I

United States Department of Commerce

I 1

^ I Technology Administration

i i National Institute of Standards and Technology

NISTIR 3981

SIMULATION OF THE MERGED SPECTRUM

TECHNIQUE FOR ALIGNING PLANAR

PHASED-ARRAY ANTENNAS, PART I

Ronald C. Wittmann

Allen C. Newell

Carl F. Stubenrauch

Katherine MacReynolds

Michael H. Francis

NISTIR 3981

SIMULATION OF THE MERGED SPECTRUM

TECHNIQUE FOR ALIGNING PLANAR

PHASED-ARRAY ANTENNAS, PART I

Ronald C. Wittmann

Allen C. Newell

Carl F. Stubenrauch

Katherine MacReynolds

Michael H. Francis

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

October 1992

U.S. DEPARTMENT OF COMMERCE, Barbara Hackman Franklin, Secretary

TECHNOLOGY ADMINISTRATION, Robert M. White, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

CONTENTS

Page

1.0 INTRODUCTION 1

2.0 BASIC NEAR- FIELD MEASUREMENT THEORY 2

2 . 1 The Spectral Representation of Radiated Fields 2

2.2 Kerns's Transmission Equation 3

2.3 Inverting the Transmission Equation 4

2.4 Sampling the Near Field 4

3.0 BASIC ARRAY THEORY 5

3 . 1 The Fields of a Planar Antenna Array 5

3.2 The Array Factor for a Broadside Beam 7

3.3 Steering the Beam 7

3.4 The Array Factor for a Regular Lattice 8

3.5 Transforming Between Array Factor and Excitation Matrix 10

4.0 THE MERGED- SPECTRUM TECHNIQUE 12

4.1 Choosing Parameters 12

4.2 Determining the Form Factor from Near- Field Data 13

4.3 Combining Beams 14

5.0 SIMULATION FLOW 14

5 . 1 Array Model 14

5.2 Simulation Examples 15

6.0 SUMMARY 16

REFERENCES 17

iii
LIST OF FIGURES

Figure Page

1. A centered rectangular lattice showing a set of fundamental (and

A A A primitive) lattice vectors. a = A(l/2 x - 3/10 y) , b = A 3/5 y 18

A 2. The reciprocal lattice for the lattice of figure 1. A = 2k x,

A A B = k(x + 5/3 y) . If the array factor is known on the fundamental

period (shaded) , then it may be determined everywhere by periodic

repetition. For example, in the open rectangle A(K) = A(K - B) 18

3. A rectangular lattice obtained from the centered rectangular lattice

A of figure 1 by adding interstitial points. a' = A 1/2 x,

A b' = A 3/10 y. Element positions are marked with x's. (a' =0, n+m

even.) 19

4. A fundamental period in the reciprocal lattice for the doubled

lattice of figure 3. A(K) may be determined everywhere from its

values on the fundamental period of the centered rectangular

sublattice (shaded). The mapping is indicated by quadrant. The minus

signs occur because the centered rectangular sublattice is offset from

the origin 19

5. The visible region for a broadside beam is the shaded disk.

Extrapolation would be necessary to determine the form factor from

this information. In the merged- spectrum technique, direct

measurement of the fundamental period (rectangle) is accomplished via

beam steering 20

A A 6. When the beam is steered to the four locations = k(±l/2 x ± 5/12 y)

marked with •

' s

,

the entire fundamental period can be measured

directly. The visible region corresponding to each beam direction is

indicated by a circle. The shaded rectangle (^ < 40.6°) is the

smallest region in which coupling products must be determined 20

7. Block diagram showing the phased-array simulation and the near-field

measurement simulation 21

IV
LIST OF FIGURES (continued)

Figure Page

8. Block diagram showing the merged- spectrum technique for determining

the phased-array excitation matrix 22

9. Schematic of the phased array showing location of active elements. ... 23

10. Schematic of the central part of the phased array showing location of

active elements and virtual elements 24

11. Relative amplitude excitations (dots) of elements along the center

row, y

= 0, and field amplitude (solid line), normalized to 0 dB

peak, at z = 20 cm. Frequency is 3 GHz 25

12. Relative amplitude excitations (dots) of elements along the center

column, X = 0, and field amplitude (solid line), normalized to 0 dB

peak, at z = 20 cm. Frequency is 3 GHz 25

13. Relative amplitude excitations (dots) of elements along the center

row, y

= 0, and field amplitude (solid line), normalized to 0 dB

peak, at z = 50 cm. Frequency is 3 GHz 26

14. Relative amplitude excitations (dots) of elements along the center

column, X = 0, and field amplitude (solid line), normalized to 0 dB

peak, at z = 50 cm. Frequency is 3 GHz 26

15. Contour plot of the array factor for the doubled array composed of

active and virtual elements 27

16 . Perspective plot of the array factor for the doubled array composed

of active and virtual elements 28

17. Principal plane of the array factor for the doubled

array composed of active and virtual elements 29

V
LIST OF FIGURES (continued)

Figure Page

18. Principal plane (k = 0) cut of the array factor for the doubled

array composed of active and virtual elements 29

19. Contour plot of the probe's receiving pattern, amplitude for openended waveguide 30

20. Contour plot of the probe's receiving pattern, phase for open-ended

waveguide 31

A 0

21. Amplitude of the coupling product D (k) assuming probe and average

element are open-ended waveguides 32

22. Phase of the factor exp(i7z) . Frequency is 3 GHz and z = 20 cm 33

23. Amplitude of the factor expCiyz) . Frequency is 3 GHz and z = 20 cm. . 34

24. Simulated near-field amplitude. Frequency is 3 GHz and z = 20 cm. ... 35

25. Perspective plot of the near- field amplitude. Frequency is 3 GHz and

z = 20 cm 36

26. High resolution contour plot of the near-field amplitude near the

center of the aperture. Frequency is 3 GHz and z = 50 cm 37

27. High resolution contour plot of the near-field phase near the center

of the aperture. Frequency is 3 GHz and z = 50 cm 38

28. Contour plot of the array factor for an excitation matrix with

sinusoidal errors in both x and y directions (3° amplitude,

15 cm period) 39

VI
LIST OF FIGURES (continued)

Figure Page

29. Perspective plot of the array factor for an excitation matrix with

sinusoidal errors in both x and y directions (3° amplitude,

15 cm period) 40

30. Principal plane (k = 0) cut of the array factor for an excitation

matrix with a sinusoidal phase error in the x direction (3° amplitude,

15 cm period) 41

31. Principal plane (k = 0) cut of the array factor for an excitation

matrix with a sinusoidal phase error in the y direction (3° amplitude,

15 cm period) 41

32. Contour plot of array factor for a steered beam 42

vii
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i.\-
SIMDIATION OF THE MERGED- SPECTRUM TECHNIQUE

FOR ALIGNING PLANAR PHASED-ARRAY ANTENNAS,

PART I

Ronald C . Wittmann

Allen C. Newell

Carl F. Stubenrauch

Katherine MacReynolds

Michael H. Francis

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, CO 80303-3328

This report describes the initial phase of a NIST study of the

merged- spectrum technique for determining the element excitations

from planar scanning near- field measurements of phased- array

antennas. Excitation data are used in adjusting phase shifters to

meet design specifications. Measurement uncertainties, steering

errors , and various analytic approximations will all introduce

errors into the alignment. The study is ultimately directed to

quantify the effect of these errors, to more fully understand the

merged- spectrum technique, and to recommend possible improvements.

The present report covers theory developed to support evaluation of

the merged- spectrum technique and gives simulation examples

illustrating calculation of near fields from array factor and

element patterns

.

Key words: antenna measurements; antenna array alignment; merged

spectrum; near -field measurements; phased- array antennas; planar

scanning

.

1 . 0 INTRODUCTION

This report describes the initial phase of a NIST study of a method for

determining the element excitations (amplitude and phase) from planar nearfield measurements of phased-array antennas. The method is called the mergedspectrum technique because the (broadside -beam) array factor is obtained from

several spectra, each measured with the array electrically steered to a

different direction [1] . (There may not be enough information in any one

spectrum to determine the array factor directly.)

1
Excitation data are used in adjusting phase shifters, which initially may

be set randomly, to meet design specifications. Measurement uncertainties,

steering errors, and various analytic approximations will all introduce errors

into the alignment. The study is ultimately directed to quantify the effect

of these errors, to more fully understand the merged- spectrum technique, and

to recommend possible improvements. The present report covers computer code

and theory developed to support evaluation of the merged- spectrum technique.

A first step, contained in sections 2 through 4, is a careful description

of measurement and theory needed for the merged- spectrum technique. Section 2

is a brief summary of the principles, relations, and notation of the planar

near -field scanning measurement technique. In section 3 phased- array antennas

are introduced and the concepts of element excitations, average element

pattern, and array factor are discussed. The relations between the lattice of

array elements and the corresponding reciprocal lattice in k space are

developed since these are critical concepts in the merged- spectrum technique.

Section 4 sketches the merged- spectrum theory. In section 5 we give some

simulation results, illustrating calculation of array factors for steered and

unsteered beams, as well as simulated near-field measurements for typical

element and probe patterns. Additionally, examples of the effect of certain

measurement errors are shown.

2.0 BASIC NEAR- FIELD MEASUREMENT THEORY

The theory of planar near -field scanning is briefly summarized in this

section. A thorough account may be found in Kerns [2].

2 . 1 The Spectral Representation of Radiated Fields

The theory of planar near- field scanning is based on the fact that

radiated fields can be written as a superposition of plane waves

E(r) ( 1 )

where an exp(-io>t) time dependence is suppressed. Here

A A A k = K + 7Z , K = k X -f- k y , X y'y

K = |K|

2
7 = k =

2 2 k - K , k > K

2 2

[ UK - , k < K

CO 00

j

dK = I

= I dk^Jj dk^ .

-00 -CO

The far-field pattern t(r) is defined so that

E(r) ~

r Ko ^ ;

exp(ikr)

ikr

The quantity t(k)/7k is called the snectrum of E(r)

2 . 2 Kems ' s Transmission Equation

(2)

The output of a probe can be related to the incident field through the

transmission equation

' A W(r) = r(k) • t(k) exp(ik • r) 7k

n A D(k) exp(i7z) exp(iK • r) dK ,

A A D(k) - r(k) . t(k)/7k .

z > 0 ( 3 )

D(k) is the coupling product . The receiving pattern r(k) is defined so that

the response of the probe to the incident plane wave

2^ Eq exp(ik . r)

is

r(k) • Eq exp(ik • tp) ,

where is the position of the probe. The definition of the receiving

pattern depends on the choice of probe calibration; however, the details of

this normalization are unimportant for the present application.

3
Basically, the transmission equation states that the probe output is the

sum of the probe responses to the plane -wave components of the incident field

eq (1).

2 . 3 Inverting the Transmission Equation

Fourier transformation of eq (3) gives

D(k) =

47t'

W(r) exp(-ik • r) dR , z > 0 , (4a)

or

1

1

D(k) expCiyz^) = —^ exp(-iK • R) dR , z > 0

47T

r = R -f ZqZ , R = XX -I- yy

00 CO

j

dR = Idx

j

dy .

-00 -00

(4b)

Thus, in principle, the coupling product may be determined if the probe

response is known on a plane z = z^ > 0.

2 , 4 Sampling the Near Field

A The application of eq (4) is greatly simplified by the fact that D(k) is

strongly spatially bandlimited because the plane -wave components become

evanescent when K > k. Thus, according to the "sampling theorem," the

coupling product may be reconstructed "exactly" from discrete probe

measurements as long as the sample interval is less than half a wavelength:

Ax, Ay < A/2 . (5)

In an actual measurement, of course, the scan area must be finite. A

rule of thumb is that every line drawn from the aperture in a direction in

which the far field is to be determined must pass through the measurement

region. (Like all rules of thumb, this is subject to experimental

verification.

)

4
With the assumption of discrete sampling and truncation of the

integration range, eq (4) may be reduced to a DFT (Discrete Fourier

Transform). By the nature of the DFT (discussed more fully in section 3.5),

measurements made at the points

r = nAx X + mAy y + z

nm 0

result in the determination of D(k) for

y _ B ZZL ^ + 5! ^

nm N Ax ^ N Ay

X y

( 6 )

where N and N are the numbers of data points in the x and y directions. The

X y

^ K space increments Ak and Ak are inversely proportional to the x and y X y

dimensions of the measurement plane.

It is often useful to zero-fill the data array by explicitly (rather than

implicitly) setting the probe response to 0 outside the actual measurement

region. This does not add any information, but it does result in evaluation

of the coupling product on a finer grid [since N and/or N have been

X y

increased in eq (6)]. Also, the FFT (Fast Fourier Transform) algorithm used

to perform the DFT is most efficient if N and N can be factored into small

X y

primes. For example, use N = 256, but not N = 251.

3.0 BASIC ARRAY THEORY

This section describes the basic elements of antenna array theory needed

for an understanding of the merged- spectrum technique. A good general

reference is Ma [3].

3 . 1 The Fields of a Planar Antenna Array

Consider a planar array made up of identical elements. The electric

field of the array is a superposition of the element fields:

E(r) = ^ a(R^) e(r - Rj^) . (7)

i

The sum runs over the positions of the array elements R^. We take

5
A A R. = x.x + y.y , ( 8 )

so that the array lies in the z = 0 plane. The excitation matrix a(R^)

determines the amplitude and phase of the element excitations. The beam is

shaped and steered by controlling a(R^) . The average element field e(r) is

not the free -space field of an element, but includes effects of mutual

interactions among elements and supports, etc. (In a finite array it is an

approximation to assume that all elements produce the same effective field

since the environment varies from location to location.)

A 0 An array element is characterized by a far -field pattern t (k) , which we

define so that

A e(r) - t®(r)

r—-*<o

exp(ikr)

ikr (9)

When an element is positioned in the array, its far- field pattern is

multiplied by a phase factor depending on the displacement from the coordinate

origin (which is the mathematical "location" of the array)

A A e(r - R. ) ~ t (r) exp (-ikr

r Hxi

. exp (ikr)

ikr ( 10 )

From eqs (7) and (10)

E(r) - ,

X" *<o

where t (r) is given by

t^(r) = t®(r) > a(R^) exp (-ikr • R^)

= t"^(r) A(kR/r) .

( 11 )

( 12 )

The far- field pattern of the array is the product of the average element farA field pattern and the array factor A(r)

According to eq (1) , the electric field of the array may be written in

terms of its far-field pattern:

6
= 2¥ J

1* t^(k) exp(ik • r) ^

=

2^ J

|* t^(k) A(K) exp(ik • r) yk ’ z > 0 (13)

> II n a(R^) exp(-iK • Rj’) • (14)

i

3 . 2 The Array Factor for a Broadside Beam

When the beam is broadside, the a(R^) must have constant phase since this

will maximize the far- field pattern in the z direction: If the broadside -beam

excitation matrix and the corresponding array factor are denoted by o:(R^) and

Aq(K), respectively, then from eq (14)

|Aq( 0)1 |a(Rj^)| a |Aq(K)1 . (15)

i

The purpose of the merged- spectrum alignment is to

of the a(R^) for an unaligned array so they may be

values. We will call Aq(K) the form factor of the

section 3.3, the array factor of a steered beam is

factor

.

determine the actual values

adjusted to the design

array. As demonstrated in

simply related to the form

3 . 3 Steering the Beam

Steering is accomplished by adjusting the phase of the excitation matrix.

When the excitation matrix is

a(R^) = a(R^) expCiK^ • R^) , (16)

then the array factor

A(K) - Aq(K - Kq) - ^ exp[-i(K - Kq) • (17)

i

A will produce a beam in the direction. With the convention eq (16) , the

phase of a reference element at the origin will be independent of Kq.

7
The electric field of the steered array is

E(r) = ^

I

Aq(K - Kq) exp(ik • r) ^ , z > 0 . (18)

Since the plane -wave components of eq (18) become evanescent when K > k, Aq(L)

plays an active role in determining the far field of the array only when L

lies within a circle of radius k = w/c centered on L = -Kq. Thus, the visible

portion of the form factor depends on the steering angle.

The steering vector can be related to the usual polar angles 6 and <j):

3.4

K,-, = k sin^ cosd) , = k sin^

Ox ^ Oy

sin^ = ^0^^ ’ tan<^ = Kq^/K

The Array Factor for a Regular Lattice

sin<^

Ox •

Every point in a planar lattice can be represented in the form

R = na + mb

nm

(19)

(a X b 5^ 0) . The fundamental lattice vectors a and b are said to be primitive

if R^ is a lattice point for all integers n and m. In this report we will assume that the array elements lie on a centered

rectangular lattice , illustrated in figure 1, where the fundamental lattice

vectors are given by

1

A 2^ a = A(^ X - Yo y) b = y ( 20 )

a and b are clearly primitive. [The choice of a and b is not unique; for

1

A 2A example, we could use a = A (-2 x -f Y) • ] The array factor eq (14) may now be written

A(K) = a exp ( - iK • R ) 1/^ ^ i/fi

( 21 )

UfL

a = a(R ) . nm nm

It is easy to show from eq (21) that the array factor is doubly periodic

8
A(K ± A) = A(K) , A(K ± B) = A(K) , ( 22 )

where

^ = 2k B = 2. = k<; + f

y) . (23)

A A A [For a rectangular lattice with a = ax, b = by, we have A = 27r/a x,

A B = 27r/b y. In general, |A| is proportional to 1/a (independent of b) and |B|

is proportional to 1/b (independent of a) .

]

A and B are fundamental lattice vectors of the reciprocal lattice shown

in figure 2, The form factor Aq(K) assumes its maximum amplitude at each

lattice point (K^ = nA + mB) . Consequently, if more than one reciprocal

lattice point is visible, then, in addition to the main beam, there may be

other beams C grating lobes ) in the far- field pattern. Figure 2 also shows a

fundamental period (shaded rectangle) for the array factor. If the array

factor is known on a fundamental period then it may be determined everywhere

by periodic repetition using eq (22)

.

(While the area is fixed, a fundamental

period may assume a number of shapes. A rectangle is convenient for our

purposes .

)

The preceding discussion assumes that R = 0 is a lattice point (that is,

there is an element at the origin) . If the array is offset so that

R = C + + mb , (24)

then A(K) exp(iK • ^) is periodic with periods A and B. Depending on the

array factor itself may have periods which are multiples of A and B, or it may

be aperiodic. For instance, when ^ = b/2 we have A(K + B) = -A(K) [see

eq (29)] corresponding to a period of 2B.

To gain the convenience of working with rectangular arrays we use the

"doubled” lattice shown in figure 3. Because the lattice vectors

a’ - |a X , b' - y (25)

are not primitive with respect to the centered rectangular sublattice, it is

necessary to include virtual elements with 0 excitation at the locations

R' = na' -I- mb' , (26) nm ^

9
when n+m is even. (The prime is used to distinguish the doubled lattice from

the centered rectangular lattice .

)

One effect of using the nonprimitive basis, a' and b' , is that the

fundamental period is doubled in size as shown in figure 4. In addition to

the periodicities

A(K ± A') = A(K ± A) = A(K) , A(K ± B' ) = A(K)

A' = A = 2k X , 10 ^ B' = t^k y ,

(27)

(28)

we have

A(K ± B) = -A(K) . (29)

The minus sign in eq (29) occurs because the centered rectangular sublattice

is offset from the origin. The fundamental period of the doubled lattice can

be constructed from the fundamental period of the centered rectangular

sublattice using eqs (27) and (29). The mapping is illustrated in figure 4.

3 . 5 Transforming Bet?ween Array Factor and Excitation Matrix

The array factor is given by

A(K) = ^

' a' exp(-iK • R' ) , ^ ^ i/fi i/fi

i/fi

where

a' = nma(R'

nm)

.

To produce the appropriate centered rectangular lattice, a^ must be 0 when

n+m is even (s^qq = 0) . For convenience in applying the FFT algorithm, we bias the summation

indices so they will be positive:

A(K) exp(-iK

P-1 Q-1

EE

l/=zQ ^=0

a' ^ „ exp(-iK R'

i/fi )

. (30)

10
Here N and M have been chosen so that a' =0 when n < -N or m < -M; P and Q nm

have been chosen so that a' „ = 0 when n > P or m > Q. In effect, a new

n-N,m-M

origin has been chosen so that the reference element coordinates are (N,M)

.

Let

K

nm

n ^

P a'

A m 2n

Then eq (30) may be rewritten

(31)

P-1 Q-1

. V -nN -mM A(K ) w w

nm r Q

, -i/n -am

a' ^ lur

i/=0 ^=0

where

,27riv /27ri. Wp = exp(—) , = exp(—

)

If we define

A /IT \ T nN mM A(K ) = F „ nm n+N,m+M P Q

, _ r: -(n+N)N -(m+M)M

nm n+N , m+M P Q

-N < n < P-l-N , -M < m < Q-l-M ,

then eq (32) becomes

(32)

(33)

(34)

(35)

(36)

F

nm

P-1 Q-1

U=0 fJL=0

f

-i/n

U) P

f

nm

P-1 Q-1

u=0 fi=0

F

i/fi

i/n

(37)

(38)

The sequences F and f are related by a two-dimensional DFT. The choice of

nm nm •' phase factors in eqs (34) and (35) allows use of the natural range for the

indices eq (36) without rearrangement of the outputs of the FFT routine.

The transformation between excitation matrix and array factor may be

represented schematically as

11
phase

factor FFT F phase

factor

For the specific case used in the simulation, the excitation matrix (here

denoted c ) is in a row and column format rather than in the cartesian format

nm

used in these notes. The relation to the broadside -beam excitation matrix a‘

is

nm

*^nm ^M+l-m,N+l+n ’

where

N = 39 , M = 70 (39a)

P > 79 , Q ^ 141 . (39b)

4.0 THE MERGED-SPEGTRDM TECHNIQUE

When the beam is steered to K^, the visible region of the form factor is

a circle of radius k centered on -Kq (see section 3.3). As figure 5

illustrates, when using only a broadside beam, extrapolation is necessary to

estimate the form factor. (Coverage does not extend even to radius k since

the near-field scan area must be truncated.) From a practical point of view

it is better to measure the array factor directly rather than to extrapolate.

In the merged- spectrum technique, direct measurement is accomplished by

steering the beam to make different regions of the form factor visible.

Information from several beams is merged to give the fundamental period.

Figure 6 shows an example (discussed below) using four beams

.

4 . 1 Choosing Parameters

Three criteria must be considered:

(1) The density of points in K space must be chosen so that the

requirement eq (39b) is satisfied.

(2) The beams should be obtainable without quantization errors in the

phase shifters.

(3) The beam directions must be chosen so that data points lie on a

common grid in K space.

12
To satisfy criterion (1) we take P = 128 and Q = 256, so the form factor

is evaluated at the points

K

nm

n

128

A' +

m

256 B' (40)

With seven-bit phase shifters, the beam may be steered, without

quantization error, to the directions when m is even. Thus, criteria (2)

and (3) may be satisfied with the four beam directions

^±32, ±32 ”8®' "" ^“2^“l2

which are shown in figure 6

.

4.2 Determining the Form Factor from Near-Field Data

It is most convenient to collect near- field data so that the DFT will

give the array coupling product D (K) at appropriate K values without

requiring additional interpolation. A comparison of eqs (6) and (31) yields

N Ax = 0.5 AP

X

(42)

Ny Ay = 0.3 AQ .

For example, let P = 128 and Q = 256 as in eq (40). The requirements eqs (5)

and (42) can be satisfied with the choices Ax = 0.25 A, Ay = 0.3A, and

= 256. The dimensions of the measurement plane are 64A by 77A,

whereas the array is roughly 42 A in diameter. To determine A^(K) over a

fundamental period, D (K) must be known [for each steering angle eq (41)] over

the central rectangle of figure 6 (^ < 40.6°). According to the truncation

criterion of section 2.4, calculated results should be valid in this region if

the spacing between scan plane and array is less than about 14A.

Near- field measurements are transformed in the standard manner (FFT) to

obtain ^^(^^) • From eqs (3) and (18) we have

A„(K - Kn) - D®(K )/D®(K ) nm 0 nm ' nm

^0

' (43)

0 where D (K ) is the coupling product between the probe and an "average

0 element." D (K^) is determined independently, and it must be known at least

13
for the central rectangular region indicated in figure 6. For each beam

direction eq (43) determines Aq(K) in a disk of radius K < k about the point

4 . 3 Combining Beams

One way to combine the spectral information from the beams used in

figure 6 is to simply let each beam determine Aq(K) in the corresponding

quadrant, perhaps averaging along the common boundaries. An alternative

method would be to construct a weighted average in the overlap regions. The

weights would be chosen proportional to some experimental measure of

confidence. The second method allows more flexibility in choosing the

location and number of beams in the merge.

Once the form factor has been determined on the fundamental period of the

centered rectangular sublattice , the data can be extended over the fundamental

period of the doubled lattice as shown in figure 4. Finally, the results of

section 3.5 can be applied to transform from the form factor to the excitation

matrix a' . nm

5.0 SIMULATION FLOW

The computer simulation, outlined in the flow charts of figures 7 and 8,

implements the theory of the preceding sections and provides a means for

testing various aspects of the spectral merge process. Figure 7 illustrates

the steps in obtaining the simulated near- field data which will be used for

the merged- spectrum analysis. The procedure of figure 8 uses the near-field

data for each of the steered beams to calculate the broadside -beam excitation

matrix from the merged spectrum. Implementation of the blocks in figure 8

will be completed in the next phase of the study. The final result depicted

in figure 8 should be the array excitation matrix input initially into the

simulation process of figure 7. Any differences are due to measurement errors

and effects of the merged- spectrum technique. Through simulation, we can

determine the most critical errors and the limits of accuracy.

5 . 1 Array Model

The array used in the simulation is shown schematically in figure 9. The

4350 active elements are arranged in 79 columns and 141 rows (in a nearly

14
circular region) on a centered rectangular lattice as shown in figures 1 and

9. To gain the convenience of working with a rectangular lattice, we add a

virtual element between each actual element along the columns as shown in

figures 3 and 10. Since virtual elements are assigned 0 amplitude in the

excitation matrix, the performance of the array is not altered. Nominal

element spacings in this doubled array are a' = 0.5A in the x direction

(between columns) and b' =>0.3A in the y direction (between rows), resulting

in an aperture approximately 42A in diameter; however, spacing can be changed,

as desired. The array is further extended with virtual elements (zero

filling) so that it is rectangular in shape with P > 79 columns and Q > 141

rows. Efficient use of the FFT dictates that P and Q should be powers of 2;

for example, P = 128, Q = 256.

5 . 2 Simulation Examples

The first simulation example is for a broadside beam with no errors in

any of the elements. Figures 11 through 14 are center line (x = 0 and y

= 0)

plots of the excitation matrix along with the corresponding field amplitude on

measurement planes at 20 cm and 50 cm. The field amplitudes, which are

normalized to 0 dB peak, were calculated using eq (13) . (The frequency is

3 GHz for all computations .

)

The broadside-beam array (form) factor was calculated using eq (30) with

P = Q = 512 and results are shown in figures 15 through 18. Figures 15 and 16

are plots of the array factor over the fundamental period of the doubled

lattice: -1 < < 1, -5/3 < k^/k < 5/3. These figures confirm that the

fundamental period for the doubled array can be determined if the array factor

is known in the subregion (shaded in fig. 4) which corresponds to the

fundamental period of the actual array without virtual elements. (See the

discussion of fig. 4 in section 3.3.) Thus "doubling" of the array, which is

done for computational convenience, does not cost anything from a measurement

point of view. Figures 17 and 18 show the array factor in the principal

planes. [The horizontal bands in the array factor evident in figure 15

apparently arise because the elements are driven pairwise in the y direction

(fig. 12). Similar vertical bands might be expected in figure 15, as well,

except that the y = 0 row (the excitations are plotted in fig. 11) is the only

row with elements driven in pairs.]

The next step in the simulation is the calculation of the measured data

on a specified measurement plane. The array factor is first multiplied by the

15
coupling product between probe and average element patterns and then by the

factor expClyz^) before transforming to the near-field with the FFT [see

eq (13)]. A typical spectrum for both the probe and element is shown in

A 0 figures 19 and 20 with the amplitude of the coupling product D (k) shown in

figure 21. Since both the probe and element have very broad patterns, they

have little effect on the spectrum of the low sidelobe array. (The major

effect occurs when K/k > 0.6 where the coupling product is reduced by 10 to 30

dB due to the combined effect of probe and element patterns.) The factor

exp(i7ZQ) is important, however. Within the visible region (fig. 22), where 7

is real, it reflects the phase progression of plane waves propagating from the

aperture to the measurement plane. Beyond the visible region (fig. 23), where

7 is imaginary, it reflects the exponential attenuation of the evanescent

components of the spectrum.

Figures 24 through 27 represent error- free data from an open-ended

waveguide probe on measurement planes 20 cm and 50 cm from the aperture of an

array with no faulty or misaligned elements. The program provides the option

of introducing various errors and example array factors are shown in

figures 28 through 31. Here, the excitation phases have been modified, by

adding a sinusoidal error in the x and/or y directions, to produce paired

error lobes in addition to the main beam. Other errors will be simulated in

future studies.

In addition to the broadside beams illustrated thus far, steered beams

can be produced by imposing a linearly varying phase on the excitation matrix

(section 3.2). An example array factor is shown in figure 32. The array

factor for a steered beam can be obtained by translating the form factor

(fig. 15) as indicated by eq (17) . Multiple beam directions are required for

the merged- spectrum simulation.

6.0 SUMMARY

Computer code and supporting analysis have been developed as the basis

for a thorough evaluation by simulation of the merged- spectrum technique for

aligning planar arrays. The actual evaluation of the merged- spectrum

technique will be described in a future report.

16
REFERENCES

[1] Patton, W.T. Phased array alignment with planar near -field scanning; or

determining element excitation from planar near- field data. In Proc.

Antenna Appl. Symp . , Univ. Illinois; Sept. 1981.

[2] Kerns, D.M. Plane-wave scattering-matrix theory of antennas and antennaantenna interactions. Nat. Bur. Stand. (U.S.) Monograph 162; June 1981.

[3] Ma, M.T. Theory and application of antenna arrays. New York: Wiley;

1974.

17
X

Figure 1, A centered rectangular lattice showing a set of fundamental (and

primitive) lattice vectors. a = A(l/2 x - 3/10 y) , b = A 3/5 y.

k X

Figure 2. The reciprocal lattice for the lattice of figure 1. A = 2k x,

B = k(x + 5/3 y) . If the array factor is known on the fundamental

period (shaded) , then it may be determined everywhere by periodic

repetition. For example, in the open rectangle A(K) = A(K - B)

.

18
X

Figure 3 . A rectangular lattice obtained from the centered rectangular

lattice of figure 1 by adding interstitial points. a' = A 1/2 x,

b' = A 3/10 y. Element positions are marked with x's.

even . )

Figure 4. A fundamental period in the reciprocal lattice for the doubled

lattice of figure 3. A(K) may be determined everywhere from its values

on the fundamental period of the centered rectangular sublattice

(shaded); The mapping is indicated by quadrant. The minus signs occur

because the centered rectangular sublattice is offset from the origin.

19
k( 1 , 5/6)

Figure 5. The visible region for a broadside beam is the shaded disk.

Extrapolation would be necessary to determine the form factor from this

information. In the merged- spectrum technique, direct measurement of

the fundamental period (rectangle) is accomplished via beam steering.

Figure 6. When the beam is steered to the four locations

Kq = k(±l/2 X ±5/12 y) marked with *'s, the entire fundamental period

can be measured directly. The visible region corresponding to each beam

direction is indicated by a circle. The shaded rectangle (6 < 40.60°) is

the smallest region in which coupling products must be determined.

20
PROCESS INPUTS/OUTPUTS

Figure 7. Block diagram showing the phased-array simulation and the nearfield measurement simulation.

21
PROCESS INPUTS/OUTPUTS

Figure 8. Block diagram showing the merged- spectrum technique for determining

the phased- array excitation matrix.

22
Position

in

cm

ACTUAL ELEMENTS IN ARRAY

200

150

100

50

-0

-50 -

-100 -

-150 -

-200 J L

-200 -150

_l 1 I I

I I 1 -100 -50 0 50

X—Position in cm

_j I i_

100 150

I 200

Figure 9

.

Schematic of the phased array showing location of active elements.
Position

in

Figure 10 . Schematic of the central part of the phased array showing the

location of active elements (n) and virtual elements (•).

24
Amplitude

in

dB

e

Amplitude

in

dB

ARRAY EXCITATION AND NEAR FIELD

Z distance 20 cm

e 11. Relative amplitude excitations (dots) of elements along the center

row, y

= 0, and field amplitude (solid line), normalized to 0 dB peak,

at z = 20 cm. Frequency is 3 GHz.

ARRAY EXCITATION AND NEAR FIELD

Z distance 20 cm

Y Position in cm

Figure 12. Relative amplitude excitations (dots) of elements along the center

column, X = 0, and field amplitude (solid line), normalized to 0 dB

peak, at z = 20 cm. Frequency is 3 GHz.

25
Amplitude

in

dB

c

Amplitude

in

dB

ARRAY EXCITATION AND NEAR FIELD

Z distance 50 cm

13. Relative amplitude excitations (dots) of elements along the center

row, y = 0, and field amplitude (solid line), normalized to 0 dB peak,

at z = 50 cm. Frequency is 3 GHz.

ARRAY EXCITATION AND NEAR FIELD

Z distance 50 cm

Figure 14. Relative amplitude excitations (dots) of elements along the center

column, X = 0, and field amplitude (solid line), normalized to 0 dB

peak, at z = 50 cm. Frequency is 3 GHz.

26
Relative

Wavenumber

10 dB Contours (0, —60)

Relative Wavenumber u = kx/k

Figure 15 . Contour plot of the array factor for the doubled array composed of

active and virtual elements

.

27
RELATIVE

AMPLITUDE

IN

dB

Figure 16. Perspective plot of the array factor for the doubled array

composed of active and virtual elements.

28
Amplitude

in

dB

^

Amplitude

in

dB

ARRAY FACTOR

kx/k

17. Principal plane (ky = 0) cut of the array factor for the doubled

array composed of active and virtual elements

.

ARRAY FACTOR

Figure 18. Principal plane (k^ = 0) cut of the array factor for the doubled

array composed of active and virtual elements

.

29
RELATIVE

WAVENUMBER

OPEN-END WAVEGUIDE SPECTRUM, AMPLITUDE

2 dB CONTOURS (0. -20)

Figure 19. Contour plot of the probe's receiving pattern, amplitude for openended waveguide

.

30
RELATIVE

WAVENUMBER

OPEN-END WAVEGUIDE SPECTRUM, PHASE

5 DEGREE CONTOURS ( 60 . 110)

Figure 20. Contour plot of the probe's receiving pattern, phase for openended waveguide

.

31
RELATIVE

WAVENUMBER

ELEMENT PATTERN * PROBE PATTERN - AMPLITUDE

Figure 21. Amplitude of the coupling product D®(^) assuming probe and average

element are open-ended waveguides.
Relative

Wavenumber,

1.6

_1,6 I I I I I I I I ’-

1.0 -0.6 -0.2 0.2 0.6

Relative Wavenumber, u

Figure 22. Phase of the factor exp(i7z) . Frequency is 3 GHz and z = 20 cm.

33
Relative

Wavenumber,

Relative Wavenumber, u = kx

Figure 23. Amplitude of the factor exp(i7z) . Frequency is 3 GHz and

z = 20 cm.

34
ARRAY NEAR- FI ELD AMPLITUDE. Z=20CM. NO ERRORS

-640 - 384 -128 128 384 640

X-POSITION IN CM

Figure 24. Simulated near-field amplitude. Frequency is 3 GHz and z = 20 cm.

35
RELATIVE

AMPLITUDE

IN

dB

NEAR- FI ELD AMPLITDUE, Z=20 CM

Figure 25.

and z Perspective plot of the near- field amplitude.

= 20 cm.

Frequency is 3 GHz

36
Position

in

Simulated Near— Field Data at z = 50 cm

Figure 26. High resolution contour plot of the near-field amplitude near the

center of the aperture. Frequency is 3 GHz and z = 50 cm.

37
Position

in

Sinnulated Near— Field Phase z 50 cm

X-Position in cm

Figure 27. High resolution contour plot of the near-field phase near the

center of the aperture. Frequency is 3 GHz and z = 50 cm.

38
Relative

Wavenumber

10 dB Contours (0, —60)

Figure 28. Contour plot of the array factor for an excitation matrix with

sinusoidal phase errors in both x and y directions (3° amplitude, 15 cm

period)

.

39
RELATIVE

AMPLITUDE

IN

dB

Figure 29. Perspective plot of the array factor for an excitation matrix with

sinusoidal phase errors in both x and y directions (3° amplitude, 15 cm

period)

.

I

>

-

40 I
Figure 30. Principal plane (ky = 0) cut of the array factor for an excitation

matrix with a sinusoidal phase error in the x direction (3° amplitude,

15 cm period)

.

QQ

-20—

O

-2.0 - 1.5 - 1.0 -0.5 0.0 0.5 1.0 1.5 2.0 ky/k

Figure 31. Principal plane (k^ = 0) cut of the array factor for an excitation

matrix with a sinusoidal phase error in the y direction (3° amplitude,

15 cm period)

.

41
Relative

Wavenumber

10 dB Contours (0, —60)

II >

Relative Wavenumber u = kx/k

Figure 32. Contour plot of array factor for a steered beam.

42
BL-114A

(5-90)

ADMAN 15.01

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBUCATION OR REPORT NUMBER

NISTIR 3981

2! PERFORMING ORCiANIZATION REPORT NUMBER

B92-0241

PUBUCATION DATE

OCTOBER 1992

4. TITLE AND SUBTITLE

Simulation of the Merged-Spectrum Technique for Aligning Planar Phased-Array Antenna^ Part I

5. AUTHOR(S)

Ronald C. Wittmann, Allen C. Newell, Carl F. Stubenrauch, Katherine MacReynolds, Michael H.

Francis

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BOULDER, COLORADO 80303-3328

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

10. SUPPLEMENTARY NOTES

11. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOGRAPHY OR

UTERATURE SURVEY, MENTION IT HERE.)

This report describes the initial phase of a NIST study of the merged-spectrum technique for

determining the element excitations from planar scanning near-field measurements of phasedarray antennas. Excitation data are used in adjusting phase shifters to meet design

specifications. Measurement uncertainties, steering errors, and various analytic

approximations will all introduce errors into the alignment^ The study is ultimately directed

to quantify the effect of these errors, to more fully understand the merged-spectrum technique

and gives simulation examples illustrating calculation of near fields from array factor and

element patterns.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

antenna measurements; antenna array alignment; merged spectrum; near-field measurements;

phased-array antennas; planar scanning.

13. AVAILABIUTY

UNUMITED

FOR OFRCIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).

X

X

ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,

WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

53

15. PRICE

AO4

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