- Axiomatix 

(NASA-CR- 271725) ENGINEERING EVALUATIONS 
 N84-15175
 

AND STUDIES. / VOLUME 2: EXHIBIT B, PART I 

Zinal Report (Axomatix, Los Angeles, Calil.)- 322 p HC AI4/MF AOl CSCL 22B 
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G3/16 43326
 

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https://ntrs.nasa.gov/search.jsp?R=19840007107 2018-08-09T17:10:43+00:00Z
ENGINEERING EVALUATIONS AND STUDIES
 

FINAL REPORT FOR CONTRACT NAS 9-16067
 

VOLUME II,EXHIBIT B f&.-$ / 

Part I
 

Technical Monitor: William Teasdale
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

Unjeng Cheng
 

James G. Dodds
 

Gaylord K. Huth
 

Jack K. Holmes
 

Richard S. Iwasaki
 

Robert G. Maronde
 

Peter W. Nilsen
 

Andreas Polydoros
 

Don Roberts
 

Sergei Udalov
 

Charles L. Weber
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8310-6
 

October 31, 1983
 
TABLE OF CONTENTS
 

Page
 

Volume I,Exhibit A
 

1.0 EXECUTIVE SUMMARY ..... ... .. .. ................. 1
 

1.1 Contract Tasks ..... ... .. .. ................ 2
 

1.1.1 Exhibit A Contract Tasks ...... ............ 2
 

1.1.2 Exhibit B Contract Tasks ...... ........... 7
 

1.1.3 Exhibit C Contract Tasks ...... ............ 11
 

1.2 Performance of the Contract Tasks ...... .......... 12
 

2.0 ORBITER INERTIAL UPPER-STAGE STUDIES ...... ............ 16
 

3.0 KU-BAND HARDWARE STUDIES ...... .. .. .............. 19
 

4.0 S-BAND HARDWARE INVESTIGATIONS ...... ................ 21
 

5.0 SHUTTLE/CENTAUR COMMUNICATION SYSTEM ENGINEERING INVESTIGATIONS. 22
 

Volume II,Exhibit B, Part 1
 

6.0 KU-BAND COMMUNICATION SYSTEM ANALYSIS ... ........ .... 23
 

6.1 Shuttle/TDRSS Sidelobe Avoidance.... . . .. . . . .. 25
 

6.1.1 Introduction ... ................. .... 25
 

6.1.2 Derivation of Sidelobe Discrimination Budgets . . 26
 

6.1.3 Conclusions ..... .... ................. 26
 

6.2 Shuttle/TDRSS Acquisition and Tracking Performance . . . . 30
 

6.2.1 Introduction ...... ... ................ 30
 

6.2 2 Dynamic-Rate Limitations ...... ..... .... 30
 

6.2.3 Flux-Density Specification Considerations . .. . 31
 

6.2.4 Summary and Conclusions ..... ............. 32
 

7.0 S-BAND SYSTEM INVESTIGATIONS ...... .. ....... .... 33
 

8.0 ORBITER ANTENNA STUDIES AND INVESTIGATIONS ..... .... .. 34
 

9.0 PAYLOAD COMMUNICATION INVESTIGATIONS ...... ... ........ 36
 

10.0 SHUTTLE/TDRSS AND GSTDN COMPATIBILITY ANALYSIS..... . . .. 37
 

10.1 TDRSS Antenna Scan for Shuttle Acquisition ... ........ 37
 

10.2 PSD of Staggered Quadriphase PN with Identical Sequences . 38
 

*1 
TABLE OF CONTENTS (CONTINUED)
 

Page
 

Volume II,Exhibit B, Part 2
 

11.0 PRELIMINARY SHUTTLE/CENTAUR COMMUNICATION SYSTEM ANALYSIS . . 39
 

12.0 SHUTTLE GLOBAL-POSITIONING SYSTEM INVESTIGATIONS . . ... .. 40
 

13.0 TELEVISION DIGITIZER DEVELOPMENT ...... .. ... .. .. 41
 

13.1 Background ..... ... .. .... .............. 43
 

13,2 Principles of Delta Modulation .......... ...... 44
 

13.2.1 Conventional Bistate Delta Modulation ...... 44
 

13.2.2 Tri-State Delta Modulation ...... ....... 47
 

13.2.3 Application of TSDM to NTSC Color TV . ..... 50
 

13.3 Breadboard System Description ...... ... ........ 52
 

13.3.1 Overall Test Setup.. .... ... . . 52
 

13.3.2 Transmitter Functional Block'Diagram'Description 55
 

13.3.3 Receiver Functional Block Diagram Discussion . 57
 

13.3.4 TSDM Digitizer/Reconstructor Implementation . . 59
 

13.3.5 Run-Length Encoder Implementation ......... 63
 

13.3.6 Implementation of the Run-Length Decoder . . . 66
 

13.4 Summary and Conclusions ...... ... .. ... ... 66
 

Volume III, Exhibit C
 

14.0 ORBITER/PAYLOAD INTERFACE INVESTIGATIONS ..... .. ...... 69
 

14.1 Spacelab ICD Revision ..... ... .. ............70
 

14.1.1 HRM Asymmetry and Jitter .......... ..... 70
 

14.1 2 Data-Dependent Amplitude Variations..... .. 76
 

14.1.3 Transition Density ...... ....... ...... 76
 

15.0 PAYLOAD/SHUTTLE DATA-COMMUNICATION-LINK HANDBOOK . .... .. 77
 

REFERENCES ... .... ... .. ................ ...... .. 78
 

APPENDIX A - MONTHLY PROGRESS REPORTS
 

11
 
23 

6.0 KU-BAND COMMUNICATION SYSTEM ANALYSIS
 

The studies and investigations of the Ku-band communication system
 

were focussed inthe following areas:
 

(1) Shuttle/TDRSS sidelobe avoidance
 

(2) Shuttle/TDRSS acquisition and tracking performance
 

(3) Ku-band TDRSS ground station, two-channel, Costas-loop tracking
 

performance for unbalanced quadriphase-shift-keying (UQPSK) with arbitrary data
 

formats
 

(4) Ku-band bent-pipe mode 2 BER performance
 

(5) Degradation due to dither of the antenna during angle tracking
 

(6) Potential Ku-band sidelobe reduction.
 

Analyses of Shuttle/TDRSS sidelobe avoidance as well as Shuttle/TDRSS
 

acquisition and tracking performance are presented insections 6.1 and 6.2, respectively. Analysis of the two-channel Costas-loop tracking with UQPSK signals with arbitrary data formats ispresented inAxiomatix Report No. R8010-8,
 

dated October 30, 1980, which is included inthis section. The analysis in
 

this report was kept general, so the results can apply to a variety of Shuttle
 

payload data formats. Specifically, the mean-square, phase-jitter performance
 

isdetermined for the TDRSS ground station, two-channel Costas loop with active
 

arm filters, synchronous or asynchronous symbol clocks, and arbitrary data
 

formats.
 

The analysis of the Ku-band return-link mode 2 BER degradation is
 

presented inAxiomatix Report No. R8102-4, "Ku-Band Bent-Pipe Mode 2 BER Performance Degradation," dated February 18, 1981. There are three inputs to the
 

FM modulator inmode 2. First, on channel 3,there isthe bent-pipe wideband
 

signal which isamplitude (peak) regulated to provide 22 MHz peak-to-peak (p/p)
 

carrier deviation. Inaddition, channel 1 has operations data from the network
 

signal processor, and channel 2 receives data from the payload recorder, operations recorder, payload low-data-rate link or the payload interrogator narrowband bent-pipe signal. Channels 1 and 2 together provide for a p/p FM frequency
 

deviation of 12 MHz. The specific case of concern inthis report isthe one
 

wherein the signal of channel 1 is a 192-kbps biphase-L (Manchester) baseband
 

data stream without thermal-noise corruption, channnel 2 has an NRZ baseband
 

data stream plus thermal noise, and channel 3 is composed of two subcarriers,
 

one at 1.024 MHz and one at 1.7 MHz, plus thermal noise.
 

PRECEDING PAGE BLANX NOT FILMEV 
24 

Table 6.1. Sidelobe-Discrimination Budget [1]
 

Weakest 

Mainlobe 

Strongest 

Sidelobe 

Cumulative 

Difference 

Pointing Loss, dB 

Flux Density, dBW/m2 

-3.0 

-126.9 

-22.0 

-113.5 

19.0 dB 

Antenna Gain, dB 37.1 37.1 

Effective Area (omnl), 

dBmC (at 13.775 GHz) -

44.2 -44.2 

Effective Area, dBm 2 -7.1 -7.1 

Polarization Loss, dB -0.2 0.0 

Received Power, dBW -137.2 -142.6 5.4 dB 

System Temperature, 

dB°K (at antenna 

terminals) 

306 28.9 

Boltzmann's Constant -228.6 -228.6 

C/N0 dB-Hz (at EA-1 60.8 57.1 3.7 dB 

input) 

Threshold Setting, dB -0.3 0.3 

Noise Immunity Margin, 

dB 

-0.6 1.2 

Effective C/N0 , dB-Hz 

(at Detector) 

58.6 1.3 dB 
25
 

Another aspect of the Ku-band investigations performed was determining
 

the degradation inthe Ku-band communication system performance due to the introduction of dither during antenna angle tracking. The dither is introduced
 

to overcome the effects of stiction and the resultant limit cycling of the antenna during angle tracking. The analysis for this performance degradation is
 

presented inAxiomatix Technical Memorandum No. M8105-1, "Degradation Due To
 

Dither," dated May 11, 1981. It is shown that the dither has no significant
 

degradation on either angle-tracking or communication performance. However, a
 

minor improvement insteady-state angle-tracking accuracy isrequired to maintain the O.3-dB allowable pointing error with dither present.
 

Finally, Axiomatix Technical Memorandum No. M8302-1, "Potential Ku-


Band Sidelobe Reduction Recommendation," dated February 16, 1983, presents an
 

approach to modify the Ku-band antenna-feed-support structure inorder to reduce the sidelobes. The relatively high sidelobes found inradar and communication systems have caused problems with acquisition throughout the development
 

stage. The high (- 20 dB) sidelobes inthe communication system continue to be
 

of concern during TDRSS acquisition. From the antenna measurements, the dominant cause of the high sidelobes appears to be due to antenna-aperture blockage
 

by the feed-support structure. By using a wedge-shaped ridge along this structure, polarizing characteristics can be achieved. The linear polarization, consisting of both RHCP and LHCP components, will then contribute some offsetting
 

aperture illumination versus none with the existing configuration. Axiomatix
 

feels that this modification can achieve sidelobes of at least -22 dB.
 

6.1 Shuttle/TDRSS Sidelobe Avoidance 

6.1.1 Introduction 

Some concern has been expressed about the possibility of sidelobe acquisition of a TDRS signal, particularly inlight of revised flux-density specifications at the Orbiter. Note that Table 6.1 [1] gives the minimum flux density
 

22
 

as -126.9 dBW/M 2 and the maximum flux density is-113.5 dBW/M 2 . The resultant
 

margin isgiven as 1.3 dB. Several assumptions were made [1] to arrive at this
 

margin, and the specified dynamic range of received flux density will be slightly increased inthe revised specification. Inthe following sections, we develop a new pair of sidelobe-discrimination budgets. One is based on the new revised flux-density specification which includes a margin at the low level, and
 

the other is based on actual expected values of flux density. Inboth cases,
 

a more realistic assumption ismade concerning polarization loss.
 
26 

6.1.2 	 Derivation of Sidelobe Discrimination Budgets
 

The following derivation of the sidelobe-discrimination budgets follows the format of Table 6.1 [1] so that the results may be compared. This
 

includes the threshold settings and noise-immunity margins. InTable 6.2 [2],
 

we present a conservative estimate of the sidelobe-discrimination margin based
 

on a minimum received flux density of -126.0 dBW/M 2 and a maximum of -110.5 dBW/
 

M2. The minimum flux includes a margin of 2.5 dB added by Rockwell to the estimated -123.5 dBW/M2, based on the Orbiter ICD [2] which includes polarization
 

loss. Table 6.3 [2, Table 4-IX] isreproduced herein and the flux-density values are calculated based on a minimum loss of -162.0 dB M-2 and a worst-case
 -
 loss of -613.5 dB M 2. In addition, polarization loss on a sidelobe should be
 

greater than on a mainlobe: typically, polarization loss increases monotonically off boresight. For our conservative estimate, Table 6.2 [2], we assume
 

that the sidelobe and mainlobe polarization losses are equal. The net margin
 

is-0.6 dB which, as we have stated, includes a 2.5-dB margin added by Rockwell
 

to the flux-density specification.
 

Table 6.4 [2] presents a more liberal estimate of the sidelobe discrimination budget without a margin added to the expected flux density. In
 

this budget, we assume a polarization loss at the first sidelobe down I dB
 

from the mainlobe, which we feel isrealistic. The result is a positive margin of 2.9 dB.
 

6.1.3 	 Conclusions
 

Based on realistic estimates of received flux density and polarization loss, sidelobe acquisition should not be a problem. Moreover, the maximum
 

flux density of -110.5 dBW/M2 ispredicated on the TDRS pointing directly at
 

the Orbiter. The TDRS Ku-band antenna full beamwidth is0.28' and isopen-loop
 

pointed at the Orbiter during acquisition. We feel that the most likely event
 

which would cause the maximum flux during acquisition isan attempted reacquisition immediately following loss of track while the TDRS isstill accurately
 

pointed toward the Orbiter. This should not be a problem but, inorder to more
 

accurately quantify the performance, the axial ratio at the most significant
 

sidelobes should be measured.
 
27 

Table 6.2. Sidelobe-Discrimination Budget [2]
 

Pointing Loss, dB 


Flux Density, dBW/m2 


Antenna Gain, dB 


Effective Area (omni), dBm2
 

(at 13.775 GHz) 


Effective Area, dBm2 


Polarization Loss, dB 


Received Power, dBW 


System Temperature, dB0 K 


(at Antenna Terminals)
 

Boltzmann's Constant 


C/N0 dB-Hz (at EA-1 Input) 


Threshold Setting, dB 


Noise Immunity Margin, dB 


Effective C/N0 , dB-Hz 


(at Detector)
 

Weakest 


Mainlobe 


-3.0 


-126.0 


37.1 


-44.2 


-7.1 


0.0 


-136.1 


30.6 


-228.6 


61.9 


-0.3 


-0.6 


61.0 


Strongest 


Sidelobe 


-22.0 


-110.5
 

37.1
 

-44.2
 

-7.1
 

0.0
 

-139.6 


28.9
 

-228.6
 

60.1 


0.3
 

1.2
 

61.6 


Cumulative
 

Difference
 

19.0 dB
 

35 dB
 

1.8 dB
 

-0.6 dB
 
Table 6.3 TDRSS-To-SSO Ku-Band Forward-Link Mode 1 Interface Characteristics [2)
 

Information Time-Division-Multiplexing Spectrum TDRSS Transmission SSO SSO
 

Operational Spreading Carrier Carrier Polari- Min. Required
 

Channel Rate TOM c 216 kbps TOM d Modulation Frequency EIRP zation G/T Prec /NO
 

Voice 1 32 kbps 

Rate: 72 kbps 

Bit Rate: 

216 kbps 

PN Sequence: 

Gold Code 

Acquisition: f 

40.0-49.5 dBW 

64.0 

dB-Hz 

Voice 2 32 kbps ± 0.001% 

Sync Pattern: Chip Rate. PSK 13.775 RCP 6.0 i 

Command 6.4 kbpsa 765714408 Format: (31/141024) a= r/2rad GHz Tracking: § Axial dB/K 

Sync 1.6 kbpsb Biphase-L ±30 

e Normal power: 

46.5-49.5 dBW 

ratio 

<1 dB 

h 

65.4 

Sync Pattern: Format: dB-Hz 

Forward 128 kbps 

Data 

N/A 

267727063548 NRZ-L 

Period: 

High power:

48.5- 51.5 dBW 

g 

Sync 16 kbps N/A 1023 chips 

Legend: a - Encoded and permuted command data
 

b - 1.2 kbps synchronization pattern plus 0.4 kbps arbitrary fill bits
 

c - Same as S-band high data rate uplink and forward link operational TOM; encrypted for transmission
 

d - Unspread capability exists
 

e - Doppler compensated in TDRSS for 17.5 kHz at SSO
 

f - Effective EIRP toward SSO during acquisition with open-loop TDRSS antenna pointing and allowing for SSO
 

of 6 dB; not to exceed upper limit 

g - With Ku-band return link acquired 

h - Based on SSO antenna temperature of 10 K(no Earth or Sun intercept) 

i - SSO link acquisition threshold - For BER of 10-5 , sub3ect to specifications in 4.4.2.1 
29 

Table 6.4. Sidelobe-Discrimination Budget [2]
 

Weakest 

Mainlobe 

Strongest 

Sidelobe 

Cumulative 

Difference 

Pointing Loss, dB 

Flux Density, dBW/m2 

Antenna Gain, dB 

Effective Area (omni), dBm2 

(at 13.775 GHz) 

Effective Area, dBm 2 

-3.0 

-123.5 

37.1 

-44.2 

-7.1 

-22.0 

-110.5 

37.1 

-44.2 

-7.1 

19.0 dB 

Polarization Loss, dB 0.0 -1.0 

Received Power, dBW 

System Temperature, dB°K 

(at Antenna Terminals) 

-133.6 

30.6 

-140.6 

28.9 

7.0 dB 

Boltzmann's Constant -228.6 -228.6 

C/N0 dB-Hz (at EA-l Input) 64.4 59.1 5.3 dB 

Threshold Setting, dB 

Noise Immunity Margin, dB 

-0.3 

-0.6 

0.3 

1.2 

Effective C/N0 , dB-Hz 

(at Detector) 

63.5 60.6 

I 

2. 
30 

6.2 Shuttle/TDRSS Acquisition and Tracking Performance 

6.2.1 Introduction 

The current scenario for Shuttle/TDRSS acquisition consists of having 

the TDRS open-loop point toward the Shuttle and radiate a minimum of 40.0 dBW
 

inthe direction of the Shuttle. TDRS pointing-loss considerations are discussed in [3]. The Shuttle Ku-band high-gain antenna then executes a spiral
 

scan to acquire the forward link. Once the Shuttle Ku-band system detects the
 

forward link and re-radiates on the return link, the TDRS can initiate closedloop tracking. When the TDRS isaccurately closed-loop tracking, the minimum
 

received flux at the Shuttle is increased due to the more accurate TDRS pointing and the Shuttle, inturn, can maintain more accurate pointing toward the
 

TDRS.
 

Hughes Aircraft Company (HAG) had recently expressed concern that
 

the Ku-band system cannot accommodate the wide variation inexpected signal
 

strength (dynamic range) and still maintain accurate pointing. This concern
 

stems from observations that the variation in received flux density at the
 

Orbiter, combined with unit-to-unit variations, thermal effects and detector
 

weak-signal suppression, add up to provide the communication-track servo with
 

an error signal having an estimated 39-dB dynamic range. The incident flux
 

density variation was taken from the Rockwell specification, Rev. B.
 

6.2.2 	 Dynamic-Rate Limitations
 

Hardware limitations affecting Ku-band performance are discussed in
 

Section 3 of this report. Briefly, a combination of manufacturing variability,
 

AGC performance and temperature effects combine with the input-signal flux variability to produce a 39-dB dynamic range at the angle-track servo input. The
 

servo was not designed to accommodate this wide dynamic range and, infact, the
 

capability of the servo isunknown at the present time.
 

HAC system engineering has partitioned the problem between the servo
 

and the detector: Investigate whether or not the servo can handle a 15-dB dynamic range and explore detector design changes and specification-relief options
 

which can reduce the dynamic range from 39 dB down to 15 dB or less. Hughes has
 

compiled a list of 10 candidate options to improve the detector performance,
 

one option issimply 11 dB of incident flux-density specification relief with
 

a slightly narrower track IFbandpass filter. The flux-density relief is needed at the low end, e.g., the minimum signal level for accurate tracking would
 
31 

have to be boosted up to 11 dB. While this seems like a prodigious amount, an
 

examination of the Rockwell specification indicates that it is possible. This
 

isdiscussed inthe next section.
 

6.2.3 Flux-Density Specification Considerations
 

HAC dynamic-range calculations using received flux density are based
 

on the Rockwell Rev. B specification. Apparently, Table 6.5, which appears on
 

page 29 of the specification as Table 3-I, was used to determine minimum flux.
 

This table is reproduced herein and gives "minimum EIRP required for autotrack"
 

as 35 dBW at the TDRS. Using a worst-case space loss of -163.5 dB/M 2 and a polarization loss of 0.5 dB, we arrive at a minimum flux density of -129 dBW/M2.
 

This isthe worst-case flux density used by HAC as a requirement for accurate
 

tracking.
 

Table 6.5. TDRS RF Interface Requirements [1]
 

Parameter 	 Return Link Forward Link
 

Center frequency 	 15.0034 GHz (nominal) 13.775 GHz ± 10 kHz 

Bandwidth (3dB) 225 MHz (transponder 50 MHz (transponder


throughput) throughput)
 

Antenna gain (55% eff.) 52.6 dB 	 52 dB
 

Antenna polarization RHC 	 RHC
 

48.8 	dBW end-of-life for
 

data transfer
 

EIRP N/A 40.0 dBW minimum for detection 	inthe direction
 

of the Orbiter
 

Maximum-allowable EIRP 60 dBW at Orbiter 55 dBW at TDRS
 

Minimum EIRP required for 30 dBW at Orbiter 35 dBW at TDRS
 

autotrack
 
32
 

A flux density of -129.0 dBW/M 2 isoverly pessimistic since the
 

Shuttle ICD [4) guarantees a mininum of 40.0 dBW toward the Orbiter, including
 

polarization loss. This isequivalent to a flux density of -123.5 dB. Moreover, this flux represents open-loop TDRS pointing. When the TDRS is illuminated by the Orbiter return-link signal, the flux density will increase to a
 

minimum of 46.5 dBW for an equivalent flux density of -117.0 dBW/M 2. The maximum TDRS EIRP toward the Orbiter is51.5 dBW. With a minimum space loss of
 

-162 dBW/M 2, the maximum flux density is-110.5 dBW/M2. The EIRP values come
 

from the Shuttle ICD. From Table 6.3, note that, when the TDRS isclosed-loop
 

pointing, the flux density at the Orbiter is12 dB higher than the worst-case
 

flux used by HAC inthe dynamic-range calculation. The maximum flux density of
 

-110.5 dBW/M2 is3.7 dB less than the -106.8 dBW/M2 employed by HAC intheir
 

dynamic-range calculation.
 

6.2.4 	 Summary and Conclusions
 

The revised flux-density dynamic range provided by using the EIRP
 

values from the ICD, which are more current than those inthe Rev. B Rockwell
 

specification, plus about 3-dB relief from a narrower bandpass filter at the
 

detector input, will allow the Ku-band system to provide adequate communication tracking. This conclusion assumes that one servo will indeed accommodate
 

a 15-dB dynamic range. Axiomatix has made this recommendation to Rockwell and
 

NASA. As a consequence, Rockwell will be revising the flux-density specification. The maximum flux density will be specified at -110.5 dBW/M 2, and the
 

two minimum flux-density numbers, including a 2.5-dB pad, will be specified at
 

-119.5 dBW/M2 and -126.0 dBW/M 2 . The former applies when the TDRS isaccurately
 

closed-loop pointing, and accurate SSO pointing will be required at this flux
 

density. The latter will be specified for acquisition only and assumes openloop TDRS pointing.
 
TWO-CHANNEL COSTAS LOOP-TRACKING PERFORMANCE FOR UQPSK SIGNALS
 

WITH ARBITRARY DATA FORMATS
 

Interim Report
 

Contract No. NAS 9-16067
 

Exhibit "B"
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

(Contract Monitor: W. E. Teasdale)
 

Prepared by
 

M. K. Simon
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8010-8
 

October 30, 1980
 
TWO-CHANNEL COSTAS LOOP-TRACKING PERFORMANCE FOR UOPSK SIGNALS
 

WITH ARBITRARY DATA FORMATS
 

By 4-

Marvin K. Simon
 

1.0 INTRODUCTION
 

In a recent pair of papers [1,2], the authors investigated the
 

carrier-tracking performance of a two-channel Costas-type loop for unbalanced quadriphase (UQPSK) signals. Such a loop evolves naturally as a
 

closed-loop system motivated by the maximum a posteriori (MAP) phase estimation approach. Although not explicitly stated, the performance results
 

presented in these papers pertain only to the case where both data channels
 

have NRZ formats. In many applications [3-5], the data formats on the
 

two channels are different; for example, the high data rate channel might
 

be Manchester coded, while the low data rate channel might be NRZ, or vice
 

versa. Alternately, both channels could be Manchester coded.
 

In this report, we generalize the results given in El] so they
 

will apply to any of the mixed-format cases mentioned above. Specifically,
 

we examine the mean-square phase jitter performance of two-channel Costas
 

loops with active arm filters, synchronous or asynchronous symbol clocks,
 

and arbitrary data formats on the two channels. Also considered will be
 

the performance of such loops with hard limiters in the in-phase channels.
 

Here again, we shall follow the approach taken in [7J, with the exception
 

that the data formats on the two channels can be arbitrary.
 

2.0 PERFORMANCE ANALYSIS OF TWO-CHANNEL COSTAS-TYPE LOOP
 

WITH ACTIVE ARM FILTERS WITHOUT HARD LIMITERS
 

2.1 System Model and Loop Equation of Operation
 

To avoid duplication of effort, we shall draw heavily upon the
 

notation, definitions and results given in [1]. Consider the two-channel
 

Costas loop with integrate-and-dump arm filters illustrated in Figure 1.
 

The input x(t) is an unbalanced QPSK (UQPSK) signal-plus-noise, i.e.,
 

x(t) = 2TPdl(t) cosCwot+e ) + 2 d2 (t)sinC&ot+eD + n(t) (1) 
I T (t)Pl (t)dt k TG(.) 

I s:(t) (k-1)T 


I Data Stream 1
 

kT2 I
 

- cc(t)P2 (t)dt
 

x(t) (k-I)T 2
 JskT2 D

s(t)P2(t)dt G)
 

(k-l)T 2 
 00
 

Data Stream 2 
 -

02
 S' kT1 1
 

kTIkT 1U
 

-
 (t )s(t )Pl(t)dt 


k-l)T 1 S mbol
 

2KO Cos (w0t + )Sync i 

G(x) =x, 
 without hard limiters


xsgn x, with hard limiters
 

Channel 1 is NRZ (i=1,2) Channel i is Manchester (i=1,2)
 

P,(t) 0 otherwise P(t) =-1 ; T/2 < t < TI
 

Fiurt e 1wpeLv otherwise
 

Figure I Two-Channel Costas-Type Loop wth Actve Arm Filters
 
ORIGINAL PACE 3
 

OF POOR QUALITY 

where w0 isthe radian carrier frequency, P1 and P2 represent the power 

inthe two quadrature components, d1(t)and d2(t) are two independent 

binary waveforms (arbitrary format) with respective data symbol rates 

Rl = I/Tl and R,= I/T2 , and a isthe carrier phase to be tracked by the 

loop. The additive channel noise n(t) has the usual narrowband process
 

representation, i.e.,
 

n(t) = VW{Nc(t) cosCwot+e) - Ns(t) sinCwot+e} (2) 

where Nc (t)and Ns(t) are approximately statistically independent, stationary, white Gaussian noise processes with single-sided noise spectral
 

density No W/Hz and single-sided bandwidth BH<m/2n.
 

The input signal isdemodulated twice by the quadrature reference signals
 

rs(t) = 2 snCot+e) , rc(t) : v2KcosCwot+eD (3) 

to yield the phase detector (gain m) Outputs 

c(t) - -x (t) r Wt 

V= e. ~ d1 Mt + Nc (t~Cos 99(t) + KIPy d2t) - Njtd sin t 

Es(t) NAm x(t) rs(t)
 

= KoKm _ d( t ) + Nc (t) sin (t)- 2 d2(t)-Ns(t)d Cos <(t)1 (4) 

where * = 0 - & isthe loop phase error which shall be assumed constant
 

(intime) over the duration T2 of a symbol corresponding to the lowest
 

data rate d2(t)*.
 

The signals cc (t)and es (t)are each passed through integrate
 

and dumpswhich are matched to the pulse shapes p t) and P2 (t)respectively
 

of the two quadrature modulations d1 (t)and d2(t) The outputs of like
 

As in [1], we arbitrarily assume that dl(t) is the higher
 

rate channel.
 
4 ORIGINAL PAGE ES 

OF POOR QUALITintegrate and dumps are multiplied, and the difference of these
 

multiplications isused to form the loop error signal. This error signal
 

isthen passed through the loop filter F(s) whose output drives the VCO,
 

generating the pair of demodulation reference signals given in (3). Without belaboring the details, we may follow the identical approach taken in
 

[1] and immediately write the linearized loop (sin 2 = 2p) equation of
 

operation inHeaviside operator notation as*
 

2o = H2(P)IsN ,5) 

where
 

H(S) S'(O) F(s) (6)
 2( - s+S'(O)F(s) 

S'(O) isthe slope of the normalized loop S-curve at the origin ((P = 0)
 

given by
 

S'(O) = KIT12 -P2 d212(t + K2T22L2 P1 d (7) 

and N(t) = Nl(t) - N2(t) is the equivalent zero mean additive noise with 

3 

N() I N(t)t); i = 1,2
 

=I
 

-N11 Ct) = K[Nc2(t) - Nsi 2(tdJ sin 29+ 2K, Nsi(t) Nci(t) cos 29 

Ni2(t)= 2KiTi{ P di(t) ?11(t)+4P2d 21(t) NSI(tJ sin 29 

+ JP-7d,,(t) Nsi(t) -4Py d2 1(t) Nci(td1cos 2Q 

Ni3(t) = KiTi2 P3-i d3 .l2(t)-d 3-1,1i2(t) sin2+2JPT-P dll(d(td 2 t)cos29O (8) 

The linearization of the equation of operation as characterized
 

by [1,'(28)-(31)] isincorrectly scaled.
 
2.2 

ORIGINAL PAGE ES 

OF POOR QUALITY 

In (7) and (8), the overbar denotes statistical expectation and
 

Al kT, d(s) p1(s)ds ; (k-1)Ti < t < kT.
 s
 d~it) T1 j Pi 

(k-l)T i i=1,2
 

Ncl(t) 6 f kTi Nc(S) P1 (s)ds ; (k-l)Ti < t < kTi
 

(k-l)T 1 i=1,2
 

Nsi(t) A f kTi Ns(s) p1 (s)ds ; (k-l)T I < t < kTi (9) 

(k-l)T i i=1,2
 

Note that (9)differs from the corresponding expressions in [1] in that
 

the channel pulse shapes p1 (t)and P2 (t)are included as weighting factors in the integrands.
 

Assuming that the bandwidth of H2qp(s) (loop bandwidth) ismuch
 

narrower than the bandwidth of N(t), the variance of 2q' can be approximated
 

by
 

2 NO IBL
 

a2 - [S,(O)] (10)
 

where BL is the single-sided loop bandwidth and No ' is the single-sided
 

spectral density of the equivalent noise N(t).
 

The next two sections are devoted to an evaluation of S'(0) and
 

N0 ' for the various data combination formats on the two channels
 

Evaluation of the Loop Signal Component S'(0)
 

From (7), we observe that evaluation of S'(0) reduces to the
 

evaluation of the two mean-square crosstalk quantities d212M and dm2T)
 

for the four combinations of NRZ- and Manchester-coded data on the two
 

channels. Such evaluations have already been performed in [6] under the
 

guise of a different notation. In particular, [6, Table 1] evaluates what
 

is herein referred to as d21 

2(t)for the case of synchronous symbol clocks,
 

and [6, Table 2] gives the corresponding results for asynchronous symbol
 

clocks. In the former case, the ratio of data rates RI/R
 2 is constrained
 
ORIGINAL PAGE 0
 

OF POOR QUALITY
 

to be an integer. Translating these results into the notation of this
 

report, we obtain the following tables.
 

Table 1.* Evaluation of d212(t); Synchronous Symbols Clocks,
 

R1/R2 = n (n an integer)
 

dI(t)
 

Manchester NRZ
 

0 ;n even ;n even 

Manchester 1 n- n 

n o n odd 

d2(t) 

NRZ 0 
 1
 

Table 2. Evaluation of d21 2(t); Asynchronous Symbol Clocks;
 

RI/R 2 = n (n need not be an integer)
 

dl(t)
 

Manchester NRZ
 

1[22 +12_6; 21 

n -F 6 3n 

Manchester R2 < R1 < 2R2 R2 < R1 < R2
 

I R1 > 2R2 1 1 R. 2 

2n ' 1 n 1' 2R2 

d2(t) 

NRZ 1 3n 

Also, from the definitions of d12(t) and d21(t)in (9), it is
 

straightforward to show that
 

Note that [6, Table 1] contains an error for the Manchester-


Manchester case.
 
ORIGINAL PAGI'I OF pOOR QUALIV 	 7 

(11)
 MI 


for all entries inboth Tables I and 2. Thus, further introducing the
 

notation [1]
 

P +P ; P2 A PTI (12)
 

12"t" 


p1 2 11 No 

the loop signal component of (7)simplifies to* 

S'(0) KTT 2 P(1+x,{ -x11~ 1 	 2 n(nx - d21 (13) 2 


where d21

2 isobtained from the appropriate entry of 	Table 1 or 2.
 

'
 2.3 	 Evaluation of the Equivalent Noise Spectral Density N0
 

' of the equivalent
 The single-sided noise spectral density No
 

noise N(t) whose components are defined in (8)isobtained from
 

(14)
 No' 2 f RN(T) dt 


where 

RN(T) = < N(t) N(t+t) > = < RN(t,t+r) > (15) 

and < > denotes the time average. Using (8)and taking the statistical 

average required in (15) gives [] 

*Herein we eliminate the notational dependence of d212(t) on t. 
ORIGINL- AG 

OF pOOR QUALITY 8 

2 

X[lkd ( P2 (t i(t t ) 

RM~tl~t - 41(1K'2 e+k 

2 T{~'(e+t~~) + T2 l 

.e -I( l c2 -\t -2 

xIRdl id 12 (tk'tk)+ P2Rd21 d22 (tk' tt RNcl Nc2((tktk) 

+ T 2T22 PIP 2 Rd ldl2(tft 2 ) Rd21d22 (tt£) _ I (16) 

where we have introduced the simplifying notation
 

2 (17)
 

Also in (16), 

2 A NoT . ak 2 £ 1=,2 (18) 

and
 tI ift5 and t
2 are both inthe interval


I2Ctl't2 = (kT
2, (k+l) T)

0 otherwise
 

k = 1,2 (19) 

Clearly then, we must evaluate, for each of the mixed data format combinations, the various nonstationary correlation functions required in (16)
 

before performing the time-averaging operation which finally results in
 

RN(r). 

We begin by presenting some general results which relate these
 

correlation functions to one another, namely, it can be shown that analogous to (11),
 
ORiUGIAL PAGE 15 9 

OF pOOR QOALfl 

Rd d12Ctl't 2) =i-Rd21d22Ctl't2 (20) 

where again n is the ratio of data rates. Furthermore,
 

RNcl Nc2 Ctl t2) ali2 Rd21d22Ctl t2J (21)
 

The above two relations are valid independent of the data formats on the
 

two channels and whether or not these data streams are time synchronized.
 

Using (20) and (21), we can determine the products of correlation functions
 

needed to evaluate (16). Thus, we obtain
 

R2 1 t NciNc2 Ctl't23 = 

4 R222 

al d21d22 Ctlt2 

a12 

Rd d Ct't2) RN N Ctlt29 = a' Rd2 Ctl't2) 

Rd2 21 d22 22Ctl't2 RNclNc2CtI't2' ci c2 = ain
2 dd2121dd22
 22 Ctl~
 t2)
 

= d21 d22 Ctl't2)l 


2
 

from which we observe that it is sufficient to evaluate only Rd21d22Ctlt2)
 

Rd1ld12 Ctl't2] d21 d22 Ctl't2 l-'E (22)
 

for the various data format combinations on the two channels. These specific relationships will now be given in summary form.
 

Channel 1 is NRZ, Channel 2 is Manchester; Synchronized Symbol Clocks,
 

n an integer
 

< < nT
 i ; 0 t1 T 0 < t2 1
 

Rd21d22 Ctl t2)= I;T< I< < tZ < 2nT 1 (23)
 

; otherwise
 
10 

ORIGiNAL PAGi 1 

OF POOR QUALITY 

Channel 1 isManchester, Channel 2 is NRZ; Synchronized Symbol Clocks; 

n an integer 

R21d22Ctlt 2J = 0 all t1 ,t 2 (24) 

Channel 1 is NRZ, Channel 2 is Manchester; Synchronized Symbol Clocks; 

n an integer
 

n even 

1 ; 0 <t I <T , 0 < t 2 < nT1 

R22id22CtI t2) R1;T 1; T1 <t C(R+l)T l , nT <t 2 < 2nT l 

21 d22 1 ; +) T < t <nT , nT 1 < t2 <2nTl 

0 ; otherwise (25)
 

n odd 

1 ; O<t 1<T l 10O<t2 <nTl 

1 ; T < t1 . (ti-)T l , nTl t2 2nTl 

d2 1 d22 Ctl2t2J = 0;(f )T, < 1(2)T l , nT1 < t 2 < 2nT1 

1; (n3)T 1 < t1 4 nTl , nT 1< t 2 < 2nT l 

0 , otherwise (26)
 

Channel 1 isManchester, Channel 2 is Manchester, Synchronized Symbol 

Clocks; n an integer 

n even
 

2Rd2Ctl,t2 = 0 all t1 ,t 2 (27) 
ORIGINAL pQ S1 

OF POOR QUALIT'V1
 

n odd
 

0 ; 0 C t I < T1 , 0 < t 2 C nTI 

0 ; T1 C< t I < ( T1 , nT1 < t 2 < 2nT 1 

d21 d2 Ctl't2) ; C t < , nT < t 2 2nT 

o ; T< t < nT1 , nT1 < t2 42nT1 , 

0 ; otherwise (28) 

Letting t1=t, t2 =t+T and averaging over t yields the following 

results 	for the stationary correlation function
 

nT-)nTl R2 (t,t+T) dt (29)
 

Channel 1 isNRZ, Channel 2 isNRZ; Synchronized Symbol Clocks,
 

n an integer
 

1~ rtC 	(n-1l)T 1
 nT1 ET1+tD-T1 

A(T) = 2n-I)T I -T; (n-)T 1 < T < (2n-l)T 1 

0 ; otherwise 	 (30)
 

Channel 1 isManchester, Channel 2 is NRZ; Synchronized Symbol Clocks, 

n an integer 

A(T) = 0 ; all T (31)

Channel 1 is NRZ, Channel 2 is Manchester; Synchronized Symbol Clocks, 

n an integer 

n even 

1 CTl+'D ; -T1 4 'r< (n-l)T 1I 

ny 2n-l)T-]; (n-l)T 1 < T < (2n-l)T 1 

0 , otherwise (32) 
12 ORIGINAL PAGE ES 

OF POOR QUALITY 

n odd 

n (T ); -Ti < 

E- In-_ c_ (Th) InnT-- T 4 r < (n-I)T I
 

A(t) R1 F2n 2)T, - ; (n-l)Tl , (_l)T, 

n-l I\3n-~)T C1 T 

1[2nl- 1 ]; T < < n-l)T 

jti L' 1 1 I2) 

0 otherwise (33)
 

1 is Manchester, Channel 2 is Manchester; Synchronized Symbol
 Channel 


Clocks; n an integer
 

n even
 

A) 0 ; all T (34)
 

n odd
 

0 ; -T1 < -< ! )Tl 

1 [(3-n)T+T]; (f2)TI <,T < (n-)T1
 

A(T) 1 , -")T- T (tT 


___ -j. (3n-3y (3n-lf 

1 

0 otherwise 
 (35)


Making the same substitutions in (16) and averaging over t
 

yields
 
ORIGINAL PAGE IS 13 

OF POOR QUALITY 

RN(T) TiJ RN(tt+T)dt 

0 2 

=41 KI2~4 aE Tf (PP ydZ)+ TL4PP2 d22.1( T 

- 4KlK2E4+c2 2 ( P)+ 12T2 2 (i A-)A-)(6 

where we have also made use of the relations in (22). Finally, performing the doubly infinite integration on T gives the desired results for
 

No ' of (14), namely
 

2 O'=83. KL 2 Tg2 ddz_­ 2TfLGR+a 2)+P 

0=k 8 V4 2 2'2\p Pd 

16IK 2 2(1+P)+T2T p+ +TTT22PP 2 
f]ICAD (37)

where
 

ICA,D 4 fA(t)d-T (38) 

All that remains to put (37) into its final form is to evaluate ICA()D
 

for the various mixed data format combinations.
 

Substituting (30)- (35) into (38) and carrying out the integrations enables us to put the results in a surprisingly simple and compact
 

form, namely,
 

ICA(T)D = nT1 d21 (39) 

where d21 

2 is obtained from Table 1.
 
2.4 

ORIGINAL PAGE IS
 

OF POOR QUALITY
 14 

Finally, then, substituting (39) into (37) and making use of
 

(l),we obtain, after some simplification,
 

N0' =2NoTI 3 IKl2Eo+2TI I+P 2 7)] 

2 + K2 01n+2nT d2 2) 

22n%2nT +n 

-K K nd 2 i2[No+2T (Pi +nP)]} 

5 2 n2 T1 1 Pl 2 R[Kl2 n d2 1 - 2 K, K2 d2 li (40) 

Although the result in (40) has been derived for the case of
 

synchronized symbol clocks, with some effort, itcan also be shown that
 

italso applies for the unsynchronized symbol clock case with the caveat
 

that d21 2 is now found from Table 2.
 

Squaring Loss Evaluation
 

The "squaring loss" of the two-channel Costas loop of Figure I 

isdefined by 

,LA 4 NBL 42TP (41) L a2(p 

where a29 isfound from (10). As is traditional in suppressed-carrier 

types of tracking loops, this loss represents the degradation relative to 

the mean-square phase jitter performance of a linear loop caused by SxS, 

SxN and NxN distortions. Using (10) in (41) allows _L to be rewritten 

inthe form 

S_ (0 _ (42) 
15 

ORIGINAL PAGE 'S 

OF POOR QUALITY 

Substituting (13) and (40) into (42) and simplifying using the definitions in
 

(12) gives the desired result, namely,
 

[Kl (1. -XdI2)+ 2~~X-_1Ix212
 

2 ;1(K,-J2 )2 -K2n3,n,) () L 3+4.±kA(K I + KI 2+2n3-2KI 2+K223 2K K2nd21-)+xd212 (I +g~ - KI-2K2 -K2n (ln))- -212 

or using the optimum* ratio of the gain constants
 

K2 P2T1 - X (44) 

K 1 PIT 2 n( 

we have
 

2
 

= 
 -
 F1- 2 x 212+nx (1+X) (45) 

S+1 
 2d 21 + nX2) +xd2 1 n+ (1-n) nX(l-nx)l-d2
 

Comparing the above result with [1, (38)], we observe that the only modification necessary to make the previous result applicable to the more general data
 

format situation isto calculate the single parameter d212 from either Table 1
 

or Table 2 and use itwhere appropriate in (45). Also observe that by letting
 

K2=0 in (43), we get the result for a single-channel Costas loop, namely,
 

= 
 , (I-Ad 2 2 )-I (46)
 L-A
 2T1 l 2c, d2122
 

which agrees with [6, C32)] ifthe appropriate changes innotation are made.
 

The word "optimum" isused here in the same sense as that used in
 

[], namely, the ratio of gains motivated by the MAP estimation approach.
 

Note that this gain ratio does not necessarily minimize squaring loss or
 

mean-square phase jitter, as demonstrated in the next section.
 
16 

2.5 

ORiGNAL PAGE 18 

OF POOR QUALITY 

Cptimization of Channel Gain Ratio to Minimize
 

Mean-Square Phase Jitter
 

Inthe previous sections, we developed a theory for calculating
 

the mean-square phase-tracking jitter based on a linear loop model as a
 

function of such system parameters as signal-to-noise ratio ineach channel, the ratio of powers inthe two channels, and the ratio of data rates
 

inthe two channels. Although the general results were derived as a function of the ratio of the gains inthe two channels, specific results were
 

presented only for the "optimum" choice for this ratio based upon MAP
 

estimation theory.
 

Unfortunately, as we shall see shortly, selection of this gain
 

ratio as motivated by MAP estimation theory does not necessarily optimize
 

the tracking performance. Infact, selection of this gain ratio to directly
 

optimize the tracking performance inthe sense of minimum mean-squared
 

phase-tracking jitter can yield considerable improvement inthis performance relative to the MAP estimation choice of gain ratio.
 

To illustrate this point, we shall begin by rewriting (41) in
 

the form
 

A 4 oBL _e 1 K(A (47) 

L2 K K 

a2 P o (l+A)Jy+B 2)+B2(+2_)] 

where
 

2
 

A = 1
- xd21
 0 


Al = n(nxi2) 

B0 = 1 + 12+ d 1 2 (1+ 2-) 

S+d 1 2 - (48) 
ORIGINAL PAGE IS 17

OF POOR QUALITY 

Clearly, from (47), minimizing the phase-tracking jitter a2(p2 is directly
 

equivalent to maximizing SL (minimizing the squaring loss). From the form of
 

(47), it is a simple matter to differentiate this equation with respect to
 

K2/K, and thus find that value of channel gain ratio which maximizes " L"
 

After some routine algebra, the solution to the equation
 

" L (49)

B(K2/KI) 

is
 

-
 (/K)lto 2A1B0 -
A0B1 (50)
 

2Ao 2 
1 1 

1+d2Vt 


_1 S(I1 \ 	 21 

K' T d2	 nld ;-+n j\-(2]2-1 _ /--E)(~21-)d21+nxkl l- 2l1 2 d2ln 2l 

(51) 

Note that the first factor in (51) 	 is the 'optimum" value oflK/K as motivated by
 

that given by (44), which is independent of
 the MAP estimation theory, namely, 

Only when have equal energy, i.e., xl/n, are signal-ta-noise the results of (44) ratio. 	 and (51) equal, the i.e., channels K2/K[ 2.

L op: _(51) an expression for the minimum squaring
 Substituting (51) into (47) results in 


loss, namely,
 

2 2A + A ) (AO2B2 + 	A2Bo- A(AI 2 B1) (52)


where A0 , Ae, Bo, B1 and B2 are found from (48).
 
18 

ORIGINAL PAQE IS
 

OF POOR QUALITY 


As an example, when both channels have NRZ data and synchronous
 

symbol clocks, from Table 1,we then have that d21 and (52) simplifies
 

to
 

aL + + i+ ()A 

On the other hand, letting d212 = 1 in (45) gives the result obtained in
 

[1], namely,
 

(I - 2x+ n:2) 2 (54)
 (l+ )I+2 1)- 2 + nA2)+ X(l + 2' i 

Figure 2 illustrates 2 as computed from (53) and (54) versus 

the power ratio A with n= 10 and l = -3 dB and 10 dB. Itis clear from 

this figure that considerable reduction of squaring loss ispossible by 

choosing the channel gain ratio so as to minimize this quantity. 

Another illustration of the squaring loss improvement obtained
 

by optimizing the channel gain ratio is given in Figure 3 Here we consider a mixed data format where the high data rate channel (channel 1)is
 

Manchester coded, while the low data rate channel (channel 2) is again
 

NRZ. Furthermore, we now assume asynchronous symbol clocks, which implies
 

-that the value of d212 to be used in (48) and, hence (52), isobtained 

from Table 2,namely, d122= 1/6n= 1/60 (for n= 10). Comparing Figure 3 

with Figure 2,we observe that the small value of cross-modulation loss 

(d2') for the mixed format case (Figure 3), as compared with a unity 

value for the case where both channels have NRZ data (Figure 2), results 

in a considerable reduction insquaring loss for both the optimum and MAP
 

estimate selections of the channel gain ratio.
 
ORIGINAL PAGE IM
 

OF POOR QUALITY
 

og 101 1 (1B 

C 10 dB 

-3 dBl 

S of (53) 

St 

I 

10 dB" 

SL of (54) 

10 " -21f 

1/n 

0 25 0.5 0.75 

S=2/P1 

Figure 2. Squaring Loss versus Power Ratio for Unbalanced QPSK, 

Two-Channel Costas Loop; n = T9/T= 10 

1.0 
100 

20
 

8 

9 

7 

ORIGINAL PAGE IS 

OF POOR QUALITY 

OI of (52)1 

L Ci 

10 dB 

-3 dB 

6 

5 

I 

3 

2 

I 

Figure 3. Squaring Loss versus Power Ratio 

for Unbalanced QPSK, Two-Channel 

Costas Loop with Hard Limiters, 

n = T2/T1 = 10, Channel 1 is 

Manchester; Channel 2 is NRZ, 

Asynchronous Symbol Clocks 

I1 

10-1 

n 

n 

=~~~ 10~ Of (45) 

=-3 dB 

0 2 0.4 0.6 0 8 1 0
 
ORIGINAL PAGE N 21 

OF POOR QUALITY 

3.0 	 PERFORMANCE ANALYSIS OF TWO-CHANNEL COSTAS-TYPE LOOP
 

WITH ACTIVE ARM FILTERS AND HARD LIMITERS
 

3.1 	 System Model and Loop Equation of Operation
 

Here again, we draw heavily upon the notations, definitions and
 

results given in [l]. The loop isstill represented by the block diagram
 

of Figure 1 except that now G(x)= sgnx-. Thus, the dynamic phase error
 

z(t) now becomes
 

z(t) = Jiz 5sl(t) sgn DyZc ct1 - zc2Ct) sgn L'RTZ 2CtJ 

= JvKZ(t) 1	 sgn [zcpt] ­ v1K Zc2(t)sgn 1zs2 t)] (55)

where
 

Zs1(t) = f skT (t) p1(t)dt ; kT
i < t < (k+l)T i


(k-l)T i i=1,2
 

zc1(t) = fc(t) pi(t)dt ; kT1 < t <(k+l)T I
 

(k-l)Ti i=1,2 (56)
 

are the integrate-and-dump outputs. Splitting z(t) into its random and
 

nonrandom parts, i.e.,
 

z(t) = E z(t) I1 + N(t,y) = S(y) + N(t,y) 	 (57) 

where S(y) isthe loop S-curve and N(t,y) isthe equivalent zero-mean
 

additive noise process, then the loop equation can be written inHeaviside operator notation as
 

= - Kv F(p) z(t) - Kv F(p) L9) + N(t,)] (58) 

where Kv isthe VC0 gain in rad/s/v. Finally, linearizing the loop by
 

letting
 
ORIGINAL PACE 0 

22 OF POOR QUALITY 

S(q) = S'(0)( ; N(t,q) = N(t,O) (59) 

we get an equivalent form analogous to (5), namely,
 

(/2 = H90(p)[r->%tO] (60) 

where
 

S'(0) K F(s) 

Hq(s) - s + S'(O)K v F(s) (61) 

Assuming, as before, that the bandwidth of H1(s) ismuch narrower than the
 

bandwidth of N(t,O), the variance of v isapproximated by
 

2 No1 8L2(62) 

0 [S,(0] 2 2 

where N0'isthe single-sided spectral density of N(t,O) and BL, the
 

single-sided loop bandwidth, is simply
 

BL IH9(Jw)12 dw (63)
 

0
 

3.2 Evaluation of the Loop Signal Component S'(0)
 

Comparing (55) with (57), we immediately observe that the loop
 

S-curve isdefined by
 

S(T) = Ji S1 () -J2 S2() (64) 

where
 

Sl((p) E{Zsl(t) sgn licl(tdk} 

OA2E{%c2(t) sgn lzs 2(t] J49 (65) 
ORIGINAL PAGE 2323 OF POOR QUALITY 

Substituting (4)into (56) and carrying out the integration gives
 

Cassuming KoKm= 1)
 

Zsi t) = [Ti V Pl d,I(t) +N1cit]sin (p - [T i[Py2d 2 i(t)+Ns,(t fCos 

zc 1 (t = [Ti N17 dl I(t)+Ni(t)cos9 + [Ti P2 d21 (t)+Ns,(t sin9; 

i=1 ,2 (66)
 

Performing the statistical expectations first with respect to the noise
 

components inzsi(t) and zc1(t), keeping d11 (t)and d2i(t) fixed, gives
 

the conditional S-curve components
 

sj(ojdIl,d2 l) = ENJNs {zs 1 sgn (zil)q5~dii~d2i} 

= 2fj 1 ldl1 sin o - T,J-Pd 2 cos J
 

x errc T1 P7 dl1 cos-l-,T RP2d sin g/% ul
 

- T1P- dl, sin 1- T,NP2 d21 cos
 

x erfc{[T' ft7 d11 cos q'+ T1 JP2 d21 sin g/w2 a,}
 

(67)
 

where erfc x isthe complementary error function defined by
 

erfc x = 2 exp-y )dy (68)

AT]xx 

Since
 

erfc(-x)- erfc(x) = 1+erf x - (I-erf x)= 2 erf x (69)
 

then
 
ORIGINAL PAGE N 24

OF pOOR QUALITV 

S,(pjd,,,d 2 TV9,d 1 2 co1er[ T dil.J~cos O+JPP.21 singj1 

(70)
 

isgiven by (18). Similarly,
 where a1 

EN 2 {Zc2 n (zs 2 )I 22} T1 dd2 os 

= T2 [P dl 2cos9 + [P-2d 22 sin(o]erf[ d 1 in9­

(7)
 Since evaluation of 


of from (62) requires only the slope of the loop
 

S-curve at the origin, we next proceed to differentiate (70) and (71) with
 

respect to 9 and evaluate these conditional results at = 0. Thus, 

" ) ' (odl1,d2 l
 

9o=0
 

AT d21 V2 ,) exp[ T1IPdldlerf(_T PTdll 2 22__ T1 d 1
 

p)( .4 j exp L- 112 TT dl erf(FTNdx 11)- -L d212(T1 

(72)
 

Similarly,
 

S2'(0Id 12,d22) = - T2 J 'd22 erf X 12) + 2 ( neL - ld 2

(73)
 
25 ORIGINAL PAGE WO 

OF POOR QUALITY 

Since, from their definition in (9), di, and d22 take on only 

±1 values with equal probability, then (72) and (73) simplify to 

Si' (0 i1 = {erf(FTIAY XAd2 d,) Ti 477.P 21
 rl+' exp(- Tl+)$

s2'(ojd12) = -nTjg {perf ( ) d 1d- i (74) 

Finally, then, averaging over d21 and d12 and recalling (11) gives the
 

desired result, namely,
 

s'(O) =.VKf Edl {S''(o0i 2l)} VRf-2E 6{2S2'(Oldi4( 

Iii~./~~erf( [ 2)1+)1 - ;X exI exp-l+7 

I~c 
 T ncll \-1

For the ratio of channel gains as specified by (44), (75) further simplifies
 

to
 

S'(0) 4r -Ti4 erf( )+ xfn-erf~fW 

2 12) l ( l 

- ) 1 x, fc / (76) 
ORIG.INIAL PAe' 26 fa 


OF POOR QUALITY2
 

3.3 	 Evaluation of the Equivalent Noise Spectral Density NO0 

To evaluate No', we must first find the correlation function of N(t,O), 

namely, 

RN(T) = < N(t,O) N(t+r,O) > (77) 

then integrate it between -- and -, as in (14). Since the value of the loop 

S-curve S(9) at qp= 0 iszero, from (57), we then equivalently have 

RN(z) = < z(t K0) z(t+T Pj=0) > (78) 

Letting 99 = 0 in (66) and substituting in (55) gives 

F LlvI-2d 2 c(t) + Ns, sgn 1 4Pld 

+\/R2 T2 4Pyd'12(t) + Nc2Ct)] Sg [T2 VPyd'22 (t) + N5 5t j (79) 

z(tjq'=0) 	= [Tw tiJ [T, 11C(t) + Nclj(tjj 

Substituting (79) into (78) and carrying out the statistical averages over the
 

various noise components and data process results in,after much simplification,
 

z(t 1 =O) z(t 2 P=O) = K, IP 2 d212 + a1 2]A (t]t 2) + K2 L 22Pl d122 + c22]A2 (tl t 2) 

(It)erf T2 2)-+ 2 l-,exp -T 22'P2)B/t,t2) 

2, nexp --- 2')B t It 

_[TPV 2lRd2 d22 \tlt 2)erfk 0 )+ 2a T2 P2iv/ t2} 

F! 	 )22 a I 2 ] 

-4IF -2Rd d22(t t 2) erf(T I+ . iexp( 22P 2) B(t 2 lt 1 ) 

202 - T21 P1 

+ exp ~ 21)2 t2tj 

(80) 
ORIGINAL PAGE ES -27

OF POOR QUALITY 

where 

f ;and t2 are both inthe interval 

Ai(tl t 2) : (kTi,(k+1)Ti) for some k 

0 ; otherwise 

=l ,2 

B(tl,t 2 ) Rdid 22 (t1 't2) (81) 

Using (ll) and (20), we are able to simplify (80) whereupon, averaging over t
 

with t2 =tI+ T , we get
 

RN(t) K,lP2 d1 1 j + iL + K I 2 Pl) d21 + 0211 __n_) 20 T2 T 

202 2 c2/ 22/ 

1 e NJ f/ P2 T2 P2 + 2cy, exp 2_2 )

VT-~u \ 2a2 T2 1r 1] 

Tl) + exp oil C + A(-z) (82) 

L2-vaI! r2 2a/ 

where A(c) isdefined in (29) and evaluated in (30)-(35). Finally, integrating on
 

T as required in (14) gives the desired result, namely,
 

N2)= 2 2N2 Ix (d 2 )+ I
 

2 -K nnx ~d21 Trk e +r -e n xpx erf + -L exp (83) 
28 

3.4 

ORIGINAL PAGE 

OF POOR QUALIT"Y 

or for K2/K1 as in (44),
 

N0- 02N1 K 17 + ) I 

1r exptf bl 

2 (W21~ rfj'X++I 7Zx 

4-f/ 2C -+ 1)L expr (84) 

Squaring Loss Evaluation
 

Analogous to (41), the squaring loss of the two-channel Costas
 

loop with hard limiters inits in-phase arms isgiven by
 

NOBL 
 (85)


2 2p
 

where o 2 s found from (62). Making this substitution gives
 

k: [S'(O)]2 
 (86)


'9 S1()P(No'/No0)(6 

Finally, substituting (75) and (83) into (86) gives the desired result,
 

namely,
 

A(cxnd 2l ) (87)
 

(]+x)B (I,,~22
 
where 

ORIGINAL PACE M 

OF POOR QUALITY 

29 

A(c 1 ,-,nd 212 4 (Ln ertf i Ir (d 2)+-- exp( 

KA2r 1 2/Vn 1+., 

+V! -nwrX erf tW I LTIX '')U 	 exp( 4 

10r
 

n	 l1 

- 4~F2n(d 2)[i r(IZ) ex( 1I+ 

x FL erf + -L exp (88) 

lFx 

TNT, + 

For K2/K1 = A/n as in (44), (88) simplifies in the same manner that (75)
 

becomes (76) and (83) becomes (84).
 

3.5 	 Optimization of Channel Gain Ratio to Minimize
 

Mean-Square Phase Jitter
 

As for the double-channel Costas loop without hard limiters, here
 

we can choose the channel gain ratio to minimize mean-square phase jitter.
 

.From (87) and (88), we observe that the squaring loss can be written in
 

the form
 

kOICl 	 (89) 

(1+2 Do+Dl vk +D2(~7~
ORIGINAL PAGE i OF POOR QUALITY 30 

where 

erf( n) e ((d n2) e2p -) 

D 2xc1I-I-\ 
 = $

DO ++1 T 

F___ nxclg 

x [/ erf( + -ex ep I 

Do = I + 2 + (90) 

From (40) and (50), it is obvious that the gain ratio that minimzes the 

phase-tracking 3itter is 

(tKK 2CI1 o - CoD l gi 

2CD 2 - C1 Dl 

with the corresponding mimmum squaring loss 

x(C 02 2 + J- (92) 

(l+x)(DoD2 2-2 

For the case of NRZ data on both channels with synchronized sym­ bol clocks, Figure 4 illustrates thas computed from (92) versus the power 

ratio x with n=lO acorre ndB 
 d 1 dB. For comparison purposes,loL

as computed from (88) with K =/n issuperimposed on thes e curves.
 
31 

100
 

9 Figure 4. Squaring Loss versus Power Ratio for 

Unbalanced QPSK; Two-Channel Costas Loop 

0get4 , \ with Hard Limiters; n = T2/T 1 = 10; 

8 P GE i Both Channels are NRZ Data with 

OF poOR QUALII'( Synchronous Symbol Clocks 

7_. =10 dB S
d of (92)
 

6-= -3 dB 

5 

4 

3 

C= -3 dB 

4~ of (88) with 

i K1
 

2 = 10 dB 21o
 

10-1 0!2 0 4 0.6 018 110
 
--

ORIGINAL PAC IS 32


OF POOR QUALITY 

Clearly, optimization of the channel gain ratio inaccordance with (91)
 

offers a significant reduction insquaring loss relative to the values
 

obtained by choosing this ratio in accordance with the MAP estimation
 

relationship of (44).
 

To further illustrate the effect of power unbalance on squaring
 

loss performance, we consider the case of equal bit energy inthe two channels, namely,
 

PITI = P2T2 A Eb (93)
 

or, equivalently, from (12) and the definition of n,
 

nA = 1 (94)
 

Substituting first (94) into (87), then letting K2/Kl =X/n= A2 gives
 

L d2 

= z (95)
 

(l+x)(d2) + 1 -2A( 21 T r r+ Lep- 2(+ 

where
 

4 l -

PIT40 1 P2oT2 F

Eb O- (9 6 )
 

Equation (95) represents the generalization of [1, (70)] for the case of
 

arbitrary data formats on the two channels and either synchronous or asynchronous symbol clocks. Figure 5 is a plot of 'L as determined from (95)
 

versus x with Eb/NO as a parameter and NRZ data on both channels.
 

The corresponding curves for the optimum channel gain ratio are
 

obtained by substituting (94) into (93) and also are illustrated in Figure 5. Unlike the two-channel Costas loop without hard limiters, the two
 

results here are not, ingeneral, equal when the channel energies are made
 

equal. Infact, for large unbalance inpower and large values of Eb/No ,
 
101 

33
 

5dB
 

ORIGINAL PAGlE I 

OF pOOR QUALITY 

0 dB 

/ 

-/ 

K2\
 

of (92) 

-Lof (88) with rF E 

i= _5 dB 

2 I 

00 2 0.4 0 6 0 8 1 0 

A =" 

Figure 5. Squaring Loss versus Power Ratio for Unbalanced QPSK; Two-Channel
 

I/ (Both Channels have Equal
 

Costas Loop with Hard Limiters, n = 

Bit Energy), Both Channels are NRZ Data with Synchronous Symbol
 

Clocks
 
34
 

much benefit isderived by optimizing the channel gain ratio. Only when
 

both the power and data rate ratios are equal to one (i.e., balanced QPSK)
 

do the two results agree exactly as one would expect.
 

Figure 6 is another example of the effect of channel power
 

unbalance on squaring loss performance when the bit energies in both channels are equal. Here we assume that channel 1 has a Manchester data format
 

while channel 2 is still NRZ data. The symbol clocks, however, are now
 

assumed to be asynchronous. Thus, the appropriate cross-modulation loss
 

factor isdetermined from Table 2 as d21z = (6n) -l = X/6. For large power
 

unbalance (small X), the cross-modulation loss issmall and, thus, we observe
 

an improvement insquaring loss performance relative to the previous case
 

where d-. = 1 for all x.
 21
 
35
 

10 dB 

9 5 dB 

8' 8" ORIGINAL POOR QUALITY /,,IOF 
 PAGE iS

7 

6]/ 
 0 dB 

I­ 5
 

of (92) K2 =
 

." of (88) with =
 

3Eb 


-5 dB
 

N0
 /
 

/
 

/ I 1I I I 

20 0 2 0 4 0 6 0.8 1.0 

, 2/PI 

Figure 6. Squaring Loss versus Power Ratio for Unbalanced QPSK;
 

Two-Channel Costas Loop with Hard Limiters, n = I/X
 

(Both Channels have Equal Bit Energy), Channel 1 is
 

Manchester, Channel 2 is NRZ, Asynchronous Symbol Clocks
 
36
 

REFERENCES
 

I. 	W. R. Braun and W. C. Lindsey, "Carrier Synchronization Techniques


for Unbalanced QPSK Signals--Part I,"IEEE Transactions on Communications, Vol. COM-26, No. 9,September 1978, pp. 1325-1333.
 

2. 	W. R. Braun and W. C. Lindsey, "Carrier Synchronization Techniques for
 

Unbalanced QPSK Signals--Part II," IEEE Transactions on Communications,
 

Vol. COM-26, No. 9, September 1978, pp. 1334-1341.
 

3. 	C. L. Weber, "Candidate Receivers for Unbalanced QPSK," ITC'76 Proceedings, Vol. XII, pp. 455-464.
 

4. 	H. C. Osborne, "Effect of Noisy Phase Reference on Coherent Detection
 

of UQPSK Signals," NTC'78 Conference Record, Birmingham, Alabama,
 

December 1978, pp. 2.2.1-2.2.6.
 

5. 	D. Divsalar and J. H. Yuen, "Performance of Unbalanced QPSK in the
 

Presence of Noisy Reference and Crosstalk," NTC'79 Conference Record,
 

Washington, D.C., November 1979, pp. 44.4.1-44.4.7.
 

6. 	M. K. Simon, "Tracking Performance of Unbalanced QPSK Demodulators:
 

Part II--Biphase Costas Loop with Active Arm Filters," IEEE Trans.
 

on Comm., Vol. COM-26, No. 8, August 1978, pp. 1157-1166.
 
KU-BAND BENT-PIPE MODE 2 BER PERFORMANCE DEGRADATION
 

Contract No. NAS 9-16067, Exh3bit B
 

(Technical Monitor: W. E. Teasdale)
 

Interim Report
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

J.K. Holmes
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8102-4
 

February 18, 1981
 
1.0 INTRODUCTION AND DISCUSSION OF PROBLEM
 

The purpose of this report is to determine the bit error rate
 

(BER) degradation for the Ku-band return link mode 2. Figure 1 illustrates
 

the overall Ku-band return link. The typical return link data sources are
 

indicated in Table 1. Inmode 2, there are three inputs to the FM modulator. First, there is the bent-pipe wideband signal which is amplitude
 

(peak) regulated so as to provide 22 MHz peak-to-peak carrier deviation,
 

which is channel 3. In addition, there is channel 1, which has operations
 

data from the network signal processor, and channel 2, which receives data
 

from the payload recorder, operations recorder, payload low data rate link
 

or low data rate payload interrogator. Channels 1 and 2 together provide
 

for a peak-to-peak FM frequency deviation of 12 MHz
 

Table 1. Ku-Band Return Link Data Sources
 

Source 


CHANNEL 1 (MODE 1/MODE 2)
 

Operations data--Network
 

Signal Processor (1,2) 


CHANNEL 2 (MODE 1/MODE 2)
 

Payload Recorder 


Operations Recorder 


Payload Low Data Rate 


Payload Interrogator (1,2) 


Low Data Rate 


CHANNEL 3 (MODE 1)
 

Payload MUX 


CHANNEL 3 (MODE 2)
 

Payload Interrogator (1,2) 


High Data Rate 


Payload 


Payload High Data Rate 


Television 


Type 


Digital 


Digital 


Digital 


Digital 


Digital/ 


Analog 


Digital 


Digital/ 


Analog 


Analog 


Digital 


Analog 


Rate or Bandwidth
 

192 kbps (biphase)
 

25.5 - 1024 kbps (biphase)
 

25.5 - 1024 kbps (biphase)
 

16 - 2000 kbps (NRZ)
 

16 - 1024 kbps (biphase)
 

16 - 2000 kbps (NRZ)
 

16 - 1024 kbps (biphase)
 

1.024 MHz (PSK or FM)
 

2 - 50 Mbps (NRZ)
 

16 - 4000 kbps
 

0 - 4 5 MHz
 

0 - 4.5 MHz
 

16 - 4000 kbps
 

0 - 4.5 MHz
 
Ch3
 Bent-Pipe Bn-peLPF:11: Ku-Band Transmitter
 A Amplitude 1 r P _____, ,__________ 

- BW =ideband IMHz Regulator
 

Signal - -l
 

NSP Data LPF 
 QS are
 ...Subcarrierr Bb = Frequency
 C2 r- Modulatorf 
 M z Modulator
 

Bent-Pipe LPF
 

Narrowband BW= Mu -Limiter
 

Signal TDRS
 

PALimiter Filter Receiver ­

00
 

Bent-Pipe L Bent-Pipe 2 

Narrowband Narrowband 

Signal Signal ' U

QPSK Processor 

- ~~Tracker, Subcarrler Dt rr

Data


FM Frequency Synchronizer NSP Data
 MReceiver BPF 
 and Detector
 Demodulator
 

Bent-PipeBent-Pipe
 

Wldeband 
 Wideband
 Signala
 

Processor 
 Signal


Figure 1. Ku-Band Bent-Pipe FM Return Link
 
2.0 

3
 

Both channels 1 and 2 are hard limited by TTL receivers
 

(essentially limiters) and QPSK modulated onto an 8.5 MHz subcarrier.
 

This quadriphase subcarrier isadded to channel 3 at baseband and frequency
 

modulated onto the carrier. Figure 2 gives a more detailed block diagram
 

of the modulation process. In Figure 2, both modes 1 and 2 are illustrated;
 

however, inthis report, we are concerned with mode 2 only, the FM mode.
 

The specific case of concern inthis report isthe case wherein
 

the signal of channel 1 is a 192 kbps biphase-L (Manchester) baseband data
 

stream without thermal noise corruption, channel 2 has a 2 Mbps NRZ baseband data stream plus thermal noise, and channel 3 is composed of two
 

subcarriers, one at 1.024 MHz and one at 1.7 MHz, plus thermal noise.
 

The power spectral density (PSD) for the case inwhich channel 1
 

is192 kbps Manchester, channel 2 is 2 Mbps NRZ and the 1.7 and 1.024 MHz
 

tones are unmodulated and are sketched in Figure 3 along with the respective lowpass and bandpass filters (into the summer) of Figure 2.
 

CHANNELS 1 AND 2 QUADRIPHASE SIGNAL
 

Inthis section, we discuss the Hughes Aircraft Company (HAC)
 

implementation of the quadriphase modulator. A digital subcarrier oscillator (SCO) running at 8.5 MHz supplies a clock and its complement.
 

Both clocks are passed through a fixed-time delay (T)corresponding to a
 

fixed-phase delay 2e0 inthe square-wave fundamental. When not filtered,
 

the result is a constant-envelope four-phase signal, as shown on the I-Q
 

diagram of Figure 4 that has a ratio of I Q of 4:1 The actual signal
 

wave isfiltered so that only the first harmonics are applied to the FM
 

modulator.
 

Inorder to understand the power relationships of the I and Q
 

-channels, we establish the form of the filtered quadriphase signal following the work of Cager [3] in the case of no noise. The digital subcarrier
 

phase modulator isdescribed by
 

s(t) 5 inCmsct+o(t)D (1) 

where
 

dl(t)d2(t) ,= - f Cd2(t)-IJ (2) 
4-I100 Mbps


Mode1 

LDR PL Digital -nvo uti°na
 

PL Recorder Data TTL ECL TMode Channel1 3
 Encoder
 OPS Recorder Data _2 I Receiver 


ChannelModes
 1,2 4-50 Mbps
 

r 
-o QPve


or 


Channel 2 Sg 


"__ 129 kbps ,
 

Return Link OPS Data - NSP 1 -Md
 

Return Link OPS Data - NSP 2 L - Receiver T MCx ­

(Narrowband) C-- 7
 

Q_ Mode 2
 

LS6_
 

r27 Attenuator- _to 

PL Interrogator 2 A 2 FM 

Data [ - S12 ,ChannelLPF od "­ - _ C-l4i MHz " Mode 2 ,To :0 

o DEA-B CU PL Interrogator 1 75o 


Data SI Termination DEA-A
 

HDR PL Analog
 

CCTV Video
 

HDR PL Digital l ­

ri Translator 9
 

Figure 2 Ku-Band Return Link Transmitter-Modulator
 
SChl 

> 

aa) 

h3 

LPF Response 

/ 

BPF Response 

I0 1I 12 (MHz) 

0 

m iE 

Figure 3. Sketch of the FM Input Spectra Along With the Associated Separating Filters 
C 

ORIGINAL PAGE IS 

OF POOR QUALITY 

6 

Digital 

Subcarrier Multiplexer 

Q 8.5 MHz 

G(Square wave) 

Q I 

Channel 1 ___ 

Channel 2 

Q (Channel 1)
 

(1,0) (1,]) 

'0(Channel 2)
 

(0,1) (0,0) 

00 = 26 60 (Nominal) 

Time Delay T---- Phase Delay 200 

Figure 4 QPSK Subcarrier Oscillator Modulator 

for Mode 2 Ku-Band Return Link 
ORIGINAL PAGE 1 7 

OF POOR QUALITY 

with d1 (t)the channel 1 baseband modulation and d2(t)the channel 2
 

baseband modulation so that the four phases of *(t) are described by
 

Table 2 below.
 

Table 2. Square-Wave Oscillator Phase Relationship
 

(t) dl(t) d2(t)
 

-8+r 1 -I 

O+t -I -I 

-e -l 1 

The value of 80 isthe value of o such that the I and Q relationship yields a power ratio of 4:1 inthe first harmonic. Expanding (1)
 

in a sine-Fourier series yields
 

s(t) _s@n t + g(t) (3) 

n=l 
 nTf


where E denotes summation on the odd terms. Expanding (3)yields 

s(t) = cosrn(t)] sin nwt + sin[n(t)] cos nwt (4) r
 n=l n 


Using (2)for *(t) yields
 

cos[np(t)] = cos ndld2 e] cosflnCd2-lDlr/2] (5) 

sin[nd t)] = sinlnd [dId zCos[nCd 2 2-gl)a/2] (6) 

since sin [nCd2-l)w/2] equals zero. Simplifying (5)and (6)yields
 
ORIGINAL PAGE 12 

OF POOR QUALITY 8 

cos[n (t)] = d2 cos no n odd (7)
 

sin[n(t)] = d, sin no n odd (8)
 

Therefore, the quadriphase signal isof the form
 

s(t) = d2t) +-cos(no)sinCnosct) + dl(t) n sin(no) cosCnwct) 

n=l n=l 

(9)
 

Let
 

U(t) = cos no sinCnwctD (10) 

n=l 

V(t) = 4 sin no cosCnoictj (11) 

n=l 

then 

s(t) = d2(t) U(t) + dl(t) V(t) (12) 

To identify the waveforms expressed by (9), note that
 

co st 5 inCwct- o) + 3 lnCwsct + oj Ut) n cos(no) slnCnct 	 2 (13)
 n=l T2 

where 3 inC sct+ e) is a hard-limited sine wave of the argument Cwsct+e). 

Since 

SinCwst-8 + 	 inGsctieD X 4 cos(-n8) slnCnwtD+sin(-nQ)cosCnwct) sc 	 n=1 nirscs 

+ 	 E 4 cos(ne) sinCnwctD+ sin(ne) cosCnctD 

n=l 

(14)
 
ORIGINAL PA3' 9 

9
 OF POOR QUALITY 

It can also be shown inthe same manner that
 

V(t) 4 sin ne cosCntsct) = inCst+e) in scte (15)

n=l 
 s2


shown
 Now the right-hand sides of (13) and (15) can easily be plotted as 


in Figure 5. Thus, we see that the digital quadriphase signal can be
 

broken down into two orthogonal basis functions that span the space of
 

Note that the sum of U(t) and V(t) yields a square­ periodic functions 


wave signal that is early (from t= 0)by 20/wsc seconds, whereas U(t) ­

V(t) generates a square wave with a delay of B/osc seconds, etc.
 

Note that the total power inthe waveform of (9)isgiven by
 

Pl 16 CcosZ(ne) + sinZ(ne)) = - E l _ s 2:l n2 2 n2 n=l n2
 

Furthermore, the power inchannel 1 isgiven by
 

W lj 16 228T 

2 121l == 
 n2 no- T > 6 '>0 (17)

Vn=l 
 niT7


and the power inchannel 2 isgiven by
 

= Pu 116 l 6cs1co2 oL 2_1j 0 (18)

PU = P2sn=l en 2
 

Notice that, when e = 0.464 radians (26 580), the power split is
 

P1 = 0.295 (19)
 

P2 = 0.705 (20) 

and P2/Pl is2.38:1 However, since the BPF at the output of the MUX
 

(quadriphase modulator) passes only the fundamental, the expression for
 

the respective channel powers become
 
U(t) 

1 

27 

to sc 

WSc 

t 

_IsHJ LI LI LI n m 

Figure 5 Two Basis Functions for QPSK Channels 1 and 2 Signal 

0 
11 

3.0 

P1 2 sin 2e (21)
 

P28 = 2 (22) 

P2 2c5 

and, for 0 = 0.464, we have that P2/PI is4:1, as was desired.
 

We note from (9)using only the first harmonics that the signal
 

can be expressed as
 

Sf(t) = 4 dl(t) sine cos[wsct) + id 2(t)cose sincwsctD (23) 

where dl(t) isthe output of the channel 1 limiter and d2(t) is the output
 

of channel 2. Thus, we see that we can model the bandpass process interms
 

of its baseband processes.
 

IMPERFECT CARRIER REFERENCE LOSSES
 

Since we have a quadriphase signal with both a low rate NRZ waveform and a high rate Manchester waveform, we must consider two cases of
 

cross-channel interference: (1) a high rate NRZ interferes with a low rate
 

Manchester data stream, and (2) a low rate Manchester data stream interferes with a high data rate data stream. The two data streams are illustrated in Figure 6. Besides the cross-channel interference, the imperfect
 

carrier reference loss also exists and is, in general, not negligible. We
 

essentially follow the approach of Osborne [8] in the following.
 

First consider the case inwhich the low rate Manchester signal
 

interferes with the high rate NRZ signal. We model the received RF quadriphase signal by
 

NRZ Manchester
 

r(t) = ,22 dz(t) sinCwsct+o0J + 2%i dI it)cosCwsct+eo) + n(t) (24) 

where Pi isthe channel power inthe ith channel, d (t) is the ith data
 

stream (i=1,2), and w., and 00 are the angular frequency and phase of the
 

8.5 MHz subcarrier frequency, and finally, n(t) iswhite Gaussian noise.
 

The pulse shapes P (t) are either NRZ or Manchester symbols. The demodulation circuit isshown in Figure 7.
 
I-- n/2 gas n/2 

Manchester F - -

- II ones --- -- n2 ones 

NRZ 10 

U0 

0 Za 

Figure 6 Data Sequence Model for NRZ and Manchester Cross-Channel Interference 
mTI 

(m-I)T 1 

. dt > 0 

r(t) 900 i ~Generator Symbol i°:3 

ian d Timing 

snCw sct+°o 

'" 

0 Q 

Imperfect 

Reference 

(n-I)T 2 

P2 Ct-(n -1)T2 

zi 

Figure 7 Quadriphase Demodulator Model 
14 

ORIGINjAL PAGE W~ OF POOR QUALIT1
 

In channel 2, after demodulation by V2 sinCwsct+e0, we have
 

r(t) v2sin(nsct+60) = jP2 d2Ct) cost - P d,(t) sin@ + n'(t) (25) 

where = eO and isthe phase error, and n'(t) is an essentially white 

Gaussian noise process. Assuming that the bit transition probability is 

50%, the conditional bit error probability isgiven by Cassuming 4(t) is 

essentially constant over max (TIT 2), 

iF!~2 NT_ 2PI,- 2P
0 

PE() [ P2T2 cos P 1T2 sin + Q cos+sin T 


N 0 0
 

(26)
 

where
 

Q(x) 1 e-t2/2 dt (27)
 

In order to evaluate (25), itisnecessary to specify the phase
 

error probability density. To do this, we assume, for lack of a better
 

model, that p( ) (for a CW loop) isTikhonov, i.e.,
 

) exp (acos ') < (28) 

To convert to a loop having an S-curve that has a period of w
 

-radians (such as a Costas loop or a quadriphase with power unbalance of
 

4 1), we let €' 24 so that
 

p(4=) exp(acos2 ) , - < (29) 

To determine the meaning of a in this modified Tikhonov density, let a be
 

large so that we can use the asymptotic expression for (28), producing
 
ORIGIA PAQ 15 OF POOR QUALIT' 

p(0)- T) 7K< (30) 

implying that the linearized phase error variance is given by
 

2 1 (31) L :4-a
 

In the same manner, loops having S-curves periodic over ±w/4 radians,
 

such as balanced quadriphase tracking loops, have a density given by
 

p(@7? Io(a) 4 = 2exp acos47) < <4 (32) 

and the linearized phase error variance satisfies
 

2 1 (33)


16-a
 

The power ratio and data rate ratio for channels 1 and 2
 

satisfies
 

P2 R = 10 
 (34)
 

Using (29) to average over (26) produces the results shown in Figure 8.
 

The parameter aE is the inverse of the linearized phase error variance
 

so that
 

aE 1
 l2- 4a (35)
 

OL
 

We now consider the case where the high rate Manchester signal
 

interferes with the low rate NRZ signal.
 

From the received signal of (24), after demodulation by
 

F2 cosC sct+ 60J, we have
 
10-2 , -T|I! I 16 

6­

4- OF POOR QUALIY' 

2­

8­

6- (FL 

10-3Co 

10-R 

'aE 

-rL 

= 143 °

D 

E = 12 dB 

4­

2­

> 

10-4_ 

8­

6­

4-

2-

Figure 8 High Rate NRZ 

Interfered by a 

Low Rate Manchester 

4 Loop 

(11 4-) 

1 -5 

8­

6­

(

4E 20 dB (5 70) 

2-

10- 6 

4 5 6 7 

aE 

8 E 

= dB 

(00) (4 61 

aE=2 2 dB 

aE= 24 dBE 

(3 

9 10 0bN 

E= 16 dB 

(9 10) 

B 

6-(7.)

I(36) 

12 13 14 
ORIGINAL PAGE [f; 

OF POOR QUALI1Y 17 

(HR) (LR) 

r(t) 42Wcos( 03sct+) = 2 d2(t)sin + PI dl(t) cost + n"(t) (36) 

where n"(t) isan essentially white Gaussian noise process and is independent of n"(t). Again we notice that, if*

= 0 identically, there is
 

no cross-channel interference.
 

When the transition probability is 50% after matched filtering,
 

the detected signal Is given by
 

F2(n2- nl =+ (2--nl)" n, (7n2) - n,2 P 

Sd[ n TTI cos* + n n PT 1 sin (37) 

where n1 denotes the number of ones inthe first half of the Manchester bit
 

and n2 denotes the number of ones inthe second half of the Manchester bit
 

and n isthe total number of ones ina Manchester bit (see Figure 6). It
 

follows that the conditional average bit error rate is given by (neven)
 

_ n/2\nl] n/2 -0 - n2 -0o\21Q / 2P1 1 l ~ / s 2Pn2 f 0 

2n/2 2n/2 n =0 (n/21 n/2)Q(_NTI n-op 22PnT 

(38)
 

However, by direct numeration, itcan be shown that a single summation form
 

of (38) isgiven by
 

P( )=.
In (n)- /n\' 2P1TI coso I-2i ,7 2P2T1 

l

sn 

s in (9()

2 =0 0 cV ) 0 

Since this is simpler, we shall use this.
 

Inour present case, n= 10 and P2 = 4Pl Evaluating (39) by 

averaging over *,i e , 

7T/2 

PE =-]2P(E1q) p( ) d (40)
 

-ir/2 
18 

4.0 

yields the results of Figure 9. Note that these curves assume that 4dt)
 

is essentially constant over maxCTi,T2D.
 

By comparing Figure 8 and 9,we see that the case of the low rate
 

Manchester signal being interfered by the high rate and high power NRZ
 

wave has more degradation than the case of the low rate Manchester interfering with the high rate NRZ signal.
 

We note that both cross-channel effects and an imperfect carrier
 

reference have been degradation sources for Figures 8 and 9.
 

CHANNEL 3 NOISE INTERFERENCE EFFECTS ON CHANNELS 1 AND 2
 

We assume in this section that noise dominates channel 3's input
 

and that the amplifer is linear (neglect clipping) since the degradation
 

is expected to be quite small anyway. The filter preceding the amplitude
 

regulator is a three-pole Butterworth filter so that the noise spectral
 

density shape is given by
 

NO N0 1
 6 7- jH(f)J2 2 1 + Cf/fo (41) 

where f0 = 5 MHz It is known that the lowpass filter (LPF) of Figure 3 has
 

a rejection of 21.6 dB at 8.5 MHz. Furthermore, the 5 MHz LPF is down
 

10 log[l+ (8 5/5)6] 14 dB at 8.5 MHz so that the total noise is down by
 

about 35.6 dB. Since channel 3 is only roughly (22/12)2 = 5.3 dB larger,
 

we conclude that the noise is down about 30 dB in the band of channels 1
 

and 2.
 

Consider this in greater detail We have, since channels 1 and
 

-2together deviate the carrier 12 MHz, that
 

=
 PCHI + PCH2 K2(12)2 (42) 

where K is the constant relating signal voltage to frequency deviation.
 

Now 

PCHI - 1 (43) 

PCH2 4 

so that
 
10-2. 19
 

8" 

6- ORIGINAL PAGE 13 

OF POOR QUALITY 

4­

2 - aE 12 dB 

°) Cr =14 3 L 

10-1 

4­ aE 14 dB 

a) 10-

2­

L 

s- 8­

< 6- cL V 

a 16 d 

° ) 4- (9.1 

2-

Figure 9 Low Rate Manchester
 

5 Interfered by a
 -
 10 High Rate NRZ- 4 Loop
 

8­

6­

(00 4- ) C 18d

E dB (7.10)­

= 2 aE 28 dB 

2(2.30) 

10 6 1_I 

3 4 5 6 7 8 9 10 11 12 13 14 

Eb/'N0 
20 

ORIGINAL PAGE 1.2
 

5 %rH K2(12)2 OF POOR QUALITY (44) 

or
 

V _Er7= 5.37 K = IS 1I+PN1 (45) 

and
 

_CH10.7 K=V!VS (46) 

(46) 2 N2 

Now the new noise degradation inchannels 1 and 2 due to channel 3 is given by
 

PI+PN3L
 

DEGR P1 3 (47) 

and 

PN + PN3L 

DEGR 2 - 3 (48) 

where L is the reduction in channel 3 noise at 8.5 MHz. From (45) and
 

(46), we have that
 

p (5.37) 2K2 (49) 

and
 

P (07)2K (50) 

N2 a,
 '37)

From (49) and (50), and (47) and (48), we obtain
 
ORIGINAL PAGE IS 21 

OF POOR QUALITY 

DEGRCHT = 1 +PS (51) 

DEGRCH 2 = lOlog 1 + p 1 (.52L 
 )

Recall that L = -35.6 dB so that, at minimum noise conditions, i.e.,
 

EB2 B1 

2 20 dB (53)
 

we have
 

EB
2
 

SNR2 -N 10 log CB2 T2 (55) 

0 

Using (53) in (54) and (55) yields 

SNR1 = 5.9 dB (56) 

and 

SNR2 = 16 dB (57) 

Using (56) and (57) in (51) and (52), respectively, yields
 

DEGRCH 1 = 0.1 dB 

DEGRCH 2 = 0.2 dB (58) 

We conclude that the degradation in channels 1 and 2 from the
 

noise in channel 3 under worst-case conditions is essentially negligible.
 
ORIGINAL PACp j22


OF POOR QUALITVY 
 22
 

5.0 
 UPLINK AND DOWNLINK NOISE COMBINING
 

In this section, we briefly discuss the effect of uplink noise
 

on the received signal. 
 The signal flow in the bent-pipe mode goes from the
 

user to the Shuttle, from the Shuttle to the TDRSS, and from the TDRSS to
 

earth.
 

The signal power to the total noise power is given by
 

S 1 
 (59)
 NoI+NO NO NO
 012 
 0 2
 

S S
 

where S denotes the signal power at the ground, No, is the Shuttle noise
 

spectral density received on earth, and No
2 is the receiver noise at the
 

earth receiver. Equation (59) can be written in the form
 

S 
 1 
 (60)
 N + N0 
 ( + N(

where NI/S is the noise power to signal power ratio at the Shuttle (K-band)
 

transmitter. 
Notice that the right-hand side of (60) is expressed in terms
 

of S/No1 on the user to Shuttle link, the SNR at the Shuttle transmitter
 

CS/N 1D, and ratio of received downlink power to the earth receiver noise
 

spectral density C(S+Nl)/N 02D. From (60) or 
(59), we see that, when S/N
0 2
 

is large, (60) approaches S/No1 ,
as would be expected Caccording to [10,
 

pages 4T7 and 4-8]D.
 

6.0 HARMONICS OF CHANNEL 3 TONES
 

We now use Appendix III to evaluate the level of the harmonics
 

of both tones from channel 3 in order to see if they will 
degrade channels
 

2 and 3 BER performance. We accomplish this task by comparing the amplitude coefficient h010 with h070.
 See Appendix III for details, particularly
 

equation (111-9). The term h070 corresponds to the seventh harmonic of
 

1.024 MHz and, as 
such, corresponds to an interfering tone at about 7.2 MHz
 

which lies in the band of channel 2 when operating at 2 Mbps.
 
ORIGINAL PAGE M2 OF POOR QUALIY 23 

From Appendix IV,equation (46), we have for K= 0,mI = 1 and 

m2 = 0 Csee Appendix III, equation (9)D that 

IF1 3-2k1 3b2
 eb2/2 Pi 1 

Sn2 2 

h~ b2-22e1 £!(+2) r (61) 

where
 

a. output clipping level of clipping amplifier
 

d" input clipping level of clipping amplifier
 

b: b 
 d
 2 A2
 

2 AL 

2 2
 

an an = NoBG
 

G: power gain of AGC inAppendix III, Figure 1
 

2
 A2 A22
 

SP2= - 2 , power in the 1.7 MHz tone 

Pl
 

Pi: PI=(n2?

2 + input SNR of tone 1 regarding tone 2 as noise
 

IF (a,b,z)- confluent hypergeometric function
 

_ P1
 

1Pl ( n2 + P
 

P 


P2 2 + 

2

2Pl)
 

To account for the AGC effects on the clipper amplifier (see
 

Appendix III), we note that the RMS power into the AGC isgiven by
 

P1 + P2 + an2 = PRMS (62)
 
24
 

ORIGINAL PAGE 13
 

Ifwe let OF POOR QUALIT 

and 2
 P1
 PRMS (63)
 

a2 ­ (64)
 PRMS 

then it iseasy to show that
 

2 SNRI (65)
 

a = + SNRl + SNR2
 

and
 

2 SNR2 (66)
 

a2 l + SNR + SNR2
 

where SNR l = Pl/On2 and SNR2 = P2/rn2. Ifwe assume a representative tone
 

power to noise power ratio of each tone to be -17 dB in the 5 MHz bandwidth,
 

then,
 

P1 =P2 = 0.019 (67) 

Set 

P + 2
 1 + P2 on = 22 = 4 watts (68)
 

and set the clipping level at the output of the clipper to be 4 volts
 

Ctwo-sigma clipping level) 
so that a=d= 4. We have by calculation
 

l = a2 = 0.183
 

n = [1- 2(0.0183)]22 - 3 854 watts (69)
 

Pl = P2 = 0.073 watts .
 
ORIGINAL PAGE iM 

OF POOR QUALITY 25 

Evaluating (61) produces
 

h010 = 0.195 (70) 

and the corresponding tone autocorrelation function isgiven by
 

R1() = 2(0.195)2 COS W T (71)
 

where
 

2 = 1.024 MHz (72) 

Now we will determine "070 We have from Appendix IV,equation (46) that
 

eb ¢ d h_ 2 pie


J

1/ h070 2 pi7/
 2 . 7/2 e-b/ 

/2
 

' h2 A b [ 

1 1 2 '2'
 

x r(2 7)(73)
 

Evaluating (73) after some algebra, we obtain
 

h070 = 3 13 x lO07 (74)
 

so that
 

2
 

-7 T
 RI(T) = 2C3 13 x 10 cos Wi (75)
 

By comparing (71) and (75), we see that the seventh harmonic is down 116 dB
 

from the original tone. We conclude that harmonics are totally negligible.
 
26 

7.0 

ORIGINAL PAGE 0 

OF POOR QUALITY 

BANDPASS FILTER LOSSES: CHANNELS 1 AND 2
 

Inthis section, we estimate the losses associated with filtering
 

the quadriphase channels 1 and 2 signals and matched-filtering them. In
 

Figure 10, a sketch indicates the block diagram including the bandpass filter under consideration. The multipole channels 1 and 2 BPF has (upper
 

left portion of Figure 10) a 3 dB bandwidth (specified) of about 8 MHz.
 

For channel 1 at 192 kbps Manchester data (approximately 384 kbps NRZ
 

data), we have that
 

BT = 21 (76) 

and 

(BT)-I 0.05 (77) 

By estimating the predetection filter losses compared in [9, Figure 7]
 

with the transmission filter degradation from the transmission filter
 

degradation of [9, Figure 6], we conclude that, for channel 1, the BER
 

degradation isapproximately
 

DEGR1 0.06 dB
 

However, the 0.5 dB ripple (filter specification) causes about 0 25 dB 

degradation based on [9, Figure 9]. The total degradation to channel 1 

is then given by 

DEGR 1 = 0.31 dB (78) 

We now consider channel 2 filter losses. Channel 2 has a
 

2 Mbps NRZ waveform so that
 

BT ; 4
 

and
 

(BT)-I = 0.25 (79)
 

so that the BER degradation, using [9, Figure 6] (seven-pole curve) and
 

[9, Figure 7] to relate predetection and transmission filter losses, is
 

about 0.4 dB. Additionally, the 0.5 dB ripple in the filter bandwidth
 

could produce about 0.25 dB degradation. Consequently, the total channel 2 bandpass filter loss isabout
 
Channel 1 

Channel 2 

Channel3 

SCO 

Mod"-'- I ---CO

(ADL) 

BF1 

BPF - --

Possible Additional 

-PoslleAddltna 

Filtering at Receiver 

Fltrnga ecie 

Channel 3 LP 

L 

(D)&D 

Lrr 

_ 

rTo 

Channel 

Demodulator 

Ch 2 00-n 

; 

0 

Figure 10 Baseband Transmission Reception Model with FM Modulation and FM Demodulation Removed 
28
 

DEGR2 = 0.65 dB (80)
 

Now we estimate the BER degradation when the additional BPF and
 

LPF shown inFigure 10 are used to reduce channel 3 tone interference of
 

channels I and 2. These filters are identical in filtering characteristics to that of transmitter bandpass and lowpass filters of Figure 2.
 

From [9, Figure 6], itisestimated that the cascade of BPF's increases
 

the BER degradation less than 0.05 dB for channel 1 and less than 0.2 dB
 

for channel 2. We conclude that the overall BPF BER degradations are
 

given by
 

DEGR1 ' = 0.36 dB (81)
 

DEGR2' = 0.85 dB . (82) 

8.0 RF-FM MODULATION/DEMODULATION LOSSES
 

The main thrust of this analysis has been at baseband; however,
 

the FM modulation/demodulation process isnot perfect, so BER degradation
 = ­ at PE I x 105 has been assumed to be
 

DEGR = 0.5 dB (83)
 

This degradation estimate includes nonlinearities as well as
 

the effect of thermal noise inthe FM demodulation process.
 

9.0 DEGRADATION OF CHANNEL 2 DUE TO CHANNEL 3 TONES
 

From Figure 10, we see that the two tones of channel 3 will be
 

j'seen" by the integrate-and-dump matched-filter detector for channels 1
 

and 2.
 

Therefore, ifthe additional filters are not included as shown
 

inFigure 10, the cross-channel interference of channel 3 onto channel 2
 

will not be negligible. InFigure 11, the two tones of Figure 3 are
 

illustrated along with the channel 2 matched-filter response. Since channel 2 has a much higher data rate, its response is affected more by the
 

channel 3 signals modeled inthe report as two unmodulated tones. The
 

channel 2 matched-filter response is given by (inboth time and frequency)
 
Channel 2 r- J 

Matched-Filter
 

Response
 

fIf
 

2
 

1 024 MHz 1.7 MHz
 

0 2 3 4 5 6 7 8 

fo 

8.5 9 

Figure 11. Response of Channel and the Two Channel 3 Interfering Tones 

.0 

00 
30
 

ORIGINAL PAGE FS 

OF POOR QUALITY T (

J ( )dt 

0 

with corresponding frequency response 

H(f) = sin(nfT) e-JwT/2 (84) 

rfT 

Define the channel 2 signal-plus-received noise as 

y(t) = V7A d(t) sin wo0t + Fns(t) sin wo0t + r nc(t) cos bot 

+ FAII sin wlt + 2A1 2 sin w2t (85)
 

where nc(t) and ns(t) are independent, bandlimited, baseband, Gaussian
 

noise processes. When this signal iscoherently demodulated by rsin n0t,
 

there remains
 

eD(t) = A d(t) + ns(t) + A1 cos Co0-t1 + Ai1
 

+ A12 Cos[@naO-w2)] + A,2Cos Lmuvfl 2)d] (86)
 

where
 

fo = 8.5 MHz
 

fl = 1.024 MHz 

f2 = 1.7 MHz (87)
 

We see that the interfering signals and noise have powers given
 

by
 

P VarCnsj+ P1 + P I + P + P (88)
 

1 12 2
 

where
 
31 

ORIGINAL PAGE Q5 

OF pOOR QUAL'T3 

VarCn) f Tf T ntfnTu) dt du (89) 

or
 

VarCns) (90)
 

with N0/2the two-sided noise spectral density and PI1 and PII corresponding to the AI interference terms in (86) and P12 and Pi2 being the
 

A12 interference terms in (86).
 

We now evaluate the channel 2 interference terms. We have
 

sin 2C7.48wrx1 6/Z2xl1O6D 

Il (7.487/2)2 P1.024 00039 P1.024 

P' sin 2(9.52 /2)p= 0.0021 P 

1 (9.52w/2)2 1.024 .024 

P sin 2(6.8w/2) p 0.0079 P
 

P2 (6.87/2)2 P1 7 1 7
 

Pi' = sin 2(10.27r/2) p 0.00037 P (91) 

12 (10.27/2) 2 1.7 .7 

Inorder to compute the total degradation, we must evaluate the
 

tone power levels Assume that channel 3 is at maximum SNR (no noise) so
 

that the sum of both tones modulate the carrier to 22 MHz peak-to-peak.
 

- Therefore, for channel 3, 

2CAI+A 2D = K 22x 106 (92) 

where A1 and A2 are the tone peak amplitudes of channel 2. Since
 

2
 A12 AI 2 

P3 += A1 (93) 
32
 

OF POOR QUALI y It follows that 

A l1K x 106 11 6 

A1 2 K' CK' KxlO (94) 

So 

PT3 = 30.25(K')2 (95) 

and the individual tone powers are 

PT31 = PT32 = 15.13 (K')2 (96) 

Now in channels 1 and 2, we have, since the filtered quadriphase is in quadrature, that
 

2 A1

2 + AQ2 = 12K x 106 (97)
 

where AI is the peak channel 1 subcarrier amplitude and A2 is the peak
 

channel 2 subcarrier amplitude. Also,
 

= A 2 A 2A (98)
 

I,Q 8 Q 
 A
 

Solving (97) and (98), we obtain
 

AQ= 5 37 K' 6 (K' Kx 10 J 

A1 = 2.68 K' 

P = 18.1 (K')2 (99) TQ
 

Now, if the SNR in channel 2 is 10 dB, then the noise power in the Q channel is given by
 
33 ORIGINAL13.GI vs 


OF POOR QUNL1'T
 

I (14.4)(K,2
 T (100)
 

Therefore,
 

(101)
 CNOB)CH2 : 1.3(K')2 


where B is the equivalent 4 MHz lowpass process from the 8 MHz bandpass
 

process. It follows that
 

S1.3(K) (102)


O)cH2 B
 

Therefore, (89) can be written using (90), (91) and (102) as (T=1/2xlO 6)
 

(PTN) = 2T 13(K')2 + 0.0039 (15.13)(K')2 + 0.0021 (15 13)(K')2 

+ 0.0079 (15.3)(K')2 + 0.00037 (15.3)(K')2 (103)
 

This can be factored as (B 4x 106)
 

PT 0.325(K')2 [1+ 0.66] (104)
 

We conclude that the degradation is given by 

DEGRCH 2 = 2.2 dB (105) 

Now we consider the same calculation utilizing the benefit of
 

the BPF and LPF in Figure 10 at the receiver to chop out the I&D filter
 

response near the channel 3 tones Figure 12 illustrates this point.
 

Equation (103) can now be written as
 

2 2 002)2
 + 0.0039 (15.13)(K')
 PT 21Bl( .0.8) 

(15 3)(K')2 (0.02)2 + 0.0079 (15.13)(K) 2 (002)2 

TNN (1.3)(K')


+ 0.0021 

+ 0.00037 (15 13)(K')2 (0.02)2 (1061 (26.8/ 
Lowpass equivalent of actual
 

bandpass filter (ADL)
 

,f f f f +f1 of1 

1 2 3 4 6 7 8 9 1 10 1
 

Figure 12 Spectra After Demodulation
 

40 
ORIGINAL PAGE i 35 

OF POOR QUALITY 

where the factor
 

(002)2 

isthe reduction infilter voltage response (squared) due to the presence
 

of the ADL BPL.
 

Evaluating and factoring as in (104), we obtain
 

P = 2.6(K')2 [l+ 0.0004J (107)
 

We conclude that the degradation due to the channel 3 tones is less than
 

0.01 dB for channel 2 when the transmitter BPF isalso used on the receiver.
 

10.0 CHANNEL 3 DEGRADATION DUE TO CHANNEL 2 NOISE
 

This source of degradation isbased on the fact that channel 2
 

noise can get into channel 3,thereby increasing the noise level. The
 

transmitter BPL, however, limits the noise interference as well as the
 

5 MHz lowpass filters into the channel 2 limiter. Itshould be noted that
 

the additional BPL and LPF at the receiver does not affect this phenomena.
 

From the ADL filter response data, the filter reduces the channel 3 noise by
 

2
 

L =10log0.02 

L = lOlog_ 82 = -32.1 dB (108) 

When channels 2 and 3 are near the same SNR's, we see that this additional
 

noise will be essentially 32 dB down, plus or minus a few decibels due to
 

the different FM drive levels. It follows that this noise has negligible
 

effects on channel 3 performance
 

Now consider the degradation to channel 1. Equation (88) still
 

applies, but now Var(n) denotes the variance of channel 1 noise. The
 

response of the Manchester matched filter isgiven by
 

H(f) = sln2(ffT/2) e-jw T/2 (109) 

(nfT/2) 
ORIGINAL OF pOOR QUALIW PAM MS 36 

Equation (91) becomes, for this case,
 

_=sin4Cir7.48x0 6/0.4xo06P
 

I (7.487r/0.4) 2 P.024 = 0.00012 p1. 024 

P' = sin 4 (9 .52r/0.4) - 0 

I1 (9.52r/0.4)2 1.024 0.000021 P1.024 

Ssin4 (6.87r/O.4) P 0
 

P2 (6.8n/0.4) 2 1.7 

P, = sin4 (10.2n/0.4) p 0.00016 

2 (10.27r/0.4)2 1.7 1 .70) 

From (99) and assuming that channel 1 is at I0 dB, the noise
 

power is then given by
 

I1 (3 59)(K')2 (111) 

Therefore,
 

CNoBDCH] = 0.33(K')2 (112) 

It follows that
 

B(l) (K') 2 CNQCNOCl CHI 0.33 (113) 

Hence, using (88), (90), (110) and (112), we have
 

(PT) - !- (0.33)(K,)2 + 0.00012(15 13)(K')2 + 0.000021(15.13)(K')2 

T CHI T 

+ 0 00016(15.13)(K')2 (114)
 

This can be factored as
 
ORIGINAL PAGE 18 

OF POOR QUALITY 37 

PN ClIi = 0.0825(K') 2 [1+0.055J (115) 

We conclude from (115) that the degradation is given by
 

DEGRCHl = 0.23 dB . (116) 

11.0 BER EB/N 0 DEGRADATION SUMMARY
 

Inthis section, we summarized the EB/N% BER degradation for
 

channels 1,2 and 3. The link budget for channel 1 isgiven inTable 3
 

and the link budget for channel 2 is given inTable 4.
 

Table 3. Channel I BER Degradation
 

No Receiver Receiver 

Filters Filters 

Component (dB) (dB) 

1. Imperfect carrier reference inciuding
 

crosstalk (assumes aL = 4"50 @ BER =
 

lx 10-5) 0.2 0.2
 

2 Channel 3 noise interference 0.1 0.1
 

3. Harmonics of channel 3 tones 0.0 0.0
 

4. Lowpass filter losses 0.31 0.36
 

5. RF-FM modulation/demodulation losses 0.5 0.5
 

6. Matched-filter response of channel 1
 

to-channel 3 tones 0.23 0.0
 

7. Channel 3 degradation due to channel 2
 

noise 0.1 0.1
 

Totals 1.44 1.26
 
38 

0 

Table 4. Channel 2 BER Degradation
 

No
 

Receiver Receiver
 

Component Filters Filters
 

(dB) (dB)
 

1. Imperfect carrier reference =
 including
 crosstalk (assumes OL 4.5' @ BER 


1x 10-5) 0.7 0.7
 

2. Channel 3 noise interference 0.2 0.2
 

3. Harmonics of channel 3 tones 0.0 0.0
 

4. Bandpass filter losses 0.65 0.85
 

5. RF/FM modulation/demodulation losses 0.5 0.5
 

6. Matched-filter response of channel 2 to
 

channel 3 tones 2.2 0.01
 

7. Channel 3 degradation due to channel 2
 

noise 0 1 0.1
 

Totals 4.35 2.36
 

These tables do not include bit synchronizer degradations or
 

relay losses or take into account that the modulation BPF centered at
 

8.5 MHz is not narrowband, so that additional losses may be present. In
 

addition, they do not include the C/N0 loss out of the limiters in channels 1 and 2 (see Appendix II). Also, numerous assumptions on SNR were
 

made to obtain the estimates, with most of them based on worst-case conditions - Therefore, the two BER degradation tables represent, for the
 

most part, a worst-case estimate for each entry.
 

Channel 3 BER degradation was found to be negligible due to
 

additional thermal noise from channels 1 and 2
 

12.0 CONCLUSIONS
 

Estimates of BER degradation have been obtained, inmost cases,
 

by assuming worst-case conditions. One exception is the four-phase loop
 

which demodulates the quadriphase signal of channels 1 and 2. There an
 

RMS phase error of 4.5' was assumed as a reasonable guess.
 
39 

It is seen that the additional filters of Figure 10 reduced
 

BER degradation in decibels almost in half for channel 2 and about 10%
 

for channel 1.
 
40
 

REFERENCES
 

1. R. H. Cager, D. T. LaFlame and L. C. Parode, "Orbiter Ku-Band Integrated Radar and Communications Subsystems," IEEE Transactions on
 

Communications, Vol. COM-26, No. 11, November 1978.
 

2. 	J. C. Springett and S. Udalov, "Communication with Shuttle Payloads,"
 

IEEE Transactions on Communications, Vol. COM-26. No. 11, November 1978.
 

3. R. H. Cager, "Phase-State/Power Division Relationships in the Return
 

Link Baseband Modulator," Hughes IOC No. HS237-782, June 14, 1977.
 

4. J. C. Springett, "Shuttle Payload S-Band Communications Study,"
 

Axiomatix Report No. R7903-1, March 9, 1979.
 

5. 	"Ku-Band Integrated Radar and Communication Equipment for the Space
 

Shuttle Orbiter Vehicle," Hughes Aircraft Co. Preliminary Design
 

Review, March 14-24, 1978.
 

6. M. Abramowitz and I.Stegun, Handbook of Mathematical Functions,
 

National Bureau of Standards, Applied Mathematics Series 55,
 

June 1964.
 

7. 	R. J. Helgeson, "PI-PSP BER Degradation Due to PI Baseband Output
 

Limiting Amplifier," TRW IOC SCE-50-79-491/RJH, September 5, 1979.
 

8. H. C. Osborne, "Effect of Noisy Reference on Coherent Detection of
 

Unbalanced QPSK Signals," NTC '78 Proceedings, Birmingham, Alabama.
 

9. J. J. Jones, "Filter Distortion and Intersymbol Interference Effects
 

on PSK Signals," IEEE Transactions on Communication Technology,
 

Vol. COM-19, No. 2, April 1971
 

10. 	 "Space Shuttle Communications and Tracking RF Link Circuit Margin
 

Summary," Lyndon B. Johnson Space Center, EH2-M/79-309, April 1979.
 
ORIGINAL PAGE if 

OF POOR QUALITY 

APPENDIX I
 

BASEBAND DIGITAL SIGNAL-PLUS-NOISE THROUGH A HARD LIMITER
 

Inthis appendix, we determine the output spectral density of
 

a baseband digital signal plus bandlimited Gaussian noise. This calculation allows one to determine the output C/N0 value as a function of the
 

input C/N0 value. The derivation of the autocorrelation function follows
 

that of Painter [1] with minor corrections in the final results. The
 

model isshown in Figure 1 below.
 

Am1 (t)+ n(t L dl(t)
 

Figure I-1. Baseband Limiter Model
 

where m1 (t) is an NRZ or Manchester binary-valued digital waveform, n(t)
 

is bandlimited, white Gaussian noise, and dl(t) isthe output binary-valued
 

waveform. Let
 

x(t) = Ad(t) + n(t) (I-I)
 

y(t) = sgn[x(t)] (1-2) 

The first- and second-order density functions are given by
 

PxCxt,tD _ exp r 2 12 Cxt-pt} (-3) 

xc2 a2 e%[ 2(l- {C 2 x t2 t 2 t2} 
1-2
 

ORIGINAiL PAGE IS 

where 	 OF POOR QUALITY 

xt = 	 x(t) 

X = 	 x(t+t) 

1t = E[x(t)] = Ad (t)6 s(t)
 

liz = E[x(t+r)] = Ad (t+T) =As(t+r)
 

a2= 	 ECxt-,)] =Ex,_ f 

a- E- (x -Qxt-ptJ] 

or 1Rni 	 O)

o 	 2 E fn(t+T)n(t)] (I-5) 

2 2 

First let us compute the mean value of our process. We will
 

temporarily drop the time subscript.
 

p = sgn(x) Px(X) dx 	 (1-6) 

which 	can be expressed as
 

Sexp (x-s)jdx - exp - (x-s) dx (I-7) 

where 	s = s(t) = Ad(t). Letting x-s = v produces Za 

2 	 2/2a e-v 2 dv = erfLt (1-8) 

0
 

Now let us determine the output autocorrelation function defined
 

by
 

Ry(t+T,t) = E[y(t+T)y(t)] sgn[xt] sgn[xT] PxCxt,x,)dxtdxT (1-9)
 
ORIGINAL PAGE M 1-3
 

OF POOR QUALITY 

Equation (9)can be written
 

Ry(t+T,T) = I PxCXt,X) dxtdxT + pPcxCt,XJ dxtdx
 

0 0 -w ­

-f °PxCXt,x3 dxtdx- PxCXt,xodxtdxT (I-1O)
 

a 0~ 

Letting x - t : ua', xT - = va produces 

T 

Ry(t+T,T) g(u,v,p) dudv + g(u,v,p) dudv 

a g(u,v,p) dudv - g(u,v,p)dudv 

aa (I-ll) 

where g(u,v,p) is the density function of the random variables u and v
 

induced from the variables xt and x.
 

Let
 

lit __ 

h=- k= 

then 

Ryy(T,t) = L(-h,-k,p) + L(h,k,p) - L(-h,k,-p) - L(h,-k,-p) (1-12) 

so using [2] 

L(h,k,p) = Q(h)Q(k) + zz(n) (h)z(n)(k) pn+l (1-13)

n=O (n+l)! 
I-4 

ORIGINAL PAGE JS
 

OF POOR QUALITY
 and 

Q(h) e-t2/2 dt 	 (1-14) 

we arrive at
 

Ryy(t,T) = Q(-h)Q(-k) + I zn(-h)zCn)c-kmn+1 

n=O (n+l)I 

zn (h)zn (k)pn+l
 + Q(h)Q(k) + 


n=0 (n+l)I
 

(-h)z n)(k)(-P)n+l
 - - z
 - Q(-h)Q(k) 


0 (n+l)I
 

(n)h (n)(-) Pn+1
 - Q(h)Q(-k) - z (h)zn(-k)(-p) (I-i5)
 

n=O (n+l)!
 

where
 

z(n) (x) - -i--e-x2/2 	 (1-16)
 

dxn 
 2-f 

Now 

Ry(t,t) = [Q(h)-Q(-h)][Q(k)-Q(-k)] 

+ 	n n+ zn)(-h)zn(-k) + z(n)(h)z(n)(k) 

n=O (n+1)! I 

n + l (_I) [(n)(-h)z(n)(k)+z(n)(h)z(n)(_k 

(1-17)
 
ORIGINAL PAGE 2 

1-5
 OF POOR QUALITY 


Now note that
 

Q(-x)-Q(x) = erfQ2) (I-l8) 

Also note that 

z(n)(x) = zn(-x) , n even 

z(n)(x) = -zn(-x) , n odd (1-19) 

So
 

)4+ z(n) (h)z(n)(k)pn+l
 Ry(t, ) = erf erf 

4n= (n+l)' 

z(n)(h)z nJk)Pn±1 (1-20) 

n=l (n+l)! 

where denotes sums over n odd and denotes sums over n even. Changing 

Indices yields 

41
 2~)v)
 R(t,T)_ = err " erf1 +2~l 2+ 

7M= (2m+2)1 

+4 2m+l (2m) (II) z(2m) (1-21) 

m=O (2m+1)' 

We have, since s(t) = Ad(t),
 
ORIGINAL PA 	 F 1-6

OF POOR QUALITY 

Ry(t,T) : d(t)d(t+T) erf2 a]
 

2 2m+2
 

-
 + 	4 d(t+T) d(t) [(2m+)( M ] [Rn ] 

m 1 0n((2+2m+I) 

(2m) A 2
 [Rn 	r)J 
 (1-22) 

m=C - - /(2m+l)! 

where we have let
 

A2 
 S (1-23) 

2N 

Now consider d(t), 

d(t) = dI p(t-iT) 

where p(t) = l,ts(O,T) and dI = ±1 with equal likelihood. Hence, the time 

and ensemble average of d(t)d(t+T) yields 

Rd(r)T <d(t)d(t+T)> : p(t-iT) P(t+T-iT 

or 

(T) = 	 -IjL] 
 NRZ waveforms

so that we have
 

c-9a
 
ORIGIMNI PAGE I8 

OF POOR QUALITY 1-7
 

R(t) R ) f+ 4 + dQ.nRw F2)m=O 1Z(4 f2 (2m+2)! 

( (2m+l)
 2 


m+-O J 2] /2m+l)! (1-24)
 

The first term isthe SxS term, second isthe SxN terms, and the last term
 

isthe NxN term. Notice at high SNR
 

Ry(T) Rd(T) 
 ( (1-25)

At low SNR
 

lim erf N-


S
 

2
 

ff4 0 (


lm 2m+( - 0 (1-26) 

Therefore, (1-24) converges at S/N= 0 to
 

lrn RT) 4 j (_ )I 2m [m 1 (1-27)
 F 2 2+) nn__ 2m+l 

S 0 m=0 (7 2 

or or 
 T 2 n(T (1-28)


-arc s2n8)
 

wh7 s 2 sn VsT] 

which isa well-known result. It is easy to show that Rd(r) isgiven by
 
ORIGINAL PAGE FS 

I-8 

OF POOR QUALITY 

RdCT) = [ ITf TIT } 
 = o ITH > T (1-29) 

Rd (T) = I < T 

Manchester 

= 4- < ITI < T (1-30) 

Therefore, at the input to the FM summer due to channels I and 2,we have 

s(t) t)t) A dt)2 A cose sinCsct + A dl(t)sine cosCmsctj (1-31) 

where d2(t)has the spectral density given by 

gjd 2(f) = .(f) rf + 49 Rd() B2m+1] ( 2 ]( 
 : }

2 


2m+( 
 + 

4t@{n9 (2m+1)I (1-32) 

with x() the Fourier transform and where d(t) has the spectral density
 

given by 


( llm=0 


I (7R 22

(

m+ 2 )1 

= pdf) ,Y/(f) erf(I) + 4r Rd(T) ] J 22m+

2m+ -33) 

46{ I 1/2m('cr] 2 (21)i (I-3 
1-9 

and where W9(t) denotes the data spectrum of that respective channel.
 

Evaluation of the SxN and NxN spectral densities ismade in
 

Appendix II.
 

REFERENCES
 

1. 	J. H.Painter, "Ideal Limiting of Digital Signals inGaussian Noise,"
 

IEEE Transactions on Information Theory, Vol. IT-16, #1,Jan 1970.
 

2. 	M. Abramowitz and I.A. Stegar, Handbook of Mathematical Functions,
 

Dover Publications, Inc , New York, 1965.
 
ORIGINAL PAGE 'S 

OF pOOR QUALITY 

APPENDIX II
 

SPECTRAL DENSITY OF SxN AND NxN TERMS OUT OF THE LIMITER
 

The purpose of this appendix is to obtain the analytical form
 

of the NxN and SxN terms at the output of the limiter when the input isa
 

baseband signal plus bandlimited Gaussian noise. We follow Helgeson's [2]
 

work inthis section and slightly extend his results to cover fifth-order
 

NxN terms. We model the lowpass filter into the limiter as an ideal LPF
 

of the form illustrated in Figure II-1 below.
 

N0 4n(f)
 

(B = 5 MHz)
 

f
 

-B 0 B
 

Figure II-1. Baseband Noise Spectral Density Model
 

We therefore model the limiter input noise spectral density by
 

- [u(f+B) - u(f-B)]
 

where nI denotes the original process and nk isthe kth-order convolution.
 

Now, using Figure 11-2, we have
 

n2(f) ff iCflD Jnf-fl) dfI (I1-2) 

- ucf1-fi-sjcw (11-3) 2(auatECfl+BD - uCf,-BD]Fu(f-f1 +BD 

Eval uating
 
11-2
 

ORIGINAL PAGE FE 

OF POOR QUALITY 2n1(f)
 

I' 5 MHz '
 

-B B 1 

jnI Cf- fI I
 

r 

II 

-l V' /t
 

f-B f B I 

Figure 11-2 Spectra for Determining Of 2 (f) 

Jn 2 (f) 

2- = 2B-f f>0 (11-4) 

2) = 2B+f f < 0 

so that
 

2 

< .= (/)[2B -Ifl] Iff 2B 

= 0 Ifl > 2B (11-5) 

.Now determine 4n3 (f) Using Figure 11-3, we have 

J'n 3 (f) = fxn 2CflDsnlCf- flD df1 (11-6)
 

So that
 

N 3 f(2Bjlf ID(uCf- + BJ + uCf-f5 - BD) df1 (11-7) 

(L2 ~ -C 
11-3 

ORIGINAL PiWE o&' 

OF POOR QUALITY 

jcfn2Cf I ) 

An1(f- flD 

-2B f-B o f +B 2B f 

Figure 11-3. Spectra for Determining efn3CflD
 

Inthe region 0 < f < B,we then have
 

n3(f) - C2B- IflIdfl + C2 -If, df1 (1I-8) 

(i) f-B 0 

Since /n3(f) isan even function, we obtain
 

(f) ( C3B2 - IfI2 0 < Ifl < B (11-9) 

Now inthe region B < f < 3B, we have 

4n(f) = f 2B If]df 

(No) f-


Since n3(f) isan even function of f,we have
 

.zn3 (f) - 9 B2 3BIfI + B < Ifl < 3B (II-0)
 

No 3 L22
 -

_ 2 CIf<81-0
11-4 

ORIGINAL? PAGE I 

OF pOOR QUALIT' 

To summarize, 	we obtain
 

o(N 0 C 2 2D0 	 < f
 

S3(f) = = B!2f 	 C<f1 

.A'L(f) = (No) B (9 
 2 _3Bl+ fB 	 < f C< 3B (11-11)

Note that the two equations are equal at f= B. Now consider fn4 (f).
 

Figure II-4 illustrates the spectra involved in the convolution.
 

.,n 4 (f) = / 3 CflDPnl~f- fl df I 	 (11-12) 

aJni(f)
 

1 	 I f]
 

-3B -B f-B f+B 	 3B
 

Figure 11-4. 	 Spectra Used for the Fourth-Order Convolution
 

of the Noise Spectra
 

Consider the case 0 < f < 2B. We have
 

or 4 (f) ( 
 2) df 4-f ( B2 Bf+ f df (11-13) 

() f-BB 

or, using symetry,
 
ORIGINAL PAGE IQ 11-5


OF POOR QUALITY 

,tS'n4(f) 16 B3 2Bf2 + } Ifl 0 < If <2B (11-14) 

4 T (N) 

In the region 2B < f < 48, we have that
 

______(2 = B~B- 3Bjfji- df 2B <f <4B (11-15) 

After evaluating, we obtain
 

4( f ) . 16 B3-2Bf 2 + f<f 1' 0 4If 2B (I-16) 

4B 3 *I 3 f 2O ff < B (11-1)
 

(T) 

Now compute ,,5(f). Figure 11-5 illustrates the spectra involved. Now
 

4(fl) 

mjnil (f- (f-f1
 . _

; fl7fl
 

-4B -2B -Bf B 2B 4B
 

f-B f+B
 

Figure 11-5 Spectra Used for the Fifth-Order Convolution
 

(0 < f 4 B) 
11-6 

OF POOR QUALITY 

ORIGINAL PAGE K 

f jn4CflDnCf- f o) dfl (11-18) 

Evaluating for 0 C f 4 B,we have 

~Atfn(f) ff+B (g B- _2Bf2 + ,- Jf,3) df (IT-1s) 

(-f-B
 

so that, after some algebra for 0 < f < B, we have
 

_____ _ 1 - 582f2 + f (11-20) 

TT1 2 4 

Now consider the region B < f < 3B illustrated in Figure I-6
 

f 

0 B/2 B 2B8
 

f-B
 

2 f4
 

Figure 11-6 Spectra Used for tne Fifth-Order Convolution
 

(B < f < 3B)
 

L( Bat- (h 21 we ) dfbBf 8t f + 2Bf 

Nf-
 2B
 

Evaluating (11-21), we obtain by symmetry
 
H-7 ORIGINAL PAGE E 

OF POOR QUALITY 

Bf3 /n5 (f)_ B + T-B3 8f 	 i6 

5 	 f - 5B2f 2 +-Bfl 3 _Ifl B< IfI<3B (11-22) 

Now consider the case when 3B < f < 5B is illustrated in Figure H1-7.
 

-by 

4cf- fl) 

-2B -B 
 B B 2B


Figure H1-7 	 Spectra Used for the Fifth-Order Convolution
 

(3B < f < 5B)
 

We have 

SZ (f) r i 

AM 

(11-23)


(No5 J ' n4 Cf,) JnCf- flJ df1 

jdf 	 (11-24) " (3' Bs-s a-fI2Befb e 

f-B
 

After some tedious algebra, we obtain by symmetry
 

(t)__ - il 87 	 4 12 f__2 ~2 B3+1f 3B<jfj 52 (11-25) 5~ 72 6 46 	 24 
ORIrINAL PAGE i-


OF pOOR QUALIT 
 I-8
 

In conclusion, for the fifth-order convolution, we have
 

5 B4 f 3 An5(f) O-+ Bf I f4 0 < IfI (3B 6
 

( ; gN 6 

2 2 

From equation (1-22), we have that the spectral density out of
 

the lumiter Es given by
 

2
 g(f) 1er2 ( ZB 12 + 0(<fI 2Cr)} BR(t)Rn

1 8 [7(3)(2 i]3 Rd(T) Rn4 Q )} + 43()( --I l <5n( 1) 

Considerhae Frmeuatedn the spectral attfo0 ha
 density
 ( B 726 1 (f) + A2F / l 5 () (11-27)

+ 2 

tsar 


22
 

Th scnantrd 4x deman r e ob vl
 

3 ) ;S 4 + 41 (0) f ] Rd(= ) Rfln()}
 

2 [z2)( n3f + () /n(f) (11-27) 

or1Z 

('M o 
11-9
 

- ORIGINAL PAGE ES 

OF POOR QUALITY F 

~~NO .4~ I z. J{L (11-29) 

a LZ 

2 2 

(Vs- N/B + 0.056 -2 No/B 

The output signal power first term of equation (11-27) to NxN
 

and SxN spectral density is evaluated in Table II-1 using equations (11-27),
 

(II-1), (II-1I), (11-26) and (1H-30).
 

Table II-I. Input and Output SNR Comparisons
 

(IN (O)IN (0)OUT (or)OUT
 

(dB) (dB-Hz) (dB) dB-Hz
 

10 77 27.3 94.3
 

5 72 10.4 77.4
 

0 67 - 1.3 65 7
 

- 5 62 - 6.2 60.8
 

-10 57 -12.15 54.85
 

-15 52 -17.4 49.6
 

Notice that the output C/N6 ratio is greater than the input
 

C/N0 ratio at 5 dB SNRIN and higher. Due to this apparent improvement
 

Cwhich parallels the bandpass limiter I/0 ratio improvement [I]D, we conclude that knowing C/N0 is not sufficient to estimate BER degradation.
 

Consequently, we leave this component out in the BER degradation tables.
 
REFERENCES
 

1. 	W. C. Lindsey, Synchronization Systems in Communications and Control,
 

Appendix I of Chapter 4, Prentice-Hall, Inc., 1972.
 

2. 	R. J. Helgeson, "PI-PSP BER Degradation Due to RF Baseband Output
 

Limiting Amplifier," TRW IOC #SCTE-50-79-491/RJH, September 5, 1979.
 
ORIGINAL PAGE F 

OF POOR QUALITY 

III


APPENDIX 


TWO TONES PLUS THERMAL NOISE INTO A CLIPPER
 

The clipper model isshown in Figure III-l. 
 Inthis section,
 

we follow the work of Davenport and Root [1], Helgeson [2], Shaft [3] and
 

Shaft [4]. We extend the analysis to the two-tone case using an approximation due to M. Gyi [5] and an extension of that approximation due to
 

the author of this report [6].
 

The input to the clipper ismodeled as the sum of two tones and
 

bandlimited thermal noise:
 

1.024 MHz 1.7 MHz 

x(t) = .Tp7 Cos (isc t+ei) +VZPicos~nsct+e 2) + nt (111-1) 

where P1 and P2 are the respective powers of the two tones. We use the
 

indirect approach [1] and form the autocorrelation function
 

RyCt1lt2D = E{lg~Ct1D] 9[xCt2j} (111-2) 

From [1, section 13-3], we can write that
 

RyCt 2
 1 t2D (21 fCw d /fCw2 ) di2 MCI )2 (111-3) 

c c 

where M Cilt2) is the joint characteristic function of xt1 and xt
2.
 

.Now for independent noise and signal processes, we have
 

MxC"In, 2D = MslCwI 'w2) MnCm1 " 2

) Ms2CJ l )

2 Ms3Co1 ,w29... (111-4) 

From Shaft [3], we can write
 
g(x) 

.a 

(Mz)(G) y(t) - - - -xd 

(a) Block Diagram (b) Nonlinearity Transfer Function Model 

00 

ou 

Figure III-1 Clipper Model Used in the Analysis 

to-0 
ORIGINAL PAGE 9,9 

OF POOR QUALITY 111-3
 

w =xCwS eP ( A2 --
 m1=0 m2=0 m 1 'm2 " mn
 

I CA,, Im2CA2 2J ...Imn CAnwn)
 

cos(mISClT) cos(m2wsc2t) . os(mnMsc) (111-5)
 

where
 

CIn 1 mi 0 

-2 mi 0
 

Now Imi(x) is the modified Bessel function of the first kind and of order
 

MI. Hence, we obtain
 

RyCtI t2D= (f) 2 fC1Df CiIffc 2D -in ImiCAll] I.nCAnml)

n
 

c c ml=0 mn=0 1 2 1 

...ImICAlw2D ... Imn CAnu2) exp - 1 2 2 

nx[~
 (0)22)1W 

x2 (111-6) 

k=O kM I C MS 

This can be written as
 

RyCtlt 2D = _Oing mn(Ok'

k=0 ml=0 man=0 ki Em1Em2 
 1 2 m
 

cos(MiSClTn ) cos(m2sc 
 .. cos(mnscnt) (111-7)


where
 
ORIGINAL PAGE E5 111-4 

OF POOR QUALITY 

22 

1 ff, 

f (j wk Im2CA2 ( 2&.-" ImnCAn (wI • e 2 dw hkm1m2.m rn n 2 I I~wj IM ' 

C (111-8) 

where c isan appropriately chosen contour inthe complex plane. Shortly,
 

we will discuss this contour more precisely.
 

An expansion for two signals is given by
 

h2 kml m Ry ) . C In 2 cos(mlSCiT) cos(m2s 

k=1,3,... ml:0 m2 =0 1 2 1m2 c2 / 

m 1+m2=0,2,...
 

+ 	 ys mE h2kmmCosbmcnscT CosbneT\ 

k=0,2,... 0 m2=0 1m2 km1m2 l 1I m2 

m1 +mi2=1,3, .. 	 (111-9)
 

Now consider the clipper amplifier shown in Figure lll-l(b) and
 

the expression in equation form in equation (III-10),
 

-a x -d 

g(x) = (a/d)x lxi <d 

a x d 	 (III-10) 

From (111-8), the hkmm 2 are given by
 

1 -ff ()wk m1 AwI2CA2 e§F i dw (III-lI) 

hkl /2P2 C 2 
 ImlN 0BGd

where A I = 22 the AI's are peak voltages and NoBG ' with i= 
G


the AGC power gain.
 
ORIGINAL PAG -	 111-5 

OF POOR QUALITY 

Let 

; gx(x) (o (111-12) g0(x) x < g g(x) x< 

and also let
 

f(W) = f(0) + f (111-13) 

with 

f( ) = g+(x) e- x dx f(W) = f g(x) e-'x dx (111-14) 

0 -

For an odd transfer characteristic (such as our clipper), we have 

f0 (M = )= -f-w) (111-15) 

Hence, from equation (III-11), we have
 

22 

hkmlm 2 : ] (M)wk ImCIAO m2A2D e 2 d 

kmS _k(2i c1 @( m ImlC Al I2 A2 e1	2 2 d 11-6 

222 

2 ImCb'l Im C e 2 doi (111-16) 7T1 J ww 

c 

where c+ and c are shown in Figure 111-2. Let -w = w' in the second term 

of (111-16) so that we get 
Figure 111-2. Complex Integral Contours Used For Evaluating hkmlm 2
 

22
 

hk 2 ( )w k I CwA1D I. CwA2D e 2 dw
 

km1m2 ri 2 2 [m l m
 

C.+ 
 2 2
 

- I (m')C-w')k(-l)m(-1) m
 TnCu'A1DI cm'A? e (+ ) dtn' 

T "'A "2 e d f "1 Dm 2 )

C+
 

(111-17)
 

k+m +m 2
 -w n 

So when (-1) 1, i.e , k+ml+m 2 is even, then 

hkmlm 2 0 (k+ ml + m2 even)
 

22
 

hkmm J 2 2 ' (m)mk ImCml) ImcmA2De d 

hkm _2 ikImC~l mCA2D 2 din 

1m2 1r 2
 

(k+ mI + m2 odd) (111-18)
 

We now compute (w). We have that
 
OF ORIGINAL POOR QUALITY(%­ PAGE s 111-7 

tp (to) F g(x) e-'x dx (III-19) 

fda xe- x fm e-W x 

- x + ad e dx (111-20) 

0 a 

or finally, 

S(1 e-dd) 
 (111-21) 

Notice that we can write
 

- - d W lim 'afe - a w e w 11-2 4dm) = Fem_> gdm (1II-22)
 

So 

i' l) 

22
 

h li]m Wk m CwA1J Im CwA9 e 2 d. 

km1m2 4 2 jfdC+ 1 2 

1(2)
 

2Ja i ed k ImlCAiWD) Im2CA 2WD) pn~)( 2 w 1123 22 a - t mo) l n m exp (n d w (111-23 ) 

W
 C+ 


for k + m, + M2 odd 

and
 

hkm-m 2 0 for k + mI +m2 even.
 

Note = .
 that hkmm2 I(1)- 1(2) Rotate axis 7r/2 radians clockwise by 

letting m= izso that (see Figure 111-3 for the new contour) 
ORIGINAL PACE w 

OF POOR QUALI1V 

111-8 

G+ 

Figure 111-3. The Contour c+After Rotation 

-

C+ 

(iz) k ImlCAliZD I CA2 iz) e 

2z2 

2 idz (111-24) 

or 

1(0) = 1Ja-e-1z Tf d 

C+ 

(iz)k-2 CAizj Im1 

Im CA2iz 

2C2 

e 

22 

az 

2 dz (111-25) 

Let 

Im (iz) = (i)mj JM (z) (111-26) 

We finally obtain that 

,(I) = 1 f()(,)k+ml+m2 -2 zk-2 

C+ 

e-lZ Jm CAzD J. CA2zD e 

22 

2 dz (111-27) 

Let 

J 

~1 ) 

0 (iz/2)2 

£0 k'r Cm1±+1D] 

(111-28) 
ORIGfNAL PA11 -9

OF POOR QUALITY 

so that
 

.k+ml+m2-2+22 Ir A ZxL 6 2 z dz
 

l f + m A= + z 2 2) 

(111-29) 

To evaluate this expression with m2= 0, we approximate J 0 CA2zJ by 

JoEA2 z) -- 2 (111-30) 

to get
 

1k+ml+m2-2+2k| l m2 "2 A 


2
 (_ a- k +k) fzk-2+ml+2k exz z- 2+A2) 2 4-,) V=r Cm, +-+l] C+ exp L2 7 

(111-31) 

From (111-31) and (111-28), we have 

Ck+ml+m22+2k2t)ml+2 +2-ml-22 

( I ) , a c 1 L2-- T'1 +

(l d 2+ k-l+m +2k

k=0 m~tD a 2+ A2L) 

-F
 F I+m'I+ _ 1 1 

A 2 

a22 2 2+2
 ,_ ,''2' 2 F 1 21


2 n t f 

(111-32)
 

where il(a,b,z) is the confluent hypergeometric function.
 
ORIGINAL OF POOR PAGE QUAL'I- 1

Define an effective input SNR by
 

A 2
 

2 21A2'
 

or
 

zmz
 

,(1) = d +n22 _/2 Pl =o
 klrLmj+k+l) Pi (1) 

+2 + 2; -,2 

X 

V2+ A 2 2k-rmn-2Z)
 J k-m,-2) 2 

(111-33)
 

Hence, from (111-23), we have for the coefficients
 

1-k
 

ml/2 p i
 

hkml hm 0 d Pa 2=0 £ ~lZl .[a2; j 1 = krmlk
 

-2A1~ 1 F 2 , ;
 

'2 V+"m km12

+2
 +2k -d' 

1- 1 2 ( 2 A 1 Fl '2 A 2) 

+ x 

24 2A
 

(111-34)
 
ORIGINAL PAGE S III-]I
 

OF POOR QUALITY
 

Note that, since r(-k)= -c, k> 0 (kan integer), when £>0, the first
 

denominator will be either ±- and hence, hkmlO= 0. Also, when £= 0 and
 

ml+k=1, then iFl(0,b,z)=1, and again the first term drops out. Since ml+k
 

must be odd, we see that the first term drops out.
 

We therefore have
 

1-k
 

vra n2+ A22) /2 1

2
 

hkml /
 _ 1+


2 2"2=
 

-0 2+ 2_ 1 L2/2 P1 P£ l
 

+ L2 

2 2 2 (n2 +
 

x (111-35) 

r[C2 -22.- k-m1 

Now note that
 

xF(Y-X, y, -z) -z
 = e 1FI(x,y, z) (111-36) 

so that letting
 

2d
 

a -2
 

we finally obtain
 
ORIGINAL PAGE IS 111-12 

OF POOR QUALITY 

1-k 

- Pi £/2 

1 + 2] mlf2 	 -b2 /2 - b0n2 Le h 0 	 2 i =o k!cm1+.)! 

F(3- k 1, b2]:4 

X (2 - k- m1) (111-37) 

r 2 

ml-+k odd 

= 0 otherwise 

where 

= Al 2 d 

P 2(( 2 Ab A22 ' n2 A22 

(111-38) 

+ +A 

In (111-29), we now consider the case m2 1 and all mI and all k. Consider
 

the approximation
 

j,(A2x) A2x 	 (111-39)
 

= 2 e\ 32T(1-9 

See Figure 111-4 for a comparison of the approximation. From (111-29), we
 

get
 

+2 t
 ,(l) a ik+ml +m2 -2+2k (ml (AQfk2+mi2z+l
 

L=O ! r Cm-++l1)c 

x exp (-iz - (n2 + dz 	 (111-40) 
l(x), 2 e-X2/6, x -x2/2 e , e/ P , 

- OF ORIGINAL POOR QUALITY PAC

££1 

1.( 

SJl(x)2 

0. 

Xx e-X2/6 

D. 

0 0.2 0.4 0.6 0.8 1.0 12 1.4 16 1.8 2.0 2.2 24 26 2.8 3.0 

(a) Case of 0 <x 30 

0.8' 

06­

04- x e-x 2/6 

02- Jl 

x 

(b) Case of 3 0 < x < 8.0 

Figure 111-4. Comparison of Jl(x) and Its Approximation 

x e-X/6 

e- / 

and Others 
ORIGINAL PACE W 

111-14
 OF pOOR QUALITY 

Going through the same calculations as before, using equation (111-28), we obtain
 

the result
 

3-k+l-m--2m A 1 

(a km+-22 

kmd 2=0 2!r (m1 +t+1 2 A2 , 2k+m +2 9 

1. ___ 2t__ ___ 3 -62_ __ 

[ + 1 , ___ ___ __ 

L 2 

wek+2mY221+l 

1 

A2 

'42 

(-k+2-m,2h) 

I2I 

(111-41) 

Letting 

Al1 

2 

Pi CF2 +A 2

2 

n 3 

and 

A2 

2 

P2) (111-42) 

we then have 
ORGIIAL PAz a 111-15 

OF POOR QUALITY 

a\ ml/2 a'2 + A Plk 

jh(2) P' tn_ km I= d \2/ 2 2=0 £irfm1 +2.+1
 

2 12=0 , [~l+1£+l
 

22 2 - --A2
 2 2 2 

2 t~n2+ - -)]2 
 n2+3
 

r(2-k-m, 2) 2 rk+7-m12] F-

(iii-43)
 

Hence, using
 

hkm 1 lim lh - lh = (111-44)
 k1 kmI1 km =d
 

along with the fact that
 

lrm iFl(a, b, s) = 1 

64*0 

yields from (111-42)
 

/kl (25/ A2) -k/2 
 £


,2=0 !rCml+t+1D
 

+ml+ -[2k 
 -,I +mi+2'+I '3 -'2

k +2+, (m1-45) 

X, 1 (2k (2-k-m---2)U' 2A 2 A ) 7n2 A__r-k+l 2 -( 2m122
 ]

(111-45)
 
ORIC ,AL PAGE IS 

OF POOR QUALITY 111-16 

Since
 

Ir(n)l n=Oo-I,-2,...
 

Hence, when k> 0 r(.)-o and contributes nothing to hkmll. Furthermore, 

when k= 0 and m1+k= 0, the first term 

1 '2k = 0 

Also, when m1+k= 2, the gamma function is and again we get zero, so that
 

we obtain for hkm I
 

2 k (an2_+ 2)-k/


hkml1 d V lm/2 2 k=O £IrCm1 +1D 

+m+2+ Y 2 

2 n2+ 

x
 

2

2 2 A -k+l-m,-2) 

(111-46)
 
6 

111-17
 

REFERENCES
 

1. 	W. B. Davenport and W. L. Root, An Introduction to the Theory of
 

Random Signals and Noise, McGraw-Hill Book Company, Inc., Chapter 13,
 

1958.
 

2. 	R. J. Helgeson, "PI-PSP BER Degradation Due to PI Baseband Output
 

Limiting Amplifier," TRW IOC #SCTE-50-79-491/RJH, September 5,1979.
 

3. 	P. D. Shaft, "Limiting of Several Signals and Its Effect on Communication System Performance," IEEE Transactions on Communications
 

Technology, Vol. COM-13, No. 4, December 1965.
 

4. 	P. D. Shaft, "Signal Through Nonlinearities and the Suppression of
 

Undesired Intermodulation Terms," IEEE Transactions on Information
 

Theory, September 1972.
 

5. 	M. Gyi, "Some Topics on Limiters and FM Demodulators," Stanford
 

Electron Laboratory, Stanford University, California, Rep SEL-65-056,
 

July 1965.
 

J. K. Holmes, unpublished report on approximate evaluation of some
 

nonlinear systems.
 
OF pOR 	QU "-If
 

APPENDIX 	IV
 

EVALUATION OF AN INTEGRAL USED IN THE CLIPPER PROBLEM
 

We start with a result due to Middleton [l]:
 

2 u ­ 1(2) fe-C2Z2 z 1 d jc- 2u r e-'UJ in
 =dz = jc r(u) e JsflnU 

c jarg cj < /4 (IV-I) 

Now
 

r(z)r(l-z) = 	 (IV-2) 

so 

r(u) slfnliu r 	 (IV-3)
 

and
 

1(2) 	 j 7(IV-4)
 

c2Ur(V-u)
 

so that
 

JexpC-c
2z2 )z2u-1 dz = Jexp(-wuJ) (IV-5)
 

c2Ur(l-u)
 

Consider
 

1(3 ) 
= .z z	 (IV-6)
 u- i av(aZ) expC-q2 2Jdz 


Now
 
ORIGINAL PACE FS 

OF POOR QUALITY 

IV-2 

v k=O kir(v+k+l ) 

so that 

1(3) 

J 

Zu- ( v . Tqzd a'V8 

02) k= (ALz)k k!r(v+k+l) 2 2 

(_lk (a) 2 k 

(av ( \2] 

k=O k!r(v+k+l) 

fzv+2k+u 

, 

2 

-q2z (IV-9) 

Using equation (IV-5), we obtain 

1 3) 

1() = 

aV ) (-)k(a 

(-) _T 

k:O kr(v+k+l) 

• 

j7 exp(jJ r[k+ 

2[k+ v (Iut2v 

q L 2F l2~ 

)-­

(TV-10) 

Now let 

then 

Cr = c(c+l)(c+2). .(c+r) (IV-1l) 

r(a-n) (,)(l)n (IV-12) 

so 

r(I- QA4Y-- k) = r 

[1k 

(IV-13) 
ORICINAL VpE V IY-3

O, pOOR QUALIv 

Note also that
 

exp (-j ) = (j)-n (IV-14) 

Hence, from (IV-IO), we find that
 

1(3) /a\V (1)k (ak 17rk1-(2k+v+u) 1 2-1k k+v (Iv-15) )

k=O k!r(v+k+1) r(1- ui9) q2k+V+U 

r(v+1)[v+l]k
 

Since
 

r(a+n) = r(a) (a)n (IV-16) 

we have
 

r(v+k+l) = r(v+l)[v+lk (IV-17)
 

and since
 

i-2k (_I) - k = (-1)k (IV-18) 

we have
 

k =-k ( -2)k (IV-19) 

Now using the identity
 

k 

v lk-a akl 

/ -=21 FI | .y+u2 2~ (IV-20)

k=0 ki1v+l~ k 4q
 
ORIGIN AL PACE F5 IV-4 

OF POOR QUALITY 

22 irilvu / \V U F f u-Jv (az) e- Zdz = rcv+i) dzr(1 -!)q-2q)(IV-21), . ,, 

Now consider
 

(IV-22)
 1(4) = fzu eELiZbOc2zI dz 

C+ 

now
 

e±izbO = cos b0z ± i sin b0z (IV-23) 

and
 

J1 2(x) 2)1/2 /22(x1/2cosx
 

so 

±ib0ze (rb 11{2 Cb0zD± I J/2 Cb0zD} (IV-25) 

Hence, using the above in (IV-22) yields
 

f 2 2 2 FgO 

4e- _ J-L/ 2Cb0zD dz 

c+
 

2z2
 f/u+1/2 -c TWb0
 -26) 

+1 e 2 J2Cbz dz (IV 

soc
 
ORIGIWAL PACG- ib 

OF POOR QUALI U TV 11-5 

7T11+1/2-u-3/2 (b 1 2 l L
 

ucU+3/2 1
 

r - (B (fl \ 2c/ 1/

2 u+2 3 -b1 

11-1/2-u-3/2 

2 r(jr( U++2 .u+ 3/2 F 2 - J (IV-27) 

Hence, we finally obtain
 

- b0 2 +2. -b02 2 2 FU 1

+ F 

u e+lZb0c 

z- e z + L (IV-28)

cC (l2u) c r(_ u) 

jarg ci < Tr/4 
IV-6 

REFERENCES
 

1. 	D. Middleton, "Some General Results in the Theory of Noise Through
 

Nonlinear Devices," Quarterly of Applied Mathematics, Vol. 5,
 

January 1948, pp. 445-448.
 

2. 	R. J. Helgeson, "PI-PSP BER Degradation due to PI Baseband Output
 

Limiting Amplifier," TRW IOC #SCTE-50-79-491/RJH, September 5, 1979.
 
OF pOOR QUALI Y ~Axiomnatix 

9841 Airport Boulevard 0 Suite 912 a Los Angeles, California 90045 * Phone (213) 641 8600 

TECHNICAL MEMORANDUM NO. M8105-1
 

TO: Bill Teasdale DATE: May 11, 1981 

FROM: Jim Dodds Copies: 

SUBJECT: Degradation Due To Dither 

NAS 9-16067 "B" 

Distribution 

1.0 INTRODUCTION
 

In order to reduce the effects of stiction in the Ku-band
 

deployed assembly and the resultant limit cycling, dither is being
 

introduced. This dither is an autonomous sine wave, located at a frequency sufficiently removed from the mechanical assembly resonance to
 

preclude excitation, which keeps the antenna in virtually continuous
 

motion. An heuristic description of dither is given in [1].
 

Not only must a possible resonance excitation be considered,
 

but the dither frequency cannot be too high or low. At very high frequencies, the mechanical assembly cannot repond and stiction isnot
 

affected. At low frequencies, the mechanical assembly excursion is
 

excessive and introduces pointing errors. Hughes has picked a dither
 

frequency of 17 Hz, which does not fall close to any known mechanical
 

resonance. In the following paragraphs, we derive the mechanical motion
 

due to dither and the resultant AM on the sum and difference channel.
 

It is shown that worst-case AM on the sum channel, within the 3 dB beamwidth, is about 0.4 dB; AM on the sum-plus-difference channel is about
 

5 dB.
 

2.0 
 DERIVATION OF ANTENNA RESPONSE
 

The two axes of the deployed assembly antenna are excited by a
 

17 Hz sine wave at sufficient current to produce the equivalent of two
 

foot-pounds of torque. The response of the antenna can be derived from
 

the angular equivalent of F= MA; L IU. The quantity L istorque, equivalent to force, I is the moment of inertia, equivalent to mass, and 6 is
 

angular acceleration. Values of the parameters follow.
 
M8105-1 ORIGINAL PAGE W 2 

OF POOR QUALITY 

L = 2 cos wt (foot-pounds)
 

= 27rx 17 (radians) 

Im = 0.4 (slug-feet2 )
 

I = 0.62 (slug-feet2 )
 

The angular excursion, e, is given by the double integral of
 

6,as shown below:
 

2L cos wt 1 2 cos 34
 

2 (341)2
 

The peak-to-peak excursion is
 

S-(34r) 4 
2 I
1 

Thus, . = 0.000876 radians, or 0 05020, and B = 0.000565 radians, or 0.0324* The vector sum of these two gives a worst-case peak-topeak angular excursion of 0.060. The vector sum isnot an unrealistic 

quantity since the two dither signals are presumably being derived from 

the same source, and the resultant motion isprobably close to being 

in phase.
 

3.0 SUM CHANNEL DEGRADATION
 

Inthis section, we derive the sum channel degradation based
 

on worst-case 0.0600 peak-to-peak antenna motion due to dither. These
 

calculations are based on the approximation to the antenna main lobe pattern given below
 

^ sin ke 

G(e) = 2Ologs , dB 

Taking a nominal 3 dB beamwidth of 1.60, we can solve 

2Olog sin 0.8k = 3 to find k = 1.74, with o indegrees and the sin(.) 0.8k
 

function in radians. Now the peak-to-peak gain variation can be computed
 
4.0 

oRnGI
 M8105-1 


OF pOOR QUA3
 

from 

sin.74 j 2log sinl.74(o-0.065 AdB = 2Olog 

Ll.74e j - L 1.74(e -0.06) 

This isplotted in Figure 1. The points plotted, which we calculated in
 

0.060 increments, appear to fall on a straight line. Since the computations were done on a calculator with built-in linear regression, itwas
 

a simple matter to compute the best straight line fit to the points in
 

the region of interest, 0.060 to D.801. In fact, the points are very well
 

correlated to a straight line, with p = 0.999. The equation is 

AdB = 0.594a - 0 030. 

Ifwe want to maintain the total pointing loss less than 0.3 dB,
 

the maximum allowable e from the (sin x)/x approximation is a= 0.260 minus
 

one-half the dither amplitude, or 0.230. That is,the maximum steady
 

state pointing error must be less than 0.230 to maintain the required
 

0.3 dB maximum loss.
 

DIFFERENCE CHANNEL DEGRADATION
 

The AM on the difference channel can be estimated using the
 

autotrack scale factor and the worst-case antenna motion of 0.0600. The
 

scale factor isdefined as
 

AGA a
 = Ae X G3 

Representative parameters taken from ADL test data are given below:
 

Km =0.5
 

Ae = 0.060, from prior calculation
 

03 = 1.6, the 3 dB beamwidth
 

Gz = 79.4, the antenna gain (38 dB) in volts/volt
 
04 

0.3 

00 

AdB -CC) 0 

0.2. r1 76- 0.06)­

74* L 

0.1 AdB 0 594e - 0 030, p = 0 999 

0.6 0.12 0 18 0 24 0 30 0 36 0.42 0 48 0 54 0.60 0.66 0.72 0.78 

e (degrees) 

Figure 1 Sum Channel AM due to Dither versus Angle Off Boresight 
M8105-1 5
 

AGA is the difference in A channel gain, 62-G I s in volts/volt. Using 

these values, we find that G2 - G1 = 1.49 volts/volt. In order to convert 

this to decibels, we need a reference value. From the test data, the 

difference channel gain at the point of steepest slope appears to be 

about 6 dB, or 2 volts/volt. Thus, G2 = 2+ 1.49 = 3.49, or 10.86 dB. The 

AM near boresight on the difference channel is then 10.86- 6= 4.86 dB. 

5.0 CONCLUSION
 

The 17 Hz AM on both channels is insufficient to degrade tracking or communications performance significantly since the angle-tracking
 

filter is so narrow and the 17 Hz is well below the data frequency. However, a minor improvement in steady state angle-track accuracy is required
 

to maintain the 0.3 dB allowable pointing error with dither present.
 
Axioratix
 

9841 Airport Boulevard o Suite 912 e Los Angeles, California 90045 aPhone (213) 641-8600 

TECHNICAL MEMORANDUM NO. M8302-1
 

TO: J. Dodds DATE: February 16, 1983 

FROM: R. Iwasaki FILE: 16067"A" Distribution 

SUBJECT: Potential Ku-Band Sidelobe Reduction Recommendation 

During the recent Ku-band meeting at Hughes, sidelobe acquisitions
 

by both the radar and communication systems were major problems which required
 

many modifications to circumvent limitations in performance. The relatively
 

high (-20 dB) sidelobes are an antenna characteristic that Axiomatix believes
 

may be reduced by relatively simple means which may or may not affect the program schedule, but should at least be brought to the attention of Hughes management as a possible alternative to consider for future STS missions.
 

First, it is to be emphasized that no major modification is being
 

considered. A simple test is proposed during a currently scheduled antenna
 

pattern measurement that will quickly determine whether or not the change is
 

even feasible. Only one additional pattern cut in the plane of the highest
 

sidelobe is required, which should not impact the tight schedule.
 

This modification is the second item recommended during Axiomatix's
 

antenna study. The primary fix--the RF fence to minimize mutual-coupling effects in the monopulse feed--reduced the sidelobes from 13-14 dB to the present 20-dB level. At the time, this improvement was considered acceptable but
 

recent experimental tests indicate that a wider acquisition dynamic range is
 

desirable. Although antenna design is still somewhat of an art and therefore
 

unpredictable, based on our antenna design experience which corrected the earlier unacceptable sidelobe problem, Axiomatix feels that an improvement to
 

-22 dB is not unreasonable. This would increase the measured acquisition dynamic range from 17 to 19 dB, thereby eliminating the acquisition threshold
 

setting problem that will exist for missions beyond STS-7.
 
Technical Memorandum No. M8302-1 
 2
 

Axiomatix proposes a test inwhich a wedge-shaped ridge of heavy-duty
 

aluminum foil is placed along both sides of the feed-support structure, as shown
 

in Figure 1, to minimize the antenna aperture blockage effects which now appear
 

to be the dominant causes of the high sidelobe levels. The rationale for this
 

modification is to reduce the deleterious asymmetrical obstruction of the aperture by allowing some radiation to propagate past the feed, thereby allowing
 

more destructive interference to occur with radiation impinging on the opposite
 

side of the antenna.
 

The wedge-shaped ridge can be pictured as having polarizing characteristics similar to those of the septum which allows electric fields that are
 

oriented perpendicular to the ridge to be transmitted, whereas the parallel
 

electric fields see an electrical short and are therefore reflected. This behavior is valid even for structures larger than the wavelengths of the electromagnetic radiation and therefore does not depend on any diffraction arguments.
 

The linear polarization, consisting of both RHCP and LHCP components, will then
 

contribute some offsetting aperture illumination versus none with the existing
 

configuration.
 

Incidentally, this technique is used in radome structural support
 

designs to minimize blockage effects of metallic ribs.
 

The exact taper of the wedge is somewhat arbitrary because the stowage envelope of the antenna must accommodate the modification but, as a general
 

rule, the longer the taper, the better the performance. If a metallized fiberglass shell is used to shape the wedge, this structure would also serve as a
 

thermal shield which will help stabilize the thermal environment surrounding
 

the sensitive RF mixers, thus reducing the need for the electrical heater.
 
M8302-1 ORIGINAL PAGE M 3 

OF pOOR QUALITY 

A'
 

Frontal View of the Side View of the
 

Parabolic Antenna Parabolic Antenna
 

A
 

A'
 

Cross Section of the Modified Wedge-Shaped Cross Section
 

reed Support of the Feed Support
 

Figure 1 The Wedge-Shaped Blockage-linirnization Technique
 

for Sidelobe Reduction
 
7.0 S-BAND SYSTEM INVESTIGATIONS
 

Task 2 of Exhibit B was oriented toward the S-band network communication equipment which had already been through the major development stages
 

before this contract effort started, but on-orbit tests with the TDRSS have
 

taken place only inthe last few months due to the delayed launch of the TDRS
 

and the initial problems of the satellite achieving synchronous orbit. Therefore, analyses of the S-band network overall system performance were limited
 

to acquisition times with the wide data filter and data off as well as with
 

Shuttle G/T and EIRP calculations. These analyses are presented inthis section. Itshould be pointed out that additional analyses related to problems
 

which developed inESTL and KSC tests are documented as part of Exhibit A,
 

Task 15, which ispresented inSection 4.
 

Analysis of the Shuttle S-band acquisition times with the wide data
 

filter and data off shows that the acquisition threshold isapproximately C/N0
 

50.5 dB-Hz.
 

A Shuttle S-band G/T of 29 dB and an EIRP of 15.6 dBW was calculated,
 

as documented inTechnical Memorandum No. M8206-3, "Shuttle G/T and EIRP Calculations," dated June 21, 1983. This memorandum was generated inorder to eliminate any possible differences of opinion regarding G/T and EIRP.
 
S-BAND SHUTTLE ACQUISITION TIMES
 

WITH THE WIDE DATA FILTER AND DATA OFF
 

Contract No. NAS 9-16067
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

Axomatix
 

9841 Airport Blvd , Suite 912
 

Los Angeles, California 90045
 

July 9, 1980
 
TO: Gaylord Huth 

cc: Q. Tu 

E. Dodd 

B. Batson 

B. Teasdale 

FROM: J. K. Holmes 

641-8600 

DATE: July 9, 1980 

SUBJECT: S-Band Shuttle Acquisition Times With The Wide Data Filter And 

Data Off 

SUMMARY:
 

The S-Band Shuttle PN code despreader mean acquisition time
 

is estimated for the case the data isturned off and the high data rate
 

bandpass filters are employed in the despreader.
 

Based on the available data,acquisition should occur at and
 

above C/No = 50.5 dB-Hz.
 

ANALYSIS
 

The intent of this memo is to estimate the mean acquisition
 

time of the Shuttle S-Band and PN code despreader with the high data rate
 

filter switched in and the input data turned off. The only available
 

acquisition time data, however, is for the high data rate mode with the
 

high data rate signal being received Thus to estimate the desired
 

acquisition time we must compute the filtering losses that occur in
 

the pre-detection filters of figure 1 for the case of the high data
 

rate signal in high data rate pre-detection bandpass filters.
 

After the filtering losses are computed, it is then a
 

simple matter to adjust the measured acquisition times from reference 1
 

by the filtering loss to yield to the desired results.
 

The S-Band Shuttle forward link PN code length is 2047 chips
 

and has a rate of 11.232 mega chips/sec. The data is Manchester encoded
 

so that ifwe assume a random data stream with the probability of a data
 

one and a data zero being 1/2, independent of the previous bits, then the
 

power spectral density is given by
 
DATRATE 

CM60 

BFACQ CHANNEL (51E 

2.d IF 

INPUT -20 d 

A 1 3 dam AT31 MHa TIMESH E P 

EV 

DE O 

00Kl 

RT 

L 

lIDMTRSSHOD 

VTA 

1/0PN 8 = 1 ftGI o0 

EA COD LOFCCNTI -J' DOPPflENSE 

0 DEPAD ouTPUT 

DESPREADCHANNEL 

F IureI S-Hand Spread Spectrum Processor Functonal Diarm 
ORIGINAL PAGE fi 

OF POOR QUALITY 

(HfT/2)L I S(f) = (PT) sin4(HfT/2)

where P is the Manchester data stream power, and T is the Manchester
 

symbol duration.
 

For zero carrier and code doppler we have that the filtering
 

loss in the bandpass pre-detection filter (assumed to be ideal) is given by
 

BW
 

L = 2PT sin4ifT/2) df

S (HfT/2)


-BW
 

2
 

where BW is the RF bandwidth which is 480 KHZ for the high data rate
 

mode. Numerically evaluating the integral yields L=O 72=-l.43 dB. We
 

conclude that with zero doppler and no data the despreader sensitivity
 

increases by 1.43 dB. Because of the inherent bias in the code loop
 

during acquisition the S-Band Shuttle despreader biased the code loop
 

frequency to -200 chips per second so that a true code doppler of
 

+400,0, and -400 chips per second respectively appears to the acquisition
 

circuitry and the code loop as +600, +200, and -200 chips per second,
 

respectively.
 

Now at a code doppler of 400 cps, corresponding to
 

about 60 KHZ carrier doppler, the filtering loss is given by
 

BW +60 KHZ
 

L =f PT sin4(fT/2) df
 (nfT/2)
 

- BW
 

_2-+60 KHZ
 

which when evaluated numerically is given by L = -1.52 dB
 
By replotting figure 9 of reference I (which is the average
 

code acquisition time versus C/No (dB-Hz).) with the filtering loss
 

subtracted from the abscissa for each acquisition time we obtain figure 2
 

which is our desired result. Notice from figure 2 that threshold appears
 

to be around 50.5 dB-Hz.
 
t
 

I­

-;zoo cp I I-~ tut 

ORIINA r,' ' I 

-4/ -20 IA aOF POOR QUAL|T" 

48t 49t Yo 5/ 5:/,fS 53 ,54­
References
 

(I) J. K.Holmes, "Predicted Mean SSP Code Acquisition Times with the
 

New Algorithm" TRW IOC SCTE-50-76-251/JKH, July 21, 1976.
 

(2) W.K. Alem, G.K. Huth, J.K. Holmes, and S.Udalov "Spread Spectrum
 

Acquisition and Tracking Performance for Shuttle Communication Links,"
 

IEEE Transactions on Communications, Part I,Nov. 1978.
 
ORIGINAL PAGE M 

OF POOR QUALIAxio aix 

9841 Airport Boulevard o Suite 912 a Los Angeles, California 90045 0 Phone (213) 641-8600 

TECHNICAL MEMORANDUM NO. M8206-3
 

TO: G. Huth DATE: June 21, 1982
 

FROM: P. Nilsen COPIES: J. Johnson
 

M. Zealon
 

SUBJECT: Shuttle G/T and EIRP Calculations
 

On June 17, 1982, I met with Mike Zealon, Glen Yates and Dick Davis
 

to review the RI methodology for Shuttle S-band G/T and EIRP calculations.
 

Also, we reviewed the RI calculation of RF feedline insertion loss. The RI and
 

Axiomatix methodologies for these calculations were essentially the same, with
 

the minor exception that RI did not include contributions to effective noise
 

temperature beyond the preamp. The formula used by Axiomatix for effective
 

noise temperature, namely,
 

(L2 - 1JT0L1 

TA + CLI- 1)T 0 + CNFpA - ToL + Teff 
 1PA
 

+ CNFXPDR - 1T 0LIL 2 

GPA
 

where
 

TA = antenna temperature (75°K) 

L1 = loss between antenna and preamp (4.34 dB)
 

L2 = loss between preamp and transponder (0.5 dB)
 

NFPA = preamp noise figure (2.6 dB)
 

GPA = preamp gain (20 dB)
 

NFXPDR = transponder noise figure (7 dB)
 

did include the temperature contribution for the cable between the preamp and
 

the transponder as well as the transponder's contribution.
 

However, we see that, after making these calculations with and without these last two terms, the difference in G/T is only approximately 0.15 dB,
 

with the more precise result being G/T = 31-2 = 29 dB for 80% coverage.
 
M8206-3 
 2 

The EIRP calculation is given by
 

EIRP = PT + GANT - LT
 

where PT is the transmitter output power (20 dBW), GANT is the antenna gain
 

(80%)(2 dB), and LT is the transmit circuit loss (6.4 dB).
 

The calculated EIRP is 15.6 dBW for 80% coverage. It should be
 

noted that, in both the EIRP and G/T calculations, the transmission line loss
 

due to VSWR was included in circuit loss values. The VSWR loss was found
 

according to the attached nomograph.
 
- -

3 M206-3 OF ORIGINAL POOR QUALITY PAGE ifS

VSWR Nomograph #2
 

POWER TRANSVOLTAGE POWER RETURN TRANS- MISSION VSWR REFL COEFF REFLECTED LOSS MITTED LOSS

% DB % DB 

130 ­

2~9±~I7t2 " S -IS21­

- -1 - 209 

-I8 

3 ' 

-r.. i .... . I97­

- I .. .I ­

4 --- --- 4- .. I I--- ­ - .0- - .IL. . -

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-I--- ..-- I_1 -- . 9 - - . ,~, / '------ . I-

- --_ _ I - '530 ' 

- -'-_ -'- - - £20
-- 1-93 - ­ - 4-

-0 

---

/ t
 --:-I j T - - -, I _.. I . i
 

t a- - i * r -- _I -- - . .i... 

ii
 

'7m i I
 

... - -..... ..------­ 193 -- I- t 

--i- - I { 2 

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I-t" i 
8.0 ORBITER ANTENNA STUDIES AND INVESTIGATIONS
 

Task 8, Orbiter Antenna Patterns, was composed of two separate areas
 

of investigation. First, were analyses to improve the performance of the Sband quad antennas. Axiomatix was heavily involved in this effort and provided
 

many significant contributions to the improved coverage performance. The second area investigated under this task was to develop techniques by which to detemine the actual Orbiter antenna gain and coverage from on-orbit measurements.
 

After an initial investigation, the results were presented inAxiomatix Report
 

No. R8009-1, "The Concept of Maximum-Entropy Filtering and Its Feasibility as a
 

Candidate for Shuttle Antenna-Gain Processing," dated September 5, 1980. Following the review of this report by NASA, Axiomatix was directed on June 1, 1981
 

to terminate further analysis in this area as there seemed to be little use for
 

on-orbit measurements derived from several ground stations which were calibrated
 

differently.
 

Inorder to investigate methods by which the S-and quad antennas
 

would meet the relaxed gain-coverage requirement of 4 dBci over half the sphere,
 

Axiomatix extended many of the concepts used in the switched two-beam quad antenna, taking advantage of the Rockwell-developed basic hardware. The outcome
 

of this study has been the evolution of a new generation of variable-gain,
 

multiple-switched-beam antennas, the principles of which can be applied to the
 

improved Shuttle Orbiter S-band quad antennas.
 

The switched two-beam quad antenna developed by Rockwell was an improvement over the earlier single-beam quad antenna built by Watkins-Johnson
 

but, due to over-optimistic computer analysis projections of the antenna patterns, the actual performance results were disappointing. The primary limitation of the gain peformance was the selection of only two elements and two beam
 

positions, the minimum required for switched-beam operation. Using the same
 

type of hardware configured in a slightly different manner, Axiomatix developed a two-dimensional cross array that could be electromechanically scanned
 

in both the roll and pitch planes as 2Mx2 N discrete-beam positions, where M
 

and N are the number of double-pole/double-throw electromechanically latching
 

switches in each plane. The advantages of using this type of switch rather
 

than electronic-phase shifters are: (1)high power-handling capabilities, (2)
 

low insertion losses, (3)absence of electrical "holding" power, and (4)the
 

binary nature of the double-throw switch which can readily be converted to a
 

binary code for a specific beam position.
 
As part of Axiomatix participation in the development of S-band quad
 

antennas, Axiomatix personnel attended the design reviews and technical exchange
 

meetings. InAxiomatix Report No. R8007-2, "S-Band Quad Antenna PDR Evaluation,"
 

dated July 23, 1980, recommendations are made to improve the gain-coverage performance since the two-beam approach did not achieve the specification. This
 

report attempted to illuminate the dilemma by suggesting, first, some concavelens design concepts for the existing randome which would broaden the dipole
 

beams in the array plane, an essential feature to correct the present two-beam
 

coverage deficiencies. If the proposed three-beam scheme was to be used instead
 

of the two-beam design, Axiomatix recommended that a switched four-beam configuration be seriously considered as an alternative system since the hardware and
 

software would be extensively modified for the three-beam scheme. Finally, a
 

novel RF fence design usign the total-reflection concept of critical angle was
 

outlined, which should substantially reduce the amount of mutual coupling and
 

deleterious interaction effects between the two dipole-array cavities for all
 

switched multiple-beam configurations.
 

Further investigations of the four-beam configuration to simplify
 

the design are presented inAxiomatix Report No. R8106-4, "Simplified Orbiter
 

S-Band Quad Antenna Planar Array Candidate Configurations," dated June 25, 1981.
 

Also, improvements to the lens concept required to broaden the beam of the
 

two-beam design are presented inAxiomatix Report No. R8112-1, "A Protruding
 

Wedge-Shaped Dipole Element Modification For The Orbiter S-Band Quad Antennas,"
 

dated December 9, 1981. Finally, the concept of the two-dimensional cross array that would be electromechanically scanned in both the roll and pitch planes
 

at 2Mx2N discrete-beam positions is developed inAxiomatix Report No. R8112-4,
 

"ALarge Switched-Beam Planar Array," dated December 15, 1981, and Axiomatix
 

Report No. R8209-3, "AVariable-Gain Multiple-Switched-Beam-Antenna Configuration,' dated September 22, 1982. To implement a variable-gain multipleswitched-beam array, the beam-switching logic must be simple. This switching
 

logic is presented in Axiomatix Technical Memorandum No. M8209-4, "Binary
 

Code Designation of Multiple-Switched-Beam Positions," dated September 30,
 

1982, as well as inAxtomatix Technical Memorandum No. M8209-5, "Simplification of Antenna Beam-Switching Logic," dated September 30, 1982.
 
THE CONCEPT OF MAXIMUM ENTROPY FILTERING
 

AND ITS FEASIBILITY AS A CANDIDATE
 

FOR SHUTTLE ANTENNA GAIN PROCESSING
 

Contract No. NAS 9-16067
 

Interim Report
 

Prepared for
 

Lyndon B Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

V. K. Murthy
 

With Contributions By:
 

A. Polydoros
 

C. L.Weber
 

P. W. Nilsen
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8009-l
 

September 5, 1980
 
1.0 	 INTRODUCTION
 

Maximum entropy filtering has been suggested by John Tranter as
 

a candidate approach for filtering the raw antenna gain measurements made
 

during early Shuttle flights. The motivation for this suggestions isthe
 

fact that this method has been shown to yield power spectral density (PSD)
 

estimates which have higher resolution than traditional periodogram (Wiener) or weighted periodogram estimators. This advantage isespecially true
 

when the measured data is sparse.
 

The following discussion offers an explanation of maximum entropy
 

filtering and presents several examples. Most important, an explanation
 

of why this technique isnot suited to the determination of the Shuttle
 

antenna gain pattern ispresented
 

2.0 	 DISCUSSION
 

The expression "maximum entropy" is a misnomer inthe sense
 

that one generally maximizes the information, not entropy. However, this
 

expression is used in connection with the channel capacity of a communication system where maximum entropy corresponds to maximum information
 

transmission rate. Instatistics and probability theory, the entropy
 

H(X) of a random variable X with probability density function (PDF) f(x)
 

isdefined by
 

HCX) =- f(x) £nf(x) dx 	 (I)
 

The entropy H(X) isclosely related to the disorder or uncertainty associated with making a realization X. For that reason, maximizing the entropy
 

is a method for finding the distributions that represent high uncertainty
 

or, equivalently, a state of high ignorance Thus, given a set of constraints, the maximum entropy method provides a candidate distribution
 

which confirms the constraints without assuming anything more,
 

For example, let the constraints be
 

/ x f(x) dx = E(X) = m0 X) 0 	 (2) 
2 

OF POOR QUALITY 

ORIGINAL PAGE IS 

Then the distribution that maximizes (1)subject to (2)is
 

f(x) = - _e 0 (3) m0
 

Notice that, since constraint (2)specifies m0, the maximum
 

entropy distribution contains no additional information and istherefore
 

maximally random.
 

As another example, consider maximizing (1)subject to
 

f x f(x) ds = E(X) m0 

(4)
 

Cx - E( X))2 f(x) d2 a2 = a02 

f 0 c x E ~ f X 0

In this case, the distribution that maximizes (1)subject to (4)is
 

f(x) 1 exp 1 2(X m 02) (5) 

f T) 0 2a0 

Notice that, since m0 and a0 are specified by constraint (4), the maximum
 

entropy distribution given by (5)for this case contains no further information on the underlying PDF of X and is thus maximally random.
 

Similarly, extending our concepts to a random process, it can be
 

shown that the maximum entropy random process with constraints on only the
 

first and second moments is a Gaussian process. Also, the entropy for a
 

stationary Gaussian process may be shown to be given by
 

-f- log f(x) dx (6)
 X:0 2 
ORIGINAL PACE [$ 3 

OF POOR QUALITY 

where 0 isthe domain inthe frequency space and X isthe spectral
 

measure of .
 

Let X(t) be the stationary Gaussian process inquestion, with 

zero mean and with covariance matrix of observations XCtl), XCt 2),..., 

XCtn) sampled at times tl, t2,...,t n given by 

Rij = E(xixj) = RjIt = ri_ 

where, for brevity, we write XCtiQ = X,, i=l,2,...,n and 

2
 rk = e K f(X) dx 	 (7) 

7r 	 Discrete 

Time PSD 

The maximal entropy spectral density estimate subject to the
 

constralnts
 

0
 rK = rK K = 1,2, . ,n 	 (8)
 

isobtained by maximizing (6)subject to (8)and will turn out to be a
 

rational function of rK.
 

Thus, in general, the maximum entropy principle works if the
 

constraints are practically noiseless or error-free and, ifthey must be
 

estimated, they should be based on a large data base of observations
 

corresponding to repeated cycles or replications The specification of
 

the constraints could also be based on proven prior knowledge arising
 

from past and cross-sectional experience.
 

We therefore resort to smoothing procedures for the antenna
 

gain data. The unacceptability of the maximum entropy method of estimating the gain at missing points will be demonstrated for the onedimensional case.
 
4 

OF POOR QUALI gy 

Assume that
 

fCxQ) - c1 , i=1,2,...,n (9) 

fCXl ) > 0 (10) 

N 

I fCx1) = 1 ,N > n (11) 

1l 

We will now estimate fCx,) at the missing points by using the maximum
 

entropy principle, i.e., choose fCx,) such that
 

N 

H(f) = - fCx1) log fCxiJ (12) 

1=1
 

ismaximized subject to (9), (10) and (11)
 

From (9), (10) and (11), itfollows that itsuffices to maximize
 

N 

Hi(f) = fCx,) log fCx1) (13) 

i=n+l
 

But maximization of (13) subject to (10) and (11) or, equivalently,
 

(11') where
 

N

3 fCx1D 1-c (11') 

i=n+l 

where c = cl +c2+.. +cn' isattained by
 

1 - c i=n+l,...,N (14) e.tN uio dit - n p

i.e., uniform distribution for'the missing points.
 
5 

This solution, while mathematically correct, is totally
 

unacceptable for the specific problem at hand since itdoes not take into
 

account the smoothness properties of the gain function. Furthermore, it
 

is easy to demonstrate by examples that, inmost cases, the missing gain
 

distribution between two measured points is not a uniform distribution.
 

In the next communication, we specifically address the problem
 

of smoothing the gain data, in the presence of noise, and establish the
 

rationale, obtain acceptable procedures and develop algorithms for
 

implementation.
 

3.0 CONCLUSIONS
 

The maximum entropy filtering technique is not suitable for
 

processing the Shuttle antenna gain measurements. Therefore, it is recommended that no further investigation of this technique be undertaken.
 

Axiomatix recommends that a more "ad hoc" approach to processing the raw gain measurements be utilized An example of such a technique
 

is the nearest neighbor interpolation to fill inmissing data combined
 

with a "pseudo-Kalman" filter for smoothing data from multiple passes and
 

multiple stations. This is the approach which Axiomatix is now pursuing.
 
S-BAND QUAD ANTENNA PDR EVALUATION
 

By
 

Dr. Richard S. Iwasaki
 

Contract No. NAS 9-16067
 

Axiomatix Report No. R8007-2
 

July 23, 1980
 
S-BAND QUAD ANTENNA PDR EVALUATION
 

July 15-16, 1980
 

Dr. Richard S. Iwasaki
 

Axiomatix
 

1.0 INTRODUCTION
 

At the S-Band Antenna Preliminary Design Review (PDR) held at
 

Rockwell/Missile Space Division inAnaheim on July 15-16, 1980, the
 

main issue was the proposed change from the initial switched two-beam
 

configuration to a switched three-beam configuration for the S-band
 

quad antennas. The other major issue of note was the poor antenna
 

patterns of the lower S-band quad antennas. Further work on these
 

antennas cannot proceed until a firm decision ismade whether or not to
 

pursue this new approach because of the required alterations inthe hardware and software of the system. The meeting concluded with a plan to
 

resolve this issue within ten days after the pertinent system personnel
 

are queried as to the ramifications of the changes.
 

The other aspects of the S-Band Antenna Program appear to be
 

progressing satisfactorily except for some specification deviations which
 

must be resolved by either waiver or additional engineering adjustments,
 

such as the axial ratios of the quad antennas. However, inlight of the
 

major impact of a new three-beam configuration and the obvious inadequacy
 

of the low quad antennas, these latter problems warrant immediate attention.
 

2.0 	TECHNICAL SUMMARY OF THE PDR
 

From the data supplied during this PDR, itappears that the three­

-beam approach was analyzed within two weeks of the last technical exchange
 

meeting held May 14, 1980. The implication is that the two-beam approach
 

was deemed unsatisfactory very soon after the preliminary antenna pattern
 

measurements were made and, since completed three-beam patterns were
 

presented, that the decision has been tentatively made and the two-beam
 

approach has been essentially abandoned.
 

The two-beam approach resulted in a 65% 4.0 dB spherical coverage,
 

for the upper quad antenna (versus the coverage specification of 85%) and
 

a modified two-beam approach which moved the aft beam forward to cover the
 
2
 

actual roll plane (by using a -660 instead of a -90° phase shift) gave
 

a 64% 4.0 dB coverage. The lower quad gave a 35.1% 4.0 dB coverage,
 

indicating serious measurement and/or test antenna design problems.
 

The corrective measures necessary to improve the two-beam 4.0 dB
 

coverage include broadening the beam to redistribute the peak 6.5 to 7.0 dB
 

gain more evenly over the specified coverage area. The technique attempted
 

by Rockwell MSD was to place a "parasitic" ring on the radome to enhance
 

generation of a third-order mode to put more radiative energy away from the
 

axis. This parasitic ring was intended to function similar to the directional
 

reradiators of the Yagi antenna, but the alternate phasing of the third-order
 

mode greatly complicates the design. The modification did not produce the
 

desired results and itappears that at this point an alternate concept
 

(e.g. three beam) was seriously evaluated.
 

3.0 	 POSSIBLE ALTERNATIVE DESIGN MODIFICATIONS
 

Some possible approaches to broaden the two beams might be considered
 

inthe future ifthe three-beam scheme cannot be realized. One isusing a
 

concave-lens configuration on the radome. Itwas mentioned that Rockwell
 

MSD has experimented with this concept earlier, but the concave surface was
 

on the 	opposite side away from the dipole, which is not the ideal situation
 

ifit is assumed that the dipole radiates quasi-spherical waves and the
 

radome surface isto be inintimate contact with the Thermal Protection
 

System (TPS) tiles. Further, the argument of high cross-polarization levels
 

for broader beams must be evaluated more closely ifthe beam is selectively
 

broadened, primarily inthe array plane where the beam isnarrowed by the
 

two-element array.
 

Another reason for using an inner radome aperture concave
 

surface isthe reduction of the cross polarization. The high dielectric
 

constant (er=4) radome behaves as a polarizer inthe present dipole configuration,and therefore unequal magnitudes of orthogonal linear polarization
 

(equivalently orthogonal circular polarization) are transmitted through the
 

radome. The reflected components of polarization create the higher VSWR
 

mismatch measurements, which also appear to exceed the desired performance
 

specifications. Impedance matching to decrease the VSWR by design isdifficult to achieve within the limited area of the aperture, but the concave
 

inner radome surface will most probably decrease both the VSWR and the
 

cross-polarization levels.
 
3 ORIGINAL PAGSE 1 

OF POOR QUALITY 

The reason for the selective polarization reflection isthe
 

optical principle of reflection at a dielectric interface or, inelectromagnetic theory, impedance matching.
 

0.40 

mi2 i 

0
 

0 0 600 W)0 

Fraction of incident power reflected from the surface of an infinite
 

dielectric slab vs. angle of incidence: (a)electric vector perpendicular
 

to the plane of incidence, (b)electric vector inthe plane of incidence:
 

n = 1.6. 

Because of the quasi-spherical phase fronts radiating from the dipole
 

element, the incidence angle varies from the central axis and selective
 

polarization transmission occurs. The same phenomenon also occurs at
 

the outer radome surface where it is incontact with the TPS tiles and
 

should be considered indetermining cross polarization effects.
 

The increased path lengths through the outer edges of the radome
 

-also distorts the quasi-spherical phase wave front which isthe basis for
 

array theory. Insummary, the radome should be analyzed as a polarizing
 

lens system.
 

By broadening the individual dipole pattern inthe array direction,
 

power redistribution can be achieved since the excess power inthe peak
 

gain (6.5 to 7.0 dB) isactually being selectively rearranged to reduce
 

the pattern-multiplication factor inthe array plane. Therefore, the beam
 

broadening essentially tries to re-establish the symmetrical single-dipole
 

pattern, but uses the two-element phased array to electronically tilt the
 

resultant pattern in the desired directions by switching the appropriate
 

phase shifts.
 
ORIGINAL PAGE 4
 

OF POOR QUALIT-Y 

Inorder to reduce the gain and broaden the antenna coverage,
 

a 
concave aperture interface isproposed. Snel's law, which bends
 

the ray closer to the interface normal ina denser medium, isnow manipulated to spread the energy outwards since more of the incident spherical
 

wavefront energy istransmitted into the dielectric at divergent angles.
 

I W,-RAY TRACING CONCAVE SURFACE 

PRESENT PLANAR RADOME 
 PROPOSED PLANO-CONCAVE RADOME
 

i, , LENS AXIS 

LOWER QUAD CONVEX OUTER SURFACE RADOME
 

The dielectric radome, presently polyimide E-glass, ismachined
 

such that the dielectric interface to the dipole antenna cavity isflush
 

with the cavity aperture. The shape of this radome, since the aperture
 

is indented to accommodate the radome~resembles a convex lens which focusses
 

incident radiation. 
This effect is even more apparent when the dielectric
 

constant of the dielectric material ishigher than anticipated, as the case
 

when the polyimide E-glass replaced the preliminary fiberglass material,
 

especially inthe case of the lower quad antenna where the outer radome
 

surface is slightly curved. The net result of this configuration isa
 

plano-convex lens which focusses radiation, or equivalently, increases the
 

gain. This focussing effect of the convex 
radome surface isfurther enhanced
 

by the quasi-spherical waves radiating from the dipole element which is located
 

inclose proximity to the radome. Inorder to compensate for this curvature,
 

(ifthis convex lens shape isindeed found to contribute to the poor lower quad
 

pattern since it isthe only difference from the upper quad antenna),a concave
 

inner radome surface will reduce the amount of focussing. Geometric ray
 
5 

tracing can be useful for determining the overall lens effect. This
 

technique can be extended to produce divergent beam broadening in the
 

lower quad with increased curvature of the inner concave lens surface.
 

This beam broadening technique need not be symmetrical. In
 

fact, the concave curvature can be tailored to the broadening of the
 

beam indesirable directions. The advantage of this type of modification
 

isthat the existing quad antennas can be used and only the dielectric
 

radome, which isreadily replaced, ismodified.
 

The lens effect of the radome slightly affects the effective
 

location of the dipole element phase centers. The convex outer radome
 

surface of the lower quad decreases the dipole spacing since the dipoles
 

are not located on the longitudinal axis of the lens, which might contribute to the poor results. For the upper quad with the planar outer radome
 

surface, this effect isless dramatic since, although the "depth" of the
 

phase centers into the Orbiter varies, the dipole spacing remains constant,
 

and the earlier analytical models are still valid.
 

4.0 RF FENCE DESIGN
 

Although mutual coupling between dipole elements is not obviously
 

a major problem, it isaxiomatic that itexists for adjacent radiators with
 

a common radome. During the recent performance tests, the mutual coupling
 

isindicated by the frequency-swept input return loss measurements where
 

-10 dB was not uncommon. A desirable design goal then isto attempt to
 

minimize this effect as much as possible, especially since the measured
 

performance isunsatisfactory This problem iscomplicated in the quad
 

antenna design because of the limited space available A technique to
 

surmount these difficulties with minimal effort is suggested here which
 

can be utilized for the switched two-or three-beam case (and even the proposed switched four beam case to be discussed later).
 

The optical basis for this RF fence is the physical principle of
 

the threshold-critical angle for total reflection which exists for electromagnetic radiation propagating from a dense (high dielectric constant)
 

medium to a less dense one, as isthe case of the radome/free-space interface
 

of the S-band antennas The threshold-critical angle 0c isderived from
 

Snell's low of refraction to be
 =
 sin ec l]
 
6 

where the angle ocis measured from the boundary normal and er is the
 

relative dielectric constant of the dense medium; in this casepolyimide
 

E glass 	with er=4 . Thus any radiation incident on this boundary at a
 

greater 	glancing angle (e>ec) is totally reflected, constituting an RF fence
 

in the classic definition.
 

The near-field radiation pattern is difficult to characterize. If
 

the radiated waves can be approximated by quasi-spherical waves and geometric 	ray tracing is employed, this type of concept of RF fence to isolate
 

the two 	dipole-radiating elements becomes very attractive since there are
 

few readily implemented isolation mechanisms that are frequency independent.
 

The design criteria for this particular configuration is to have all
 

incident rays greater than 300 from the boundary normal so that total reflection occurs. The separation between the two dipole cavities is small,
 

but the 	use of the smaller diameter iris or circular aperture provides sufficient 	room to mill the triangular groove into the radome, restricted primarily 	by the thickness of the radome and the requirement to retain structural
 

integrity since the groove is prone to fracture. To minimize crack initiation,
 

the production model should have a minimum milling radius at the tip to reduce
 

stress concentrations. The depth of this milling groove is the primary factor
 

in determining the degree of isolation, but even a shallow groove will disrupt
 

surface 	waves at the radome/ground plane boundary that result in mutual coupling.
 

5.0 	 SWITCHED FOUR-BEAM CONFIGURATION
 

The switched three-beam configuration proposed by Rockwell/MSD
 

attempts to alleviate the apparent design problems associated with obtaining adequate 4.0 dB coverage. By adding the third beam to cover the gap or
 

-"cusp" 	between the initial switched two-beam configuration, the coverage
 

problem 	isessentially solved, at the expense of added complexity. At the
 

PDR the 	consensus appeared to favor this approach as the one with the highest probability of successfully meeting the coverage specification. The
 

impact of such an abrupt design change, however, has not yet been fully
 

assessed, and a firm decision to change to the switched three-beam system
 

was delayed until the responsible system and software engineers could be
 

queried.
 
7
 

The implications of such a design change require further examination, at least in principle, to ensure that engineering requirements are
 

optimally satisfied, especially for the added complexity. Besides the
 

appropriate phase shifts, an additional identical RF switch isinserted
 

inseries with the switched two-beam configuration to provide the zero
 

phase-shift differential for the broadside array generating the center
 

beam. However, one RF switch position isredundant and unused The
 

question arises whether or not this switch position can be employed in
 

a useful manner, which brings up the point of the feasibility of a switched
 

four-beam configuration with the identical hardware and similar software
 

altqrations.
 

The proposed switched four-beam would have the primary advantage
 

of a larger area coverage with higher gain, taking full advantage of the
 

peak gain which is approximately 6.5 to 7.0 dB (even higher at broadside).
 

The differential gain over the 4.0 dB coverage specification of 2.5 to 3.0 dB
 

over the additional fourth switched beam area may be critical for some marginal communication links, and, if it was available for little extra software
 

development (which has to be revised anyway), the consideration of the switched
 

four-beam approach might be justifiable.
 

A logical switched-beam configuration might use an arbitrary center
 

line slightly offset from the roll plane (90*) so that one of the two central
 

beams provides adequate coverage inthis critical region. The other central
 

beam would then be electronically and mechanically tilted slightly forward
 

Since these inner two beams are closer to broadside, they would have higher
 

gains and narrower beamwidths inthe array plane than the outer beams (yet
 

experience some beam broadening for broader coverage than the third
 

.broadside center beam of the switched three-beam configuration). The outer
 

two beamwidths can therefore be positioned for the desired coverage, with
 

a bias for more coverage towards the nose of the Orbiter. The additional
 

flexibility provided by the fourth beam simplifies the coverage "filling in"
 

process, an important consideration for the lower quad pattern results where
 

obvious problems exist.
 

Two switch settings of one RF switch then might be "fore" and "aft"
 

of the imaginary center line and then the other two switch settings of the
 
8 

other RF switch would be "inner" and "outer", denoting the relative
 

locations of the beam. The appropriate phase shifts for the various
 

beam settings can be readily determined. The availability of the four
 

distinct switched beams from the two serial RF switches is not conceptually unreasonable and makes maximum use of the proposed hardware modification.
 

6.0 CONCLUSION
 

There are two possible solutions to the switched-beam S-band quad
 

antenna problem. One is to continue to pursue the original switched twobeam quad-antenna concept which was initially selected for its straightforward simplicity. The problem with the two-beam approach is that the performance specifications have not been achieved since only 65% of the 85%
 

4.0 dB coverage requirement has been measured. Without new design concepts
 

to broaden the dipole patterns and therefore the coverage, the two-beam
 

effort was abandoned in favor of the more promising three beam scheme which
 

which provides the necessary flexibility at the cost of added system complexity, which is presently being evaluated. Obviously the three beam scheme
 

is more likely to be successful and has nearly attained the desired 85%
 

coverage for the upper quad, according to the 78.5% coverage of the initial
 

three-beam measurements.
 

This paper has attempted to illuminate the dilemma by suggesting
 

first some concave lens design concepts for the existing radome which would
 

broaden the dipole beams in the array plane, an essential feature to correct
 

the present two beam coverage deficiencies. This concave surface also reduces
 

the reflection and polarizing properties from a dielectric boundary, which
 

minimizes polarization effects and therefore decreases the axiall ratio. In
 

.the likely event that the more promising proposed switched three-beam scheme
 

isselected instead, it is recommended that a switched four-beam configuration be seriously considered as an alternative system since the hardware
 

and software for the three-beam scheme conversion have to be extensively
 

modified anyway. The addition of another beam position provides higher gain
 

levels in that particular coverage area (2 to 3 dB of communications link
 

margin) and therefore obviously increases the 4.0-dB coverage area. And
 

finally, a novel RF fence design using the total reflection concept of critical angle has been outlined which should substantially reduce the amount of
 
9 

mutual coupling and deleterious interaction effects between the two
 

dipole-array cavities for all switched multiple-beam configurations.
 
SIMPLIFIED ORBITER S-BAND QUAD ANTENNA
 

PLANAR-ARRAY CANDIDATE CONFIGURATIONS
 

Contract No. NAS 9-16067
 

Exhibit B
 

Interim Report
 

Prepared for
 

NASA Lyndon B Johnson Space Center
 

Houston, Texas 77058
 

Technical Monitor: William Teasdale
 

Prepared by
 

Axiomatix
 

9841 Airport Blvd , Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8106-4
 

June 25, 1981
 
A recent presentation of more breadboard evaluation configurations
 

for the Orbiter S-band quad antennas considered the mechanically switched
 

three-element/four-beam system and two electronically phased eight-element/15­

beam and 16-element/20-beam systems. -The primary reason for proposing thesesystems isthat the current two-element/two-beam system gives only 51%, 4-dBci
 

coverage instead of the 85% coverage specified.
 

As stated earlier, Axiomatix feels that the present two-element/twobeam configuration has not been optimized since 2 dB excess mainbeam gain indicates that the radiated energy isnot uniformly distributed to the 4-dBci level.
 

However, since Rockwell deems any further suggested modifications to the present
 

inadequate system as infeasible, by definition, a configuration change istherefore mandatory iffurther performance improvement isdesired.
 

Using the available experimental pattern measurements of the existing
 

dipoles, the Axiomatix-proposed three-element/four-beam mechanically switched
 

configuration satisfied the 85%, 4-dBci coverage. However, itappears that
 

even higher gain coverage isdesired since the phased-array approach has been
 

suggested. "Are the phased-array configurations proposed truly satisfactory
 

from a practical viewpoint?' then becomes a reasonable question.
 

Since hardware and software complexity are directly related to the
 

number of elements inthe array, we first consider the simplest two-element
 

case. Ifone element isdirectly fed and the other has the phase-shifting capability, the array will scan. The two-beam position is implemented by a mechanical switch interchanging a discrete phase shift between the two elements.
 

This isthe present S-band quad configuration.
 

Ifmore than two positions were achievable, higher percentage coverages would be attainable. A multiple-value phase shifter capable of high-power
 

operation was therefore required. But, ifmore positions are available, itbecomes more desirable to have higher gain since little performance improvement
 

occurs with a wide beam, which has a high degree of beam overlap between positions. For this reason, the three-element array was proposed and, with two
 

mechanical switches in series with the appropriate phase shifts, analysis indicates that 85%, 4-dBci coverage isachievable. Since the center element is
 

directly fed, some compensation for the additional switch loss occurs. This
 

configuration satisfied the quad-antenna-coverage specifications, but was unacceptable due to the higher cost and delayed schedules involved. However, if
 

higher gains are desired, it is still not totally impractical to consider using
 
2 

another mechanical switch inseries to obtain eight beam positions with this
 

configuration. Although the 4-dBci percentage coverage will not be substantially improved, the higher (>4 dBci) gain coverages will be greatly increased.
 

However, the planar-array approach allows more design freedom inachieving the
 

high-percentage-gain coverage.
 

The emphasis on higher than 4-dBci gain coverage appears to be the
 

motivating factor for suggesting the use of planar arrays. Extending the previous arguments, we find that higher gains from more elements and more beam positions, inboth pitch and roll planes, will result inhigher percentage, higher
 

gain coverage. There is no question that this is a valid assumption; the questions now really are: "has the gain-percentage coverage performance requirements
 

changed?" and "what isthe feasible limitation on the array complexity?"
 

Application of a two-dimensional planar array creates greater flexibility ingain coverage than simply adding beam positions ina linear array.
 

The beamwidth inthe orthogonal dimension isdecreased, with a corresponding
 

increase ingain. The simplest configuration for a planar array might be a
 

triangular arrangement, as shown inFigure 1,where the two lower elements are
 

identical to those presently being built.
 

Dipoles 

Mechanical 

Switch 

* 

Electronic 

Phase Shifter 

Figure 1. Triangular Planar Array
 
3 

Iftwo-beam switching can now be incorporated with respect to the
 

third element, is is then possible to generate four beams, with the amount of
 

beam overlap determined by the degree of phase shifting. Note that this particular three-element configuration ismost readily implemented by using an
 

electronic phase shifter inthe third element, with mechanical switching still
 

feasible between the other two elements.
 

With the mechanical-switching arrangement used here for explanatory
 

purposes, it is more feasible to consider using a four-beam/four-element array
 

set in a square configuration, as depicted inFigure 2.
 

Note that a tradeoff exists between the number of beam positions and
 

the number of elements since there ismore beam overlap for more beam positions
 

using the wider beamwidths from the smaller number of elements. Inorder to
 

increase the gain coverage, another possible configuration would be another
 

three-element array superimposed orthogonally on the first three-element array,
 

as shown inFigure 3. The center element isstill directly fed, with the four
 

outer elements determining the beam positions by appropriate phase shifting.
 

This configuration isthe largest one considered here since any additional elements provide illumination taper and higher gain, but at the expense of considerably higher costs and complexity.
 

Using the earlier arguments for the linear array, it is now possible
 

to consider a four-beam planar array using the earlier switched two-beam concept
 

inboth orthogonal directions with two mechanical switches, as illustrated in
 

Figure 3(a). The gain coverage issubstantially higher than before because of
 

the narrower beamwidth in the orthogonal direction. Similarly, since the threeelement linear array was suitable for four beam positions, it is feasible to
 

consider 16 beam positions using a total of four mechanical switches, as shown
 

inFigure 3(b). Since the center element isdirectly fed and each orthogonal
 

linear array isfed inparallel, the switch losses are not as significant.
 

The phase-shifting mechanism has not really been specified except
 

that, since mechanical switches have been utilized inthe present configuration, ithas been retained inprinciple. Mechanical switching is preferable
 

to electronic switching because ithas low-loss characteristics and possesses
 

the required high-power-handling capabilities. Inaddition, being the bistate
 

"latching type," no holding current (as inthe case of PIN diode phase shifters)
 

is required. Since rapid scanning isnot a critical factor, the mechanical
 

switch is suitable for these applications. The only situation wherein the
 
OF pOOR QUALU" 

Dipoles 

Mechanical Switch
 

Figure 2. Square Planar Array
 

6 
 0.6


0.6 0.6
 0.6 0.6 


.00 1.0 

0.6 
 0.6


1 @2 @3 04 1 102 ' 44 

;P5 11 *67I@18 

(a) 
 (b)


Figure 3. Five-Element Planar Array
 
5 

electronic-phase shifting might be seriously considered inthe planar arrays
 

discussed here isfor the triangular three-element array.
 

Inconclusion, ifhigher gain levels at 85% coverage are desired for
 

the S-band quad-antenna system, a five-element/16-beam mechanically switched
 

system isproposed. The microwave circuit losses would be about the same (1.8
 

dB) as the three-element/four-beam linear-array system proposed earlier since
 

the switching systems are inparallel and therefore identical. The only additional new component is a power splitter used to distribute the power inthe
 

orthogonal direction, which iseasily incorporated into the input stripline.
 

Because the number of beam positions isnearly identical to the eight-element
 

(4x2) phased array proposed by Rockwell, it is anticipated that the 85% gain
 

coverage would also be about 6.5 dB. However, since the microwave circuit
 

losses are significantly lower (1.8 dB versus 3.6 dB) because mechanical
 

switches are used, the overall 85% gain coverage should be accordingly higher.
 

Also, since there are no electronic-phase shifters, no electrical "holding"
 

power (24W - 48W inthe electronic-phase configurations proposed) is a definite
 

advantage inspace operations. This configuration also has the advantage that,
 

since only three linear elements form the beam, the beamwidth isquite large so
 

that extreme pointing accuracy isnot required, and there is a great deal of
 

pointing margin before a communication link islost.
 

Although it is considered somewhat impractical, by adding another
 

switch inseries with both feed lines (which adds only 0.2 dB more circuit
 

loss), it is conceivable to have a 64-beam/five-element array which will definitely have the highest gain-percentage coverage--if that isindeed the primary operational requirement.
 
A PROTRUDING WEDGE-SHAPED DIPOLE ELEMENT MODIFICATION
 

FOR THE ORBITER S-BAND QUAD ANTENNAS
 

Interim Report
 

Contract NAS 9-16067, Exhibit B
 

Contract Monitor: William Teasdale
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

R. Iwasaki
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8112-1
 

December 9, 1981
 
1.0 INTRODUCTION
 

A meeting was held at the Rockwell facility inAnaheim on
 

November 24, 1981 to discuss some alternative R&D modifications for the
 

Shuttle Orbiter S-band quad antennas. The present 4-dBci gain percentage
 

coverage of approximately 50% is substantially below the 85% specification, with the coverage at one frequency, 2.217 GHz, measured to be 38%.
 

This frequency sensitivity is somewhat disconcerting since the production
 

flight models are fabricated identically, but there is optimism that the
 

coverage degradation might be "tuned" out by mechanical adjustments. The
 

cause of the frequency sensitivity is not known and therefore cannot be
 

easily remedied without extensive testing. The low-percentage coverage
 

problem, however, is more amenable to improvement and, after extensive
 

discussions, a preliminary modification program, funded internally by
 

Rockwell, was formulated, with the antenna measurements to be conducted
 

at JSC. This interim report attempts to document the results of this
 

meeting such that the approach and justifications are clearly delineated.
 

2.0 PROBLEM
 

The prototype S-band quad antennas have a 4-dBci percentage
 

coverage of only 50% instead of the desired 85% coverage. The selection
 

of the two-element/two switched-beam quad configuration early in the program arose from some unrealistic projections on reducing circuit losses
 

as well as misunderstanding the interpretation of the data, specifically
 

the gain levels. Since the configuration cannot be readily altered, the
 

obvious question is: can any suitable modifications be suggested which
 

might improve the percentage coverage? The extent of improvement is difficult to assess, of course, but some ideas were aired that appeared to
 

have a valid basis for improving the percentage coverage, off-axis axial
 

ratios and radiation efficiency.
 

3.0 RECOMMENDATIONS
 

The flush-mounted dipole elements appear to have an intrinsic
 

limitation on the off-axis axial ratios because of the small electric
 

field component radiated normal to the plane of the aperture. In order
 

to circumvent this problem, a protruding-antenna concept was introduced
 
2
 

which extends the radiating elements to the thermal tile interface, or
 

even beyond, by recessing the tile. This radical departure from the present conventional design will indeed improve the axial ratio if properly
 

implemented.
 

The radiating-element design was addressed next. The existing
 

crossed-dipole design was found to be the most suitable because the circular polarization characteristic could be readily controlled and the
 

criterion of minimal modification was satisfied. Shaping of the radome,
 

as described earlier, was deemed unfeasible and the related concept of a
 

dielectric rod antenna was similarly dismissed due to the difficulty in
 

controlling the propagation modes and, therefore, the radiation properties.
 

The "wedge" dipole concept, basically a curved dipole as sketched
 

with dashed lines on the present dipole configuration in Figure 1,was then
 

introduced. This design attempts to broaden the beam of the individual
 

crossed-dipole elements and, therefore, the array beamwidth by physically
 

altering the radiation pattern of the dipole by geometric means. Generally, dipoles are not considered in a three-dimensional manner. The idea
 

here is to exploit the maximum radiation intensity normal to the radiating
 

current by imposing a symmetric angular relationship to the crossed-dipole
 

elements, thereby broadening the beam. The basis for this idea is that
 

the peak gain of the array is approximately 6.5 dBci, as illustrated in
 

Figure 2, and this excess gain can be usefully employed by spreading out
 

the energy by broadening the beam, as depicted in Figure 3. Any increase
 

inthe beamwidth is substantial with regard to the percentage coverage
 

since the area about the perimeter increases rapidly for relatively small
 

beamwidth additions.
 

The iris aperture restriction, which earlier broadened the beam
 

by decreasing the aperture area, was discussed. Itwas felt that this
 

planar iris would affect the wege dipole design by maintaining the same
 

aperture fields which determine the final radiation pattern. Incorporation of the protruding-antenna concept, however, permitted extension of
 

the wedge dipole past the circular iris into the radome, as sketched in
 

Figure 4, which appeared to be a reasonable compromise of the conflicting
 

requirements. Therefore, the basis of the modification program was
 

conceived.
 
ORIGINAL PAGE IS 3 

OF POOR QUALITY 

Thermal T Ies D l e C iuration .--, //////Radome "/ 

Figure 1. Present Dipole-Element Configuration
 
4 

Oftfh'AL PA, -,
 OF POOR QUALISS
 

4 dBcl 

gA 2
 

Figure 2. Excess Radiation (>4 dBci) 
to be Redistributed
 
5 

ORIGINAL PAGE ' 

OF pOOR QUALITY 

. . .- 6 dBci 

Fu 3 etu4 dBc 

Figure 3. Redistributed Radiation by Beam Broadening
 
6 

ORIGINAL PAGE [9 

OF POOR QUALITY 

Figure 4. Proposed Wedge-Shaped Protruding Dipole
 
7
 

The angular relationships of the wedge dipole cannot be easily
 

assessed but, using the previous gain values and standard dipole radiation
 

equations, an angle of 30' isfree space isnot unreasonable. Iftoo large
 

a wedge angle is used, it is conceivable that the peak gain would be reduced below 4 dBci, which iscounterproductive. However, an R&D effort
 

could establish the optimum angle using this baseline design wedge-dipole
 

concept once itwas demonstrated. The effects of the radome, however,
 

must be considered since some refraction might occur. The angular interface, using geometric optics, now has a tendency to further broaden the
 

beam and, thus, this lens phenomena must be compensated for by reducing
 

the free-space baseline angle. Therefore, an angle of 20' is not unreasonable for any initial R&D attempt to verify the concept.
 

The limitation on the percentage coverage measurement isthat
 

only JSC has the facilities to measure and analyze the data. Rockwell,
 

however, can measure the principal plane dipole-element patterns, then
 

analytically use pattern multiplication to predict the array beamwidths.
 

This assumption allows Rockwell to assess the modification results in a
 

preliminary manner to determine ifthe percentage coverage was indeed
 

improved.
 

Since the dipole shape would be modified, the other recommendation made to Rockwell was to correct what was felt to be a design deficiency inthe shorting strap of the crossed dipoles. There was a failure
 

inthe connecting solder joint during earlier vibration acceptance tests
 

which was attributed to a "blind" connection where inspection was not
 

feasible. By separating the crossed shorting straps onto two separate
 

disbs, visual inspection was made possible, avoiding a recurrence of the
 

problem. However, itwas noted that the positions of the shorting straps
 

-were above the plane of the crossed dipoles, which creates an uneven radiation pattern since the radiating current is "stepped," especially inthe
 

critical central region where the current density is highest. Although
 

itwas felt that the shorting strap functioned only as a bridge to connect the dipole elements, itwould, by definition, affect the radiation
 

pattern since it carries current. Therefore, itwas suggested that the
 

dipole connection points be indented, as shown in Figure 5,to accommodate
 

the shorting strap on the same level as the dipoles to retain the classical dipole configuration. The wedge design of the dipole complicates this
 

indented dipole strapping arrangement, but itmight be modified as shown
 

inFigure 6 to incorporate the angular configuration.
 
8 

ORIGINAL PAGf I
 

OF pOOR QUALTV
 

Shorting Strap at High Current
 

High Current Density
 

Insulation
 

(a)Present Cnnfiguration with Stepped Shorting Strap
 

(b) Proposed Indented Shorting Strap
 

Figure 5. Indented Shorting-Strap Concept
 
ORIGW4MAL ?AGE [a 

OF POOR QUALITY 

Figure 6. Indented Shorting Strap for Wedge Dipole
 
10 

4.0 CONCLUSIONS
 

Although it is somewhat late to introduce new concepts which
 

attempt to improve the gain percentage coverage performance of the Orbiter S-band quad antennas, it is still feasible in that the modifications
 

required to upgrade the design, once it has been satisfactorilly demonstrated, has been intentionally restricted to be minimal. At the present
 

time, there are definite questions as to the reproducibility of pattern
 

coverage results, even though they do not satisfy the existing gain percentage coverage specification. Rather than simply accepting the present
 

design and, therefore, the inadequate performance, these possible modifications have been introduced which may ameliorate the poor performance.
 

Simply tuning to obtain impedance matching for selective frequencies in
 

order to reduce losses isoften unsatisfactory if large discrepancies in
 

predicted performance arise unexpectedly. Itshould be noted that antenna
 

development is not a textbook problem and the antenna performance isoften
 

dictated by minor modifications to correct obvious deficiencies which
 

result from the initial design.
 

The wedge-dipole design which has been introduced here will most
 

certainly broaden the beamwidth and, therefore, increase the gain percentage coverage. Inretrospect, this wedge dipole is a descendent of the
 

earlier radome lens concepts since itphysically implements the ray bending that was to be achieved by refraction effects from a concave lens
 

surface. Infact, some design compensation must be incorporated for the
 

radome lens effect since ithas a tendency to broaden the beam even further Itmust be recognized that this modification has the specific goal
 

of increasing the gain percentage coverage by broadening the existing
 

antenna pattern at the cost of reducing the peak gain. Ifgain was the
 

essential criterion, as might be required for some marginal communication links, other more complex solutions would be required.
 
A LARGE SWITCHED-BEAM PLANAR ARRAY
 

Interim Report
 

Contract No. NAS9-16067, Exhibit B
 

Technical Monitor: William Teasdale
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

Dr. Richard S. Iwasaki
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8112-4
 

December 15, 1981
 
There has been a recent emphasis on high-gain antennas for both
 

the Shuttle Orbiter S-band quad antennas and the Centaur vehicle. An
 

earlier Axiomatix report (R8106-4, June 25, 1981) outlined a two-dimensional
 

switched-beam array using five elements. However, since high gain is of
 

interest, this report continues the development of larger switched-beam
 

arrays. Although the basis of these concepts utilizes electromechanical
 

switches, this technique is by no means limited by this constraint since
 

PIN diode phase shifters can also be readily substituted. The justification
 

for this report is to expand the available options by describing the use
 

of switched beams rather than phased arrays for applications where high
 

power and no electrical holding current are desired at relatively slow
 

switching rates. Since the switches are utilized in parallel, little
 

overall loss (0.2 dB for a switch and discrete phase element) occurs. A
 

systems trade-off using PIN diode phase shifters can then be made using
 

the baseline configurations for comparison purposes.
 

The next step in increasing the gain beyond the basic threeelement linear array described earlier is to add two additional outer
 

elements with the appropriate illumination taper. The five-element
 

array is sketched in Figure 1 with the additional switching arrangement.
 

Note that there is no additional loss compared to the three-element array
 

since the switches are incorporated in parallel. The primary difference
 

inthe new set of switching circuits is that the discrete phase shifts
 

are twice those of the corresponding three-element set to account for
 

the elements being at the extremes of the array.
 

- The proposed Centaur biconical array is now composed of three
 

elements, but higher gain can be achieved (with no additional circuit loss)
 

by adding this second set of switching circuits. The beam position which
 

is described by a binary code, as described in an earlier report, is identical for both the three-element and five-element arrays since the switch
 

positions are the same for both sets of switches.
 

This concept of extending the linear array can be continued
 

further by adding additional pairs of outer elements but, for explanatory
 

purposes, a large planar array will be developed using the five-element
 

linear array as a basis.
 
2 ORIGINAL PAGE W 


OF POOR QUALITY
 

o !_ 

3 2 

1!0I 

3 + -2 

L 

3 

o + 

3 + 2 

Beam 

1 

S 

Element i 

40 - 6@ 

@0 

3~ 

4 _0 _o 

Element 2 

@0 - 3_ 

0 -

~ 

Element 3 

_ 

a 0 

20 0@ 3

o0 

Element 4 

__ 0 + 3@ _

0 -2 

2 30 

o 

Element 5 

_0 + 6 

L0 + 4 

30 + 2 

4o 

6 + 4@'o 2 0 

00 

400 24 __ 0 

-

- 4@ 

Figure 1 Seven-Beam/Five-Element Switched-Beam Array 
3
 

The last planar array described was a five-element array sketched
 

in Figure 2. Note that the central element isdirectly fed, and the four
 

outer elements have circuit losses of 0.6 dB (0.2 dB per switch) for a
 

49-beam (i.e., 7x 7)configuration with redundant broadside beams. A 225­

beam (i.e., 15x 15) configuration would have only 0.8 dB of loss but, for
 

the present, let us only use three switches inparallel as the baseline
 

design.
 

There is a definite trade-off between the gain desired and the
 

number of beam positions since these two parameters constitute the gain
 

percentage coverage. Naturally, the higher the gain, the more complex
 

the system, but the number of beam positions can be approximately doubled
 

by adding only another switch (and 0.2 dB of loss) to each set of switches
 

This feature isespecially important when the gain percentage coverage is
 

emphasized since, effectively, the percentage coverage isdirectly related
 

to beam positions
 

To continue with the large planar array development, iftwo of
 

the five-element linear arrays are orthogonally positioned, a nine-element
 

planar array with extremely high gain isdeveloped by the two-dimensional
 

pattern multiplication concept. Again, note that no additional circuit
 

losses are involved since the radiated power isredistributed and the
 

switching sets are inparallel The essential features of this nineelement array are sketched in Figure 3 Again, the beam position isdetermined by a binary switching code which, for this two-dimensional case with
 

three switches ineach arm, iscomposed of six binary numbers for the 49
 

beam positions (assuming redundant broadside beams). The 225-beam configuration is correspondingly characterized by eight binary numbers.
 

The next step in increasing the gain involves filling inthe array,
 

as shown in Figure 4,with another four elements. This 13-element configuration still has 49 beam positions with no additional circuit loss even though
 

four more elements are incorporated The addition of these four additional
 

elements, however, greatly increases the complexity since these elements are
 

shared by the two orthogonal linear arrays so that the appropriate phase
 

shifts must be computed by an algorithm. For example, ifthe phase shifts
 

for a given beam position are characterized by C0 ± moCp0 ± nqD, the phase
 

shifts for these four elements are then described by [ 0 ± (±m±n)P] so that
 

the complexity is substantially increased.
 
4 

ORIGINAL PAGE 19
 

OF POOR QUALITY
 

Set B 

Set A 

Figure 2. 49-Beam/Five-Element Switched-Beam Array
 
ORIGINAL PA4GE 5 iFS OF POOR QUALITY 

II 

BEAM B1 B2 

Al A]BI A]B2 

A2 A2BI A2B2 

A3 A3B1 A3B2 

A4 A4B1 A4B2 

A5 A5B1 A5B2 

A6 A6BI A6B2 

A7 A7B1 A7B2 

B3 

AIB3 

A2B3 


A383 


A4B3 


A5B3 


A6B3 


A7B3 


1
 

B4 

AIB4 

A224 


A384 


A4B4 


A524 


A6B4 

A7B4 


B5 


AIB5 


A2B5 


A315 


A4B5 


A5B5 


A6B5 


A7B5 


Set B 

B6 B7 

A1B6 AIB7 

A2B6 A2B7 

A3B6 A3B7 

A4B6 A4B7 

A5B6 A5B7 

A6B6 A6B7 

A7B6 A7B7 

Figure 3. 49-Beam/Nine-Element Switched-Beam Array
 
6 

ORIINAL PAGE- is 

OF POOR QUALITY 

0
 

l_0 0 0 

0
 

0 

1) C i g t 

a STc2 4 Figur 

+ 2 - -0-

+ 71 2 1

T - 2 lI 

1 1 - 2 44 

to
 ofg 


wthdBa 


4-em1-lmn
 FXiu4 

7
 

Note that the set of switches required to fill in the array is
 

one more than that used for the other sets; inthis case, four switches
 

instead of three because of the (m+n) factor which doubles the number of
 

phase shift increments required. For example, for beam position ABi
 

which, for element 2 of both sets, is & 0 -3*DC,0 -30D, the mutually adjacent fill-in element should then have a phase shift of CO0 -6J, which
 

requires four switches for integral values of 0. Because of the added
 

complexity, however, a system analysis isrequired to justify filling in
 

the array to obtain the higher gain values.
 

Insummary, techniques to implement a large switched-beam planar
 

array have been described. Higher gain has been achieved by adding more
 

switches and elements to increase the effective aperture. The circuit
 

losses remain constant except for any incurred for redistributing the power,
 

which should be small. The primary value of this approach isthe large
 

number of beam positions available, which approximately doubles for every
 

switch added to a set, at a cost of only 0.2 dB of additional loss per
 

switch and discrete phase element. For two orthogonal three-switch sets,
 

for example, 49 beams are achievable. Even more dramatic are two orthogonal four-switch sets which can generate 225 beam positions. With the
 

capability of precise pointing, the peak gain of the array can be fully
 

utilized and the net gain percentage coverage is thus accordingly higher.
 

Another advantage isthat the switches are electromechanical latching
 

switches which can handle high-power levels and require no electrical holding currents. The discrete phase shifts are lengths of cable or microstrip
 

sections which are not temperature-dependent. Finally, the selection of a
 

particular beam position can be reduced to a six-binary-number sequence
 

for the three-switch set (49 beams) and an eight-binary-number sequence 

- for the four-switch set (225 beams), thereby simplifying the software 

required for implementation.
 

These advantages should be compared with comparable electronic
 

phased arrays with PIN diode phase shifters to select beam positions.
 

This electromechanical switched-beam approach is used on the Shuttle S-band
 

quad antennas for two-beam positions, but its capabilities for large planar
 

arrays were not fully investigated or appreciated previously. The intent
 

of this series of interim reports has been to discuss and develop the
 

necessary techniques to exploit this technology as an alternative option
 

in selecting future antenna configurations for the high-gain percentage
 

coverage applications required for the new generation of communication
 

requirements for space vehicles.
 
ADDENDUM
 

TO 

"A LARGE SWITCHED-BEAM PLANAR ARRAY"
 

Further consideration of the 49-beam/13-element switched-beam
 

array has resulted in an improved version which completely eliminates
 

the software problem of determining the proper phase settings for the
 

four "fill-in" elements. The improved configuration is shown in Figure 5, where again there are only two switch sets to consider. The modification which permits this simplification is feeding the "fill-in"
 

elements off of switch set A, which determines the proper phase shifts
 

for one orientation. Switch set B then determines the proper phase
 

shifts for the orthogonal orientation. The primary advantage of this
 

configuration is that the two switch sets designated B have identical
 

binary codes so that the beam position notation is greatly simplified.
 

Fewer switches (six versus eight) are required but, because of the power
 

distribution, additional circuit loss is encountered (although, because
 

of the weighting factors, the actual amount is small).
 

This improved version of the 13-element array completes the
 

basis for the development of a large planar switched-beam array since
 

the basic mechanism is clearly outlined. Further modifications for even
 

larger arrays can be developed from the techniques outlined, but the conceptual feasibility has been demonstrated for this new type of array.
 
ORIGINAL PGE W 

OF POOR QUALITY 

p 

o
 s
 .o0 


Set Be
 

Set A
 

Figure 5. Improved 49-Beam/13-Element Switched-Beam Array
 
A VARIABLE-GAIN MULTIPLE-SWITCHED-BEAM-ANTENNA CONFIGURATION
 

Interim Report
 

Contract No NAS9-16067, Exhibit B
 

Technical Monitor William Teasdale
 

Prepared for
 

NASA Lyndon B. Johnson Space Center
 

Houston, Texas 77058
 

Prepared by
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8209-3
 

September 22, 1982
 
1.0 OF p0OR QUIALIV INTRODUCTION 


The difficulty of achieving the original percentage coverage gain
 

specification of 85% 4 dBci, which has been relaxed to 50% 4 dBcl, became apparent during the Shuttle Orbiter S-band quad antenna program. The purpose of this
 

specification was to ensure that adequate communication link margins were available about the Orbiter. As a result of monitoring the S-band antenna program
 

under NASA Contract NAS 9-16067, Axiomatix extended many of the concepts used
 

in the switched two-beam quad antenna, taking advantage of the Rockwell-developed
 

basic hardware. The outcome of the study has been the evolution of a new generation of variable-gain multiple-switched-beam antennas whose principles can be
 

applied to the improved Shuttle Orbiter S-band quad antennas and Centaur biconical arrays, as well as to proposed future Space Station antenna systems.
 

This switched two-beam quad antenna developed by Rockwell was an
 

improvement over the earlier single-beam quad antenna built by Watkins-Johnson
 

but, due to overoptimistic computer analysis projections of the antenna patterns,
 

the actual performance results were disappointing. The primary limitation of the
 

gain performance was the selection of only two elements and two beam positions,
 

the minimum required for switched-beam operation. Using the same type of hardware configured in a slightly different manner, Axiomatix developed a two-dimensional cross array that could be electromechanically scanned in both the roll and
 

pitch planes at 2

M x 

2N discrete beam positions, where M and N are the number of
 

double-pole/double-throw electromechanical latching switches in each plane. The
 

advantages of using this type of switch rather than electronic phase shifters are:
 

(1)high power-handling capabilities, (2)low insertion losses, (3)absence of
 

electrical "holding" power, and (4)the binary nature of the double-throw switch
 

which can readily be converted to a binary code for a specific beam position.
 

Higher gain can be achieved by adding additional radiating elements to
 

the basic configuration, the only complexity being a doubling of the differential
 

phase shifts of the outer elements. The binary code for the beam positions remains the same since the switching configuration used for the inner elements is
 

identical to that of the outer elements.
 

Since it is possible to physically switch out various combinations of
 

radiating elements to form an antenna pattern of the desired shape using this
 

same electromechanical switch, an even more unique feature arises from this
 

switched-beam configuration in that low, medium and high gains can be independently attained ineach plane by simply activating two switches, with the beam
 
ORIGINAL PACE I5 

OF POOR QUALITY 2 

position remaining constant and without phase transients. This variable-gain
 

capability in both planes greatly relieves the tracking requirement since the
 

pattern shape can be adjusted to accommodate the relative motion between the
 

antenna system and the "target" vehicle. For example, if a free flyer was passing the Shuttle Orbiter (i.e., being in a lower orbit), the low/high-gain configuration combination would result inan elongated pattern which follows the
 

path of the free flyer, thus eliminating the tracking function.
 

The variable-gain multiple-switched-beam antenna takes advantage of
 

the trade-off between gain and percentage coverage. Ifhigh gain isrequired,
 

the full antenna-array aperture isutilized, and this switched-beam array can
 

position the narrow beam anywhere in its sector of coverage. Ifhigh gain is
 

not required, the outer radiating elements are switched out and the larger resultant beamwidth associated with the lower gain relaxes the tracking requirement. Finally, iflow gain from an omnidirectional antenna isneeded for a
 

nearby target, only the central radiating element isused by switching out the
 

inner two elements, which also automatically removes the outer elements, as
 

will be shown later.
 
2.0 

3
 

MULTIPLE BEAMS
 

The Rockwell proposal to develop an improved S-band quad antenna
 

using the Axiomatix "cross" five-element configuration prompted a more detailed
 

evaluation of the design. As a result, the switched-beam phase relationships
 

were reexamined and a significantly better scheme evolved, one which was not
 

immediately obvious. Since a similar switched-beam scheme has also been proposed for the Centaur vehicle, documenting the results herein is considered
 

appropriate.
 

This modified configuration avoids the "double-angle broadside gap"
 

previously described inAxiomatix Report No. R8112-4, where the two adjacent
 

near-broadside beams were pointed #4 for an angular separation of 20. In order
 

to avoid this problem, itwas suggested that the one-beam position be sacrificed while having a redundant broadside beam. With this improved phase relationship scheme, the beams are equispaced and separated by the same angle, 4' =
 

20/3, without losing a beam position. Note that the near-broadside beams are
 

now ± 4/3. The only design criteria is then the selection of the number of beams
 

in each plane and, therefore, the number of switches required.
 

In order to dramatically demonstrate the desirability of adding
 

another switch in series, which doubles the number of beam positions at the cost
 

of only 0.2 dB additional loss, the phase relationships for the two-, threeand four-switch configurations are shown in Figures I-3 for four, eight and 16
 

beams in each plane, respectively. It should be noted that a similar switching
 

arrangement exists in the orthogonal plane so that various combinations of beam
 

positions, up to a maximum of 256, are possible for four switches in each plane
 

for each of the four quad antennas. Although this might be considered "overkill
 

the points to be emphasized are that the simple addition of another switch in
 

each circuit greatly increases the potential advantages of switched-beam technology by doubling the number of beams, and the resultant availability of a large
 

number of beam positions easily compensates for the unpredictable performance
 

which has plagued the quad antenna program due to the curved Orbiter shape.
 

The high-gain version of the switched-beam array is shown in Figure 4,
 

where two additional elements create a five-element linear array whose phase
 

lationships for the 16 beam positions are observed to be similar to the threeelement array except that the differential phase shifts for the outer elements
 

are doubled. The binary code for the beam position is identical since the
 

switching configuration for the five-element array is directly related to the
 
A B Beam Element A Element B 

0 1 "0 *0 + 0 

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8 

three-element array. Note that, since the additional four switches for the
 

outer array are in parallel, negligible additional switching insertion loss occurs. And, if four more elements are added in the orthogonal plane, high gain
 

can be attained in that plane also, and the total number of possible beam positions is then 256.
 
3.0 

ORIGINAL PAGE ES 

OF POOR QUALITY
 

VARIABLE GAIN
 

Further progress in the switched-beam-array technology has resulted
 

in the conceptual development of a variable-gain antenna which can be adjusted
 

for particular mission requirements. For example, on the Centaur five-element
 

biconical array, there may be occasions when high gain is unnecessary for a distant target and it is desirable to have a wider beamwidth with medium gain in
 

order to simplify tracking. In this case, a scheme to switch out the two outer
 

elements has been developed using the same double-pole/double-throw electromechanical latching switches, as shown in Figure 5. Further, extension of the
 

variable-gain technique applies for the low-gain or omnidirectional mode where
 

only the central element radiates. Thus, by adding two switches, it is possible
 

to have low-, medium- and high-gain capabilities with the same array, with very
 

little additional insertion loss.
 

For the improved version of the Shuttle Orbiter S-band quad antennas,
 

this technique can similarly be used for variable-gain capabilities in both dimensions, as shown in Figure 6. Thus, it is possible to alter the pattern to a
 

desirable shape, such as using high gain in one dimension and low gain in the
 

other in order to form an elongated beam that accommodates a moving target and
 

therefore satisfies the tracking function without any antenna switching.
 
INNER OUTER
 

ELEMENTS ELEMENTS
 

Switch 4 Switch 4'
 

Switch D _
 

00


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Forward 


Log-Conical-Spiral 


Antenna 


"a0
 

)
 

-- Aft Log-Conical-Spiral Antenna M
 

If 
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SSwitch
 B 

Input
 

CD 

Figure 5 Multimode Antentia-Switching Arrangement 
OUTER INNER
 

ELEMENTS ELEMENTS
 

Switch Y4 0 Switch Y4 

INNER 

ELEMENTS 

Switch X4EX4 

OUTER 

ELEMENTS 

Switch C 

Switch 0D0 

A / -­ 0­o T 

Input 

Figure 6. Two-Dimensional Multimode Antenna-Switching Arrangement 
ORGINAL PAGE gg 

OF POOR QUALITY, 

9841 Airport Boulevard e Suite 912 oLos Angeles, California 90045 0 Phone (213) 641-8600 

TECHNICAL MEMORANDUM NO. M8209-4
 

TO: Don Eggers DATE: September 30, 1982 

FROM: Richard Iwasaki FILE: NAS 9-16067"B" 

SUBJECT: Binary Code Designation of Multiple-Switched-Beam Positions 

One of the important considerations in implementing a variable-gain
 

multiple-switched-beam array is the simplicity of obtaining a desired beam position. Itwas noted earlier that the electromechanical double-throw switches
 

have a binary nature which can be described by a binary code. For four switches,
 

a beam position in one plane can be characterized by a unique four-digit binary
 

number and, for the two-dimensional case, by eight binary digits in sequence
 

Inactuality, only 15 memory locations need to be allocated for this selection
 

process since the orthogonal plane can use the identical binary codes.
 

As an example, Figure 1 shows the switching arrangement necessary to
 

achieve beam position 4. Using the notation that a "0" represents a "straightthrough" switching position and a "1"denotes a "cross-over" switching position
 

and the binary code begins with the switch closest to the radiating elements,
 

the binary code for beam position 4 is 1010. Note that the switching arrangement is identical for the inner and outer elements so that, if the high-gain,
 

five-element array is used, no modifications will be required Table 1 lists
 

the remaining binary codes for the other beam positions
 

For the two-dimensional cross array, the beam position is specified
 

first by the X-plane beam position, then by the Y-plane beam position, as depicted in Figure 2, which has a total of 256 beam positions. If the crosshatch beam position designated by 4,15 is desired, the appropriate binary code
 

is simply 10100100, where the first four digits characterize beam position 4
 

in the X-plane and the last four digits characterize beam position 15 in the
 

Y-plane.
 

Copies NAS9-16067"B" Distribution
 

H. Ellis, RI
 

S. Wong, RI
 
Beam Element A Element B Element D Element E Binary C 

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 T 17T TT-T3T
 
M8209-4 ORIGINAL PAGE M 

OF POOR QUALITY 

Table 1. Binary Codes for the 16 Beam Positions
 

Beam
 

Position Binary Code
 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

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2 

3 

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6 

7 

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Y-Plane 8 

10 _ 

11 _ 

12 ___ 

13 

14 

15 

_ 

4,15 ----------- 10100100 

-­

16Bood 

Figure 2. 256-Beam-Position Selection Using Binary Codes
 
ORIGINA1L P-AC- tS 

OF pOOR QUALI- Axiomatix 

9841 Airport Boulevard 9 Suite 912 e Los Angeles,California 90045 e Phone (213)641-8600 

TECHNICAL MEMORANDUM NO. M8209-5
 

TO: Richard Iwasaki DATE: September 30, 1982
 

FROM. Jim Dodds FILE- NAS 9-16067"B"
 

SUBJECT: Simplification of Antenna Beam-Switching Logic
 

The switched-beam array described in Axiomatix Technical Memorandum
 

No. M8209-4 requires that the beam position be mapped into a binary code in
 

order to control a set of phase-shifting elements. This can be effected by
 

using one of the following methods: (1) a microprocessor to access one of
 

16 memory locations, each of which contains a four-bit code, (2) a programmable logic array or, (3) hard-wired logic. The format of the conversion from
 

four input bits to four output bits is particularly simple in this application
 

and can be implemented with the EXCLUSIVE-OR gates, as can be shown by inspection, or by using Karnaugh maps.
 

Table I of M8209-4 is reproduced below, with the ordinal beam positions represented by four-bit binary sequences bO , bI, b2, b3. The binary
 

output is represented as a four-bit code ao, a,, a2 , a3.
 

INPUT OUTPUT
 

Beam Binary Code Binary Code
 

Position b 
 b1 b2 b3 a0 a1 a2 a3 

1 0 0 0 0 0 0 0 0
 

2 0 0 0 1 1 1 0 0
 

3 0 0 1 0 0 1 1 0
 

4 0 0 1 1 1 0 1 0
 

5 0 1 0 0 0 0 1 1
 

6 0 1 0 1 1 1 1 1
 

7 0 1 1 0 0 1 0 1
 

8 0 1 1 1 1 0 0 1
 

9 1 0 0 0 0 0 0 1
 

10 1 0 0 1 1 1 0 1
 

11 1 0 1 0 0 1 1 1
 

12 1 0 1 1 1 0 1 1
 

13 1 1 0 0 0 0 1 0
 

14 1 1 0 1 1 1 1 0
 

15 1 1 1 0 0 1 0 0
 

16 1 1 1 1 1 0 0 0
 
ORIGINAL PAGE 93
 

M209-5 	 OF POOR QUALITY 2 

The phase shift logic can be implemented as shown below for both the 

X-plane and Y-plane logics, with a0 = b3 , a1 = b2 ®b 3, a2 
= bl®b 2 and a3 = b0iEb1 .

As can be seen, only three mod-2 (EXCLUSIVE-OR) adders are needed per plane.
 

X-Plane ba 

Beam a2 X-Plane 

b 

Select al Phase 

Logic b o Shifters 

Y-Plane a 

Beam b, Y-Plane 

Select b2 

a2 

Phase 

Logic b o Shifters 

Copies 	 NAS 9-16067"B" Distribution
 

Haynes Ellis, RI
 

Sam Wong, RI
 
9.0 

ORIGIAL PAGE U0
 

OF pOOR QUALITV 

PAYLOAD COMMUNICATIONS INVESTIGATIONS
 

The major thrust of Axiomatix investigations of payload communications
 

centered on the payload signal processor (PSP) bit-synchronizer problem. During
 

ESTL tests, itwas found that the TRW PSP bit synchronizer would false lock under
 

high SNR conditions. This pnenomenon was subsequently duplicated at RI, ADL and
 

TRW.
 

Axiomatix efforts to analyze this problem included numerous conversations with Brett Parrish of JSC, Larry Stein and Rich Helgeson of TRW, and Mike
 

Rueterman of RI, as well as attendance at meetings and hardware demonstrations.
 

Axiomatix has analyzed the algorithm lock-detector-threshold parameter for optimality over SNR's ranging from low to high as well as carefully evaluating
 

the entire algorithm. A recommendation was made to change both the threshold
 

parameter and the filter bandwidths, computer simulations later indicated an
 

improvement inperformance with the new parameter values.
 

The technical memoranda which follow document the analyses performed
 

and meetings attended by Axiomatix personnel on this problem.
 
Axiomatix
 

9841 Airport Boulevard * Suite 912 o Los Angeles, California 90045 e Phone (213) 641-8600 

TECHNICAL MEMORANDUM NO. M8206-1
 

TO: G. Huth 	 DATE: June 18, 1982
 

FROM: J. Holmes
 

SUBJECT: PSP Bit Synchronizer NRZ False-Lock Status--Interim Report
 

REFERENCES: (1)	Telephone Conversations with Brett Parrish (NASA/JSC),
 

Larry Stein (TRW), and Rich Helgeson (Consultant)
 

(2) Abbreviated Flow Chart for the PSP Bit Synchronizer
 

Algorithm, by Brett Parrish
 

1.0 INTRODUCTION
 

During recent ESTL tests, it was found that the TRW PSP bit synchronizer
 

would false lock under high SNR conditions. This phenomenon was subsequently
 

duplicated at RI, ADL and TRW Axiomatix has begun to analyze the problem.
 

This effort has included numerous conversations with Brett Parrish of JSC, Larry
 

Stein and Rich Helgeson of TRW, and Mike Reuterman of RI, as well as attendance
 

at meetings and hardware demonstrations. Axiomatix is analyzing the algorithm
 

threshold parameter for optimality over SNR's ranging from low to high as well
 

as carefully evaluating the entire algorithm. This interim report documents
 

the findings to date concerning this problem.
 

2.0 STATUS 	SUMMARY
 

(1) Brett Parrish has determined that the summation used in the
 

filter computation of the mid-phase filter* overflows at high SNR such that
 

the algebraic sign of the phase update control changes and therefore provides
 

a stable lock point at nominally 1800 away from the true-lock point (a halfbit error) Brett has also demonstrated at JSC that, by prohibiting the overflow and associated algebraic sign change, the false-lock condition at -180
 

will not occur This fact has been verified by Rich Helgeson with the TRW
 

program simulation
 

FSTB2+PRTB2 doesn't have overflow protection on page 2 of Brett's
 

brief algorithm.
 
M8206-1 
 2
 

(2) Rich Helgeson and others have determined that a "lack-of-atransition" detector must be added (on page 2 of the brief flow chart). This
 

is because, for example, if the data were turned off, no transitions would
 

occur so that it is impossible for the lock detector to enter the out-of-lock
 

state and, therefore, to return to the acquisition mode. According to Larry
 

Stein, this detector, which counts nontransitions, can be implemented without
 

a great deal of trouble. Most probably, it is also a problem when the subcarrier or the carrier drops out temporarily.
 

(3) The offset in both the true-lock and false-lock points can be
 

attributed to the fact that, at high SNR, the A/D converter outputs its maximum value of ±3 (inbinary-twos complement) so that 64 samples of the maximum
 

value limit at about two thirds of the way across a presum, or about 43 samples.
 

The remaining samples are, in effect, discarded. Although this problem is disconcerting, the author does not believe that itmust be fixed immediately, if
 

at all, since the main effect is to induce a static phase (or timing) error at
 

either the true-lock or false-lock point. Larry Stein has indicated that it
 

is possible to fix the problem by allowing only 16 samples per presum to accumulate, then scaling the sum of the presums so that it will not overflow in
 

an eight-bit register
 

(4) Manchester data does not produce false phase lock Possibly,
 

this is because, with NRZ data and a ~1800 error offset, the sum of the midphase signal can roll over and change sign, providing a stable -180* error
 

whereas, in the Manchester case, there is always a transition at (true) midphase so that, at least for transition densities not equal to zero or 100%,
 

Manchester data is tested for -1800 false lock in the algorithm and would be
 

rejected
 

(5) The.automatic gain control (AGC) used for the A/D converter is
 

done in a 5-MHz bandwidth, but the input is then filtered by a 250-kHz filter
 

which makes this signal poorly AGC'd since the signal component will increase
 

at high SNR's in such a way as to produce a level larger than the A/D range.
 

(6) Itappears that the "very small timing error" test performed in
 

the bit synchronizer algorithm (which is indicated on page 3 of Brett's short
 

algorithm just below the NTSPC transition count from the last phase correction)
 
M8206-I 
 3 

is not required and could be eliminated if necessary in order to implement the
 

changes required in items (1) and (2). This check bypasses a phase update correction if the error is small enough; otherwise, it does nothing.
 

2.0 CONCLUSIONS
 

It is the opinion of the author (and others) that it is important to
 

fix only items (1) and (2) in order that the bit synchronizer will not exhibit
 

false lock, namely, change the mid-phase summation of FSTB2 and PRTB2 so that
 

it will not roll over to the opposite algebraic sign, then add a nontransition
 

counter to limit the number of nontransitions which can occur before the algorithm declares that the loop is out of lock Neither "fix" appears to be a major
 

problem. It would be desirable to fix item (3), the saturation in the presum,
 

but this item is not related to false-lock avoidance. Also, item (5) is not
 

an important item and probably should not be changed if its space in the microprocessor is not needed.
 

Any additional comments, observations or corrections should be indicated to the author so that a revised memorandum can be written prior to the
 

next meeting at TRW scheduled for early July.
 

Distribution-


R. Helgeson, Consultant
 

J. Johnson, NASA/JSC
 

S. May, TRW
 

B. Parrish, NASA/JSC
 

T. Pederson, TRW
 

M Reuterman, Rockwell
 
# MAxiomatix
 

9841 Airport Boulevard * Suite 912 * Los Angeles,California 90045 * Phone (213)641 8600 

TECHNICAL 	MEMORANDUM NO. M8206-4
 

TO: 	 G. Huth DATE: June 30, 1982
 

FROM: 	 J. Holmes
 

SUBJECT: 	 PSP NRZ Lock Detector and Loop Control Signal
 

Phase Error Range of Operation at High SNR--Interim Report
 

1.0 	 INTRODUCTION
 

This is the second technical memorandum/interim report that concerns
 

the PSP bit synchronizer false-lock problem. The causes of the false lock are
 

considered based on analyzing the bit synchronizer and lock detector, but not
 

considering the DC bias removal algorithm It appears that, at least at high
 

SNR, the lock detector constant (6.5) should be increased. In order to enable
 

us to choose the best value of the lock detector constants, an analysis is continuing and documentation will be forthcoming.
 

2.0 	 SUMMARY
 

The error control signal and the lock detector test for the PSP S-band
 

bit synchronizer has been analyzed at high SNR for the 8-kbps data rate case by
 

hypothesizing various fixed phase errors and evaluating the inphase and midphase
 

filter outputs. It is to be noted that DC bias removal was not considered in
 

this analysis.
 

It was found that the lock detector indicates an in-lock condition
 

anywhere within a phase error of zero to about ±75' (±21% of a bit, which is
 

considered too large) Furthermore, the lock detector will indicate an in-lock
 

condition between about 1040 and somewhat less than 1300 (and between -104' and
 

- -130') and also between 1500 and 2100.
 

The error control signal of the bit synchronizer loop pushes the loop
 

in the wrong direction when the phase error is between 900 and 1500, which is
 

due to the unprotected overflow of the midphase addition.
 

This control signal reversal causes the false-lock condition The
 

in-lock indication between a phase error of 1500 and 2100 can be removed by prohibiting the saturation of the presums.
 

It is expected that the above two problems will occur at the other
 

data rates as well, although they have not yet been analyzed in detail.
 
M8206-4 2
 

3.0 RECOMMENDATIONS
 

Based on this preliminary study, it is felt that, at least at high
 

SNR, the lock detector constant of 6.5 is too small and should be increased
 

since this value allows large timing errors (±21%) to be declared in lock.
 

Itmight be desirable to provide different constants for different data rates
 

as well as for low or high SNR conditions. Further analysis aimed at recommending values for this constant is currently underway.
 

Also, overflow protection must be provided for the midphase summation
 

and the presums should be modified to prevent limiting.
 

4.0 ANALYSIS
 

We consider the lock detector test given in [1] as
 

AVIN > 6.51AVMII (I)
 

where AVIN and AVMI are lowpass filtered versions of the inphase and midphase
 

summers, respectively. Basically, the idea of the lock detector test given by
 

(1)is to announce when the PSP bit synchronizer loop is in or out of lock.
 

In Figure 1, a simplified block diagram illustrates some details of
 

the bit synchronizer loop. First note that all presums and adders outside the
 

microprocessor are combined with two's-complement arithmetic with overflow protection. Internally, the inphase (AVIN) signal is formed with two's-complement
 

combining with overflow protection, whereas the midphase combining is done with
 

two's-complement combining without overflow protection'
 

The loop uses a filtered version of the output error to correct the
 

timing of the loop, with the addition of a rate-aiding component which attempts
 

to remove small doppler errors (rate tracking).
 

In order to analyze the lock detector, it is necessary to analyze the
 

value of AVIN, AVMI and TRIND, which is the variable that indicates whether a
 

positive, negative or lack of transition has just occurred We now consider
 

various phase error angles Table 1 lists the data rate, samples per presum,
 

fractions of a bit for a presum, samples per bit, and whether or not the number
 

of samples saturates the presum.
 
Loop 

Three-bit r-


A/D MICROPROCESSORo
 

IC.
 

Timing
 

4Presummer Dump Count
 ICorrection
 O-N4


II AvLogrc
 

I T mun
 

8 bt8btLast torI
 Rate


Neet, HalfBit Distrlbu-
 I 

I ton Rrr I 

e1 

Presum , Over NBits i 0

I FtIntegral S Dait O1 T L I Mldphase U 0 

-|M delay e 

8 ltk IDMidphase 0 r. 

Averaging to 

Presm PrsmFilter C: i ddl-lel 

Add 

bit 11Avraing 0

­

al 8 

-- d l Integra Inphase I :
 

S8 bit (not use I& Transition
 

Presum 
 16 bit
 

I~~ First = Itbit 8 bit Half-Bit Inphase
 

S -Integral iAveraging
 

Oldest I 
 FilterI
 

Presum 
 i -

Figure 1. PSP Data Transition Tracking Loop for MRZ Data 
 -

4 

ORIGINAL PAGE ES 

OF POOR QUALITY 

Table 1. Samples Per Bit Versus Data Rate Matrix
 

Rb Samples/ Fraction of Samples/ Presum
 

Sum a Bit Bit Saturated?
 

16 kbps 16 1/4 64 No 

8 kbps 32 1/4 128 No 

4 kbps 64 1/4 256 Yes 

2 kbps 64 1/8 512 Yes 

I kbps 64 1/16 1024 Yes 

4 1 0 = 0', 8 kbps
 

Denote Fj as two's-complement addition (or subtraction) with overflow
 

protection and I as two's-complement addition without overflow protection.
 

Based on Figure 2 and Table 1, we can write
 

AVIN (n+l ) = LIAVIN(n) + 196 El 961 (2)
 

since the two's complement has a maximum of 127 and a minimum of -128. Therefore,
 

n 

AVIN (n~l ) = 127 1 L'I 3
 

1=0I
 

Solving (3)yields
 

S n+lf 

AVIN ( n + l ) =127 11- L--I (4) 

For large n, AVIN (n ) converges to
 

127
 AVIN(n) 


1 - LI
 
S 

CD 

I 

AVIN 

FSTB2 LSTB2 

H+ 

PRTB2 FSTB2 

0 c 

AVMI 

'0­

01 
ORIGINAL PAGE iq M8206-4 
 6 OF POOR QUALITY 6 

For the lowpass filter defined by (2), LI = 0.99 so that
 

AVIN (n) -- 12700 or 12800 (5) 

or an average of
 

AVIN (n ) = 12750 (6) 

The bit transition decision variable (TRIND) takes on the value of 1, -1 or 0
 

according to whether the bit goes from negative to positive, positive to negative, or does not change. Now consider the midphase integration (AVMI) for the
 

situation depicted in Figure 2. We have (TRIND = 1)
 

AVMI (n+l ) = LM AVMI(n) + IFSTB2 +LSTB21 (7)
 

Now, when centered at zero phase error, the value of FSTB2 =127 and PRTB2 =-128
 

so that we have
 

AVMI(n+ l) = LM AVMI (n) = (-128 m] 127) (8) 

Solving (8)yields (-128 R] 127 = -1) 

AVMIn) - _CLM) , L= 0.95 (9) 

1- LM
 

Hence, for large n,we have
 

AVMI(n) -1 - -20 (10) 

I - LM 

The lock detector makes the comparison
 

>
 

AVIN > IAVMII (11)
 
M8206-4 7
 

OF OR!Gv's-p POOR QUALtTV Pg~ f

which, in our present case, leads to
 

1 

12,750 > 20(6.5) = 130 (12)
 

Consequently, the loop is declared in lock.
 

4.2 = 450, 8 kbps
 

Consider Figure 3, which illustrates a 450 phase error (1/8-bit
 

timing error). We find that
 

AVIN(n+1) = LI AVIN(n) + 196 1i1(96[96)I (13)
 

or
 

AVIN (n +l) = LI AVIN (n) + 127 (14)
 

where, for convenience, we have assumed a positive bit in the AVIN interval*
 

Solving, we obtain
 

AVIN (n ) - 12700 (15)
 

Since TRIND =1, we have 

AVMI (n + 1) = LM AVMI(n) + [(-96E]-96) M (-48+489n96)] (16) 

or 

AVMI(n+1 ) = LM AVMI(n) - 32 (17) 

Solving, we obtain 

AVMI n -n-) 32 640 (18)
 

We use this assumption inwhat follows.
 
co 

S 

4=, 

AVIN 

AVMI 

00 

0),, 

to.1J 

Figure 3 Phase Error Relationship for 4=450 (1/8 of a Bit) 
M8206-4 	 9
 

The bit synchronizer lock detector comparison is given by
 

?
 

12700 > 640 (6.5) = 4160 (19)
 

co the lock detector again declares an in-lock condition.
 

4.3 	 = 67.50, 8 kbps 

Now consider Figure 4, which illustrates the case when @ = 67.50. 

For the inphase component, we have 

AVIN(n+ l ) = LI AVIN (n ) + IC(-72+24) E]96) E](127)1 (20) 

or 

AVIN (n+1) = LI AVIN (n ) + 11271 (21) 

or solving 

AVIN n

n ) -- 12700 (22) 

The midphase recursion relationship is given by (TRIND = 1) 

AVIN(n+1) = LM AVMI (n) + [-128 M(-72+24) EJ96] (23) 

This yields 

AVMI (n+1) = LM AVMI (n ) - 80 (24) 

Solving, we obtain
 

AVMI(n) -80 = -1600 (25)
 0.05
 

The lock detector test is given by
 

?
 

12700 > (1600)(6.5) = 10,400 (26)
 

Hence, the lock detector declares that the bit synchronizer loop is still in
 

lock.
 
co00 

C>, M

AVIN
 

I I i "i 

00 

I' A

AVMI !Il U 0 

0 

Figure 4 Phase Error Relationship for 4= 67 50 (0 188 of a Bit) 

CD 
M8206-4 11
 

4.4 ,= 750, 8 kbps 

Figure 5 illustrates the case inwhich the phase error is 750. Again
 

it can be shown that
 

AVMI(n) -- 12700 (27) 

and, further, that TRIND = 1 so that, for the midphase signal, we have 

AVMI (n+l) = LM AVMI(n) + 1-128 MC(-80+16) E]96)1 (28) 

which can be evaluated to be 

AVMI(n+ l ) = LM AVMI (n ) - 96 (29) 

Therefore, for large n, we have 

AVMI(n) -96 -1920 (30)
 0.05
 

Checking the lock detection equation, we have
 

7
 

12700 > 1920 (6.5) = 12480 (31)
 

which indicates that the lock detector is almost ready to declare the bit synchronizer out of lock] Itcan be shown that the out-of-lock condition for a
 

positive transition occurs at 760
 

4.5 ' = 90', 8 kbps 

Now we consider the case when ' = 900, as shown in Figure 6. It can 

again be shown that 

AVIM(n) -+12700 (32)
 

so we now consider the midphase component We have
 

AVMI(n+ l) = M AVMI (n ) + [(-96 R]-96) M(-96[1]96)] (33) 
:o 

coS

C)

0) 

V I I 

AVIN 

--

AVM I 

00 

Figure 5 Phase Error Relationship for d= 750 (0.21 of a Bit) 

n 2 
0 

:oS

cf 

AVIN
 

AVMI
 

Figure 6 Phase Error Relationship for q=900 (1/4 of a Bit)
 

to
 
M8206-4 14
 

so that
 

AVMI(n+ l) : Lm AVMI(n) + E-128] (34)
 

Solving, we find that
 

AVMIn) -128 = -2560 (35) 0.05
 

Checking the lock detector function, we obtain
 

?
 

12700 > (2560)(6.5) = 16640 (36)
 

and the lock detector still declares that the loop is out of lock.
 

4.6 4 = 1350, 8 kbps 

We now consider the case when 4 = 135'. Consider Figure 7. We have 

that (TRIND = 1) 

AVIN(n+ l ) = LI AVIN(n) + (-96 E r0)[](96 F196)I (37) 

or
 

AVIN(n+ l ) LI AVIN (n) + 31 (38)
 

Solving, we obtain
 

AVIN (n)-+3100 (39)
 

Now AVIN(n) satisfies
 

( n ) AVMI(n+1) = LM AVMTI + C(-96r]-96) M (-96Gmo)D (40) 

or 

AVMI (n+1) = LM AVMI(n) + (-128 M-96) (41) 
co 

C, 0

AVIN
 

l I "00o 

AVMI
 

Figure 7 Phase Error Relationship for 4=1350 (0 375 of a Bit)
 
M8206-4 16 

Simplifying, we have
 

AVMI (n +1) = LM AVMI (n ) + 32 (42)
 

Solving (42) produces -128 m-96 = +32 since the subtraction is not protected.
 

In the limit, we have
 

AVMIn) 0.0-5
 AIn= 640 (43)
 

The lock detector test yields
 

7
 

3100 > 6.5 (640) = 4160 (44)
 

which implies that the loop is declared out of lock at @ = 13501 Additionally,
 

the error control increases the phase error since the algebraic sign of the
 

correction isnow positive. In fact, a little reflection will indicate that
 

the error signal is positive between 90+' and 1350; furthermore, the lock detector is satisfied in a region between 1040 and -130'.
 

4.7 = 1450, 8 kbps
 

Consider the case when 4 = 1450, which is illustrated in Figure 8.
 

Considering AVIN, we have that
 

AVIN(n+l) = LI AVIN(n) + (-96 Ej (6 (96[]96j (45) 

or
 

AVIN(n+1) = LI AVIN (n ) + 1-117 + 1271 (46)
 

so that
 

AVIN(n+1) = LI AVIN (n ) + 10 (47)
 

Solving,,we obtain
 
00 

CD 

AVIN
 

AVMI 

.TlN
 

-~ 00 

koI 

Figure 8. Phase Error Relationship for 4= 1450 (04 of a Bit)
 
M8206-4 18
 

AVIN(n) ) 10 1000 (48)
 0.01
 

Since TRIND = 1,we have 

AVMI(n+1) = LM AVMI(n) + ((-96 F-96) r(-96rEI-96 5-35)) (49) 

Evaluating, we obtain
 

AVMI(n+1) + LM AVMI(n) + 11 (50)
 

so that
 

AVMI 0.05 = 220 (51)
 

The lock detector test produces
 

7 

1000 > (220)(6 5) = 1430 (52)
 

which indicates that the loop is out of lock.
 

4.8 	 , = 1500, 8 kbps
 

In the same manner as that computed above, we find that
 

AVIN (n ) -	 100 , TRIND = 0 ; AVMI (n ) + 0 (53) 

so that there is no update signal, however, the lock detector signal is satisfied and declares that the loop is in lock.
 

4.9 	 4 = 1580, 8 kbps
 

In the same manner, we can show that
 

AVIN (n ) -	 100 ; TRIND = 0 , AVMI(n) . 0 (54) 

and again there is no update signal and the lock detector test declares a
 

locked condition. Itappears that AVMI(n)=o out to 4)=1800 and beyond to 2100
 
M8206-4 19
 

beyond which the lock detector is not satisfied, but the control is to push
 

the error back to 4 = 1800, where the lock detector again declares that the
 

loop is in lock. This would explain a false-lock mode inboth the co~ierent
 

and noncoherent tests. The coherent test would yield a very stable false lock
 

whereas the noncoherent test would not appear to be as stable.
 

5.0 CONCLUSIONS
 

We have considered the case in which the data rate is 8 kbps and the
 

input SNR is very large; therefore, the signal level into the A/D converter
 

yields a signal which is either +3 or -3 (seven-level output). The result
 

found in this case was that the lock detector indicates an in-lock condition
 

up to a phase error of ±750 (±20% of a bit). Furthermore, the lock detector
 

will indicate an in-lock condition between about 104' and less than 1300.
 

Again, in the region 1500 to 210', the lock detector declares an in-lock
 

condition.
 

Itwas further determined that the error control signal changes its
 

algebraic sign when the phase error is between 900 and 1500, so that phase false
 

lock potential exists in the region between -150' and ~210'.
 

Based on the error control and lock detector functions, itappears
 

that false lock could occur if the signal was turned off, then turned back on
 

when the phase error was between 90' and 2500. By symmetry, the loop is also
 

pushed inthe wrong direction between -90' and -135'.
 

By preventing the midphase adder from overflowing, itappears that
 

the loop will no longer produce a control signal in the wrong direction. The
 

only remaining point is that, since the lock detector will be satisfied only
 

in the region between 1500 to 2100, drift would tend to pull the loop in. But,
 

by removing saturation this lock detector false indication would be prohibited,
 

thus causing the loop to enter the acquisition mode and hasten acquisition.
 

References
 

1. Abbreviated Flow Chart for the PSP Bit Synchronizer Algorithm, by Brett
 

Parrish
 

2. J. K. Holmes, "PSP Bit Synchronizer NRZ False-Lock Status--Interim Report,"
 

Axiomatix Technical Memorandum No. M8206-1, June 18, 1982
 

Distribution
 

R. Helgeson, Consultant B. Parrish, NASA/JSC
 

J. Johnson, NASA/JSC T. Pederson, TRW
 

S. Mayo, TRW M. Reuterman, Rockwell
 
0Axiomatix 

9841 Airport Boulevard e Suite 912 a Los Angeles, California 90045 * Phone (213) 641 8600 

TECHNICAL MEMORANDUM NO. M8207-2
 

TO: B. Parrish, J. Johnson DATE: July 23, 1982 

FROM: J. Holmes, P. Nilsen 

SUBJECT: Recommendations on PSP Bit Synchronizer Design Changes 

Axiomatix has been involved in a review and analysis of the PSP bit
 

synchronizer false-lock problem. This has entailed the following effort:
 

(1) Attending technical discussions, meetings/telecons
 

(2) Witnessing laboratory tests
 

(3) Analyzing algorithms and hardware performance
 

Axiomatix has not yet completed this activity, especially with respect to (3)
 

above However, Axiomatix does feel it appropriate to make certain design
 

change recommendations in light of its findings to date. The design changes
 

which Axiomatix recommends at this time are considered by Axiomatix as the
 

minimum changes required in order to achieve a viable, satisfactorily performing payload communication capability. These changes are as follows
 

(a) Correct the microprocessor midphase addition overflow (and
 

sign change)
 

(b) Modify or reparameterize the lock dectector algorithm
 

(c) Add a bit transition counter.
 

Without these changes, there appears to be a reasonable probability
 

that, given certain operational situations which result in signal strength
 

fluctuations, i.e., antenna pattern modulation of signal amplitude, the bit
 

synchronizer will false lock and data will be lost. Furthermore, the present
 

lock detector does not reliably indicate this condition. Also, the present
 

PSP design may necessitate the SSO crew reinitializing the PSP several times
 

in order to achieve initial acquisition. At present, the only reliable indication of successful PSP bit synchronizer acquisition is the successful frame
 

synchronizer indication.
 
M8207-2 
 2


Inaddition to the three design issues C(a)-(c)) cited previously,
 

there are several other related design issues which have surfaced during the
 

investigation which have the potential to affect communication performance.
 

These are:
 

(d) Saturation of the PSP A/D converter due to a gain change made
 

between the 250-kHz filter and the A/D converter
 

(e) Saturation of the presum integrators
 

(f) Potential DC bias-removal algorithm problems.
 

Axiomatix has analyzed the second item (e)and feels that the present presum
 

design, while being undesirable, does not cause the false-lock problem to occur;
 

however, correction of this problem is a relatively simple software change.
 

Because the DC bias-removal algorithm (f)has been substantially changed,
 

we feel that this should be reviewed by all interested parties in order to
 

verify the accuracy of the new algorithm. We also believe that the gain between the 250-kHz lowpass filter and the A/D converter (d)should be reduced
 

to the original design value or changed to a new, more nearly optimum value
 

in order to avoid clipping the signal plus noise.
 
Axiomatix
 

9841 Airport Boulevard e Suite 912 e Los Angeles,California 90045 * Phone (213)641-8600 

TECHNICAL MEMORANDUM NO. M8208-1 

TO: P. Nilsen DATE. August 9, 1982 

DISTRIBUTION: R. Helgeson 

FROM: J. K. Holmes G. Huth 

J. Johnson 

SUBJECT: Observations of NRZ and Biphase-L S. Mayo 

PSP False Lock at Rockwell B. Parrish 

M. Reutermann 

Both NRZ and biphase-L false lock induced by RF power interrupts were
 

observed at the PSP tests conducted at Rockwell ADL. Details of this observation are provided herein.
 

Testing of the PSP false-lock phenomenon was observed at Rockwell
 

(Downey) on August 2-3 by the author, representing Axiomatix, Mike Reutermann,
 

Michelle Knowlden, Pete Berndsen and Barry Mathias of Rockwell (who actually
 

performed the tests), Sal Mayo of TRW, and Jack Rivers and Bob Vermillion of
 

NASA
 

Both NRZ and biphase-L false lock was observed at high SNR (-60 to
 

-80 dBm). Three types of simulated RF dropouts were used, as follows:
 

(1) Fast switching between two antenna ports with negligible
 

differential signal level, except for the approximate 3 ms switch time
 

(2) Slow switching from antenna port to dummy load and back to
 

the antenna dwelling on the load for about one second
 

(3) Manual attenuation to simulate an antenna fade, with maximum
 

attenuation held for about one second
 

The last two switching procedures were used since the first did not
 

induce false lock. Biphase-L false lock was different from NRZ false lock in
 

that it was data-pattern dependent and NRZ was not Also, both lock detector
 

and frame synchronization discretes are low in biphase-L false lock. Practically speaking, since data would tend to be more or less random, the biphase-L
 

false-lock problem may not be "real" although it should be verified with actual
 

payload data. Table 1 summarizes the differences between NRZ and biphase-L
 

false lock. Some of the test data are included at the end of this memorandum.
 
M8208-1 
 2 

Table 1. False-Lock Summary
 

NRZ 


1. Only at high SNR (-60 to -80 dBm) 


(At 16 kbps, need > -80 dBm) 


2. Not data-pattern dependent 


(ability to false lock) 


3. In false lock: 


Bit synchronization - high 


Frame synchronization - low 


4. 	Changing data pattern does not 


break false lock 


5. 	False locks at all data rates 


6. False locks on switching from 


antenna to load to antenna and 


when using the attenuator to 


increase attenuation to simulate 


a fade 


7. Not dependent on having wire on 


PSP clock pin to cause false lock 


Biphase-L
 

1. Only at high SNR (-60 to -80 dBm)
 

(At 16 kbps, may need > -80 dBm)
 

2. Data-pattern dependent (ability
 

to false lock)
 

3. In false lock:
 

Bit synchronization - low
 

Frame synchronization - low
 

4 	Changing data pattern breaks
 

false lock
 

5 False locks only at 1 and 16 kbps
 

(2 kbps not yet tested, does not
 

false lock at 4 or 8 kbps)
 

6. 	False locks when switching from
 

antenna to load to antenna and
 

when increasing the attenuation
 

by means of an attenuator to
 

simulate a fade
 

7. Not dependent on having wire on
 

PSP clock pin to cause false lock
 
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Axiomatix
 

9841 Airport Boulevard 0 Suite 912 e Los Angeles, California 90045 e Phone (213) 641-8600 

TECHNICAL 	MEMORANDUM NO. M8208-2
 

TO: 	 P. Nilsen DATE: August 27, 1982
 

FROM: 	 J. K. Holmes
 

SUBJECT: 	 Recommended Changes to the Tracking Mode of the
 

PSP Bit Synchronizer Lock Detector for NRZ Data
 

REFERENCE: 	 B. Parrish, "PSP Bit Synchronization Algorithm Flowchart Summary
 

for the NRZ Mode," JSC document, undated (about July 1982)
 

1.0 	 INTRODUCTION
 

This technical memorandum documents the recommended parameter value
 

changes which were previously transmitted to TRW by telephone. The recommendations are to narrow the in-phase and midphase filter bandwidths and to change
 

the lock detector test ratio in the NRZ mode of the PSP bit synchronizer lock
 

detector By making these changes, it is believed that the problem of the lock
 

detector not indicating loss of lock, sometimes not until minutes after the
 

signal is removed, will be eliminated. These recommended changes will require
 

only that the bit synchronizer algorithm parameters be modified, and no algorithm or hardware changes will be involved.
 

2.0 	 RECOMMENDED CHANGES
 

It is recommended that both the in-phase and midphase filters be reduced in bandwidth so as to increase their respective output signal-to-noise 

ratios (SNR's) and thereby improve the reliability of the lock detector test. 

Specifically, it is recommended that (i) the L-value of the in-phase filter be 

changed from 0.99 to 0.999 (if feasible to implement digitally) so that the
 

"gain" goes from 100 to 1000 and, (2) the L-value of the midphase filter be
 

changed from 0.95 to 0.995 so that this "gain" goes from 20 to 200. With the
 

narrow bandwidth digital filters, it is possible to change the in-lock bit
 

synchronizer test from
 

AVIN > 6 51AVMII
 

to
 

AVIN > 501AVMII
 
M8208-2 
 2
 

The result of this change would be to greatly improve the probability
 

of declaring that the bit synchronization loop isout of lock when the signal
 

is removed. Presently, the loop can display an in-lock condition, sometimes
 

until minutes after the signal has been removed. This would be confusing to
 

flight operation personnel.
 

The analysis which led up to the recommendations of narrowing the
 

digital filters and changing the in-lock test isbased on a rather involved
 

analysis, which will be the subject of a future memorandum.
 

DISTRIBUTION
 

R. Helgeson, Consultant
 

J. Johnson, JSC
 

S. Mayo, TRW
 

B. Parrish, JSC
 

T. Pederson, TRW
 

M. Reutermann, RI
 
Axiomatix
 

9841 Airport Boulevard a Suite 912 * Los Angeles, California 90045 * Phone (213) 641-8600 

TECHNICAL MEMORANDUM NO. M8209-2 DATE: September 3, 1982 

TO: P. Nilsen COPIES: NAS 9-16067"B" File 

FROM 

SUBJECT: 

J. K. Holmes 

PSP Bit Synchronizer Lock Detector 

Parameter Value Changes for the 

NRZ Data-Tracking Mode 

R. Helgeson, Consult. 

J. Johnson, JSC 

S. Mayo, TRW 

B. Parrish, JSC 

T. Pederson, TRW 

M. Reutermann, RI 

REF. (1) Technical Memorandum No. M8208-2
 

(2) B. Parrish, "PSP Bit Synchronization Algorithm Flowchart Summary
 

for the NRZ Mode," JSC document, undated (about July 1982)
 

1.0 INTRODUCTION
 

This memorandum makes a "final" recommendation for the PSP bit synchronizer lock detector parameters in the NRZ data mode based on the current
 

state of the microprocessor available memory and processing rate. Due to the
 

lack of additional memory storage space, Rich Helgeson has pointed out that,
 

with the present schedule and memory constraints, it is feasible only to reparameterize the lock detector algorithm at this time.
 

The recommended recursive L parameters of the respective one-pole
 

in-phase and midphase filters are given by
 

254
 

LI = = 0.9922
 

L = 248 = 0.9688
 

and the lock detector test becomes
 

AVIN > 13*IAVMII
 

for NRZ data.
 

2.0 BASIS OF NEW RECOMMENDATION
 

Based on a recent telecon with Rich Helgeson, it was determined that
 

it was impossible to implement the desired values of L=0.999 for the in-phase
 

filter and L=0.995 for the midphase filter of the bit synchronizer lock detector
 
M8209-2 
 2
 

recommended inref. 1. Basically, the allowable values of the recursive filter
 

parameter L must be of the form
 

TI (1)
 

where n is an even integer less than 256. Based on this constraint, the maximum recursive filter L value is then L=254/256 =0.9922 and, hence, 0.999 cannot
 

be achieved.
 

In ref. 1, it was recommended that the lock detector for the NRZ case
 

have separate in-phase and midphase lock detector filters which are much narrower than the bit synchronizer loop In-phase and midphase filters; however,
 

in light of the above-mentioned limitations, it is recommended that both the
 

loop and the lock detector use the same filters. Thus, the in-phase and midphase filters should be modified from
 

LI = = 0.984375
 

= 2-42 = 0.9453 

with the lock detector test in tracking specified by
 

AVIN > 6 5*IAVMII
 

to
 

LI 256
 = 2254_56= 0.9922 0 92 

= 
 = 2 0.9688
 LM 256
 

with the lock detector test intracking now being specified by
 

AVIN $ 13*IAVMII
 

where the coefficient 13 is close to the desired value of 15 and is a simple
 

shift of one digit, resulting in a multiply-by-two of the current value of 6.5.
 

The above changes should provide the same probability of detection
 

value when in lock as those of the existing parameters but, inaddition, they
 

will decrease the probability that the lock detector declares that the signal
 

is present when it is not. As a result, the lock detector light should not
 

remain on for long periods of time when the signal drops out; this is a problem
 

in the original algorithm.
 
Axiomatix
 

9841 Airport Boulevard * Suite 912 * Los Angeles, California 90045 oPhone (213) 641 8600 

TECHNICAL 	MEMORANDUM NO. M8210-1
 

TO: P. Nilsen DATE: October 4, 1982
 

FROM: J. Holmes FILE: NAS 9-16067"B"
 

SUBJECT: PSP Lock Detector Performance in the Tracking Mode for NRZ and
 

Biphase Data
 

1.0 	 SUMMARY
 

In this technical memorandum, we analyze the lock detector performance
 

of the Shuttle PSP (intracking only) and recommend improvements and/or modifications to minimize the time required to indicate out-of-lock status when the
 

input signal is removed. Because of microprocessor limitations in both memory
 

size and speed, it was determined that the digital filters used for the inphase
 

and midphase bit synchronizer tracking loop and lock detector must be the same,
 

forcing a compromise of sorts. Consequently, with this constraint and the fact
 

that only a limited number of one-pole digital-filter-recursion parameters were
 

available in the microcode, the following results were obtained.
 

* NRZ Data
 

A. 	Present parameters
 

Inphase parameter: LI = 252/256 = 0.9844 

Midphase parameter: L = 242/256 = 0.9453 N?
 

Lock detector criteria: AVIN > 6.5 IAVMII
 

B. 	Recommended parameters
 

Inphase parameter. L, = 254/256 = 0.9922
 

Midphase parameter: LM = 248/256 = 0.9688
 

Lock detector criteria- AVIN > 13 jAVMIj
 

* Manchester Data
 

A. 	Present parameters
 

Inphase parameter. LI = 252/256 = 0.9844 

Midphase parameter- LM = 252/256 = 0.9844 

c d?
 Lock 	detector criteria: AVIN > 5 JAVMIJ
 
2.0 

M8210-1 
 2
 

B. Recommended parameters
 

Inphase parameters: LI = 254/256 = 0.9922 

Midphase parameter: LM = 254/256 = 0.9922 

Lock detector criteria: AVIN : 15 fAVMIj
 

The mean time to indicate out-of-lock with the new parameters when the
 

signal is removed reduces to one-half the time in the NRZ case and about onetenth the time inthe Manchester (biphase-L) case, compared to existing parameters. Since the analysis and system modeling were very involved, approximations
 

were made inthe analysis; however, it is believed that the answers are representative of the actual system performance.
 

During the course of this work, numerous helpful discussions were held
 

with Rich Helgeson (consultant) and Brett Parrish (NASA/JSC) concerning the operation of this bit synchronizer.- Inaddition, observations of the false-lock
 

phenomena on the actual hardware were observed at both Rockwell and TRW.
 

INTRODUCTION
 

Shuttle-launched payloads will communicate with the Space Shuttle
 

Orbiter payload communications system over a range of tens of nautical miles.
 

Command data is transmitted to the payloads while telemetry data are received
 

at the Orbiter on multiple S-band frequency channels. Figure I illustrates the
 

Space Shuttle Orbiter receiver block diagram for payload signals, which is part
 

of the Payload Interrogator (PI). The PI sends a 1.024-MHz subcarrier PSK modulated by either NRZ or biphase signals at date rates of either 1, 2, 4, 8 or
 

16 kbps. Signals from the PI are sent to either the NASA PSP, the DOD PSP or
 

the Ku-band signal processor (KuSP). This memo is concerned with the performance of the NASA PSP which performs subcarrier demodulation, bit and frame
 

synchronization, as well as matched-filter detection of the data, as shown in
 

Figure 2.
 

This memo concentrates on PSP bit synchronizer lock detector performance during tracking. Performance specifications of the PSP bit synchronizer
 

loop require less than 1.5 dB implementation loss over a signal level corresponding to an Eb/N 0 range of 2-11 dB at the minimum average bit transition
 

density of 12.5% (64 bit transitions in 512 bit periods). Acquisition must
 

occur within four seconds with 0.9 probability for Eb/N 0 values and data formats
 

listed in the specification amendment of March 25, 1982, shown as Appendix D in
 

this memo. A more detailed (but still simplified) block diagram of the original
 
PAYLOAD INTERROGATOR 

A nvider ITracking 	184/31 MHz IF Power Carrier-	 -nI O0 0

Antenna D~uble Loop (PLL) 	 oo 

0 

Swtch/RF Pre Downconversion PD 


Fillter Assembly Mixer/IF/AGC 	 NASA to­ 1.024 Mz 	 IPSP C7 

To 

DOD
 

MoI tlfrequency 
 LPF 	 pSP PSP
 

iSynthesizer 
 ignal
 

, ._.._Ku-Band
 

Processor
 

Figure 1 Shuttle Payload Communications Receiver
 
ANALOGSUBCARRIER TRACKING LOOP 

Data-Rate 1 A= l Adapt ive 

CROPROCESSOR 

Dump Command (delayed 

one-half b it) 

D D C Timing 

Correction 

a nd eragi c-e 

7 

Ban 

tom 

F 

Dta-Ratno 

ki F Ba n k urLP NASA P P C 

I I 

Fr 

LPF iDump 

Count 

ii-

• 

F~rom 

Ir 

_ 

L icrooessorl __ 

F D..NT rI3 G -ITAL 

COMPONE 

T 3- i i h - i 

1 

'--r " [ Pres ummerI 

f g a i 

~ 

I &D 

II 

-a"0 

Threshold 

+iTransi 

tion 

eetrI 

NRZ-L 

Frame 

Conversion 

Synchronzation 

M Frame 

Data Sync. 

0 

r 

' 

4 

1.024 MHz 

Sampl ing Clock 

Figure 2 NASA PSP Conf igurat on 
5 

3.0 

M8210-1 


bit synchronizer is shown in Figures 3 and 4. Basically, the data is demodulated
 

by the 1.024-MHz subcarrier (Costas) loop and is sampled at 1.024 Msamples per
 

second, with each sample being quantized to three bits. These samples are stored
 

in presums that are added together in eight-bit registers to form the last, middle, and first half-bit integrals, as depicted in Figure 5.
 

The inphase averaging filter is driven by the magnitude of the sum of
 

the first and last half bits, and the midphase filter is driven by the sum of
 

the first half bit and the previous last half bit (delayed one bit)--all multiplied by the transition indicator [1,2].
 

THE THREE-BIT QUANTIZER
 

Inorder to proceed with our analysis, certain assumptions were made
 

so that the analysis would be manageable. First, the effect of the second-order
 

tracking component was neglected because it was designed for a very limited
 

correctional capability. Second, the DC restoration algorithm was not modeled
 

because the new algorithm was not available at the time of the analysis for this
 

writing and, also, it was felt that, in tracking, the DC restoration was not an
 

important factor in the loop operation. Inaddition, the scaling algorithm was
 

not incorporated since it does not have a major effect at low SNR's. Finally,
 

it was assumed that the revised algorithm for acquisition and tracking will not
 

cause either limiting or register rollover due to unprotected registers This
 

has been verified by simulation, calculation and laboratory measurements. An
 

abbreviated flowchart is presented in Appendix A which was constructed by Brett
 

Parrish, NASA/JSC.
 

Table 1 lists the number of samples per bit at each data rate with
 

the new microcode.
 

Table 1. Samples Per Bit at Each Data Rate
 

Samples/Presum Samples/Presum
 Rb Samples/Bit (Current System) (Proposed System)
 

16 64 16 16
 

8 128 32 16
 

4 256 64 16
 

2 512 64 16
 

1 1024 64 16
 
From subcarrier loop
 

LPF B = 25D0 kHz
 

MICROPROCESSOR
 

Three-Bit Io
 

A/D
 

S8-bit 
 10 Timing
 

Presummer ! Dump Count Correction
 

Logic 00
 

8-bit 	 Lat 

Lastv
 

PresumO' p ADD D = Midphase ' 0 P 0.P 	 1-ID& lpae m


i- DO (nt usd[np hase I
 

8 SNote 0 P. s)eoerfowvpo
 -' iures 3.8PSP -Tt 	' oT0'( u s e	 Io
 & D) Tra n s it io n


8 - i 
 aD

o R )A 	 DD 
 e t ector
 

Bit 	 8' it First I Inphase I I > M 

n + I n l-Half -bit ' iAveraging (Loc 1eetr1 

I n t e g r a l I Olrdest L I ,, Filter (Lc eetr
 

P. 0 P 
 Lock Indication


Note-	 0 P. = overflow protection
 

Figure 3. PSP Data-Transition Tracking Loop and Lock Detector for NRZ Data in the Tracking Mode
 
Carrier-Tracking
 

Loop
 

I 

4 1MICROPROCESSOR
 1

Three-Bit I1
 

A/D
 

- I 

8-bt I 
 Dump Count 
 Timing
 

Presummer 0 0Correction 


O.P.Loi
 

, S-bit Last ;
 

Newest iltHalf-bit I


Presum D Integral
 

0PMidphase
 

8-bIt 
 Modphase nMatched 


-b t
 -F"--- Ha l f I F1 e
 

O.p. . ADD ne ral In h F l e 

8-bNt vfMatched poetn Filter 


FgPresum OPP S Di i TransLt eonr f
 

Avra gl ng
 ADDbt Integral e 

SOldest iFiler
 

Presum 0PLc n
 

0OP 0 mokIniain
 

Note O.P = overflow protection
 

Figure 4 PSP Data-Transition Tracking Loop and Lock Detector for Blphase Data in the Tracking Mode
 
M8210-1 8 

ORIGINAL PAGE ES 

OF POOR QUALITY 

Inphase Integral
 

FSTB2 LSTB2 

1i 

Midphase Integral' 

PRTB2
 

NOTE-	 FSTB2 - first half-bit
 

LSTB2 - last half-bit
 

PRTB2 - previous last half-bit
 

(a) NRZ Data
 

Inphase Integral
 

FSTB2
 

PRTB2 LSTB2
 

MITB2
 

(b) Biphase Data
 

Figure 5 Time Blocks in One Bit Period for NRZ and Biphase Data
 
9 

ORGINAL PAGE M
 

M8210-1 OF pOOR QUALM 


Let N denote the number of samples per bit which is either 64, 128,
 

256, 512 or 1024, as indicated in Table 1. The sample rate is 1.024 Msps and
 

the baseband signal plus noise is filtered to 250 kHz. In Figure 6, a model of
 

the deadzone A/D converter is illustrated. Notice that, if the signal were
 

less than one-half of a cell in peak amplitude, there would be zero output.
 

Consequently, AGC action isnecessary to keep the peak amplitude greater than
 

one-half of the cell size. The analog baseband signal to the A/D converter is
 

given by
 

y(t) = A d(t) + n(t) 0 < t < T (1) 

where A is the signal amplitude, d(t) isthe baseband NRZ data signal, and n(t)
 

is modeled as bandlimited white Gaussian noise. Each bit integral is approximated by the sampling of y(t) over one bit time to yield
 

N N N 

FSTB2 + LSTB2 = d. + n.+ qi (2) 

i=l1 i=1 =1 

Based on the new implementation, we assume that saturation cannot occur in any
 

of the registers. In (2), di is the ith sample of d(t), ni is the ith sample
 

of n(t), and qi is the ith sample of q(t), where q(t) is the quantization noise
 

introduced by the A/D sampling process. Although ql is not generally independent of di and ni, we will assume that it is in order to simplify the following
 

analysis. Now we determine the level of the signal and the thermal noise out
 

of the 250-kHz lowpass filter We have
 

E b () (3)
 

where B is the lowpass filter noise bandwidth and R is the data rate, R = lI/T,
 

with T being the bit duration. The signal power to noise power is then given by
 

Eb R
 

SNR : NO0bB (4)
 
M8210-1 10
 

ORIGINrAL PAGE 0
 

OF POOR QUALITY
 

3 

-2
 

0 

2 

}-3
 

Figure 6 Model for the Deadzone A/D Converter 
ORIGINAFL PAGE M 

M8210-1 OF POOR QUALITY 

For example, at R = 8 kHz and Eb/NO = 4 dB, we have
 

(5)
 8 x 10 -13dB 5
 SNR = 1.58 


250 x 103 
 -3d


Thus, we tend to have rather negative sample SNR's. Now the sample signal voltage to noise standard deviation is given (from (4))
 

0- 2 (6)
 

When the loop is at the correct timing position, (2)becomes
 

b- N N 

FST2 + LSTB2 = AN B n1 + q (7) 

i=I i=1 

the ± sign is determined by the algebraic sign of d(t), te(O,T). Now, since 

the samples are taken at a sample rate of 1.024 Msps and since the lowpass filter
 

preceding the sampling is 250 kHz wide, the noise samples are not independent.
 

Hence, we assume that sample pairs are independent and each sample has the same
 

value in the pair. Using this mode, we have
 

+=bR N2 N/2 

FSTB2+LSTB2 = ±NV a+ 22 n + 2> q (8) 

where, since we have assumed that the noise samples are pairwise independent,
 

we make the same assumption for the quantization samples
 

Letting n denote the noise sample sum in (8)produces
 

N/2 

n = 2 n (9) i=11 

E[n] 0 (10)
 

N/2 N/2 

Var[n] = 4 i I E[nmnj] = 2N c 2 (11)

1=I j=l
 
M8210-1 ORIGmIAL PASS [S 
 12 

OF POOR QUALITY 

If we let nq denote the quantization noise, we have
 

N/2 

nq = 2 > q1 (12) 11
 

Hence, by symmetry,
 

EEIj= 0 (13) 

N/2 N/2 2
 Var[nq] = 4 1 EE=q = 2N qf i=1 =1 1i2 Varqi a (14) 

Ifwe model the quantization noise as having a uniform distribution
 

and let L denote the width of the quantization cell, then the quantization noise
 

is given by
 

2 _1 fL/2 2 L2 

q L x dx 12 (15) 

-L/2 

and 

EEnqJ = 0 (16) 

Ifwe let the load factor* be denoted by LF, then the optimum LF should be
 

about 2.28. From Figure 6 and the definition of the load factor, we have
 

2.28 - 3L (17) 

or
 

L 0.77 a (18)
 

From (15), we obtain
 

2 2 

aq = 0.05 a (19)
 

The A/D converter load factor is the ratio of peak output to RMS
 

noise input.
 
M8210-1 ORIGINAL PAC" 13 

OF POOR QUALTr1 

For arbitrary load factors, we obtain 

q = 0.0092 (LF)2 a2 (20) 

Using (10), (11), (15) and (16) in
 

IbR N/2 N/2
 

FSTB2 + LSTB2 N O-- ua + 2 Z n i=1 + 2 Z q (21)
 VWo i=1 


yields for the mean and variance
 

E[FSTB2 + LSTB2] : ±N - (22) 

and 

Var[FSTB2 + LSTB2] = 2Na 2 + 0.1N a = 2.1Na2 (23) 

Since the magnitude of the inphase term isrecursively filtered, we now investigate the statistics of JFSTB2 + LSTB21 that are needed before we can analyze 

the filtered version. Let x denote FSTB2 + LSTB2 Since the major noise is 

thermal noise, we shall approximate FSTB2 + LSTB2 as a Gaussian random variable 

with mean p and variance 12. From [3], itcan be shown that 

Exl2 = exp() ,Ierf1 (24) 

where
 

erf(x) _ 2 f e2 dt (25)


0
 

and where
 

(26)
 p = N f r 

32 = 2.1N 2 (27) 
ORIGIAL PAe - 14 M8 210-1 OF POOR QUALIrr 

Now consider the variance of Ixi. We have
 

Var[xI] = ExG - EEx 

or 

1 Varx 2 I Z2 + -2/2 e-2Z2 + - erf 2 (28) 

When (p/s) 2 = 4 dB, we have 

VarjFxl] 	 = Z2 2 (29) 

Hence, for (b/s)2 > 4 dB, we have that 

FVar[x]
 2

va x 	 EFx IE[xI for 2 > 4 dB (o 

Hence, if 	 we assume that d(t) is +1 in every bit interval*, we have EEx;=E[x]. 

Now note that 

N = R s 	 (31)
 

with Rs the same rate and R the data rate. Hence, 

N Eb R (32) 

NH
0


or 

N R R C (33) 

Since Rs/B = 4, we have 

= 2ATT 	 (34) 

Henceforth, we assume that 1 is always positive and drop the
 

magnitude signs for simplicity.
 
4.0 

M8210-1 	 ORIGINAL 0 15 

OF POOR QUAL1T 15 

It is worthwhile to compare the performance of an analog matched
 

filter with the digital version that we have just analyzed. The digitized SNR
 

is given by
 

SE2 [FSTB2 +LSTB2] 	 (35)
 SNRD Var(FSTB + LSTB2)
 

or 

Eb 

4N 2 Eb 

SNRD = N0 = 1.9 b (36) 

2.1 N a 

Since the ideal analog matched filter would yield an SNR of 2.OC=Eb/NOD, we see
 

that, within our approximations, the sampling and quantization noise have produced a loss of 0.2 dB.
 

BIT SYNCHRONIZER DIGITAL FILTER OUTPUT STATISTICS
 

WITH SIGNAL PRESENT--NRZ DATA
 

We now present some of the main elements of the bit synchronizer and
 

lock detector algorithms. An abbreviated flowchart developed by Brett Parrish
 

is included in Appendix B.
 

Basically, two one-pole recursive filters are used in the bit synchronizer and lock detector. The first is AVIN, the inphase bit integral filter (Figures 4 and 5); the second is AVMI, the midphase bit integral (Figure 4).
 

The former is defined by
 

x(n) 

x A 

(n+1) kn) (n) (n)
 

AVIN = L AVIN + IFSTB2 + LSBT2 J (37)
 

where n denotes the nth bit time The midphase filter has an output given by
 

(n'+I) (n') / (n') (n)\(n') 

AVMI = LAVMI + \FSTB2 +PRTB2 )TRIND (38) 

where the n' denotes the n'th bit transition. Figure 4 illustrates FSTB2,
 
M8210-1 ORIG.'MAL PAG u_ 16 

OF POOR QUALITY 

LSTB2 and PRTB2. The term TRIND takes on the value 0, 1 or -1 according to
 

whether no bit transition occurs or whether a positive or negative transition
 

occurs. Therefore, the last term of (38) is a proportional measure of the bit
 

timing error whenever a bit transition occurs.
 

To determine if the bit synchronizer should be in the tradking or acquisition mode, a lock detector is utilized by making the following comparison:
 

?
 

AVIN > 6.5 IAVMII (39) 

so that, when AVIN is larger than 6.5 AVMI, the lock detector declares that the
 

bit synchronizer loop is in lock.
 

Consider AVIN first. Using the approximations of (30), we have that
 

x n) 

2V -1 b/a with probability Pt 

(40)
 (n+l) (n) nn
 AVIN LIAVIN +n) 
 +n q()+,(0


2VKW o with probability C1 -p
 

where Pt is the bit transition probability. The term j is the number of samples 

inwhich the timing is in error so that j/N is the discrete measure of phase 

error. Note that, when no bit transition occurs, there is no signal loss, however, when a transition does occur, the signal component is reduced by the factor Ci - 211j /ND, with j being the number of samples (out of N samples) in 

error. For convenience inthe analysis, we shall approximate the discrete timing 

error by a continuous phase error and let 

N - ­ (41)
 

where is the continuous-valued phase error. Using (41) in (40) leads to
 

:b X( n ) (n)(n), ,E 

x(n) n(n) + nq(n ) + 2VN, 1 - a X (42) 

and
 

AVIN(n+l) = LIAVIN(n) + x(n) (43)
 
PAGE 17 

OF POOR QUALTV 

wi th 

X( n ) 

M8210-1 ORIGINAL Q 

= 1 -!) with probability pt 

1 with probability (i- Pt44 

where Pt is the bit transition probability. Since the phase error is normally 

quite small, we will approximate 4 by a zero-mean Gaussian random variable with 

2 variance a, *I 

First, we compute E(nI T. We have from (44) that 

EE(n] = 2VN-1 NKar( - E[--) Pt + (1- Pt4 (45) 

For a Gaussian random variable, it can be shown that 

E[II] = (46) 

Using (46) in (45) produces
 

E[xn2v / a K < ' (47) 

To evaluate the variance of X(n), we first compute ERx~n))2j. Using (42),
 

we have
 

E nx(n)2j E~n(n)+nq()) 2

] + 4E In(4)+nq(n)) vF- , x(n)i 

+ 4 NEb a2 Ef (48)
 

Evaluating, we obtain (since the second term is equal to zero)
 

EF(x(n))f] 2.1c+2. 4N fo+ E b 2 k 2 2~ 24) z p 1 (1 -t pt) (49) 72T 
M8210-1 	 18
 

or 	 ORICAL. PAGE ES 

OF POOR QUALITI 

ERX(n))fI = 2.No 2 +4N %2 32o,-T } C]) 

Now 

2F(n Eb 200_ ( t -

12a4 Pt + 2 2pt1 ELX 4N No

0 a - if _7o 	 (1 

T 

Since 

VarE(n? = E(x(n))J- E2 [Cn)] 	 (52) 

we obtain
 

> Var (n = 2 IN a +4NO N a t- pt], 1 > pt 0 (53) 

Itfollows from (A-5) and (A-13) that C0 < Pt < 1J 

2.lNa 2 + 4 Eb 2( 2 2 

Var [VIN(nI =1 1I- LI 2 t (54) 

and
 

2 f 7-T p 

EAVIN(n) LI 

Now we consider the inphase one-pole digital filter The output of 

this filter is characterized by the following recursion, as indicated in (38) 
M8210-1 	 ORIqINL PAGE [ 
 19 

OF pOOR QUALWY,
 

AVMI~n+1) = AVMI(n')LM 	+ (FSTB2(n)+PRTB2(n '))TR IND(n ') (56) 

recall that n denotes the nth bit and n' denotes the n'th bit transition. Only
 

=
 where pt 1 does n=n'. In the following analysis, we will model TRIND (n ') as
 

a perfect estimator of the algebraic sign of the bit transition*. Bit transitions from -1 to +1 produce a value of TRIND = 1 and bit transitions from +1 to
 

-1 produce a value of TRIND=-I. Again writing the sum of (FSTB2+PRTB2)TRIND
 

in the form of thermal noise, quantization noise and loop phase error control
 

signal, we have as our recursion (positive bit transitions considered)
 

AVMI(n '+l)= LMAVMI(n') + nT(n) - 2j Eb R 	 (57) 

where 0OR/B a is the signal level out of the A/D converter and, therefore,
 

the le input n 
 signal nn') level (n') to the midphase recursive one-pole filter, and we have


(
 (n') =n n' for convenience.
 

From (31) and the fact that Rs/B =4, we have the alternate form
 

AVMI(n'+I) = LMAVMI(n') + nn, 
- 4( f f-a 	 (58)

Further equating the discrete phase error j/N to the continuous equivalent
 

phase error /2w, we can express (58) in the form
 

x(n') 

'
 AVMI(n +l) = L AVMI(n') 	+ nT') 2r /:Xa (59)
 

Since the rightmost two terms are of zero-mean value, we see from (A-5) that
 

E[VMI(n'+1 = 0 	 (60) 

Now consider the second moment We have
 

Since we are dealing with low bit-error rates, this assumption
 

causes a negligible error in the results.
 
20 

5.0 

ORIGINAL PAGE U 

M82 10-1 OF POOR QUALITY 

E[(x~n')) 21 = E n(n') _W Lk 10 E[knT)I En 2 2'(1 TfN 0 T 21Uo 

From (61) and (A-13), it follows that
 

E2
 

1N2 + 4 N2O ( 

Var AVMI(n' = O (62)

S2 (62)
 

Also, from (60) and (A-5), it follows that
 

E[AVMI(n'I = 0 (63) 

BIT SYNCHRONIZER DIGITAL FILTER OUTPUT STATISTICS
 

WITH NOISE ONLY--NRZ DATA
 

In the last section, we characterized the output of the inphase and
 

midphase filters when signal and noise were present and Eb/N 0 > 4 dB. At lower
 

values of Eb/No, the inphase model estimate becomes poorer and one is forced to
 

reconsider the problem when the signal is not present. Since the SNR in the
 

5-MHz filter bandwidth is,at most, -12 dB at the highest data rate, decreasing
 

the signal will have a negligible effect on the noncoherent AGC gain. We neglect this negligible gain change between no signal and the 4-dB Eb/No value.
 

Since there are no additional approximations in the midphase case,
 

we may let Eb/NO equal zero in (62) and (63) to obtain for the NRZ noise-only
 

statistics
 

VarAVM(nH A _ 2 1 N 2 -LM (64) 1 


and
 

E[AVMI(n'I = 0 (65) 

Now consider the inphase digital filter output statistics. From (37) 

with Eb/N 0 = 0, we have that 
21 M8210-1 ORIGINAL PAGE ES

OF POOR QUALITY 

x( n ) + nq( n ) = n( n) (66) 

since it is no longer possible to employ the approximations of (30), i.e., 

E[ ( n ) +nq (n - E[I n(n) +nq(n) . Var n(n) +nqf(n)7J = VarE(n) +nq(n (67) 

Ifwe again model the thermal noise plus quantization noise as Gaussian, we 

then have
 

E[x(n = 2 f x /22 (68) 

n( n ) +n n ) where 2 isthe variance of . Evaluating (68) and using (27) leads 

us to 

E1nn+nq(7)f 2-. = 2N2-- (69) 

as isclear from (24). Further, the variance isgiven by (28) with p=E[xn]=
 

and z defined by (27) so that
 

Varn(n)+nq (n j = - 2)(2.1N a2) (70) 

Itthen follows that 

E 1-VIN(n ILN 
 (71)


and
 

Var Vm(nf1 - L 2 (72) 
M8210-1 	 ORIGINAL PAGE E* 22 OF POOR 	 QUALITY 

6.0 	 BIT SYNCHRONIZER DIGITAL FILTER OUTPUT STATISTICS
 

WITH SIGNAL PRESENT--BIPHASE DATA
 

The biphase inphase digital recursive filter input when no timing
 

error exists is identical to the NRZ data case. When a timing error exists,
 

the output differs somewhat. We have that
 

AVIN(n+1) = LIAVIN(n) + IFSTB2 - LSTB21 (73a) 

Assuming 	that Eb/N 0 > 4 dB allows us to write Cwhen Jl/N < 1/4)
 x(n)
 

n l n LIAVIN~ I n~ + n q(7b

(n) + X(n) (73b)
 AVIN(n+1) LAVIN) (n) 

where
 

Xjn) 	 2MfV/E (1 4 1lI) with probability Cl-PtD (74) 

and 

Ebvw-N(1 - 21 ) with probability pt X( n ) = 2 N N (I75) 

Again approximating the discrete variable j/N by the continuous-phase error @ 

Ci.e., 131/N = II/2 D, we have 

-	 pt 2 S= 1-2 ) with probability Ci ­

and (76) 

X(n) = 2F ( with probability Pt 

Averaging 	x(n ) of (73), we obtain
 

E[(n_ 2 AT. (1­
M8210-1 23
 

ORIGINAL PAGE ES 

OF POOR QUALITY so that 

EAVIN - 1 - L) (78) 

Evaluating ERKX~n))f. we obtain 

=2 lXn)+] % 2[$a2+4N a +4:)+('o, 2)a 2 (79)

0 7r1rItfollows from (77) and (79) that
 

= 2 Var (n 2.1N u + 4N EP_ b 2L4; o (4w-8) 2 (37 2 2P (80) 

Therefore,
 

2.lNa -2 2 +4N b 2i4 / 7T2(y a +(4r-) 2 (@2 2pt 2 -- aG t (3____ 2 2 2 2] 

Var VIN(nl = L 7 it (81) 

1 - LI
 

(n ) Now we consider AVMI , the midphase filter output In the biphase 

case, the integral over the transition isfor a duration of one-half a bit time, 

not a whole bit time, as in the NRZ case.
 

The midphase filter is specified by
 

'
 x(n) (82) AVMIMn+I) LM*AVMI (n) + MITB2*TRIND 


Now the signal-plus-noise component, again neglecting bit errors, is given by
 

n ( n ) P.. ,b x n + n) 


), +qe1-1,2,-2,.. j (83)
 

By (31), we have
 
M8210-1 ORIGINAL PAGE M 24 

OF POOR QUALITY 

R Rs 4 

B NB (84) 

so that (83) becomes
 

).)2 

(n 


(n ) 
=n (n ) + n ( _ C2fN- b (85) 

if we again approximate the discrete timing error j by an approximately equivalent zero-mean Gaussian phase error 4. We use (41) in (85) to obtain
 

( n ) = n(n) + nq (n)- 2 y b-v- (86) 

Again, since 4 is assumed to be zero mean, we obtain 

EE[nj(n 0 (87) 

so that 

E[AVMI(n 0 (88) 

To obtain the second moment and, therefore, the variance, consider (n(n) =n(n)+ 

nq(n)) 

ElxGn) ) 2 = ELT(n) -2 v+4E = E(nT(n)) 21 +4ET(n) bo 

+ 4 __NO N -Y (89) 

The second term is zero if nT(n) and * are uncorrelated Since 4 depends on 

at least 20 or more bits, the correlation is very small; hence, we assume zero 

for the second term. 
7.0 

M8210-1 OF ORGMLPG' pOOR QUALITY2525 

Hence, (89) becomes Csince E[x(n)] =Oj (N'=N/2)
 

E2 (0 )] ~ ~~4Na 

Var (n = 1.05Nc 2 +___ 0EL 2 

go-HU (90) 

From (A-13), we obtain
 

1.05N 2 + 4Na 2 Eb 

Var [vMI(n - T No (91) I - LM2
 

Thus, (88) and (91) characterize the discrete one-pole filter output when the
 

distribution isassumed to be normal.
 

BIT SYNCHRONIZER DIGITAL FILTER OUTPUT STATISTICS
 

WITH NOISE ONLY--BIPHASE DATA
 

In this section, we determine the one-pole digital filter output
 

statistics for the noise-only case with biphase data. Since the inphase
 

EVIN(nH filter has the same statistics for noise only with both NRZ and bLiphase data, we therefore obtain from (71) and (72)
 

E[AVIN(n =1 I -L (92)
 I N ( 2 

and
 

- (1 - ) 2.lNa 2 

VarLVINJ - 2 (93) 

Now the midphase filter statistics with noise can be obtained from (88) and
 

(91) with Eb/NO set equal to zero:
 

EEVMI(n = 0 (94) 

and
 

Var AVMI(n _ 1.05N2 (95) 2
 1-LM

8.0 

M8210-1 ORIGINAL PAGO iR 
 26

OF POOR QUALITY 

PROBABILITIES OF THE LOCK DETECTOR TEST
 

From (29), we see that the current lock detector test is based on
 

comparing the inphase filter output with 6.5 times the magnitude of the midphase filter output. Ifthe inphase filter output is larger, lock is then
 

declared; otherwise, an out-of-lock condition is indicated. We consider the
 

slightly more general test
 

AVIN r IAVMIZ (96)
 

where r is a real-valued number greater than zero. Let pn denote the probability that the inequality of (64) is satisfied so that
 

Pn = P(AVIN(n) > r AVMI (n)) (97)
 

Denote the probability density* of AVMI(n) and AVIN(n), respectively, by
 

Pn(AVMI(n)) and p1(AVIN(n)); then we obtain
 

P= P(IAVMI(n) P(AVIN(n ))dIAVMI(n)Id AVIN(n) (98)
 

0 rjAVMI~n~j
 

Let u = AVMI and y = AVIN, then, from (54), (55), (62) and (63) and 

using the Gaussian assumption, we have 

p(u) I e MI Vu (99)
 YTraMI 

and
 

2 2 

p(y) 1 e-y /2a1N Vy (100)
 

Y2rcIN 

We assume that AVMI(n) and AVIN(n) have continuous-density functions
 

by virtue of the assumption that the filter outputs are Gaussian.
 
ORIGINAL PAGE 'q 

M8210-1 OF POOR QUALITY 27 

where Var[AVIN() aIN 2Var[AVMI(n)] =aM1 E[AVIN)] 'INand E[AVMICn)]=o. 

If we let x =lul, we then have 

p(x) 2 ex2MI2 x > 0 (101)
 

so that x=IAVMII. Using (96) and (97) in (94) yields 

2 

2 -X2/2ai2 12IN 

22 e I e dydx (102)
 

0 VFT-M I rx IN
 

A change of variables leads to
 

Pn P f 1 erFF-2TaM1 ERFC~ra IN1t dx (103) 

0 MI I 

where
 

ERFC(t) 2e dt (104)
 

4t
 

First consider the NRZ case. The original parameters for the one-pole
 

digital filter were
 

L 256 =42= 0.9453 ; AVIN 5' 6.5 IAVMII (105) I T520 4 LM 256 

and the new recommended values, to be discussed below, are
 

LI=254 =248=?
 

L 2 75 0 9922 LM = 24 0 9688 ; AVIN > 13 IAVMII (106)
 

The most desirable fix to the lock detector would utilize new narrower band
 

filters for the lock detector which would be distinct from the loop inphase
 
M8210-1 28
 

and midphase filters; however, the modified version (corrections, etc.) of the
 

microprocessor does not have excess memory or the speed necessary to implement
 

the additional lock detector filters. Consequently, single filters will be a
 

compromise between the lock detector and the bit synchronizer loop requirements.
 

It should be noted that Richard Helgeson has tried these NRZ parameters values
 

(from (106)D and found them to be quite satisfactory. The Manchester mode of 

the bit synchronizer had original parameters of 

252 = .84;L=252? 

L 5 0.9844 M 256 0.9844; AVMI 5 VMI (107) 

The new recommended values are
 

254 
 L 254 = I = 2-= 0.9922 ; LM = 0.9922 AVIN > 15 JAVMIj (108) 

Note that LI applies to both the AVIN and AVMI filters.
 

As mentioned previously, these filter L values were compromises since
 

they were constrained to be of the form N/256, where N is an even positive integer. Various values of r were tried for the new parameter values. Due to the
 

approximations in the analysis and the computer program involved, itwas determined that, for given "L"values, pn would saturate at around four 0.9999z,
 

where z is the fifth digit and is between 1 and 9. Consequently, by narrowing
 

the digital filters and increasing r, itwas possible to keep the probability pn
 

high (0.9999z) when the signal was present and simultaneously reduce the probability pn when only noise was present. This will reduce the time to indicate
 

lost lock due to signal dropout. Inother words, the detection probability could
 

be maintained and the false alarm reduced, thereby reducing the mean time to
 

indicate the out-of-lock condition when the signal drops out
 

The results for the NRZ and biphase cases are indicated inTables 2
 

and 3,with Pd denoting the value of pn when the signal is present and PFA is
 

the value of pn when the signal is absent for all data rates. Equation (103)
 

was used to evaluate these probabilities.
 
29
 

OF pOOR QUALI"
 

Table 2. NRZ Data
 

M8210-1 ORIGINAL F&pgr iS 


E Performance Performance
 

with with
 

0 Original Recommended
 -b) 


Parameters Parameters
 

4 0.999921 0.999940
 

6 0.999921 0.999940
 

7 0.999921 0.999940
 

PFA{ Noise Only 0 98850 0.9478
 

Table 3. Biphase Data
 

Performance Performance
 Eb with with 

-O (dB) Original 

Parameters 

Recommended 

Parameters 

4 

Pd 6 

7 

0.999940 

0 999940 

0 999940 

0.998923 

0.999921 

0.999954 

PFA( Noise Only 0.98783 0.769926 

Inorder to determine the mean time to indicate the out-of-lock state
 

when the signal is not present, we must evaluate
 

TOL TMF - PFA+ 2(1-PFA)PFA + 3(1 -PFA)PFA 2 + (109) 

where PFA isthe false-alarm probability and TM isthe "memory time" of the
 

inphase digital filter output*.
 

From (108),
 

(110)
 OL s ( PFA)MeT 2 FA + 3 FA. + 

*The memory is approximately Tb/C1 -LID. 
M8210-1 30
 

ORIG{iAL PAG- S 

and, since OF POOR QUALITY 

2a + 3a2 + 4a3 + - C--1) 

we have the simple result:
 

Ta TM 

S TPFA (112) 

Evaluating TOL for the four cases of Tables 2 and 3 yields the results presented
 

inTable 4 for both the NRZ and biphase cases.
 

Table 4 Mean Time to Indicate Out of Lock (Noise Only)
 

Case TOL (Seconds)
 

16 kbps 8 kbps 4 kbps 2 kbps 1 kbps
 

NRZ - Original Parameters 0.348 0.696 1 392 2.784 5 568
 

NRZ - Recommended Parameters 0.154 0.307 0.614 1 228 2 456
 

Biphase - Original Parameters 0 329 0.657 1.314 2.628 5.256
 

Biphase - Recommended Parameters 0.0348 0.070 0 139 0.278 0.556
 

Thus, we see that the recommended parameters for the filters and lock detector
 

reduce the time to indicate the out-of-lock condition to one-half of the original case for the NRZ mode and about one-tenth for the Manchester mode
 

Itshould be emphasized that the loop performance could have been
 

considerably improved ifseparate filters were utilized for the lock detector
 

since they would have been made narrower Due to the changes and additions in
 

the microprogram for saturation, etc., however, the additional room for the
 

extra filters was not available. Furthermore, the constants relating AVIN and
 

IAVMII were constrained to be of the form 2n x 6.5, where n =1, -1,2, -2, etc
 

for the NRZ mode and, for the biphase mode, were constrained to be 5, 10, or
 

15. Consequently, only approximately optimal values could be used It should
 
M8210-1 	 31
 

also be emphasized that, because the DC bias-removal and scaling algorithms
 

were unavailable, their effects could not be included in the analysis presented here. It is believed they will not affect the analysis during tracking;
 

however, when the signal is removed, the DC bias-removal algorithm could conceivably cause the lock detector to indicate the out-of-lock state more quickly.
 

REFERENCES
 

1. 	S. Houston, D. Martin and L. Stine, "Microprocessor Bit Synchronizer for
 

Shuttle Payload Communications," IEEE Trans. on Comm., Vol. COM-26, No. 11,
 

November 1978.
 

2. 	B. Parrish, "PSP Bit Synchronizer Algorithm Flowchart Summary for the NRZ
 

Mode," JSC Document, undated (about July 1982).
 

3. 	Jack K. Holmes, Coherent Spread Spectrum Systems, Chapter 12, Wiley-


Interscience, 1982.
 

4. 	R. J Helgeson, "PSP Bit Synchronizer False-Lock Problem," a presentation
 

to the Shuttle S-Band Meeting of-September 15, 1982.
 

5 	 SCTE Payload Signal Processor Preliminary Design Data Package, Vol I,
 

TRW internal report.
 
M8210-1 ORIGINAL PAG F@ Al 

OF POOR QUALITY 

APPENDIX A
 

EVALUATION OF THE MEAN AND VARIANCE OF
 

THE RECURSIVE ONE-POLE DIGITAL-FILTER OUTPUT
 

In this appendix, we derive the steady-state mean and variance of the
 

recursive one-pole digital filter used for both the inphase and midphase filters.
 

Let z(n ) be the value of the filter output at time n. We then have
 

z(n+l) = L z(n) + x(n) 0 < L < I (A-i) 

where x(n) is the nth bit sample of the input digital matched-filter output.
 

In (36), we showed that its SNR is 1.9 Eb/NO. Solving (A-i), we obtain
 

n 

z(n+l) - x1 Ln-i (A-2) 

i=O
 

To obtain the mean EE(n)J, we average both sides of (A-1) to achieve
 

Ez n + l
] = LI E[z(nJ + E x(n) (A-3) 

In steady state,
 

E[z(nl = E(n+l? (A-4) 

so that (A-3) yields
 

E[z(nJ = I - L (A-5) A5


Now we consider (n ). 
 the variance of z First, we square and take the ensemble
 

average of (A-i) to yield
 

E[z( n+1) = L2 E(z(n)) + 2LEC(n)z(n I + Ef(x(n) (A-6) 
M8210-1 A2
 

ORIGINAL PAGE FS 

Insteady state, OF POOR QUALITY 

E[Zn')1 = Ezn)(A-7) 

so that
 

Ez(n))I 2L E[x(n)] EEnlL + E x(n))2] (A-8) 

By definition, 

Var(z(n)) EFZ(n)) - E2[z(nl (A-9) 

Therefore,
 

Varz(n) _ 2LEx(n)]E(n)] + Ef(x(n))j _ (A-10) En 1- L(2 (1- L)2 

or 

- 2(1-L) LE2 L(n) + (1-L) 2 E[x(n))\ (1-L2) E2F(n)I 

C1 - L2 )C1 - LD2 (A-11) 

After simplifying, we obtain
 

Vr(n)] = E x(n))f - 2 (A-12) 

Var 1Z (n I - L2 (-2 

or, finally,
 

Var (n)[CiL§ 

VarE(nj V 1- (n (A-13)
 

Equations (A-5) and (A-13) constitute the main results of this appendix
 
M8210-1 	 B1
 

APPENDIX B
 

ORIGINAL PSP BIT SYNCHRONIZER ALGORITHM
 

The following documentation is a slightly updated form of an abbreviated flowchart of the PSP bit synchronizer algorithms for NRZ and biphase-L
 

data. They are not considered to be exact and can be compared to the "final
 

version" given in Appendix C. The NRZ algorithm includes a transition check
 

which was not in the original algorithm, plus the changes of the lock detector.
 

The biphase-L algorithm has not been updated due to the lack of a complete algorithm; however, the lock detector portion does have the new parameters. This
 

flowchart was developed at Johnson Space Center by Brett Parrish independently
 

of any flowcharts that TRW might have used tn the development of their bit synchronizer program. The PSP bit synchronizer "machine-code" program (as supplied
 

by TRW) was disassembled and flowcharts were generated from it.
 

B 1 NOMENCLATURE
 

Each bit of incoming data is sampled at a rate of 1.024 MHz and integrated (summed) in hardware. At the end of the bit period, the hardware interrupts the bit synchronizer computer (which iscaught in a loop at program location PEXEC) and the "bit-time interrupt" routine is performed.
 

The bit-time interrupt routine polls the hardware for three partial 

integrals, as shown in Figure B.1 (assume that the bit synchronizer is locked). 

When the bit synchronizer is locked, the hardware takes 1.024x 106/DATA RATE = 

Ns samples per bit. 

The current microcode program utilizes DMPNM, as described below.
 

The new program was modified to avoid saturation but, due to time limitations,
 

has not been introduced in the discussion to follow. Whenever a phase correction is required, the software changes the nominal dump count by a small amount
 

±e.
 

DMPCU = DMPNM + DMPCP + DMPCD
 

(nominal) (A@) (rate trackinq)
 

where
 

DMPCU = 	current value of dump count to be output to hardware (number
 

of samples/4 to be taken during the next bit period)
 
M8210-1 B2

ORIGINAL OF pOOR QUALU PC ' V 

d--Bit Period 

MITB2 

NRZ Input 0- -7 --

FSTB2 LSTB2 

NOTES FSTB2 - first half-bit integral 

MITB2 - middle half-bit integral (used inbiphase mode only) 

LSTB2 - last half-bit integral 

PRTB2 ­ last half-bit integral of previous bit 

PRTB2 FSTB2 LSTB2
 

NOTES For NRZ data formats, the software implements two filters:
 

jFSTB2+LSTB21 -- inphase filter
 

(PRTB2 + FSTB2)*TRIND midphase filter
 

TRIND is the transition indicator and has a value of
 

+1 or -1 or 0 The MITB2 value is not used in the
 

NRZ mode
 

Figure B 1 Half-Bit Integrals Used in the Algorithm
 
M8210-1 
 B3


DMPNM = nominal dump count = Ns/4 

DMPCP = dump count correction due to phase-tracking loop
 

DMPCD = dump count correction due to data-rate tracking loop
 

Bit-time interrupt routine (executed after hardware integrates over
 

one bit period):
 

1. Outputs previous bit decision to frame synchronizer circuit
 

2. Takes current LSTB2 and puts it in PRTB2
 

3. Inputs FSTB2 from hardware
 

4. Inputs MITB2 from hardware
 

5. Inputs LSTB2 from hardware
 

6. Outputs new DMPCU to hardware
 

7. Sets DMPCU equal to DMPNM
 

8. Set NDATR flag
 

9. Returns to PEXEC loop in main routine.
 
B4
 ~B.2. NRZ Algorithm
 BA
 PEXEC
 Loop here while 


hardware is 'A ORIGINAL PAGE IS, sampling and. 
 OF O POOR P QUALITY

computing 


integrals No NDATR N-NDATR flag is set during bit-time 

N Flag interrupt routine (after hardware 

Before bit-timeB-j Set has integrated one bit, it inter- ?/ rupts the computer)
 

interrupt occurs, Yes
 

the computer is
 

looping here 1 bsN
 

esYes 4 or No DC offset 8kbps 

/ routine- VIntegration by parts 

For 16 kbps data rate, DCOFF 1 or 2 kbps skip DC offset routine D o | for I I 

JD os et routine 

I Frame synchronization Check frame 


synchronizer F Check Hamming
 

and data rate Couatn
 correction 


SeCounts to 4096 when CUDRC =0 Flag CCID - 8 -> CCID 

t YNoldoppler Time for correction 

• -cor ect on Ti e for if underflow
 

NOTES- correction
 1 0I No in acquisition < Data Rate Yes 4% 

1 W, ?/ 
 Iut contents of
 

2. Executes once No CUDRC in DMPCD
 

each bit period
 

, No CUDRC is
 

set to 0 

Signal format SUC TF lgdrn


.check (New rate-tracking NRTIN routine
 

interval required)
 

NRZ Data\ Biphase 

NRZbranch Bpase-L branch
 
ORIGINAL PAGE 0 

OF POOR QUALIYY + B5 

Inphase averaging filte ) FSTB2 and LSTB2 are integrals 

LI 05u'"computed by the hardware and
 =
 LI 0.95 during acquistion 

L, = 0 99 during steady AVIN = LI*AVIN+ FSTB2 + LSTB2I updated every bit time 

state 

Data 3-dB point (Hz) Pos STB Neg Bit decision
 

Rate SS T4ax. also stores
 

(kbps) Aqq. Gain (TRK) Gain Set bit forltin
 

1 8.16 20 1.6 100 Decision Decision
 

0
 2 16 33 20 3.2 100 

-4 32 65 20 6.4 100
 8 65.31 20 12.8 100 ran-transto n
 

16 130.6 20 256 100 Yes sition No
 

Averaging Filter:T
 

SM = 10 fFiltered update
 

L = 0.90 during acq AVMI= LM*AVMI + (FSTB2 +PRTB2) *TRIND signal triggered
 

... TRK x<^. Cby transitions onl
 ..... 


Gain = 10 during acq ock N Lock flag test
 

LLM =0,95 during lock ACQ
 

= 20 during lock Flag ime delay: use inacquisition


Set lock flan 4096
 

Steady-State Yes transiions
 

= Bit Synchronizer Light - AVIN >
 

Test to see "are we 
 s ?i


' " - . . still in lock ? I. .
 

Lost locl', No V° IN>13 *jAVMI
 CerAO
 

IClea rLock Fag 
 YeIIs^


2 Clear-Hardware Lock Ind[ /102P
 

3 Set to Acquisition Mode i . n-tran­

! . Phase-tracking Ye It 1 
ons in


iIPCI I correction interval I ' ,a row/
 

c^.MFT,,Is ollwsin acquisition Avoids hang-up in ?'
 

Set" "l Cas " f".. low-sT 
 .. AK lock mode when out
 "No


ate (KpS) MPTV -PT-K- of lock
 

4 4
 

8 2
 

16 -
 See Next Page
 

Ilear AVIIN, AVMI
 
Enter once ) B6 

kevery 4096 bits NTI
 e 
 NOTE. 
 Rate tracking
 

Clear RTIFL Flag inhibited during
 

acquisition
 Switch steady-statey
 to 


parameters- No Stay es 

S Set loop to SS<
 

Set hardware lock ind I -*--Bit synchronzer lock 

I indicator to 0/1 MOM 

Set to lockmode: = 0.99 for inphase filter 

LM 0.95 for flter ORIGINAL PAGE E 

OF POOR QUALITY 

Set phase-tracking correction
 

interval as follows: 

kpbs Interval (see NTSPC)
 

1 1 

2 1 

4 2 

8 4 

16 8 

I 

Set phase-tracking correction va~lue
 

to I (if kbps, set it to 2)AMPTCV d 


AI FRlter "gatn has changed, n 5 * AVIN so normalize current value
 [AVM1 IMakes acquisition parameter
 

2 AVMII .gains" the same as tracking
 

"gaCns" (avods transent)
 

JACORI ) ABOR Magnitude dump count of over accumulated 4096 bits 

Poneg
 

I I Rate 

(kbps) NSPB8
 

Do'tcoretClarAC9 


rant c re ct 1 128 

rato 2 64 

Noes 4 32 

l ~BO > NSUP'B8,/I 8 16 

S 0 + CUDRC 37816 8 

32768 CCID B CCID 

to PEXEC and 2-Go 

~wait for interrupt
 
ORIGINAL OF pOOR QUALITlB PAGE 19 	 B

T Phase Updating
 

(During Acquisition. NTSPC=1 During SS
 

(#Traniti ons) Ckbps) 

corcto Pe4 
 8


8 16
 

IC
S
 No- Hav NTNo 


YsAVIN >65.O*IAVMI YsAVIN >156*jAVMI 
 MPTCV 
-Maximum
 

Phase-Tracking Correction Value
 

ACQ SS TRK 

Phase correction kbps MPTCV kbps MPTCV 

1 16 1 2 

Not Ps AM 

Pso 

oare e2 

e4 

8 

4 

2 

4 

1 

1 

-MPTCV +DMPP MPTCV+DMPCP 8 2 8 1 

PTRK1 

or retard I uame

fixed amount according 

u I n ( 

to data rate
 

ACOR =tACOR +vDMPCP+ aDMPCD l
 ACR is the accumulated dump count 

S correction over 4096 bits 

DMPCU~T DMPCUN) DMPCDn +DMPCP(n) 	 Hardware dump count ismade up
 of nominal dump count plus
 

"dump count correction" (phase)
 

PXCClear DMPCD, DMPCP plus dump count correction
 

(data rate)
 

New rate-tracking
 

No TIL interval required flag 

PEXEC and wait e 

for interrupt 

NRTIN r_ Do once every 4096 bits
 

(To Next Page)
 
ORIGINAL PAGE 1 

OF POOR QUALITY B8 

Bit Time Interrupt Routine (Executed after hardware integrates one bit)
 

Output previous bit decision 	(0or 1) from bit sync computer Output

I Idata! 

Copy value of LSTB2 to PRTB2 

lInput new value for LSTB2 from integration hardware
 

lInput new value for MITB2 from integration hardware
 

lInput new value for MSTB2 from integration hardware
 

Data rate plus) 

update sent to°Fa ut put DMPC U t oi n t e g r a t ion hardware I 1Every 

bit time 

hardware 

4 Updating value 

lestore DMPCU to its nominal value DMPNM 

!i 

etuArnFa
 
B.3. Biphase-L Algorithm B9
 

Loop here while
 

hardware is '*-

sampling and 


computing No 


integrals 


Before bit-time IYe
 

interrupt occurs,
 

the computer is 


looping here 


For 16 kbps data 


rate, skip DC
 

offset routine 


Check frame
 synchronizer 


and data rate? 


correction 


Signal " 

format check 

PEXEC (Executes once each bit period)
 

t 

PNtDATR flag is set during bit-time interrupt


NDATR routine after haredware has integrated one bit,


Flag it interrupts the computer.
 

No 

16 kbpS No 

es Yes 4 o N 

routine _ Integration by parts I
 

for 1 or 2 kbps
 

DC offset routine
 

- -,
 

FSYNC Check Hamming 

Computation
 

Counts to 4096 when CUDRC = 0
 

Flag CCID - 8 -)CCIP 'K ,;z{Timefor correction 

?corecionin im for if underflow 

No acquisition Dat Rate "Yes 

" . lPut contents of
 

No CUDRC in DMPCD
 

Every 4096 bits CUDRC is
 

SCK set RTIFL flag set to 0
 

(New rate-tracking or 1 or -1
 

interval required) during


NRTINE routine
 

bBiphase-L branch
 
1 MITB2BI BiD 

Biphase Mode 0
 

Inphase FlZe, - ,- :: FSTB2 and LSTB2 are 

AVIN= L*AVIN+IFSTB2- LSTB2, integrals computed by
 LI 0.99 SS or Acq. 
 the hardware
 

Data 3-dB Point Pos FSTB2 Neg
 

Rate (Hz)
 

(kbps) (Hz) Gain
 

1 1.6 100
 

2 3.2 100 Bit Decision =I Bit Decision =0
 

4 6.4 100 -1 -+TRIND +1L-- TRINDI 


8 9 8 100 - Se
 16 25 6 100 -


TRIND > -±bit, or
 

no bnt transetton
 AVI = L*AVMI +MITB2 *TR7IND G 

CL I = Mldphase Filter " - 0.99, Gan=10 Alternate 

inphase filter
 oc Nof 4e 

(L=0.95, Gain =20) -Acq Fag ALIN =L*ALIN +IPRTB2 -FSTBZ 1 

SS 


OF Problem
 

(L 0 99, Gain 100)A- Se O 


e IN I>512
8-[for
 Sttefo 52bits j
 Testose?
 Yes 
 in a o N VIN >A L 
 Y e
 

are we still,. 

in lock? o A lN 5 I VM 
 e a Hardware __
 

Integrator Set Lock Flag
 1/2 Besi- Clear ACOR
 

LSTLK Ye, 4095 CNTR
 

SwthAIN
 

and AI
 

ISame as for NRZ Model ----


ORIGINAL PAGE US 

" OF POOR QUALITY 
ORIGINAL PAGE- E Bli 

OF POOR QUALITY 

NRY :PTRK1 PTRAK NRZ D6uring Acquisition, NTSPC = 1 During SS 

r NTSPC kbps

/1 1 

Have NTSPC11 

Notransitlons occure 

sic atpae2 3 

latpae _.icectioNTSPC kbps 4 16 

1 2 

2 4 

4 8 Bi phase Lock SS
 

o F1 a8 Yes 8B1 aNTSPC kbps 

lg4 1
 

St8 2
 

Bpae16 4 

32 8
 

AVIN>65.0*IAVMI AVIN >156*IAVMT 
 64 16
 

No0
 

ACK SS
 

kbps MPTCV kbps MPTCV
 

Po VMNg2 8 (1) 2 1 (1)
 

os o e 4 4 (1) 4 
 1 (1)


Iz 8 2 (1) 8 1 (1)
 

I-MPTCV + DMPCP IMPTCV DMPCPj 16 1 (1) 6 1 (1) 

PTRK1
 

ACOR = ACOR + DMPCP + DMPCD JACOR is the accumulated dump count 

Icorrection over 4096 bits 

Hardware m
ina l dump count is made up of dump count plus "dump count
 

correction" (phase) plus dump count
 

IDMPCU =DMPCU + DMPCD noMCP 


( orrection (data rate)
 

PEXEC Clear DMPCD, DMPCPI
 

o back to PEXEC MPTCV Maximum phase-tracking 

and wait for correction value 

interrupt No TIFL New rate-tracking 

Flag interval required flag
 

INRTI Do once every 4096 bits
 

(To Next Page)
 
Enter once) B12
 

(every 4096 btsl RTIN
 

NOTE: Rate tracking
 

inhibited during
 

SictoClear
 RTIFL Flag acquisition
 Switch t
 

steady-state


parameters--,& No Stay-Ys 

Set loop to SS? 

Set hardware lock ind Bit synchronizer Ilock 

--7 
 indicator to 0/1 MOM

Set to lock mode:
 

0.99 for inphase fIter
 

LM = 0.95 for midphase filter 

II ORIGINAL PAGE 9S 

Set phase-tracking correction OF POOR QUALITY 

interval as follows: 

kpbs Interval (see NTSPC)
 

1 1 

2 1 

4 2 

8 4 

16 8
 

[S-et MPTCV phase-tracking to I (if 1 kbps, correction set it value
 to 2)I 

AVIN = 5 AVIN F 1 ter "gain has changed, 

so normalize current value. 

JAVMI 2 A Makes acquisition parameter 2,gains" the same as tracking
 

"gains" (avoids transient)
 

O(AMagnitude of accumulated
 

SIACORI ABOR dump count over 4096 bits
 

Poo CR Neg
 

neg -1ne+CUROC Data 

Rate 

Clear ACOR (kbps) NSPB8 

Don't correct 1 128 

2 64 

4 32 

0o )-CUDRC I ABOR > NSPB8 I32768 °o o 8

16 

16 

168 

8 

-,-Go to PEXEC and
 

wait for interrupt
 
B13 

Bit Time Interrupt Routine (Executed after hardware integrates one bit)
 

BTIMI
 

Output previous bit decision (0 or 1) from bit sync computer Output
 

data!
 

SCopy value of LSTB2 to PRTB2
 

Input new value for LSTB2 from integration hardware
 

Input new value of MITB2 from integration hardware
 

Input new value for MSTB2 from integration hardware
 

Data rate plus 
 T

update sent tO " Output DMPCU to integration hardware 4Every bit time 

hardware 

Updating value
 

Restore DMPCU to its nominal value DNPNM
 

IReturn
 
M8210-1
 

APPENDIX C
 

NEW PSP BIT SYNCHRONIZER ALGORITHM
 

This new algorithm was made available to the author by Brett Parrish.
 

The changes in the original algorithm leading to this algorithm were made primarily by Rich Helgeson and Brett Parrish. This program iscurrent as of
 

October 22, 1982.
 
M8210-1 C2
 

Power-up Routine
 

Latch 00111111 into Sample Counters
 

II 

Set CFGWD Data Type = Biphase
 

Data Rate = 16 kbps
 

Set Frame Synchronizer Word Length to 32 Bits
 
ORIGINAL PAGE IS 

M8210-1 OF POOR QUALITY .C3 

( INIPM 

r Clear PI Flag (Disable Interrupts) 

Clear Memory From DCMNU to CFGWD-1
 

Clear NRZD, LKIND, FSYI, FSYC, QBCLK
 

[Clear BITH1, BITH2, BITH3, BITH4, FSYNI, FSYNB
 

SMask TLM Data Type From CFGWD, Right Justify it adput i in 


Rate NULEN NMLEN CRCEN SEMMT SEMIT SEFST SFTB2 KBPS
 

1 00100000 00100001 00100010 00100100 00101000 00101100 00110000 00000001
 

2 00100000 00100001 00100010 00100100 00101000 00101100 00110000 00000010
 

4 00100000 00100001 00100010 00100100 00101000 00101100 00110000 00000100
 

8 01000000 01000001 01000010 01000100 01001000 01001100 01010000 00001000
 

16 01100000 01100001 01100010 01100100 01101000 01101100 01110000 00010000
 

Set up the [ 00001111.o DMPNM .-Will be changed to 14 

following constants 1 (00001110) 

Rate NSPB8 MPTLA OFBTIPB BTIPQ
 

1 10000000 (T000010000 00100
 

2 01000000 00001000 00001000 00000100
 

4 00100000 00000100 00000100 00000010
 

8 00010000 00000010 00000010 00000001
 

16 00001000 00000001 00000001 00000000
 

TEN*KBPS RSMPY
 

Clear BTICT
 

Clear BTIQT
 

Clear QBICT
 

Set upthese constantsfMask Number of Frame Sync Words From FSYLC
 

No of Frame Sync Words FSYNL FSYNP * (8*FSYNL) 

1 I 0000001 00001000 

2 00000010 00010000 

3 00000011 00011000 

4 00000100 00100000 
M8210-1 ORIGINAL PAGE i9 C4 

OF POOR QUALITY 

Number of Frame Sync Bit Errors
 

Number of Frame Sync Bit Errors
 

-Set up these constants, for Inverted Data
 

No. of Frame Sync Words NALDT-"*NALDI
 

1
 

2 is
 

3 23
 

4 2 30
 

Point HMGWA at Hamming Word 1, 2, 3 or 4 according to the number of
 

Frame Sync Words
 

No of Frame Sync Words HMGWA Points to
 

1 HMGW1
 

2 HMGW2
 

3 HMGW3
 

4 HMGW4
 

Frame 

Sync 

Length 

8 16 24 32 

SYNCi -* PSYN4 SYNCI PSYN3 SYNCI+ PSYN2 SYNCi + PSYNi 

SYNC2 + PSYN4 SYNC2 SYNC3 - PSYN3 +PSYN4 SYNC2 SYNC3 

SYNC4 

+ PSYN2 

+ PSYN3]

PSYN4 

Out DMPNM + Hardware 13 

DMPNM DMPCU 

Out NMLEN to Hardware 12 

Out NULEN to Hardware 12 

I
 
M8210-1 ORIGINAL PAGE UC
 

OF POOR QUALITY INIPM (Cont't)
 

121 AMIC ­

126 SMIDC ­

126 AINPD ­

126 SINPD ­ 5 ACDRA ­ 1 SSDRA -


KBPS/2 or PTCIA ­

1 if KBPS=A} 


4*KBPS PTCIS ­

1 MPTCA ­

1 LPDRA ­ 9 CUDRP ­

, ~phse Forma NRZ
 

1 
 Avihase 

Midphase Average Filter Decay Constant (ACQ) 115
 

MIdphase Average Filter Decay Constant (SS) 121
 

Inphase Average Filter Decay Constant (ACQ) 121
 

Inphase Average Flter Decay Constant (SS) 127
 

Filter Decay Constant Ratio (ACQ) 2
 Filter Decay Constant Ratio (SS) 5
 

Phase-Tracking Correction Interval (ACQ) I
 

' KBPS/2 or
 

Phase-Tracking Correction Interval (SS) I if KBPS=1
 

Max. Phase-Tracking Correction Value (ACQ) As set before or
 

114 ifKBPS 1
 

ACQ. to SS Normalization Multiplier 5
 

Current Filter Decay Constant Ratio 5
 

SINICP 

[ AMIoC MIDDC I
 

LAINPD >.
INPDC 

MPTCA +MPTCV
 

PTCIA-1 PTCIN
 

PTCIA-1 NTSPC 

0- CITDU, L
 

0 + ABORN, L 

i, 
 f

409610 Interval Counter
 

I-t PIFLG (Enable Interrupts) 

ID 
M8210-1 ORIGINAL PAGE V5 C6
 

[, 

CFGTO 

75 js Delay 

CFGWA No. of CFGWD 

I 

OF POOR QUALIT'Y 

Point CFGWA at CFGWD 

No 

-)- PIFLG 

DA 

-Enable Interrupts 

P =RBNToLSTB2 

i ARSMPY ,, =10 LT2 UH

FSTB2 = FEIBIU, FHBIL *2* RSMPY 

LSTB2 = LHRIU, LHRIL6*2* RSPIPY
 

Maximum value Z Neg ver Pos NZ N 

of FSTB2 or Fo 

LSTB2 or MIT2 FSTB2I = FTB2 -12 HBIU, HBIL *2* RSMPY 

is±120
 DC256
 FSN
 DOFTB LF2 + OF
 
M8210-1 

ORIGINAL PAGE kV 

OF POOR QUALITY 

( FSYNC 

No Lo 

TYes 

Shift One Bit 

HAM (BITH4 + PSYN4) -------- b- HMGW1 

HAM 

HAM (BITH3 + PSYN3) + HMGWl HMGW2 


SHAM 
 (BITHi + PSYNi) + HMGW3-2 HMGN4

FI 


HAM (BITH2. + PSYN2) + HMGW2 HMGW3
 SFCK
 

C7
 

Total Errors
 

Past 8 Bits
 

Past 16 Bits
 

Total Errors
 

Past 2 Bits
 

Past 324 Bits
 

Total Errors
 
NRZ RBiphase 
 c 

HRZIDurlng 
 acquston
 AVN, 5 +FT2LT( a =aIPNPMU, IPM FSB 
 'SB

2XINPDC + SB+SB i { INPDC = 121 0 IINPMU.L = ,T2-LSTB2
 

AVINUL 
 a, =a1(0 95)+IFSTB2+LSTB2B
 

AVINU,L IDuring steady state
 

NRZTR INPDC = 126.0 

t a,=a -1(O 99 )+ IFsTB2+LsTB2 1 CUDEC + PRDEC 

Put current bit decision into 

previous bit decision 

Clear transition indicator i 

Pos 

STB 

LSTBS Pos Ne -I 

CUDEC 

TRIND 

l s i -IvCUDEC 

1 TRIND 

NLIBIT1r oSet e Set NLBIT= 1 1 I 

Set current bit decisio et current bit decisio BDCVT Check 

I=1 X,Y 0 -Assembly 

No- Transton Counts EC Yes C


rnt es Tr-an-sTransi toonless No
 

Yeses
 

RZ oYes 
 mio


ToYe NRZMP 
 BIMDE


To(RM RT AK)
 
AVNIU,L*2*MIDDC +(PRTB2+LSTB2)*TRIND + AVMIU, AVMIU,L*2*MIDDC + MITB2*TRIND AVNIU,L 

256 

C) 

ABIUL , 

R = ?BIP 

Yes CDR* o AVIULIAVIU,L N Yes UP*Lo VIUF[FTLT2 

NCL YesN 0N IN INP IN +1 AVINU, IUL*2*INPDC vmIYeI~ A Nooj No 

27 

2* 0 1 PIN NU -F STB 2
 ,L =A I ,26 PD + IP R DB 2

N t AVIUI 

NoNo
 Ye Yee 

Set Loc Fla~l-1 4 SHFTH
 

l INClear INCTUL+11Ce CT LN 

Out Sh ft 1/ - iBC
 

LTKPRCommand L B
 

0 SHFTH
 
M8210-1 


SBCKLK CIO
 

Ys cORIGINAL PACE.gq 

FaOF POOR QUALITY 

INC BOICTUL 

Yes CU 

BCKL1
 

-, ]Yos
ICU,L
 

409 CNR
 

0~ '+ BICTUL
 

-1 SHFTH 
 Ye~s A
 

KBSNo 
 No
 

Bit Command
 

[Clear SHFTH
 

0 INCTU,L
 
M8 210-1 	 Cli ORIGINAL PAGE [S 

OF POOR QUALITY
 

Clear hardware lock indicator
 

I 

Clear lock
 Clear loop
 

Clear RCOR in hardware
 

Note If lock is lost,
 Clear NCOR in hardware


set parameters to
 

acquisition mode. I Clear FSYNI
 

Clear FSYNB
 L 

INPDC = AINPD
 

MIDDC = AMIDC 

I PTCIN = PTCIA-l I Phase track correction 

NTSPC = PTCIA-l interval 

MPTCV 
= MPTCA 	 Maxmum phase track 

correction value 

Clear AVINU,L
 

Clear ALINU,L
 

Clear AVMIU,L
 

Clear LVLCU,L
 i 

Clear RTIFL
 

o + CTTDU,L 

o * ABORU,L 

PEXEC
 
( PTAK C12
 

18210-1 

?YsOF POOR QUALITY 

SNTSPC:= PTCIN I 

OS VMI Ne9g Fo I 

MPTC V WMCPt"- Posiive/ MPTCV DMPCP I 

CITDU,L = CITDU ,L+ABOR 1
 

SCUDRC DMPCD
 

(RTRAK1
 

Yes 
 Dn
 

4096 CNTR? 

-1 RTIFL N
 

0' 0oRFIFL JACORU,L =ACORU,L+DMPCP+DMPCD I 

0ClTDU,L I jDMPCU =DMPCU + DMPCP + DMPCD 

0 ABORU,L I 

I Clear DMPCD 

DMPCP 

TClear 
M8210-1 
 C13


( NRTIN 

Rate tracking flag OF poOR QUALI Clear RTIFL RTIFL = 

4 Yes FLoopet
 

SSet BS lock indicator
 

in hardware
 

I + Loop I 

INPDC = SINPD Set phase tracking
 

MIDDC = SMIDC correction interval to SS
 

PTCIN = PTCIS-I Set maximum phase
 

Scorrecton value to SS
 

. NRZ
 

I CUDRP = 1 3 1 SCUDRP =5 1 

0CR,PosU, = AACUUL=Acumula 1eABORU,L = 1ABOR 

Pos CUABOReU 

CUDRC- uc-


NRT13) 

t I ACORU,L = Accumulation of
 

SACO]RU L = 0 .. dump count corrections
 

rBR
 

( PxBc 
M8210-1 C14
 

ORIGMAL PAGE IS 

DCOFF OF POOR QUALIr 

INP1UINP1L INP2UINP2L 

INPHUINPHL - INPIU,INP1L
 

(FHBIU,FHBIL + LHBIU,LHBIL) + INPHU,INPHL
 

No transition 
 YsTID=0
 

DCMNU,DCMNM,DCNNL
 

+ INP2U,INP2L
 

+ INPIUINP1L 

DCMNU,DCMNM,DCMNL 

+ 1


DCOBC
 = 

L DCOBC 

No DCOBC =64
 

Yes
 

DCMNUDCMDCMNM 
 L *2*RSMPY] 

OFFCO = -256 

Flear DCOBC
 

Clear DCMNU,DCMNM,DCMNL
 

o::FSYNC 
Iinput Current Last-Ha f Interval 

I 

from Hardware 'CULHI' I 

Out Current Dump Count to Hardware 

[ DMPNM DMPCU 

X SAVB4, Y + SAVB5 

Set 8uarter-Bt Clock Line to 0 

Yes 

'DMPCUJ' 

U 

8 kbps 

',_ 

1BCLIK'5 

Input FQ 

, P 

UI 

QSecondNo 

Arl QB1aLCUFHI 

I t 

A 

LHBIU!LIUL=-F L TU 

IQ22 sN 

---

BTI07 

?AHBI/,L AQB2LI+AQB , I

oYeHFTU1 -) Clear SAVB SHFTHSAVB X AT

+TCULet Clear BTIQT BiCo L, S o 0 ASB KR 

Return1 

FHBIUL BT1UL3I+AQBIU, 

+CUFHI 

or , HI LCLI 

SHFTH ,J C eaSHT 

BTQT ,S AVB +ABA AXB X

,HTHN 

2U 

CAQB2UL 

Q2 
M8210-1 B C16
 

AQ5IU,L 

"'i IS+ CUR-lI ORIGINAL PAn"V2' + CULHI 

OF poOR QUAL ' f iw+U I 

No TIQT+ BTIPQqYes
 

ISet Quarter-Bit Clock Line to 1 LDQBCLK'
 

a QBTI9
 

I
 

T BTC BT--
 ICT


es


7BTIC 

LDCKFF
 N o ~es 

IBTIQT = BTIQT + 11
 

? C2 


SYes
 

No IAQBIU, L AQB4U,LI
 

Clear AQBIU,L
 B C 


Clear BTICT
 QBIU LAQB3U,Lj

No es? ClearIQBICT - --TBL LG I 

B 1CT =I 


'AQ BU,L AQB I,LI ',.IAQB LL AQ ,LI 
 IClea AQBI',' I ND T

A
 IClear AQBIU,LI [la UL
 [Clear AQBIU,L 


N H
 

Cear SHFTH
 

Cear BTICT
 

Clear QBICT
 

C 


BTI1
 
M8210-1 	 ORIGINAL PAGE ISC1 

OF POOR QUALITY
 

Bit time interval for 16 kbps only
 

BTIMI 	 Output a bit to 

frame synchronizer 

F _XR SAVXB, A SAVBI
 4
 

Set TLM Data Flag 'NRZD' in Hardware
 

Equal to NLBIT
 

Input Last Half-Bit 	Interval 'CULHI'
 

Input Middle Half-Bit Interval 'MITB2'
 

Input First Half-Bit Interval FSTB2'
 

Output Current Dump Count
 

'DMPCU' to Hardware
 i
 

DMPNM DMPCU
 

1- NDATR I

i
 

[SAVBI -- A,SAVXB + XR 

Q tun
 
M8210-1 


Echo Configuration to Hardware 


SRDCMO ) 

xR SAvxR, 0 AZERO 

CFGWD + 11 

Output FSYLC + I
 

to -- 1
 Hardware 	 SYNG2 1 

SYNC3 + 11 

SYNC4 + 1 

0 Il
 

+SAVXR
 + 

Return4 

C18
 

Hardware issues PROGMI Interrupt
 

six times to set configuration
 

C PROGM 

0 PIFLG
 

II Loc. pointed by CFGWA
 

CFGWA = CFGA+1 

<e
 

No1PM
 

SPoint CFGWA 

at CFGWD
 

Reset Stack
 

CIE 
M18210-1 
 C19


Interrupt to check
 

SCKSYN frame synchronizer
 

FXR -) SAVXF, A -> SAMF 

No Loop
 

Yes
 

B .SAVF2
 i
 

A = HMGW1 or HMGW2 or HMGW3 or HMGW4, depending on frame 

synchronizer word length 8, 16, 24 or 32, respectively. 

0 FSYNI
 

FSYN1= ) Y o s FSYNB Y N o 

S 
 IA
 esY 
 e No 
 FS N
 

FSYNI -1 ­

1( FSYN2YN
 

T /RCOR 

FSYNI LDFSYI r _L.. -
 FSYNB +.LDFSYC NCORU
 

I SAVF2 -,B
 

SAMF + A
 

SAVXF + XR
 

CReturn
 
M8210-1 D1
 

APPENDIX D
 

AMENDMENTS TO BIT SYNCHRONIZER ACQUISITION TIMES
 

The bit synchronizer shall acquire in four seconds with a probability
 

of at least 0.9, with the Eb/N 0 value specified in Table D.1 below for the given
 

data format.
 

Table D.I. Data Format Versus Data Rate Eb/N 0 Values
 

Eb/NO (dB) Required
 

Format 

1 kbps 2 kbps 4 kbps 8 kbps 16 kbps 

NRZ 9.5 6 5 4.5 2.5 5 5 

L, M, S 

Biphase 

L, M, S 

9 5 

I 

7 5 5 5 3 5 5 5 

I 
10.0 SHUTTLE/TDRSS AND GSTDN COMPATIBILITY ANALYSIS
 

While Task 5, Space Shuttle/TDRSS and GSTDN Compatibility Analysis,
 

was directed at the overall system compatibility, many of the analyses were
 

performed as part of the individual communication subsystems, such as the Sband network equipment and the Ku-band communication system. Under this task,
 

Axiomatix attended all TDRSS design reviews, evaluated review material and,
 

together with NASA JSC personnel, generated Review Item Dispositions (RID's).
 

The final TDRSS design review which Axiomatix attended was the TDRSS Ground
 

Segment Final Design Review at TRW, September 22-24, 1981. At this design review, Axiomatix was still concerned that the ground segment decoder could not
 

achieve branch synchronization with the all-zero sequence. An action item was
 

submitted at the design review to resolve this problem. The final resolution
 

was to inform the users in the payload ICD (NASA JSC ICD No. 2-19001) that
 

branch synchronization is not achieved until the number of transitions is
 

greater than 64 and the maximum number of consecutive bits without a transition is less than 64 within any sequence of 512 bits.
 

During the design review, TRW stated that the Ku-band open-loop
 

pointing budget would be revised. While relaxation of the open-loop pointing
 

budget could be troublesome for Shuttle Ku-band acquisition, it was learned
 

that TRW intends to tighten this budget. Axiomatix reviewed the current pointing budget and found that no changes had been made during the past two years.
 

An action item was written in order to prompt TRW to update the pointing budget
 

in light of current knowledge regarding the system parameters.
 

This section discusses two of the analyses performed to establish
 

compatibility between the Shuttle Ku-band communication system and the TDRSS.
 

These analyses involve (1)the TDRSS Antenna Scan for Shuttle Acquisition and
 

(2)the Power Spectral Density of Staggered Quadriphase PN With Identical Sequences. For additional analyses concerning Shuttle/TDRSS and GSTDN compatibility, refer to sections of this report dealing with the individual communication subsystems.
 

10.1 TDRSS Antenna Scan for Shuttle Acquisition
 

Axiomatix was asked to review and comment on an investigation funded
 

by Goddard Space Flight Center regarding the feasibility of effecting a TDRS
 

scan during Shuttle acquisition. This study was conducted inresponse to a
 
concern voiced by NASA/JSC and Axiomatix as to the ability of the TDRS Ku-band
 

to open-loop point to within 0.220 of the Shuttle.
 

The study explores the possibility of using externally generated
 

(NASA) or internally generated (TDRSS) scan patterns. In the former case, a
 

NASA-supplied processor inputs updated state vectors to the TDRSS from the NASA
 

Network Control Center (NCC). The disadvantage of this approach is that, in order to account for TDRSS/NASA propagation delays of the state vectors and signal
 

status, the scan must be slowed down. The latter approach uses a TDRSS computer
 

to generate a scan, which conceivably could result in a faster scan since the
 

NASA/TDRSS interface is eliminated. Since it corrects on-track errors only, the
 

use of a At adjustment is not considered an acceptable solution.
 

Axiomatix finds no fault with the conclusions reached in the Goddard
 

study; however, the fundamental systems question--how to implement an acquisition procedure--was not addressed, i.e., "what signal isthe TDRSS supposed to
 

find?" It is unlikely that the Shuttle will be able to open-loop point to the
 

TDRS accurately enough to provide a strong signal, and we cannot expect both
 

the TDRS and the Shuttle to scan simultaneously.
 

10.2 	 Power Spectral Density of Staggered Quadriphase PN
 

with Identical Sequences
 

Axiomatix Report No. R8010-7, "The Power Spectral Density of Staggered Quadriphase PN With Identical Sequences," dated October 27, 1980, analyzes some of the TDRSS waveforms to determine the power spectral density and
 

the effect on processing gain against interference.
 
THE POWER SPECTRAL DENSITY OF STAGGERED QUADRIPHASE PN
 

WITH IDENTICAL SEQUENCES
 

Contract No. NAS 9-16067
 

Interim Report
 

Prepared by
 

M. K. Simon
 

Axiomatix
 

9841 Airport Blvd., Suite 912
 

Los Angeles, California 90045
 

Axiomatix Report No. R8010-7
 

October 27, 1980
 
THE POWER SPECTRAL DENSITY OF STAGGERED QUADRIPHASE PN
 

WITH IDENTICAL SEQUENCES
 

By 

Marvin K. Simon
 

1.0 INTRODUCTION
 

Staggered quadriphase PN is a direct-sequence spread spectrum
 

technique which employs a complex spreading waveform. Such a communication system offers twice the processing gain of the corresponding system
 

employing a real spreading waveform since, in the former, the interference energy will be distributed equally in the two quadrature channels
 

while, in the latter, all the interference energy appears in the data
 

channel.
 

More often than not, the two PN sequences will be independent
 

of one another and, thus, the power spectral density (PSD) of the spread
 

spectrum waveform is determined by the waveform of either sequence In
 

this report, we examine the effect on the PSD using the same PN sequence
 

on both channels (except for the 1/2-chip delay between the two).
 

2.0 CALCULATION OF THE POWER SPECTRAL DENSITY
 

A staggered quadriphase PN spread spectrum signal may be written
 

in the form
 

x(t) = VW R{m(t)s(t) expCjwot)} (1) 

where S is the average signal power, m(t) is a unit power digital data
 

modulation, w0 is the carrier frequency in rad/sec, and s(t) is the complex spreading waveform, namely,
 

s(t) [PNlt) + i PN2(tJ (2) 

Here we consider the case where PNI(t) = PN(t) and PN2(t)= PNI(t- A/2),
 

where A is the PN chip interval. Assuming for simplicity that m(t) is
 

a real modulation, e.g , BPSK, combining (1)and (2)then gives
 
2
 

x(t) a FSm(t) PN(t) cos Wot- PN(t-A/2) sin wnt] (3)
 

Furthermore, since the data rate of m(t) is typically much lower than the
 

PN chip rate (large processing gain), we may ignore m(t) in our power
 

spectrum calculation. Thus, we are interested inthe PSD of the normalized
 

signal
 

y(t) = PN(t) cos w0t - PN(t- A/2) sin iot (4)
 

The autocorrelation function of y(t) is easily seen to be
 

Ry() A

= <y(t) y(t+T)> 

= RpN(T) cos W 0 r + -PN(T+A/2) - RpN(r-A/2)] sin w0r (5) 

where < > denotes time average, the overbar denotes statistical average,
 

and RPN(T) is the autocorrelation function of the PN sequence. Taking
 

the Fourier transform of (5), we get
 

S(")= I ,mm *pC~ 

y 2IISPN Cw-w + SPN Cw+mSJd 

+ f Rp(T+ A12) sin oT e-J O dT 

- RP A/2) sin tOT e -j OT dT (6) - RpN (T-


Letting x = T+ A/2 or T- /2 in the second and third terms of (6), the 

sum of these two terms becomes 
OF ORIGINA4L POOR QUALI3
 PAGE iS 

TJY T PN(x) [ei -c-) XA/2) e-C+w)(X-A/2 dx 

- '

2 ZPN(X)[&exC A 2 - ea + Q x+ / § dx
 

= SpNCtw&wo sinLCf(OOJAft - 2PNC(Ow0 sin -1)+)A (7) 

Finally, using (7)in (6) gives the desired result, namely,
 

y2 PN -- $ +si L@ OJ 

+ 1-SPmwo I + sin +ThdAI2](' (8) 

The corresponding result for independent PN sequences PN1(t)and PN2 (t)
 

would be simply
 

Sy(W) = S (9)
 

Note that (8)and (9)are valid independent of the shape of the PN pulse.
 

As an example, consider the common case of a rectangular PN
 

pulse. For long codes, we may then describe the PSD of the sequence by
 

its envelope, namely,
 

A (10)
 PN~w) sin 2 = l L 2

L (10) 
4 OF pOOR QUALTY PAGE 0

Subtittin (i) nto(a)yiedsORIGINALt
 Substituting (10) into (8)yields
 

+2 6 2 

JJJwd
 A/2]
 

+ i C+oi OA/2] 2 sin c+ WmA/2]j (11) 

For simplicity,
 Figure 1 is an illustration of Sy (w)as described by (11). 


we show the result for positive w only with the understanding that SyCm)
 

Also shown in dotted lines isthe corresponding
 is an even function of w 


result for independent PN sequences obtained by substituting (10) into (9).
 
5 ORIGINAL PAGE I3 

OF POOR QUALITY S ( ) 

1.4
 

1.2
 

1.0 

0.8 I I 

I 

0.6 /

! I \ 

0.4 /

/ 2 

o./ 
 !,


/ 1, 

1T
 

Figure 1. Power Spectral Density for Staggered Quadrlphase PN