perform error

correction of data blocks by retransmission of data transferred

on the 56-kb/s duplex circuit. The ARA will attempt retransmission of an unacknowledged block one time only.

The ARA accepts and transmits either 1200-bit or 4800-bit

data blocks on the available circuitry. The circuits are utilized

as follows:

(1) All 1200-bit data blocks will be sent on the 56-kb/s

duplex line. They cannot be sent on the simplex line(s).

(2) All 4800-bit data blocks are normally sent on the simplex circuits. When necessary and if loading permits, all

4800-bit data traffic from a specified link (a link is

usually equivalent to a DSS) may be moved from the

simplex line to the duplex line.

Based on these facts and on the configuration possibilities

discussed earlier, the following conclusions are possible:

(1) While the maximum GCF DSCC to JPL rate is 280 kb/s

(56 + 224), no single link-originated 4800-bit stream

may exceed 224 kb/s, and no single 1200-bit stream

may exceed 56 kb/s.

(2) Error correction is normally applied to the low rate

1200-bit traffic from the DSCC.

In addition to these functions, the ARA writes Original

Data Record (ODR) tape files of data transmitted. The maximum recording rate is restricted to the maximum transmission

rate: 280 kb/s. Data from each DSS.is normally recorded on

its own tape drive although data from multiple DSSs can be

combined on a single tape if desired.

C. Error Correction and Switching (ECS) Computer

The ECS is the DSN computer component resident at the

JPL CCT that interfaces the communications circuits from the

DSCCs. There are five ECS computers configured identically

for support. Four serve a prime data handling function and

one is designated as a backup. Responsibilities include servicing serial interfaces to Remote Mission Operation Centers

(RMOC), Remote Information Centers (RIC), the Mission

Control and Computing Center (MCCC) and the Network

Operations Control Center (NOCC).

Capabilities of the ECS include the following:

(1) Error Correction of data blocks in cooperation with

ARA at the DCCCs

(2) Multiple routing of data blocks to the NOCC, GCF

Data Records and to any two other processing centers

(3) Delivery of data blocks predicated on the source originating the data

(4) A multilevel real-time dump facility enabling full

traceability of data blocks through the program.

164
III. Analog Intersite Communications

The Mark IVA implementation at Goldstone left the

DSS 12 antenna and front-end equipment in place approximately 13 miles from the Signal Processing Center (SPC) at

the DSCC. In addition, the NASCOM STDN 26-meter antenna

(DSS 16) at the Goldstone site was itself about 7-8 miles from

the SPC of the DSCC. In order to transfer baseband signals

to/from these two antenna sites, it was necessary to expand

and enhance the existing Intersite Communications Subsys.

tern. Microwave links were used to connect DSS 12 to SPC-10,

and a fiber-optics link was installed to provide the baseband

conduit between DSS 16 and SPC-IO.

The existing microwave channels between DSS 12 and

SPC-10 were expanded from 8 to 16 duplex channels. A channel is here defined as a communications link in one direction.

The following are provided:

(1) Two receiver baseband channels

(2) Two command modulation channels and one confirmation channel

(3) One full duplex frequency and timing channel

(4) One frequency STD channel

(5) One FEA-12 surveillance TV channel

(6) Two test modulation (SIM) channels

(7) "Two full duplex diversity multiplex channels that

support voice, 56-kb/s wideband circuits, LAN, gateway circuits, 230.4 kb/s simplex circuit and low speed

and control circuits.

At DSS 16 _t was necessary to connect the front-end

equipment at the receiver with the telemetry and command

equipment at SPC-10 in order to provide DSN/NASCOM

cross support capability for projects beginning with AMPTE.

This capability was provided with the completion in July 1984

of a six-fiber underground installation of a fiber optics system

with optical transmitters and receivers located both at DSS 16

and at SPC-10. At the present time, up to 3 baseband signals

may be transmitted in each direction by multiplexing FM

modulated carriers over the fiber optic link. Currently, there

are two fibers in use for telemetry and command baseband

signal transfer, with additional usage planned for voice and

ultimately digital transfers when the 26-meter network is fully

integrated into the DSN.

IV. Data Records Subsystem

The Data Records Subsystem is comprised chiefly of the

three prime Data Records Generator (DRG) computers and

one backup computer resident at the CCT. Functional components of this subsystem reside also at the DSCC in order to

provide gap "fill" information for data not received in realtime. The principal function of the DRGs is tO create tape

Intermediate Data Record (IDR) files for project delivery.

The DRG software checks selected data streams for correct

source code, spacecraft I.D., User Data Type, time tag, block

serial number and error status code. The DRG detects gaps

in the data and outputs real-time statistics providing the percentage of good data received. At the end of a pass, a recall

operation may be initiated requesting data from the DSCC

in order to improve the percent of good data on the IDR.

V. Voice Subsystem

The Mark IVA voice assemblies at the DSCC_ were c6nfigured from the tactical intercom assemblies (TICS), Communications junction modules, and station switch assemblies

that were previously in use. There were no changes made to

the equipment at the JPL CCT end (Fig. 8). At the Signal

Processing Center (SPC) of the DSCC, new communication

panels were developed for use at the front console. New

circuit boards were developed to bring the obsolescent TIC

panels more in line with current technology.

VI. Implementation Activities

A DSN system level approach is being taken to redesign

the GCF to meet the communication necessities of the 1990s.

Both Local Area Networks and wide area networks will form

part of this system. A software layering architecture shall be

employed in these networks in order to simplify user interfaces to the system and allow modification to lower level

structures without affecting the upper echelon layers. Key

changes during this implementation include megabit/second

communication rates from the DSCCs, automatic recall of

data before the tracking pass is complete, processing and

handling of variable size data blocks from the user and Gateway interfaces to external networks located both at the

DSCC and the CCT. Most of these upgrades will be operational by 1989.

A. High-Rate DSCC Communication

The current maximum data rate from each DSCC will be

increased from the present 280 kb/s to 1,544 kb/s. The design

calls for this capability to be modular so that the Communications capacity at any DSCC may be incrementally adjusted

to fit the combined requirements of all active projects in any

given time frame. Error correction of data blocks will be

extended to encompass the entire bandwidth of data transmitted from the DSCC.

165
B. NearReal-TimeRecall

The spare bandwidth available on the DSCC communication circuits shall be utilized in order to allow recall of data

blocks before the tracking pass is complete. By this method,

the data that could not be successfully transmitted in realtime because of line outages or transient overloads will be

shunted to a temporary data file. This data will subsequently

be retransmitted when loading and line conditions allow on

the excess available bandwidth from the DSCC to JPL. Independent of any operator intervention, the CCT data records

function will organize the playback .data into separate files

and merge the blocks with the original real-time stream in

order to provide the most complete data file.

C. Variable Size Data Blocks

Perhaps the most revolutionary innovation to the GCF

System will be the capability of accepting variable size data

units from all originating computers. Within specified minimum and maximum length constraints, the data originator

shall be free to deliver to the receiver data blocks that are

sized to match perfectly the application being supported.

This will contribute to overall network efficiency and minimize the overhead labor required to prepare data for transmission. This is also consistent with packetization.

D. Gateway Interfaces

Gateway interfaces will be developed at the DSCC and at

the CCT in order to permit data communication with networks external to the DSN. The architecture for gateway

interfacing will be installed when the GCF system is upgraded.

Actual gateway software will be developed upon identification of the requirement to exchange data with a particular

network. The location of gateway interfaces at the overseas

sites will permit circuit economies ensuing from the proximity of the external network to the DSN access point at the

DSCC.

1,

.

.

References

Evans, R. H., "DSN Ground Communications Facility," TDA Progress Report 42-65,

pp. 9-18, July and August 1981, Jet Propulsion Laboratory, Pasadena, Calif., Oct. 15,

1981.

Mortensen, L. O., "GCF Mark IV Development," TDA Progress Report 42-70, pp.

161-172, May and June 1982, Jet Propulsion Laboratory, Pasadena, Calif., August 15,

1982.

Crowe, R.A., "Design Issues in the GCF Mark IV Development," TDA Progress

Report 42-75, pp. 132-139, July and September 1983, Jet Propulsion Laboratory,

Pasadena, Calif., November 15, 1983.

166
s_:. __J

(DIGITAL) J

SFC I0

FEA 12

(WIDE BANDWIDTH)

ANALOG)

SIGNAL PROCESSING CENTER ..I

I 21Zil I

I I

AREA 10 J STATION

ASSEMBLY ROUTING i DIGITAL J--J COMM

10_1_ COM

CENTRAL COMM TERMINAL

CENTRAL

DIGITAL

COMM

i [

H ERROR NETWORK

CORRECTION COMM

• SVATCHING EQUIPMENT

_NOCC

GOLDSTONIE0J "-"

MICROWAVE l

f

_..__ JPLGOLDSTONE

MICROWAVE

,' .

J TACTICAL 10_)J _"J J

I NTERCOM

I

I

STAT 1(3N

VOICE

SWITCH

6O21

JCENTRAl. DISTRIIUTIO N

I

I

STATION

TELETYPE CENTRAL JTELETYPE

CENTRAL

COMM

MONITOR

i

I

DATA

RECORDS

GENERATION

RMOC,

MCC,

VLBI

(RT DIGITAL)

DIGITAL COMM

_ MCC

- (VOICE)

INTERSITE ANALOG

VOICE

TELETYPE

PROJECTS

L (IDRs)

MONITOR AND CONTROL

DATA RECORDS

Fig. 1. Mark IV-A era GCF subsystems

167
uu
0ss PERSONNEL TEXT MSG°s

DSCC _ _ 111ANSMISSIO N _

CCT (JPL)

TELETYPE

• _ MACHINES

Ae COMPUTER

ACCESS

TERMI NALS

• TRANSMISSION, Ii0

AND 300 BAUD

• MESSAGE SWITCHING

• TTY MACHINES

4Ee AUTO DIAL

/

\

TEXT MSGS

SUPPORT

DATA

JPL

PERSONNEL

_ ADMIN

CO,V_UTER

DSS _VOICE

PERSON N EL '_OORDI NATIO 7

FACSIMILE !_

INFO /

VOICE

• TERMINAL AND

I NSTRUMENTS

&e CIRCUIT SWITCHING

Ae FACSIMILE

MAC H INE

• TRANSMISSION • CIRCUIT

SWITCH

A• FACSIMILE

MAC HI NE

_VOICE J_

_jOORDI NATIOn//

\

FACSIMILE

INFO

POC's

RIC's

JPL

PERSONNEL

GCF DIGITAL /V I_ COMM, OICE, TTY,

TTY, ASSEMBLIES VOICE _jDIGITAL DATA/

SPC 10AND /I"V

FEA 12 , TIMING

SUBSYSTEMS _MD, TLM, SIM/

INTERSITE ANALOG

• MULTIPLEXING

• TRANSMISSION

FORMATTING

Ae LAN EXTENSION

• MICROWAVE .

TRANSMISSION

Ae FIBER-OPTICS

TRANSMISSION

• MULTIPLEXING

• TRANSMISSION

FORMATTING

ANEW FUNCTION OR MODIFICATION

VOICE, TTY,

_DIGITAL DATA/

Fig. 3. Mark IV-A end-to-end teletype, voice, and intersite analog functions and interfaces

GCF DIGITAL

COMM,

TTY, VOICE

ASSEMBLIES

169
-- SIGNAL PROCESSING CENTER CENTRAL COMM TERMINAL

I

B

o--_ AREA

ROUTING --

LAN o_ ASSY

I

i

I

I

i.

I

L

STATION

DIGITAL

COMM

DIGITAL COMM SUISYSTEM

VLBI

GSFC

MCCC

CENTRAL

DIGITAL

COMM

Sw

I _;CC

NE,WOR_ _1_I N$_

COIV_

ERROR I L EQUIP

CORRECT RTMs

AND

sw

I

IDR _ SW

,il ,tI_-_' su._._

I " I i rATA 3 " J

] 1

Fig. 4. Digital communiations subsystem overview of GCF digital configuration

170
I

I

r--- I

L]__ _1:: __ I_Ku,

-,-_..Jt -i

_oc_soR !

Fig. 5. Data transmission, station digital communications

171
I MUX I DIVERSITY I_

3-56

1 - 224

12 - VOICE

SPC 10 = Jl" GCF 10 INTER FAC E ,- I NTERFAC E

PANEL PANEL

] -it

• SPARE'_

• SPAREJ

• COMMAND"

• FTS-1 PPS

• FTS-5 MHz

• SIM I

DIVERSITY J MUX

3-56

I - 224

12 - VOICE

{.sX BASEBAND BASEBAND

f! COMMAND

FTS-1 PPS

TV1

TV2

Fig. 7. Intersite microwave configuration, general design

t TO

JPL

TO

FEA

12

I I I

TIC BAY

(50 STA)

(PEDESTAl.)

TIC BAY TIC BAY

(50 STA) (50 STA)

(SPC) " (SPC)

TIC PANELS,

64-m ANT.

TiC PANELS, _r

OTHER

ANT.

I

TPANELS,FEA AREA I I SPC AREA 1

_ 5VOICE/ I

r_uu ' _ "I STATION I

I MODULE SCAMA " _'_"_'" --

I sP¢ I PUBLIC DIGITAL

I CONSOLE

PANELS

(6/DSCC) • NEW

* WILL BE MICROWAVED IN THE CASE OF FEA-12

+ SCAMA SERVICE AVAILABLE AT OVERSEAS CONSOLES ONLY

MAY REQUIRE AMPLIFICATION/POWER FOR PANELS IN DISTANT ANTENNAS

Fig. 8. DSCC voice, general design

172
TDA Progress Report 42-82

Nss-lo4oo

April-June 1985

SETI Radio Spectrum Surveillance System

B. Crow

Communications Systems Research Section

A. Lokshin

Space PhysicsendAstrophysics Section

M. Marina and L. Ching

Radio Frequency and Microwave Subsystems Section

The SETI Radio Spectrum Surveillance System (SRSSS) will provide a data base for

assessing the RFI environment for SETI and minimizing RFI disruptions during the

search. This article describes the system's hardware and software and discusses the sensitivity of the system.

I. Introduction

The Search for Extraterrestrial Intelligence (SETI) Radio

Spectrum Surveillance System (SRSSS) was built to provide a

data base to assess the RFI environment for SETI. Initially

the system will be used to study the RFI environment at the

Gotdstone Complex. A survey will be conducted over the next

year to cover the frequency range from 1 to 10 GHz. The

SRSSS is sufficiently portable so that it can be moved easily

to other radio astronomical sites (see Figs. 1 and 2). Later, the

SRSSS will be used as a coincidence detector with the SETI

search instrument.

The SRSSS was designed as a stand alone system capable of

detecting and recording RFI signals in an unattended mode.

Periodic system calibration is achieved using a noise diode to

provide a known input power. Detected signals are stored on

floppy diskettes and will be processed with the SETI/Radio

Astronomy VAX 11/750 at the Jet Propulsion Laboratory

(JPL).

The SRSSS data base will provide guidance to the SETI

project on how to search the frequency, time and space dimensions. The objective is to minimize the probability of RFI

disrupting the search.

This article gives an overall description of the system,

including hardware and software, and a discussion of the system sensitivity.

II. Hardware

The SRSSS occupies one standard rack of equipment and

has a one-meter antenna, antenna rotator and associated RFmodule assembly mounted externally. The rack contains a

Tektronix 494p Spectrum Analyzer under the control of a

Tektronix 4052A controller. The controller also controls the

antenna position, amplifier selection and data storage on a

dual-floppy disk. A hard copy unit is available to record

"interesting" spectra. The data from the floppy disk will be

173
processed in a VAX computer at JPL. An overall block diagram is presented in Fig. 3.

A. RF Module

The signal from the antenna is amplified in one of seven

GaAsFET amplifiers. The set of seven GaAsFET amplifiers

is required to cover the spectrum from 1 to 10 GHz. Specifications of the GaAsFET amplifiers are given in Table 1.

Each GaAsFET amplifier is followed by a transistor amplifier with a minimum gain of 20 dB to eliminate subsequent

cable loss from contributing to the system noise temperature.

The measured gain and noise temperature for the RF assembly

are shown in Figs. 4 and 5.

The selection of different sets of amplifiers for the specific

frequency ranges is controlled by the 4052A Tektronix computer through HP actuators. A directional coupler is used at

the input of the coaxial switch to inject noise into the system

for the purpose of system calibration.

Calibration of the system is performed by turning the

HP346B noise source on and off and calculating the system

temperature from the y-factor measurement in the 4052

controller. This noise source has a 15.2 dB -+0.2 dB excess

noise ratio (ENR) over a range of 10 MHz to 18 GHz. The low

SWR of the noise source in both the on and off condition

reduces a major source of measurement uncertainty. The

ENR is related to the effective noise temperature of the

source (Tne):

ENR = 10 log _[ne-290)

290

B. Relay Actuators"

- i

The HP relay actuators provide the control of the microwave coaxial switch and the liP 346B noise source via control

lines to the RF unit located in the antenna rotator pedestal.

C. Antenna Control

The antenna elevation is manually set at a fixed elevation

angle (adjustable from 0 to 30 degrees). The azimuth position

and rate of rotation are controlled by the RPM antenna controller through the Tektronix 4052A computer via an

IEEE-488 Bus.

The azimuth position of the antenna may be controlled in

local mode with the front panel control. The rate of rotation

is displayed in RPM on the RPM LED indicator. The rotator

angular position is simultaneously displayed on the azimuth

DEGREES LED indicator. In a remote controlled operation,

rate or position command signals from the IEEE-488 Bus may

be used to control the rate of rotation of the rotator from

-15.5 RPM to +15.5 RPM in 0.5 RPM increments, or to cause

the rotator to point at any azimuth with a 0.1 degree resolution and 0.15 degree accuracy.

D. Spectrum Analyzer

The 494P Tektronix Programmable Spectrum Analyzer is

controlled by the 4052A Tektronix Computer through the

IEEE-488 Bus to select the desired frequency range, frequency

span/division and resolution bandwidth.

This spectrum analyzer has a frequency range of 10 KHz to

21 GHz and a minimum resolution bandwidth of 30 Hz. The

sensitivity is -121 dBm at 30 Hz resolution bandwidth.

E. Digital Clock

The HP 59309A digital clock supplies the system time.

This clock can be set under local control or by remote commands received from the IEEE-488 Bus_

F. Dual Disk Drives

The double density dual floppy diskette storage facilities

can hold up to two megabytes of information which translates

to 50,000 messages (assuming 40 characters per message).

These messages typically will contain the following data:

time, frequency, azimuth, threshold, number of hits above

threshold, operating mode, etc. At low threshold setting,

which may "result in 1,000 messages per hour, the SRSSS can

operate unattended for a full weekend. At a more conservative threshold, say 60 messages per hour, the SRSSS can

operate unattended for a full month.

G. Hard Copy Unit

The 4631 Tektronix Hard Copy Unit provides hard copies

of any spectra displayed on the 4052A CRT. A sample display

is shown in Fig. 6.

III. Software

The software that runs on the SRSSS is written in Tektronix

BASIC. The program allows a user to build and save a customized observation program on the cartridge tape. This

schedule will later direct the system when and where to point

the antenna, how to set the threshold, frequency range and

resolution bandwidth by issuing detailed hardware commands.

There are five main modules in the SRSSS software. The

EDITOR allows the operator to build and edit observation

schedules. The RUN module runs observation according to

174
the prerecorded schedule. The MAINTENANCE and UTILITIES modules aid in software development and hardware

diagnostics. Figure 7 is the top level flow diagram of the

SRSSS. The RUN module is further expanded in Figs. 8, 9,

and 10. The EDIT module flow diagram is shown in Fig. 11.

An observation schedule is a set of tasks (up to 20). The

tasks of a schedule are separated into 10 Sequential Time

Events (STE) tasks and 10 Absolute Time Event (ATE) tasks.

The STE tasks are executed by the SRSSS system one after

another. An ATE task starts and stops according to its predetermined absolute time schedule, and it interrupts any

currently running STE task, An interrupted STE task resumes

execution as soon as the interrupting ATE task is completed.

Any task, whether an STE task or ATE task, fails into one

of three categories:

1. Data Only

2. Calibration Only

3. Data and Calibration

In a Data Only task, the SRSSS determines and records hits

above some predetermined threshold. In a Calibration Only

task, system calibration operations are performed. In a Data

and Calibration task, the SRSSS interleaves calibration operations with hits collected. Figure 12 shows schematically the

azimuth-frequency coverage for a DATA/CALIBRATION

task.

The SRSSS was intended for unattended data collection,

but its software provides some convenient functions for an

observer to get information in real time. By request, the

system can work in a stepped, frame-by-frame mode, reporting

hits on the CRT, plotting current spectra on the screen and

making hard copy.

The SRSSS software utilizes the fact that the spectrum

analyzer contains an internal controller which allows some

parallel processing. This allows the spectrum analyzer to

coUect the data, while the 4052 controller processes that data

from the last spectrum.

IV. Future Plans

Currently, the JPL SETI team is working on building

data base software on the VAX 11/750 for RFI signal storage

and analysis. This data base will provide an estimate of how

many RFI events may be expected for a given SETI run.

Based on this information an optimal pattern of sky/frequency

coverage in SETI SKY Survey will be developed. A year long

survey is planned to assess the RFI environment at Goldstone.

This RFI data base should be of general interest for all potential DSN users.

175
Table 1. Specifications of the GaAsFET amplifiers

Amplifiez Frequency Range, GHz Minimum Gain, dB Noise Figure,K

ARI 1.1 - 1.7 44.5 I00

AR2 1.7 - 2.3 37 100

AR3 2.3 - 3.1 38 120

AR4 3.1 - 4.3 36 120

AR5 4.3 - 5.8 40 160

AR6 5.8 - 7.7 42 225

AR7 7.7 - 10.0 37 390

176
poo_ _UAL,_T_

Fig. 1. SRSSS antenna assembly

Fig. 2. SRSSS electronics assembly

177
ROTATOR ASSEMBLY J

m7

_l 10dB

- DIRECTIONAL

J COUPLER

t

3468

NOISE

SOURCE

INPUT

COAX

SWITCHo.._ __

G_s FET

G = 35

1.1-1.7 GHz

GaA6 FET

7.7-10 GHz

OUTPUT

COAX

_ SWITCH

RSSS-RF _ ._l

CO NTROL LINES

I _rlUBA_E_J_ IEEE.-488

r

I ROTATOR CONTROLLER _ IEEE-_I,

• DIGITAL IEEE.-4_CLOCK

4631 ]

HARD COPY _

UNIT RS- [oo_.s. _,vms __jo._ I- -I CONTROL'_R _.

Fig. 3. Radio Spectrum Surveillance System (RSSS) block diagram

494F

SPECTRUM I

ANALYZER I

4052A

:TEKTRONIX

iCOMPUTER

178
z_"

<

O

66

64i--

62--

60--

58--

56--

54-

0

I i I I I I I I

1

] I I I i I I

2 4 6

FREQUENCY, GHz

I

m

I I I

8 10

Fig. 4. Gain vs frequency for the SRS3 amplifier

1200

1000

80O

4OO

2OO

0

I I I I I I I I

I I I I I I I I

2 3 4 5 6 7 8 9 10

f, GHz

Fig. 5. The SRSSS system temperature vs frequency, April 1985

I I ! I

-50

-60--

-10(1

- 120 I I I I

.2 1.3 1.4 1.5 1.6

I I I I

AZIMUTH, deg: 60

TIME/DIVISION, sec: 1.000

CENTER FREQUENCY, GHz: 1.7228

RESOLUTION, KHz: i

SPAN, KHz: 100

REFERENCE LEVEL, dBm: -50

THRESHOLD, dBrm -83/10

I

! .7 1.8

f, GHz

I I I

1.9 2.0 2.1 2.2

Fig. 6. The SETI Radio Spectrum Surveillance System

179
i

I INITIALIZE

INITIALIZE AND

CHECK HARDWARE

AND SOFTWARE

PROMPT OPERATOR:

"1. RUN

". 2. EDIT

3, MAINTENANCE

4. INITIALIZE

5. UTILITY"

[--;OOE_J--_

Fm'r A I I ENTER I _ INITIALIZE |

I RUN A / I ;;-;J=_,,,= I I MAINT I I AND CHECK /

FUNCTION KEY INTERRUPT )

Fig. 7. The main program of SRSSS

180
1.4

C RUNt

J DISPLAY

RUN MODULE

VERSION NUMBER

f

J AR DIM=21

JL

T

J DISPLAY PRESENT

SCHEDULE NAMES

FROM TAPE

READ SCHED

READ

SCHEDULE

FROM TAPE

.@.,

J WR'DISK (1, LUN)

WRITE TO

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SET LTASK TO FIRST

PERTINENT ATE

(ABSOLUTE TIME EVENT)

TASK BASED ON CURRENT

TIME OF DAY

t

COiL SPEC DATA

COLLECT

SPECTRAL

THIS SPECIFIES ARRAY

SIZE FOR SCHEDULE

FILES

3

J WRITE FILE HEADER TO

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(LUN IS THE LOGICAL

UNIT NUMBER)

7

FAILURE TO SELECT I

A SCHEDULE FILE

WITHIN 30sec J

CAUSES AUTOMATIC

SELECTION OF

SCHEDULE FILE /1

1.4.4

( READSCHED )

-1!

PROMPT OPERATOR:

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TO EXECUTE

(0 CAUSES EXIT):"

1

J UP TO 7 SCHEDULES

ARE CONTAINED ON

A SINGLE MAGNETIC

TAPE CARTRIDGE

y

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SELECTED SCHEDULE

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_ 4

PROMPT OPERATOR: J

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Fig. 9. READ schedule flowchart

Fig. 8. RUN mode flowchart

181
o,

t

1.4.8

( COLLSPECDATA _

1 ATE FLAG IS INITIALLY

I ATEFLAG.0 1 RESET HANDLING IN READINESS STE TASKSFOR

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Fig. 10. Coiled spectral data flowchart

182
1.5

I I

EDITOR MODULE

VERSION NUMBER

2

I DISPLAY PRESENT ]

SCHEDULE NAMES

• FROM TAPE

l PROMPT OPERATOR:

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183
I

I

I

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I

CALIBRATION

I-"1 DATA COLLECTION

f

TIME

TIME

CALIB DATA CALIB DATA

Fig. 12. Time history of system calibration is • function of azimuth

184
TDA Progress Report 42-82

N86- 1 0401

April-June _1985

Mobile VLBI and GPS Measurement of

Vertical Crustal Motion

P. M. Kroger and J. M. Davidson

Tracking Systems and Applications Section

E. C. Gardner

KalamazooCollege,Michigan

Mobile VLBI and GPS geodetic measurements have many error sources in common.

Calibration of the effects of water vapor on signal transmission through the atmosphere,

however, remains the primary limitation to the accuracy of vertical crustal motion measurements made by either technique. The two primary methods of water vapor calibration currently in use for mobile VLBI baseline measurements have been evaluated:

radiometric measurements of the sky brightness near the 22 GHz emission line of free

water molecules and surface meteorological measurements used as input to an atmospheric

model. Based upon a limited set of 9 baselines, it is shown that calibrating VLBI data

with water vapor radiometer measurements provides a significantly better fit to the

theoretical delay model than calibrating the same data with surface meteorological measurements. The effect of estimating a systematic error in the surface meteorological calibration is shown to improve the consistency of the vertical baseline components obtained

by the two calibration methods. A detailed error model for the vertical baseline component indicates current mobile VLB! technology should allow accuracies of order 5 cm

with WVR calibration and 10 cm when surface meteorological calibration is used. A

statistical analysis of the results of repeated measurements of the 336-km baseline from

Big Pine, CA, to Pasadena, CA, shows the scatter in the vertical baseline component to be

consistent with this model. A similar model for a hypothetical GPS baseline measurement

is presented. A covariance analysis based upon this model shows current GPS technology

to be capable of accuracies in the vertical baseline component comparable to present

mobile VLBI measurements. Expected improvements in system components for both

technologies should allow reduction of the uncertainty in the vertical component to less

than 3 cm by 1989.

I. Introduction

The measurement of vertical crustal motion with microwave techniques such as Very Long Baseline Interferometry

(VLBI) or ranging to Global Positioning System (GPS) satellites presents difficulties not encountered in the measurement

of horizontal motion. Inherent in the measurement of the

vertical baseline component is a geometric dilution of precision (GDOP) arising from the fact that signals from the sources

(quasars or satellites) can only be viewed along positive values

of this coordinate whereas both positive and negative views of

the sources are possible in measurements of horizontal base185
line components. In addition to this inherent geometrical

weakness, the vertical baseline component is much more

sensitive to errors in the calibration of tropospheric effects,

particularly that portion due to atmospheric water vapor.

This is especially true for the regional baselines measured by

the mobile VLBI units. The relatively short length of these

baselines causes essentially all of the tropospheric errors to

map into the baseline vertical component. In the case of longer

intercontinental baselines, a greater fraction of the tropospheric error would map into the horizontal components.

To a large degree then, the precision which can be achieved in

the measurement of vertical crustal motion rests upon how

accurately we can calibrate the effects of the troposphere on

the microwave signals.

Two techniques are currently in use for the calibration of

the wet troposphere: a direct measurement of the 22-GHz

emission of free water molecules along the antenna line of

sight with a microwave radiometer and the use of an atmospheric model whose parameters are determined from surface

meteorological measurements to infer the amount of water

vapor along the line of sight. These two calibration methods

have been compared using data from several recent mobile

VLBI baseline measurements. In all cases it was found that

calibrating data with WVR measurements provided a better

fit to the theoretical delay model than did calibrating the same

data with surface meteorological based estimates of atmospheric water vapor. As expected, the major differences in the

values of the estimated baseline parameters for the two different calibrations were in the vertical baseline component

where differences as large as 20 cm were seen.

The major sources of error in GPS-based vertical measurements are identical to those encountered in VLBI measurements. An important error source not found in VLBI measurements, however, is the uncertainty of the GPS satellite orbits.

Described here is how data from a fiducial network of GPS

receivers located at a priori well-known sites can be used to

allow accurate determination of these orbits. Finally, the

results of a covariance analysis of a simulated GPS baseline

measurement in the Caribbean region are presented. These

results show that the uncertainty in the wet troposphere

calibration is the dominant source of error in the baseline

vertical component just as in the case of mobile VLBI

measurements.

II. Mobile VLBi Measurement of

Vertical Motion

The essential components of a mobile or any other VLBI

system are a pair of radio telescopes, high-precision frequency

standards (hydrogen masers) at each antenna, and a special

purpose computer known as a correlator. During a baseline

measurement, the broadband emission from an extragaUactic

radio source is recorded on some magnetic medium at each of

the stations. The tapes from each station are then brought to

the correlator which computes a cross correlation of the

quasar signals recorded at the two stations. The resulting

interferometric fringe pattern allows the difference in arrival

times of the quasar signals at the two stations (delay) and its

rate of change (delay rate) to be calculated. These delays and

delay rates obtained from many separate quasar observations

over a 24-36 hour period are used as input to a multiparameter least-squares estimation program that fits a detailed earth

model to the observables from which the baseline coordinates

and other parameters are extracted. More complete discussions

of the interferometry technique can be found in Refs. 1-4

and of mobile VLBI measurements in Ref. 5.

III. Sources of Error in Mobile VLBI Vertical

Measurements

Any deficiencies in either the earth model or the methods

of calibration will ultimately degrade the accuracy of the estimated baseline components. Because baseline results are

expressed in an earth-fLxed system, the earth model must

include effects due to the orientation and motion of the earth

with respect to the coordinate frame of the quasars. These

include the rate of earth rotation (UT1), precession, nutation,

and polar motion. As has already been mentioned, the vertical

component is particularly sensitive to the effects of propagation media on the delay observable. While the introduction of

dual.frequency recording systems has essentially removed the

effect of the ionosphere from mobile VLBI measurements,

the calibration of the tropospheric delays, particularly that

due to atmospheric water vapor, remains the single most

important source of error in the measurement of the vertical

baseline component. This is especially true for the relatively

short regional baselines where tropospheric delays map almost

entirely into this component. In addition to earth orientation

and water vapor calibration, the measurement of the vertical

baseline component is also subject to errors from several other

sources including: instrumental phase calibration, receiver

noise, and errors in the survey tie from the ground monument

to the mobile antenna. Table 1 summarizes the error budget

for the vertical baseline coordinate as it presently stands and

how it may be expected to improve within the next few years.

The error budget for the length coordinate is included for

comparison.

A. Earth Orientation

Because we have detailed theoretical models which describe

the orientation of the earth in space, it is, in principle, possible

186
to use the delay and delay rate observables to estimate these

parameters along with the baseline parameters as part of the

overall least-squares estimation process. Because the mobile

VLBI baselines are" relatively short, however, they would not

provide as accurate a determination of earth orientation

parameters as is presently available from external sources using

results obtained with much longer baselines (Ref. 6). Presently

earth orientation data from Ref. 7 are used, which are derived

from a combination of Project POLARIS VLBI results (Ref. 8),

VLBI measurements made by the antennas at the Deep Space

Network stations (Ref. 9), and the TEMPO project at JPL

(Ref. 10), and data from lunar laser ranging experiments

(Ref. 11). Data from. these sources are combined with a

v_).,_.., fiJter based ,,p,,., o),,a;). ,,¢ .,._,,.pho.;. _¢¢_.), _,. th,

earth's angular momentum budget. The values of UTI determined from this analysis are input explicitly to the parameter

estimation program. Present accuracies of these UTI values lie

in the range of 0.I to 0.6 milliseconds. In the case of polar

motion data, presently obtained from BIH Circular D

(Ref. 24), accuracies of 0.007 arc seconds were assumed for

both the x and y components. The resulting uncertainties

in the estimated vertical coordinate from these two sources

are shown in Table I for a typical r_)ional baseline of 300 to

500 km in length.

Expected improvements in both polar motion and earth

rotation data should reduce the contributions of these sources

to the levels shown in the second column of Table 1 by 1989.

These improvements will result from the upgrades to the VLBI

systems of the POLARIS project, addition of' new lunar laser

ranging stations, and incorporation of earth orientation data

obtained from laser ranging to the Lageos satellite.

B. Troposphere Calibration

It is evident from Table 1 that the tropospheric portion of

the delay model represents the most serious concern for estimation of vertical baseline components with mobile VLBI

measurements. Although it is possible in principle to estimate

the tropospheric delay (using the mobile VLBI data) as part of

the overall parameter estimation process, it is undesirable for

several reasons. First, there may be large spatial and temporal

inhomogeneities in the troposphere which are virtually impossible to model realistically. Second, for the relatively short

regional baselines monitored in the Mobile VLBI Program,

the troposphere and baseline vertical parameters are strongly

correlated, resulting in greatly reduced precision in the estimation of the latter. Third, correlations notwithstanding, the

requirement that the data be used to estimate the tropospheric

delay will seriously weaken its strength for baseline estimation.

For these reasons, the effect of the troposphere on the quasar

signals must be removed prior to parameter estimation by an

external means of calibration.

The effect of the dry component of the troposphere (i.e.,

all nitrogen and oxygen) on the VLBI data is calibrated using a

combination of surface meteorological (SM) measurements

used as input to an atmospheric model (Ref. 12). Errors in this

calibration approach arise principally from the facts that the

atmosphere is not in hydrostatic equilibrium, that there may

be large horizontal gradients in barometric pressure, and that

the line-of-sight delay must be inferred from the zenith delay

using a model-dependent mapping function (Ref. 13). Under

extreme conditions, these could combine to contribute a systematic error of up to 3 em in the baseline vertical, although

more typically the contribution is of order 1 cm (Ref. 5).

T-he calibration of the wet troposphere poses a more serious

problem. Surface humidity measurements may bear little relation to the distribution of water vapor at altitude. Furthermore, since azimuthal symmetry in the distribution of the

water vapor cannot be assumed, accurate mapping to the

antenna line of sight cannot be guaranteed. For these reasons

considerable effort and resources have gone into development

of water vapor radiometers (WVRs) to directly measure the

quantity of water vapor along the antenna line of sight

(Refs. 14-16). "

The WVRs currently in use for mobile VLBI measurements,

fabricated by the NASA Crustal Dynamics Project (CDP)

(Ref. 17), are the result of this ongoing effort. These devices

measure the intensity of microwave emission near 22 GI-Iz

from free water molecules along the antenna line of sight.

This data is then used to calculate an equivalent zenith path

delay with an accuracy of 1 to 2 cm. Because the radiometer

can be pointed with the telescope, the effects of spatial and

temporal nonuniformity are removed. Future developments in

WVR technology (Ref. 18) are expected to reduce the contribution of the wet troposphere to the vertical baseline uncertainty to less than 1 cm.

To illustrate the beneficial effect that use of water vapor

radiometer calibration can have on VLBI measurements, nine

baselines were compared where both surface meteorological

and water vapor radiometer data were available. The mobile

VLBI data were recorded during the CDP observing sessions

of June 6, June 29, and August 28 of 1983. The Chao model

(Ref. 12) was used to calibrate path delays using surface

meteorological data, and the method of Resch (Ref. 25) was

used to calibrate path delays from the WVR data. The VLBI

data were then processed twice, using each of these calibrations to produce independent solutions. Table 2 contains a

comparison of the rms scatters of the post-fit residuals resulting from the least squares fit to the delay model from these

two sets of analyses. As this table shows, the use of the WVR

data results in a better fit to the delay model (and hence a

187
lower rms scatter) for all 9 baselines in the comparison. A

simple analysis of the improvement indicates that for 6 of the

9 baselines, the use of WVR data represents a statistically

significant improvement over surface meteorology data as a

means of wet troposphere calibration. It should be noted that

this analysis assumes no correlation between the rms scatters

for the two calibration methods. Since, in fact, they are highly

correlated (due to the fact that they have all but one error

source in common), these levels of significance represent lower

limits.

The effect of wet troposphere calibration on the actual

values of the estimated baseline parameters is shown in Fig. 1.

As can be seen, the largest effect is in the vertical baseline

component where differences of up to 20 cm are seen between

the SM and WVR approaches. As expected, the method of

calibration is shown to have a significant effect on the estimated values of this component. In the horizontal components, the impact of changing the calibration method is much

less severe.

Finally, we note that for longer baselines, the correlation

of the troposphere and the baseline vertical is considerably

reduced and estimation of an error in the zenith path delay

may improve the results by accounting for any systematic

error in the troposphere calibration. This may be especially

true when surface meteorology calibration of the wet troposphere has been used. In this work, station zenith errors were

estimated for several baselines from the set of 9 used in the

above comparison (see Table 2), and it was found that the

estimation of a constant zenith error for results calibrated with

surface meteorology data does produce a significant improvement in the consistency of the estimated baseline parameters.

This is especially true for the case of the baseline vertical

component. Figure 2 shows the effect that the estimation of

this additional parameter has on the length and vertical

components for six baselines from our original comparison

set. In all cases the estimated baseline parameters tend to move

toward the results obtained with WVR calibrated data indicating that the average difference between the surface meteorology and WVR-based results is significantly reduced when this

additional parameter is estimated. If the WVR.based solution

is assumed to represent the "true" baseline, then this estimation procedure is seen to improve repeatability by a factor of

over 2 in both the length and vertical components. An unfortunate (but expected) consequence of estimating a zenith

troposphere error is that the formal uncertainty in the vertical

baseline component is increased by a factor of between 2

and 3. This is a result of a still high correlation between the

baseline vertical parameter and the estimated troposphere

parameter and a result of the simple fact that the same amount

of data is now required to estimate a larger number of parameters. Figure 3 shows how the formal uncertainties in the

length and vertical parameters increase when this constant

zenith offset parameter is estimated.

The calibration techniques described in this subsection are

all suboptimal in various senses. The WVR-based calibration is

dearly superior to the SM'based calibration. But the direct

application of WVR calibration makes no allowance for the

effect of WVR errors on the VLBI baseline results. The alternative approach of troposphere estimation improves the consistency of the VLBI results, but it also results in a serious

loss of precision. A third alternative which combines the

strengths of both the above is the hybrid approach in which

WVR calibration is applied to the raw VLBI observables and a

constrained calibration error is estimated, based on a realistic

model of that error. The constraint is applied in the form of an

a priori covariance, which limits the size of the error model

parameters, in accordance with the demonstrated inherent

accuracy of the WVR instrument. This hybrid approach is

under development at JPL (Ref. 19) and testing with data is

expected to begin soon.

C. Computation of Uncertainties

In lieu of an operational hybrid approach to computation

of uncertainties, as described in the preceding subsection, it

has not been a practice to attempt to estimate any sort of

systematic errors in the troposphere calibration of the mobile

VLBI data. However, since all calibration is subject to systematic error, we add additional uncertainty to the post-fit estimated baseline parameters in order to provide more realistic

values for their uncertainties. In the cases where water vapor

radiometer data is available, we add 2.0 cm in quadrature to

the uncertainty in the local vertical coordinate for each

station. To reflect the larger error expected when surface

meteorological data is used for wet troposphere calibration,

an uncertainty equal to one half the mean wet path delay is

added to the local vertical for that station. Typically this

results in an additional error of from 2.5 to 5.0 cm in the local

vertical coordinate. The resulting uncertainties in the vertical

baseline coordinate for a typical regional baseline are shown in

Table 1. Expected improvements in water vapor technology

should allow reduction of the contribution of the wet troposphere to less than 2.0 cm provided a sufficient number of

these new instruments are made available for use at all observing stations.

Table 1 also contains the contributions from several other

error sources which have some effect on the accuracy of the

vertical baseline component including an estimate of the

effects of unmodeled elements of the delay observable such

as ocean loading and antenna flexure. The combined effects

of these additional error sources, however, represent only a

small fraction of the contributions from earth orientation and

wet troposphere calibration to the uncertainty in the vertical

188
component. A more extensive discussion of these and other

sources of error in mobile VLBI measurements can be found

in Ref. 5.

In order to test the validity of this error model for the

vertical baseline component, the results of repeated measurements of a single baseline have been statistically analyzed.

The 336-km baseline from the 40-meter telescope at the

Owens Valley Radio Observatory located near Big Pine, CA

to the mobile VLBI site at the Jet Propulsion Laboratory in

Pasadena, CA has been measured 19 times since the first

measurement in January 1980. From these 19 measurements

we have taken a subset of 15 measurements consisting of 4

groups, each of which conta_s measurements made witbj_n, a

six month period. This was done to minimize the possible

effects of tectonic motion on these tests of consistency for

this baseline which crosses the San Andreas fault. The results

of these measurements are shown in Fig. 4 in the form of error

ellipses for baseline length and vertical components. All results

are expressed as shifts from a nominal reference baseline. It is

clear from the figure that the repeatability in the vertical coordinate is much worse than in the baseline length. This is not

unexpected since the baseline length component is entirely

unaffected by uncertainties in the earth orientation and only

slightly affected by troposphere calibration errors. It should

also be noted that some of these baselines used surface meteorological data to calibrate the wet troposphere and that in

these cases no attempt was made to estimate a systematic

error in the manner described earlier. This, as well as inadequacies in the current dry troposphere mode_, may be responsible for the poor repeatability in some cases.

From a ×2 analysis of these results, shown in Table 3, it is

evident that the error model for the vertical baseline component is adequate for all but the second group of measurements

where the scatter in the measurements is somewhat larger than

would be expected on the basis of the formal errors. This is

surprising, because all measurements in this group used water

vapor radiometers to calibrate the wet troposphere delay.

This may indicate that there exists a substantial systematic

error in some of the WVR measurements which might be

removed by estimating a zenith path delay error as was done

for the surface meteorological data with some success.

Beckman (Ref. 19) has proposed such a model, which is

currently being evaluated at JPL.

The results contained in Table 3 also serve to illustrate the

general improvement in consistency of repeated measurements

since 1980. This is due to many factors including improvements in receiver technology, theoretical models, data processing software, as well as improved wet troposphere calibration.

A more extensive discussion of these improvements is contained in Ref. 5.

IV. GPS Measurement of Vertical

Crustal Motion

A. Measuring Baselines with GPS Receivers

By 1989 the full 18 satellite constellation of GPS/NAVSTAR

satellites is expected to be in orbit (Ref. 20). These satellites

will broadcast phase modulated signals at two L-band frequencies (1575.41 MHz and 1227.60 MHz) and are expected to

have many civilian users for navigation, precise orbiter tracking and geodesy. Among the applications being investigated at

the Jet Propulsion Laboratory is a GPS based system to measure geodetic baselines with accuracies comparable to those

attainable using present mobile VLBI systems.

The basic observable obtained from all GPS measurements

is the distance from the receiver to the satellite, called the

"pseudorange." It is not exactly equal to the true satellitereceiver distance because of offsets between the satellite and

receiver clocks and the effects of the troposphere and ionosphere on the broadcast signals. The covariance analysis

described here, however, does not use the pseudorange directly

but forms a new datum by taking the difference between the

pseudoranges from two receivers to a single GPS satellite.

Use of this differential pseudorange has the advantage of

removing the clock offset term between the receiver and satellite. This still leaves an offset between the two receiver clocks,

but it is also possible to form a second difference between

two differential pseudoranges to two different GPS satellites.

This second difference has the effect of canceling out the

clock errors between the two ground receivers, thus removing

all errors due to clock offsets from the GPS range data. When

there are more than two receivers involved, a more elaborate

linear combination of the pseudoranges is actually used to

remove the clock offsets (Ref. 21). It is this more general

double differencing technique which has been used in the

covariance analysis presented here.

B. Fiducial Network Concept

Although the double differencing process removes all clock

errors from the data, the uncertainty in the satellite ephemeris

would limit accuracy to a decimeter or greater for baselines

longer than 100 kin. For this reason the concept of a fiducial

network has been developed (J. L. Fanselow and J. B. Thomas,

personal communication, 1983 and Ref. 22). The fiducial

network consists of three or more GPS receivers whose locations have been accurately established by an independent

technique such as VLBI or satellite laser ranging. During a

GPS baseline measurement, receivers at a priori unknown

locations observe the satellites simultaneously with the

receivers located at the precisely known fiducial station loca.

tions. In the subsequent least squares analysis, the range data

from the fiducial station receivers allows accurate values of

189
the GPS orbits to be estimated along with values for the

unknown station locations. Without this ability to estimate

accurate GPS orbits, centimeter-level baseline accuracies for

regional baselines longer than 100 km in length would not be

possible. The schematic diagram in Fig. 5 shows a GPS measurement scenario using a 4-station fiducial network in conjunction with 2 mobile GPS receivers. Initially, fiducial sites

will be located at Ft. Davis, TX, Westford, MA, and Richmond, FL. Other fiducial sites being considered are Quite,.

Ecuador; See Paulo, Brazil; Santiago, Chile and Cayenne,

French Guiana (Ref. 23).

V. Sources of Error in

GPS Vertical Measurements

The primary sources of error in VLBI vertical measurements described earlier are also present in GPS-based measurements of vertical crustal motion. GPS measurements have the

additional error source of uncertainties in the GPS orbits.

Use of a fiducial network of observing stations in the manner

described in Ref. 23 eliminates most of the error resulting

from this source for baselines of less than 1000 km in length.

This, of course, requires that we have adequate models of the

satellite orbits which include effects of solar radiation pressure, higher order gravitational terms, uncertainty in the value

of the gravitational constant and that we properly account for

any relativistic effects. Table 4 contains the major elements

of the error model assumed in our covariance analysis.

Table 5 shows the results of this analysis for the 370-km

baseline between San Juan, Puerto Rico and Santo Dorningo,

Dominican Republic for both the current and the 1989 system components. As was the case for mobile VLBI vertical

measurements, it is the troposphere calibration which dominates the uncertainty in the vertical baseline component. The

approximately 8-era uncertainty for the current GPS system is,

in fact, comparable to the accuracy now achieved with mobile

VLBI systems. It is clear from these results that improvements

in the accuracy of the wet troposphere calibration are essential if vertical component accuracies of 2 to 3 em are to be

achieved with GPS technology. It is encouraging to note that

errors associated with the determination of GPS orbits do not

contribute significantly to baseline errors for baselines of this

length.

VI. Conclusions

Four years of baseline measurements with mobile VLBI

systems have shown that baseline vertical components can be

measured with a precision on the order of 5 era. The major

sources of error in these measurements are uncertainty in the

earth orientation, especially in the position of the pole, and

errors in the calibration of the effects of water vapor in the

troposphere. The wet troposphere calibration presently represents the most serious error source for mobile VLBI measurements of vertical crustal motion. Future improvements in

WVRs promise to reduce the contribution of water vapor

radiometer calibration errors to a level of 2 cm or less. Along

with improvements in the knowledge of the earth's polar

motion, this will allow baseline vertical components to be

measured with a precision of 2 to 3 era. The introduction of a

WVR error model into the parameter estimation process may

allow the effects of systematic errors in the wet troposphere

calibration to be- reduced even further, to the point that the

calibration of the dry troposphere, including the effects of

dynamic terms and horizontal gradients, may become the limitation on accuracy.

Baseline measurements using GPS satellites and receivers

have the potential of achieving VLBI level accuracies in both

the horizontal and vertical coordinates. The measurement of

vertical motion with GPS technology will also require the use

of water vapor radiometers to remove the effects of the wet

troposphere from the satellite/receiver range measurements.

Present accuracies in the vertical coordinate are predicted,

on the basis of a covariance analysis, to be comparable to the

present accuracies of mobile VLBI measurements. Improvements in receiver technology, WVR instrumentation, and GPS

orbit determination could conceivably reduce the uncertainty

in vertical measurements to the order of 2 to 3 cm by 1989.

190
References

1. Thomas, J. B., An Analysis of Long Baseline Radio Interferometry, JPL Technical

Report 32-1526, vol. 7, 37-50; Jet Propul. Lab., Pasadena, Calif., 1972a.

2. Thomas, J. B., An Analysis of Long Baseline Radio Interferometry, II, JPL TechnicalReport 32-1526, vol. 8, 29-38; Jet Propul. Lab., Pasadena, Calif., 1972b.

3. Thomas, J. B., An Analysis of Long Baseline Radio Interferometry, JPL Technical

Report 32-1526, vol. 16, 47-64; Jet Propul. Lab., Pasadena, Calif., 1974.

4. Shapiro, I. I., Estimation of Astrometric and Geodetic Parameters from VLBI Observations, in Methods of Experimental Physics, M. L. Meeks, ed., 261-276, Academic

Press, New York, 1976.

5. Davidson, J. M., and D. W. Trask, Utilization of Mobile VLBI for Geodetic Measurements, IEEE Transactions on Remote Sensing and Geoscience, Special Issue on

Geodynamics, 1985 (in press).

6. Allen, S. L., Earth Orientation Effects on Mobile VLBI Baselines, TDA Progress

Report 42-78, pp. 202-206, Jet Propul. Lab., Pasadena, Calif., 1984.

7. Eubanks, T. M., J. A. Steppe, M. A. Spieth, The Accuracy of Radio Interferometric

Measurements of Earth Rotation, TDA Prog. Rept. 42-80, 229-235, Jet Propul.

Lab., Pasadena, Calif., 1984.

8. Carter, W. E., D. S. Robertson, J. E. Pettey, B. D. Tapley, B. E. Schutz, R. J. Eanes,

and M. Lufeng, Variations in the Rotation of the Earth, Science, 224, 957-961,

1984.

9. Sovers, O.J., J.B. Thomas, J.L. Fanselow, E.J. Cohen, G.H. Purcell, Jr.,

D. H. Rogstad, L. J. Skjerve, and D. J. Spitzmesser, Radio Interferometric Determination of Intercontinental Baselines and Earth Orientation Utilizing Deep Space

Network Antennas, J. Geophys. Res. 89, 7597-7607, 1984.

10. Eubanks, T. M., M. G. Roth, P. B. Esposito, J. A. Steppe, and P. S. CaUahan, An

Analysis of Tempo Earth Orientation Results, Proceedings of Symposium No. 5:

Geodetic Applications of Radio Astronomy, edited by William Carter, U. S. Department of Commerce, National Oceanic and Atmospheric Administration, 152-162,

1982.

11. Dickey, J. O., and J. G. Williams, Earth Rotation from Lunar Laser Ranging, Astron.

Astrophys. Suppl. Set., 54,519-540, 1983.

12. Chao, C. C., A New Method to Predict Wet Zenith Range Correction from Surface

Measurements, DSN Progress Report 32-1526, vol. 14, pp. 33--41, Jet Propul. Lab.,

Pasadena, Calif., 1973.

13. Lanyi, G. A., Tropospheric Delay Effects in Radio Interferometry, TDA Prog. Rept.

42-78,152-159, Jet Propul. Lab., Pasadena, Calif., 1984.

14. Claflin, E. S., S. C. Wu, and G. M. Resch, Microwave Radiometer Measurement of

Water Vapor Path Dela.y: Data Reduction Techniques, DSN Progress Report 42-48,

pp. 22-30, Jet Propul. Lab., Pasadena, Calif., 1978.

15. Wu, S. C., Optimum Frequencies of a Passive Microwave Radiometer for Tropospheric Path Length Correction, IEEE Transactions on Antennas and Propagation

AP-27, No. 2,233-239, 1979.

191
16. Resch, G. M., M. C. Chavez, and N. I. Yamane, Description and Overview of an

Instrument Designed to Measure Line-of-Sight Delay Due to Water Vapor, TDA

Prog. Rept. 42-72, 1-19, Jet Propul. Lab., Pasadena, Calif., 1982.

17. NASA, Application of Space Technology to Crustal Dynamics and Earthquake

Research, NASA Technical Paper 1464, NASA Office of Space and Terrestrial

Applications, Washington, D. C., 1979.

18. Janssen, M. A., A New Instrument for the Determination of Radio Path Delay

Variations Due to Atmospheric Water Vapor, IEEE Transactions on Remote Sensing

and Geoscience, Special lssue on Geodynamics, 1985 (in press).

19. Beckman, B. C., A WVR Error Model, 1EEE Transactions on Geoscience and Remote

Sensing, Special lssue on Geodynamics, 1985 (in press).

20. Brady, W. F. and P. S. Jorgensen, Worldwide Coverage of the Phase II NAVSTAR

Satellite Constellation, Navigation 28(3), 167-177, 1981.

21. Wu, J. T., Elimination of Clock Errors in a GPS Based Tracking System, Paper

presented at the AIAA/AAS Astrodynamics Conf., AIAA-84-2052, Seattle, Washington, August 20, 1984.

22. Davidson, J. M., C. L. Thornton, B. C. Beckman, P. M. Kroger, S. A. Stephens,

and S. C. Wu, Covariance Analysis for Geodetic Measurements Utilizing the Global

Positioning System Satellites (in preparation), 1985.

23. Kroger, P. M., C. L. Thornton, J. M. Davidson, S. A. Stephens, and B. C. Beckman,

Sensitivity of GPS Caribbean Baseline Performance to the Location of a Southerly

Fiducial Station, Proceedings of the First International Symposium on Precise

Positioning with the Global Positioning System, Rockville, MD, April 15, 1985

(in press).

24. Feissel, M., Annual Report for 1982, Paris, BIH (Bureau International l'Huere),

1983.

25. Resch, G. M., Inversion Algorithms for Water Vapor Radiometers Operating at

20.7 and 31.4 GHz, TDA Progress Report42-76, Jet. Propul. Lab., Pasadena, Calif.,

1983.

192
Table 1. Summary of mobile VLBI error sources in length and vertical coordinates a

Error Source

Contribution to Uncertainty

in Baseline Length and Vertical Coordinate, cm

1984 System 1989 System

Length Vertical Length Vertical

Earth Orientation

UT1 - UTC 0 0.5 0 0.2

Polar Motion 0 1.5 0 1.0

Propagation Media

Dry Troposphere 0.8 1.5 0.5 1.0

Wet Troposphere b 0.6 4.0 0.3 2.0

Wet Troposphere c 1.2 9.0 1.2 9.0

Miscellaneous

Source Positions 0.3 0.3 0.3 0.3

System Noise 0.2 0.2 0.2 0.2

Mobile Antenna Survey Tie 0.3 0.5 0.3 0.5

to Monument

Various Unmodeled 0.5 1.0 0.2 0.5

Elements of Delay Model

Total (RSS) 1.2 b 4.7 b 0.8 b 2.6 b

1.7 c 9.3 c 1.4 c 9.1 c

aThe contributions to uncertainties in the length and vertical baseline components shown in this

table are meant to represent those expected for a typical regional baseline of 300 to 500 km in

length measured with one mobile and one fixed antenna. Actual values will depend upon baseline length and orientation, and the performance of all system components.

bWater vapor radiometer data used to calculate wet path delays.

CSurface meteorological data and atmospheric model used to calculate wet path delays.

193
Table 2. Values of post-fit residual delay scatter

RMS Scatter, ps Probability

Experiment Stations a Baseline That RMS Are

Date (UT) Length, km WVR Calibration Surface Met. the Same, %b

06/09/83 HAT CREEK

GRAS 1933.5 87.3 139.2 0.04

GRAS

WESTFORD 3134.9 115.6 167.0 0.10

HAT CREEK

WESTFORD 4032.8 124.3 167.8 1.20

06/29/83 OVRO 130

HAT CREEK 484.3 48.8 104.6 4.0 X 10 -6

OVRO 130

MOJAVE 245.3 61.0 163.5 1.0 × 10 -s

HAT CREEK

MOJAVE 729.1 42.2 106.7 4.0 X 10 -7

08/28/83 OVRO 130

MOJAVE 245.3 104.0 120.5 36.0

MOJAVE

DSS 13 12.6 197.9 204.2 84.0

OVRO 130

DSS 13 257.6 151.4 159.1 68.0

aDSS 13 - Venus Antenna, NASA Deep Space Network Complex, Goldstone, CA

GRAS - George R. Agassiz Radio Astronomy Station, Ft. Davis, TX

HAT CREEK - Hat Creek Radio Astronomy Observatory, Cassel, CA

MOJAVE - Mojave Base Station, NASA Deep Space Network Complex, Goldstone, CA

OVRO 130 - Owens Valley Radio Observatory, Bishop, CA

WESTFORD - Haystack Observatory, Westford, MA

bCalculated assuming that residual scatter is Gaussian and that the values from the two different

calibration methods are independent. Because this is not true, these numbers represent an upper

limit.

194
Table3.Owens Valley --, JPL baseline precision a

Dates Degrees of

Freedom

RMS Scatter, cm x2

Length Vertical Length Vertical

5/80 - 11/80 3 2.2 10.2 0.33 0.55

8/81 - 11/81 3 0.8 12.6 0.28 2.29

10/82 - 2/83 3 1.1 5.9 0.83 0.17

6/83 - 11/83 2 0.8 5.6 0.40 0.17

aThe data from which these remits are calculated are displayed in Fig. 4.

Table 4. Inputs to GPS baseline covariance analysis

Input 1984 System 1989 System

Data Type Double differenced Double differenced

integrated Doppler range or integrated

Doppler

Data Noise 3.0 cm 1.0 cm

Fiducial Network _ Ft. Davis, TX Ft. Davis, TX

Westford, MA Westford, MA

Richmond, FL Quito, Ecuador

Considered Parameters Uncertainty

Relative Position 3 cm horizontal 1 cm all components

of Fiducial Station 9 cm vertical

Troposphere Calibration 2.0 cm (zenith) 0.75 cm (zenith)

Error

Solar Radiation Pressure 5% @CR = 0.5

Geopotential 10% of GEM 6 - APL50

GM 10 -8 of the total value

Estimated Parameters A Priori Uncertainty

1000 m (integrated Doppler only)

l0 m in each component (X, Y, 2")

0.1 cm/s in each component (Vx, Vy, Vz)

20.0 m in each component (X, Y, Z)

Range Bias Parameters

NAVSTAR Position

NAVSTAR Velocity

Mobile Station Location

195
Table 5. Covariance analysis results for San Juan to Santo Domingo baseline s

Error Source

Contribution to Uncertainty

Baseline Vertical Coordinate, cm

1984 System 1989 System b

Range Measurement Accuracy 3.4 1.4

Sola_ Radiation Pressure << 1 << 1

Geopotential << 1 << I

GM << I << 1

Relative Position

of Fiducial Stations 0.8 0. I

Location of Geocenter << 1 << 1

Troposphere Calibration 6.9 2.6

Total (RSS) 7.7. 2.9

aThe values in this table are based upon the results for a single baseline of 370 km between

San Juan, Puerto Rico and Santo Domingo, Dominican Rep. The values for other baselines

will depend to a degree upon baseline length and orientation.

bThese results correspond to the integrated Doppler dam type. Results for the carrier range

data type are virtually identical for this short baseline.

196
f i t 06/09/SS

|

I

E

u 0

-10

-20

• WVR

O SM

EXPERIMENTS

Fig. 1. The effect of different wet troposphere calibrations on estimated baseline parameters is illustrated for 6 baselines from two

mobile VLBI experiments completed in June 1983. Estimated values

of the length, transverse, and vertical beseiine components for both

WVR and surface metaoroiogicai calibrations are shown. See Table

2 for station names and baseline lengths.

5

u

151

10

-5

-10

I I I I I I

WVR

w

• WITH ERRORESTIMATION -

-15

-2O

AVENGE NC) EST WITH EST

IVERTICALI 10.4 _ 4.2

LENGTH 2.1 _ 0.9

-25 I I I I I I

0 I 2 3 4 5 6 7

LENGTH, cm

Fig. 2. The effect of the estimation of zenith troposphere error on

length and vertical coordinates is shown. The difference between

the WVR and SM values is substantially reduced by estimating an

error in the SM calibration.

197
7

6-

5 5-

4

U

3

2

I

0

i i

O NO ERROR ESTIMATION

• WITH ERROR ESTIMATION

I 1

0.5 1.0

LENGTH ERROR, cm

1.5

Fig. 3. The effect of the estimation of zerilth troposphere error on

formal uncertainties in the length and vertical coordinates is shown.

The accuracy of these components is substantially increased by the

estimation of an error in the calibration.

U

3O

3O

-30

JUN 1980- AUG 1981-

NOV 1980 NOV 1981

i ! •

OCT 191B2- JUNE 1983-

FEB 1983 NOV 1983

i i l i i _ i L _ h L k i I i L

o 10 0 10

LENGTH, cm

n

-10

Fig. 4. Length and vertical coordinates of baselines from Big Pine,

CA to Pasadena, CA are shown as shifts from a nominal reference

baseline. Note how the rms scatters and the formal errors decrease

with time (see Table 3).

-J(..- QUASAR • FIDUCIAL STATION

- 0 MOBILE STATION

-. _ELLITE

IIi "_

ii

II

/ I

/ I

Fig. 5. The use of s fiducial network for GPS baseline measurements is shown. The fiducial station data enable the accurate estimation of the GPS satellite orbits. This, in turn, enables the accurate

determination of the mobile station locations. VLBI observations

establish the fiducial station baselines and tie the GPS results to the

inertial frame of the quasars.

198
TDA Progress Report 42-82

NSS-lo4o 

ApnI-June 1985

Improved Mapping of Radio Sources From VLBI Data

by Least-Squares Fit

E. R. Rodemich

Communications Systems Re.arch Section

This article describes a method for producing improved mapping of radio sources from

VLBI data. The method described here is more direct than existing Fourier methods,

is often more accurate, and runs at least as fast. The visibility data is modeled here, as in

existing methods, as a function of the unknown brightness distribution and the unknown

antenna gains and phases. ICe want to choose these unknowns so that the resulting function values are as near as possible to the observed values. If we use the RMS deviation to

measure the closeness of this fit to the observed values, we are led to the problem of

minimizing a certain function o.fall the unknown parameters. This minimization problem

cannot be solved directly, but it can be attacked by iterative methods which we show

converge automatically to the minimum with no user intervention. The resulting brightness distribution will furnish the best fit to the data among all brightness distributions of

given resolution.

I. Introduction

This article describes a method for producing improved

mapping of radio sources from VLBI data. The use of VLBI

data has led to high resolution maps of radio sources in the

sky [Refs. I-4]. The data provide values of the visibility function, which is the Fourier transform of the brightness distribution. The problem of Finding the unknown brightness distribution can accordingly be expressed as the problem of

Finding an inverse Fourier transform. The methods currently

used depend on approximate inversion methods for a Fourier

transform which is known on an irregularly spaced set of

points.

An additional complication is that the signal received at

each antenna can have an unknown gain and phase offset,

depending on conditions at this antenna as well as on atmospheric conditions. This introduces unknown multiplicative

factors into the visibilities which must be eliminated before

inverting the Fourier transform. Iterative methods have been

developed for this which use an assumed map to recalibrate

the data, get a new map from these data by Fourier inversion,

then repeat the procedure starting from the new map. These

iterative methods require considerable user interaction as well

as computer time. They are also biased in favor of certain

types of brightness distributions in the resulting map.

The method described in this article is more direct than

existing Fourier methods, is often more accurate, and runs at

least as fast. The visibility data are modeled here, as in existtug methods, as a function of the unknown brightness distribution and the unknown antenna gains and phases. We want to

choose these unknowns so that the resulting function values

are as near as possible to the observed values. If we use the

199
RMS deviation to measure the closeness of this fit to the

observed values, we are led to the problem of minimizing a

certain function of all the unknown parameters• This minimization problem cannot be solved directly, but it can be attacked

by iterative methods which converge automatically to the

minimum with no user intervention. The resulting brightness

distribution will furnish the best fit to the data among all

brightness distributions of given resolution•

II. The Method

Preprocessing of the data, which we are not concerned with

here, furnishes time averages of the visibility function over

intervals on the order of several seconds to a minute, together

with estimates of standard deviation due to noise• We get a

set of values, E(; ), for the antenna pair p, q and the r/th time

interval. For a gwen value of n, only some of the possible p,

q pairs may occur, either because the source is not visible from

all antennas, or because some data were lost.

Each data value, E (n), is modeled by a function, F (n), of

• Pq q

the unknown brightness distribution, I, and the unJ_nown

gains, A (n), A (n), with noise added. The form ofF (n) is given

in AppePndix ,_. Using the standard deviations, a_f, qfrom the

preprocessing, we set up the function

(2= E. t2.,

Qn = Z IF(;q)-E(l_q) //2/o(n)pq 2 •

p,q

Each F__n) is linear in the Im's and contains the product A_(n)

A q(n--"_, w_ere the bar denotes a complex conjugate. An iterati_ve

method of minimizing Q is used which repeats step (1) followed

by step (2):

(!) Minimize Q by varying the In's, holding the Afn)'s

fixed. P

(2) Minimize Q by varying the A(n)'s, holding the 1 's

p m

fixed.

The nature of these steps is quite different. In (1), we want

to solve a system of simultaneous linear equations in a large

number of unknowns. In (2), this linearity is lost, but the

problem breaks up into the minimization of each Qn separately,

which only involves a small number of variables.

As this iterative procedure is carried out, the value of Q

always decreases, and the unknowns approach values for which

no further decrease is possible. Hence the procedure must

converge.

III. Results and Conclusions

The method was applied to two sets of real data and one

set Of simulated data. The two real data sets are based on

observations in Dec. 1982 at 6 cm. The resulting maps are

shown in Figs. 1 and 2. Source III is a simulated source consisting of two point sources convolved with a circularly symmetric Gaussian distribution, with 1% noise added. The output

map is shown in Fig. 3.

The computer time needed for each source on the Caltech

computer "PHOBOS" was 10-15 minutes, which compares

favorably with the time for conventional methods. This is

possible because the new method can give good maps with a

coarser grid than other methods. This is shown by Fig. 4,

which displays the map of Source I derived in the usual way

with a fine grid, and the effect on this map if a 32 × 32 grid is

used for the final inversion and "CLEANing" ("CLEAN" is a

Caltech program).

The method described here has been shown to be a practical alternative to existing methods. It can construct a map

without any user intervention. The resulting map is free of the

biases introduced by interpolation before inversion and by the

CLEAN program.

200
Acknowledgment

I am indebted to Dr. T. J. Pearson of the Caltech Astronomy Department for help in

providing data and access to the Campus/JPL Correlation Facility located in Robinson at

Caltech.

References

1. Readhead, A. C. S., and Wilkinson, P. N., "The mapping of compact radio sources

from VLBI data," Astrophysical Journal, 223, pp. 25-36 (1978).

2. Cotton, W. D., "A ....... l,cmuu of mapph,g compact _h_t,ctu_e in radiu sources using

VLBI observations," Astronomical Journal, 84, pp. 1122-1128 (1979).

3. Readhead, A. C. S., Walker, R. C., Pearson, T. J., and Cohen, M. H., "Mapping radio

sources with uncalibrated visibility data," Nature, 285, pp. 137-140 (1980).

4. ComweU, T. J., and Wilkinson, P. N., "A new method for making maps with unstable

radio interferometers," ilion. Not. R. Astro. Soc., 196, pp. 1067-1086 (1981).

201
z

O

,u.,

u

• r i ,

16

o

o 8 12

MILLIARCSECONDS

Fig. 1. Source 1807+698

16

12 0

0

0 4

MILLIARCSECONDS

Fig. 2. Source 2021+614

z

0

.Y.

_n

16

12

I i i i i x _ I

L _ h 1_2

MILLIARCSECONDS

Fig. 3. Simulated source

lt6 i 2O

202
"ql!

E

20

10

ot

' I ' ' ' ' I ' i 1 i r

1807 + 698 1807 +

<)

g,

Fig. 4. Output of CLEAN: (a) 128 x 128 resolution; (b) 32 x 32 resolution

P

0s

0

G

! I

2O3
DATA FILE IN

1

INDATA

VLFIT

BRIGHTNESS

DI STRIBUTI(3 N

OUT

Fig. 5. Relation of the programs

READ

INPUT

l

RESCALE J

I (NAP = 0)

ADJUST

GAINS

(NAI = O)

ADJUST

BRIGHTI'_SSES

WRITE 1

OUTPUT

FILE

Fig. 6. The program VLFIT

204
Appendix A

Derivation of the Cost Function

In determining the brightness distribution of a radio source,

preliminary reduction of VLBI data yields corrupted values of

the visibility function

f f I(x,y) exp [21ri(ux + vy)] dxdy (A1) r(u,v)

[Refs. A-1 and A-2]. Here I(x,y) is the brightness distribution,

expressed as a function of rectangular coordinates x_v in the

The visibility function V(u,v) is its two-dimensional Fourier

transform.

The core of the problem of finding the brightness distribution is the inversion of this transform. This is complicated

by the fact that only values of V(u,v) on a restricted, irregularly spaced set of points (u,v)are contained in the data.

Suppose there are NS antennas. Then the quantities u , v

are associated with the pta antenna, 1 _ p _ NS, as fol_ow_.

Let Xp be the vector from the center of the earth to the

pta antenna, and X be the wavelength of the radiation received.

Let K be the facto_ by which values of x_v must be multiplied to convert to radians (typically, x and y are measured in

milliarcseconds, and K = 7t-10-6/648). Then up,vp are the

components of KXp/X parallel to the x and y axes.

The only values of V(u,v) which enter into the data are

Vq = V(up - uq, vp - vq), 1 <p,q<NS.

These quantities are functions of time, since up, vp change due

to the rotation of the earth. The data are averaged over time

intervals sufficiently small that there is no significant variation

of Ul:,,vp over an interval. This reduces the set of visibilities

to a finite set of values V(n), where n is the index of the time

interval. For a given value of n, some pairs p,q may not occur,

because the source was not visible from certain antennas at

that time, or because some data were lost.

To determine l(x,y) numerically, it is approximated by a

series of delta functions on a rectangular grid in a restricted

region of the plane,

MF •

l(x,y) = E I 8(x- xm)80'- Y,n ).

rti = l

Then equation (A1) is reduced to

MF

pq,ra m

(A2)

where

pq,m

.. . .. [n'l (n_x . / (n_ (n'l_ • "1

= exptzrtzttu_,' - Uq')x m ," tv_,"- vb")y,,.rj •

Unfortunately, there are additional complications in the

problem. Each antenna has a gain, Mp, and a phase offset,

_p, which are unknown functions of time. The effects of noise

must also be taken into consideration. IfA (n)p = Mp exp(iCp),

then the quantities actually given by the data are

(a3)

where the bar denotes a complex conjugate, and n_(_) is the

contribution of the noise [Ref. 3).

Our problem is to determine the values I , given the quanta ,,_

tities E(n). If we start with assumed values _'(n),/,., we can

construct

MF

and

F(n) = _(n) 2(n) p(n) (A4)

Pq P q Pq •

n (n) ill The noise terms uq (A3) are assumed to be unknown, so a

criterion for the goodness of the assumed values is how close

the F (n) are to the E (n). The averaging process which furnishes

Pq Pq

the data E_) also provides estimates, o(n), of the standard

deviation oI the noise. Thus we are led to consider the cost

function

12/o '(n)2 . (A5)

n p,q

Minimization of Q gives the least-squares fit of the F(_ ) to the

E(").

pq

205
References

A-1. Pachoczyk, A. G., Radio Astrophysics, Freeman, San Francisco (1970).

A-2. Fomalont, E. B., and Wright, M. C. H., in Galactic and Extragalactic Radio Astronomy (eds. Verschunt, G. L. and Kellermann, K. I.), Springer, New York (1974).

206
Appendix B

Iterative Minimization of the Cost,Function

As stated in Section II, this minimization is accomplished

by iterating two steps.

I. Variation of |m

Let z m ffi Tm . Then, in its dependence on the zm, Q is

a positive definite quadratic form

MF MF

Q f m_, = 1BmkZmZk-2EDmZm+R'm=l (B1)

where

C(-) "//o(n)2_

IZ -- (n)2}

and

n p,q[ Pql /pq

This quadratic form is minimized by the values of the z m at

which all the partial derivatives with respect to the z m are zero:

MF

E Bmk zk =Dm ' 1 < m _gMF (B2)

k=l

Direct solution of this system is not practical because of

the large number of unknowns. An iterative method which

converges to the solution is the Gauss-Seidel method [Ref. B-l],

which consists of the following: Start with any assumed values

of the unknowns. For m = 1 to MF, make the replacement

(B3)

When applied repeatedly, this procedure is easily shown to

converge to the minimum of Q (for fixed .4(f)). The step (B3)

/,,

gives the minimum of Q when only the one unknown z is

m

allowed to vary.

II. Variation of the Ap(n)

Here each of the subsums of (A5),

n) 2 (n)2

P,q

involves a separate set of variables, so they can be minimized

separately. Let ap = A_ln).p Then

where

Gq ffil _.> =/o<.>:

I Pq '

_q pq % ,

and

P,q

This function is minimized by the gradient method [Ref. B-1 ].

Let

b = ;_Qn = 2 Z -(Gpqlaql2 ap +I-Iq aq). -a--_

P q

For v > 0, the quantities ap + vbp are directed away from ap

along the negative gradient of Qn (in the 2-NS-dimensional

space of the real and imaginary parts of the ap's). The iterative

step here is to replace ap by ap + vbp in the expression (B4)

for Qn, then choose v to minimize the resulting fourth degree

207
polynomial in v. Using the minimizing value, ap + Vbp gives

the new value of ap.

In Step (2), as in (1), convergence is guaranteed since each

step decreases Q. More generally, the steps of (1) and (2) can

be intermingled in any systematic iterative procedure which

varies each unknown infinitely often, and Q will approach the

minimum. It is advantageous, however, to carry out Steps (1)

and (2) separately to the point where the minimums of the

separated problems are approached, because this reduces the

amount of time spent in computing the coefficients in the

formulas (B1) and (IM).

Reference

B-1. Burden, R. L., Faires, J. D., and Reynolds, A. C., Numerical Analysis, Prindle,

Weber & Schmidt, Boston (1978).

208
Appendix C

Description of the Computer Programs

The method described here was implemented on the VAX

computer PHOBOS in the Caltech Astronomy department. A

series of three programs is used (see Fig. 5):

(1) The program INDATA reads the data from an existing

data File in the MERGE format currently used at

Caltech. This data is transferred into new files, with

some conversion, and some of the quantities to be used

in the minimization process are pre-computed.

(2) The program VLFIT carries out the minimization

procedure of Section II.

(3) The program MODOUT reads the file output by

VLFIT and builds a new File MOD.DAT which can be

used by the Caltech program MODPLOT to draw a

contour map of the source.

I. Program INDATA

This program reads an input file prepared by other existing programs. The first section reads a collection of header

records, saving some of the information and converting some

to a more convenient form. This information includes the

astronomical location of the source (right ascension and

declination), the staging time of the data, and names and

locations of all the antennas. The time origin is chosen to be

the start of the day on which the data begins, in Greenwich

mean time. The right ascension is converted to PLO_, the

longitude of the source at time I_. The vector from the earth's

center to the pth antenna, (STX(P), STY(p), STZ(p)), is

rescaled so that its components in the xy-plane at the source

are the quantities 2zrup, 2zrvp of Appendix A.

Next the data section of the file is read. The data points

are stored sequentially in an array of length 1P. The antenna

pair (/,J) is denoted by one index L = L(I,J) which is related

to I and J by the arrays 11B, JIB set up in the program. We

build the arrays

T(N), N=I ..... NT

• and

E(K), SGM(K), LP(K), KT(K), K=I ..... IP,

where T(N) is the time of the n TM time interval in minutes, and

E(K) = E (N) ,

Pq

SGM(K) = O(p_)2 ,

LP(K) = L(p,q),

with

KT(K) = N, in the n TM time interval.

tyata wln_lt contain no .... *"' m,urm_tuun"-r- -'"- are _th-ninat_.-l" • .^.4

Next the program reads two lines Of input data which

specify parameters of the map. These are:

(1) NX, NY: Number of grid points in the x andy directions.

(2) XL, YL: Half-width of the map in x and y directions, in

milliarcseconds.

The total number of real unknowns is now known to be

2NS.NT + MF (where MF = NX.NY). The number of real

conditions in the data is 2.1P. The ratio 21P/(2NS.NT+MF),

the redundancy, is evaluated here. If this value is less than 1,

there are not enough conditions to determine the unknowns.

If this happens, we may still get a good map, but there are

other solutions to the minimum prbblem.

The next section of the program computes the quantities

2rrup, 27rvp for each time interval, and forms the auxiliary

complex arrays CLqt(p, Ar), CLX(P,iV), CLY(P,N'). These are

used in VLFIT to generate C(p_),m as follows: The x and y

values at the grid points are

x m = x ° +(1- 1)DX, 1 <_I<_NX,

YM = Yo + (J- 1)DY, 1 <_J 6NY,

where M = J + (I ° I)NY. If we put

CL(P,M,N)= CLO(P,N). CLX(p,NY-! • CLr'(pdVY- ,

then

C(p2v) = CL(p,M,N) . CL(q,M,N).

q,M

Finally, the brightness array AI(M) and the complex gain

array AP(I,N) are initialized with values 1, and all the arrays

needed by VLFIT are written in Fries.

209
II. Program VLFIT

This program reads the files prepared by INDATA, and two

other lines of input: (i) NAP and NAI, and (ii) DSC (see Fig. 6).

The gains are not adjusted if NAP = 0. The brightnesses are

only rescaled if NAI = 0. DSC is a parameter used in image

enhancement. For DSC > 0, the peaks of the brightness distribution are sharpened twice during the iterative solution for

the brightnesses AI(M).

First the brightnesses are rescaled so that for the given gains

and shape of the brightness distribution, the fit to the data is

as good as possible.

Next (if NAP#:0) step (2) of Appendix B is applied, with

twelve iterations in each time interval.

Next (if NAI:#0) step (1) is applied. The values of AI(M),

M=I to MF, are adjusted fifty times. After the 6th time and

the 26 th time, image enhancement is applied, by replacing

each AI(M) by

AI(M) - DSC • (max (,41)- AI(M))

and then replacing any negative values by zero.

Finally, an output file of values of gains and brightnesses

is written, in the same format as the input file. This allows

the application of VLFIT any number of times, using each

output file as input the next time.

III. Program MODOUT

This program converts the output of VLFIT into a form

useable by the program MODPLOT. The brightness distribution is characterized by a series of components, in this case

delta-functions at the grid points, whose position is given in

polar coordinates. These are listed in order of decreasing

strength, terminating after at most MAXC components, where

MAXC is a number read as input by the program.

210
TDAProgressReport 42-82 April-June1985

Referees

The following people have refereed articles for The Telecommunications and Data

Acquisition Progress Report. By attesting to the technical and archival value of the

articles, they have helped to maintain the excellence of this publication during the past

year.

D. A. Bathker C.E. Hildebrand J.R. Lesh A.M. Ruskin

J. J. Bautista R.L. Horttor R. Levy R. SfeJr

D. W. Brown D.L. Johnson K.M. Liewer H.M. Shao

R. W. Burt J. Katz G. Lorden R. Stevens

A. G. Cha M.J. Klein R.J. McEliece L. Swanson

R. B. Crow C.E. Kohlhase S.N. Mohan R.N. Treuhaft

L. J. Deutsch R. Kumar B. Moultrie • C. Wang

J. L. Fanselow C.R. Latuneyer H.P. Phillips R.A. Winkelstein

E. C. Gatz D.B. Lame F. Pollara-Bozzola S.S. Wissler

R. M. Goldstein F.L. Lansing G.H. Purcell, Jr. M. Wyatt

C. A. Greenhall G.E. Lanyi M.P. Quirk C.F. Yoder

R. W. Hartop J.W. Layland D.L. Robinson

NU_SA--JPL--Co_. -A..c_. 211