NASA Technical Reports Server (NTRS) 20080004543: Phase array calibration orthogonal phase sequence

14 Pages·0.93 MB·English

by  NASA Technical Reports Server (NTRS)

United States Patent [19] [ill Patent Number: 5,861,843
Sorace et al. Date of Patent: Jan. 19,1999
[45]
PHASE ARRAY CALIBRATION 5,455,592 1011995 Huddle .................................... 3421359
ORTHOGONAL PHASE SEQUENCE 5,530,445 611996 Wachs et al. ........................... 3421174
5,677,696 1011997 Silverstein et al. ..................... 3421360
Inventors: Ronald E. Sorace, Torrance; Victor S.
Primary Examiner-Thomas Tarcza
Reinhardt, Rancho Palos Verdes;
Assistant Exarninerqao L. Phan
Clinton Chan, Chino Hills, all of Calif.
Attorney, Agent, or Firmaeorgann S. Grunebach; M. W.
Sales
Assignee: Hughes Electronics Corporation, El
Segundo, Calif. [571 ABSTRACT
Methods and systems for calibrating an array antenna are
Appl. No.: 997,078
described. The array antenna has a plurality of antenna
Filed: Dec. 23, 1997 elements each having a signal with a phase and an amplitude
forming an array antenna signal. For calibration, the phase
Int. C1.6 ....................................................... HOlQ 3/24 of each element signal is sequentially switched one at a time
U.S. C1. .......................... 3421372; 3421174; 3421373; through four orthogonal phase states. At each orthogonal
3421374 phase state, the power of the array antenna signal is mea-
Field of Search ..................................... 3421174, 372, sured. A phase and an amplitude error for each of the
3421373, 374 element signals is determined based on the power of the
array antenna signal at each of the four orthogonal phase
References Cited states. The phase and amplitude of each of the element
signals is then adjusted by the corresponding phase and
U.S. PATENT DOCUMENTS
amplitude errors.
5,063,529 1111991 Chapoton ........................... 3641571.02
5,248,982 911993 Reinhardt ................................ 3421375 20 Claims, 4 Drawing Sheets
SET THE PHASE AND AMPLITUDE OF EACH J 40
ANTENNA ELEMENT SIGNAL TO FORM AN ARRAY ANTENNA SIGNAL
1 1
SEQUENCE THE PHASE OF AN ANTENNA ELEMENT
SIGNAL THROUGH FOUR ORTHOGONAL PHASE STATES
1 1
MEASURETHE POWER OFTHE ARRAY ANTENNA SIGNAL
AT EACH OFTHE FOUR ORTHOGONAL PHASE STATES "
DETERMINE PHASE ERROR FORTHE ANTENNA ELEMENT SIGNAL 14 8
DETERMINE AMPLITUDE ERROR FORTHE /- 50
ANTENNA ELEMENT SIGNAL
1
ADJUST THE AMPLITUDE OF EACH OF THE ANTENNA ELEMENT
SIGNALS BY THE CORRESPONDING AMPLITUDE ERROR
US. Patent 5,861,843
Jan. 19, 1999 Sheet 1 of 4
--
2222 LLOOAADD
//66
11
\\ \\
1
> I (5
1 1 (5
I
20+ FIG. 1
CONTROLLER
1 I
FIG. 4
US. Patent 5,861,843
Jan. 19, 1999 Sheet 2 of 4
SET THE PHASE AND AMPLITUDE OF EACH K 40
ANTENNA ELEMENT SIGNAL TO FORM AN ARRAY ANTENNA SIGNAL
I 142
I
SEQUENCE THE PHASE OF AN ANTENNA ELEMENT
SIGNALTHROUGH FOUR ORTHOGONAL PHASE STATES
1
MEASURETHE POWER OFTHE ARRAY ANTENNA SIGNAL
AT EACH OFME FOUR ORTHOGONAL PHASE STATES
I
48
DETERMINE PHASE ERROR FOR THE ANTENNA ELEMENT SIGNAL
4
1
DETERMINE AMPLITUDE ERROR FOR THE
AMENNA ELEMENT SIGNAL
-
ADJUSTTHE PHASE OF EACH OFTHE ANTENNA ELEMENT
SIGNALS BY THE CORRESPONDING PHASE ERROR
56
STANDARD DEVIATION OF PHASE CORRECTION MEASUREMENT
30 Y
PHASE
DEVIATION
PEG)
FIG. 5
US. Patent 5,861,843
Jan. 19, 1999 Sheet 3 of 4
PHASE ITERATION SOLUTION
12 SAMPLES, SNR = 35 dB
-
0.00
i
PHASE -4 5 10 15
-0.20
ERROR
PEG) -0.40
-i-
-0.60 I
-0.80 Q DIFFERENTIAL
I
*ABSOLUTE FIG. 6A
NO. ITERATIONS
PHASE ITERATION SOLUTION
12 SAMPLES, SNR = 55 dB
PHASE :0.00 :0 : 5I I 410 1I5
ERRORI
L .
PEG) 1
Q DIFFERENTIAL
I
0.60 -D ABSOLUTE FIG. 6B
NO. ITERATIONS
PHASE --0 10 210
ERROR -2
PEG)
-6
ABSOLUTE
-C, FIG. 6C
NO. ITERATIONS
PHASE ITERATION SOLUTION
24 SAMPLES, SNR = 15 d
+--
0
PHASE 0 5 ei l
ERROR
----
PEG) --12
-3 - --
FIG. 6D
-441
NO. ITERATIONS
US. Patent 5,861,843
Jan. 19, 1999 Sheet 4 of 4
\ \
180
0 0 0
0 0.0
70
/
GROUND - - -
AID POWER
ANTENNA FILTER PTE
TERM INA L CONVERTER DETECTOR
\
\ ,72 84’
\ \
CALIBRATION
\
PROCESSOR
FIG.7
FIG. 8
5,861,843
1 2
PHASE ARRAY CALIBRATION three thousand six hundred measurements must be made for
ORTHOGONAL PHASE SEQUENCE calibration of the array, and iteration may be required to
improve accuracy. Scanning each element through all of its
GOVERNMENT RIGHTS phase values is suboptimal in a noisy environment and has
s the disadvantage of potentially large interruptions to service.
The present invention was made with Government sup-
Accordingly~a need has for a quicker and more
port under contract number [Secret Classification] awarded
efficient method which requires fewer measurements for
by the National Aeronautics and Space Administration
an array antenna.
“NASA.” The Government has certain rights in the present
invention.
SUMMARY OF THE INVENTION
10
TECHNICAL FIELD It is an object of the present invention to provide an
orthogonal phase calibration method for an array antenna.
The present invention relates generally to phased array
It is another object of the present invention to provide a
antennas and, more particularly, to a method of calibrating
calibration method for an array antenna which determines
a phased array antenna.
phase errors based on power measurements made at
BACKGROUND ART orthogonal phase states.
It is a further object of the present invention to provide a
An array antenna an array Of antenna calibration method for an array antenna which determines
for transmission or reception of electromagnetic signals. The amplitude based on power measurements made at
antenna elements are fed with one or more signals whose 20 orthogonal phase states,
amplitudes and phases are determined to form a beam, i.e.,
In carrying out the above objects and other objects, a
an array antenna signal in a specified direction. Typically,
method of calibrating an array antenna element having a
the relative amplitudes of each element signal are fixed by
signal with a phase and an amplitude is provided, The
attenuators set at appropriate levels to shape the beam, while
method includes sequentially switching the phase of the
phase connected to the are adjusted for 25 antenna element signal through four orthogonal phase states.
changing the phases of the signals to steer the beam.
At each of the four orthogonal phase states, the power of the
To Precisely control the beam, the actual Phase response array antenna signal is measured. A phase error for the
Of each Phase shifter must be known. However, Phase antenna element signal is determined as a function of the
response of a phase shifter is subject to unavoidable errors 3o power of the array antenna signal at each of the four
and variations due to manufacturing discrepancies and to orthogonal phase states. The phase of the antenna element
various changes occurring as a function of time and tem- signal is then adjusted by the phase error,
perature. Thus, calibration is required to provide phase
Further, in carrying out the above objects and other
correction for each phase shifter. The phase calibration data objects, a method for calibrating an array antenna provided
can be stored and used during steering operations to correct 35 with a plurality of antenna elements each having a signal
phase response errors.
with a phase and an amplitude forming an array antenna
The Of the fed to the are signal is provided. The method includes sequentially switch-
adjusted with attenuators connected to the elements. The ing the phase of each antenna element signal one at a time
attenuators are also subject to errors and variations. Thus, through four orthogonal phase states, At each orthogonal
calibration is required to provide attenuator calibration data 4o phase state the power of the array antenna signal is mea-
for each attenuator. The attenuator calibration data can be sured, ~~h~~~ for each of the antenna element signals
stored and used during steering operations to correct attenu- is then determined, The phase for an antenna element
ator response errors. signal is a function of the power of the array antenna signal
Previous methods of phased array calibration have relied at each of the four orthogonal phase states. The phase of
on scanning each element of the array through all of its 45 each of the antenna element signals is then adjusted by the
phase values relative to the other elements and measuring corresponding phase error.
the Power of the array antenna signal at each Phase value. Still further, in carrying out the above objects and other
The measured Phase value corresponding to maximum objects, the present invention provides an array antenna
Power is compared to the ideal Phase value. The ideal Phase system. The array antenna system includes an array antenna
value is the Phase value CorresPonding to maximum Power 50 provided with a plurality of antenna elements each having a
when there are no phase errors or variations. Thus, the signal with a phase and an amplitude forminga n array
difference between the measured Phase value CorresPonding antenna signal. A calibration processor is operable with the
to maximum Power and the ideal Phase value is the Phase array antenna to sequentially switch the phase of each
error, or phase offset, for that element. antenna element signal one at a time through four orthogonal
This procedure is repeated at least once for each element 5s phase states and measure at each orthogonal phase state the
of the array. After the phase offsets for each element have power of the array antenna signal. The calibration processor
been determined, the phases of the element signals are is further operable to determine a phase error for each of the
changed by their respective phase offsets to effect the antenna element signals. The phase error for an antenna
calibration. Consequently, the errors are, at least currently, element signal is a function of the power of the array antenna
taken into account. 60 signal at each of the four orthogonal phase states. The
A problem with scanning each element through all of its calibration processor is further operable to adjust the phase
phase values is that this requires a large number of mea- of each of the antenna element signals by the corresponding
surements. For instance, phase values fall within the range phase error.
of 0” to 360”. Thus, if the phase settings for each element The provided methods and system of the present inven-
were quantized in increments of lo, then three hundred and 65 tion further determine an amplitude error for an antenna
sixty phase values must be scanned. If the array has a large element signal as a function of the power of the array
number of elements, for example, one hundred, then at least antenna signal at each of the four orthogonal phase states.
5,861,843
3 4
The amplitude of the antenna element signal can then be In the receive mode, antenna elements 12 provide signals
adjusted by the amplitude error. received from an external source through respective phase
The advantages accruing to the present invention are shifters 14 and attenuators 16 to power load 22. Power load
numerous. The present invention circumvents the need for 22 may include a combiner (not specifically shown) for
scanning each element through all phase states in search of s combining the received signals into a single signal. Con-
extrema. The use of four phase settings as opposed to troller 20 is operable with phase shifters 14 and attenuators
scanning all possible phase states reduces the time required 16 to change the phase and the amplitude of the signals
for calibration and, hence, the potential impact on an array received by antenna elements 12. Controller 20 sets the
antenna system. The measurement of power at four orthogo- phases and the amplitudes to form a reception pattern in a
nal phase states provides adequate information for a maxi- io specified direction. Controller 20 then changes the phases
mum likelihood estimate of errors. Such an estimate is and the amplitudes to steer the reception pattern, form a
optimal in an adverse environment. different reception pattern, or the like. Typically, each of
attenuators 16 are set approximately at a common level such
These and other features, aspects, and embodiments of the
that each of antenna elements 12 feed power load 22 equally.
present invention will become better understood with regard
is However, these levels may also be varied for beam shaping.
to the following description, appended claims, and accom-
panying drawings. Referring now to FIG. 2, an illustrative phased array
antenna 30 is shown. Phased array antenna 30 has a plurality
BRIEF DESCRIPTION OF THE DRAWINGS of antenna elements 32 arranged in a MxN array. Each
antenna element 32 is coupled to a plurality of phase shifters
FIG. 1 is a schematic block diagram of an array antenna
for use with the present invention; 20 34 and a plurality of attenuators 36. Each phase shifter 34 is
arranged in series with a respective attenuator 36. Each
FIG. 2 is a diagram of a multiple beam array antenna for
serially arranged phase shifter 34 and attenuator 36 pair is
use with the present invention;
arranged in parallel with two other serially arranged phase
FIG. 3 is a flowchart representing operation of an array shifters and attenuators. All of the pairs of phase shifters 34
antenna calibration method according to the present inven- 2s and attenuators 36 are connected at one end 38 to a respec-
tion; tive antenna element 32.
FIG. 4 is a block diagram of an array antenna signal power Antenna elements 32 are fed with or receive one or more
measurement system for use with the calibration method of signals whose phases and amplitudes are determined to form
the present invention; a beam in a specific direction. In FIG. 2, as an example, three
FIG. 5 is a graph of the standard deviation of phase 30 signals are fed to or received from each antenna element 32.
correction; The signal fed to each antenna element 32 is the sum of three
FIGS. 6(a4 illustrate the convergence of an estimation signals with phase shifting and attenuation dictated by the
process of the calibration method of the present invention; desired direction of the beam for each of the radiated signals.
FIG. 7 is a block diagram illustrating array antenna Thus, phased array antenna 30 may have three different
system connections for transmit calibration with a satellite 3s beams. The signal received by each antenna element 32 is
based array; and divided into three signals with each signal phase shifted and
attenuated as desired.
FIG. 8 is a block diagram illustrating array antenna
system connections for receive calibration with a satellite Because accurate pointing of a beam of a phased array
based array. 4o antenna demands precise control of phase and amplitude,
exact knowledge of the phase and gain response of the phase
BEST MODES FOR CARRYING OUT THE shifting and attenuator electronics is essential. However, as
INVENTION stated in the Background Art, the parameters of the phase
shifting and attenuator electronics vary with temperature and
Referring now to FIG. 1, an illustrative phased array
drift with time. Thus, periodic calibration of the phased array
antenna 10 is shown. Phased array antenna 10 includes a 4s
antenna is necessary to ascertain phase and amplitude cor-
plurality of antenna elements 12. Each antenna element 12
rections for each antenna element.
is coupled to a corresponding phase shifter 14 and a corre-
sponding attenuator 16. Each antenna element 12 may Referring now to FIG. 3, a flowchart 40 illustrates the
transmit and receive electromagnetic signals such as radio procedure of the present invention for calibrating a phased
frequency (RF) signals. so array antenna such as array antenna 10 having a plurality of
antenna elements. Each of the antenna elements have a
In the transmit mode, a power source 18 feeds signals
through respective attenuators 16 and phase shifters 14 to signal with a phase and an amplitude. The antenna element
signals form an array antenna signal. Flowchart 40 begins
each antenna element 12 for transmission of an array
antenna signal. Power source 18 may include a splitter (not with block 42 setting the phase and amplitude of each
ss antenna element signal to form a test beam. The phase values
specifically shown) for splitting a single signal into the
signals fed to antenna elements 12. A controller 20 is of the antenna element signals are typically different.
operable with each of phase shifters 14 and attenuators 16 to However, regardless of the actual phase value, the phase
values of each of the antenna element signals for the test
change the phases and the amplitudes of the signals fed to
antenna elements 12. Controller 20 sets the phases and the beam position are regarded as the 0" phase state. In the test
amplitudes of the signals to form a transmission beam 6o beam position, the 0" phase state is the reference or nominal
having a given radiation pattern in a specified direction. phase state.
Controller 20 then changes the phases and the amplitudes to The amplitudes of the antenna element signals are typi-
steer the beam, form a different beam, or the like. Typically, cally the same. Thus, the attenuators connected to the
each of attenuators 16 are set approximately at a common antenna elements are set approximately at a common level.
level such that each of antenna elements 12 are driven by 65 Subsequently, block 44 sequences the phase of one
power source 18 equally. However, these levels may be antenna element signal through four orthogonal phase states.
varied for beam shaping. The four orthogonal phase states consist of the reference
5,861,843
5 6
phase state (0') and the phase states corresponding to BO",
90°, and 270" relative to the reference phase state. The M
phases and amplitudes of all the other antenna element r(t) = C u,cos(wt + 6,) + n(t)
m-l
signals remain constant while the phase of the one antenna
where,
element signal is being sequenced.
o is the transmitted frequency,
At each of the four orthogonal phase states (OO, 90°, BO", 6, is the phase offset of the mth element relative to its
and 270") block 46 measures the power of the array antenna nominal value,
signal. The power measurements Po, P,,,, P,,, and P,,, a, is the RF voltage from the mth element, and
correspond to phase states $ ,, ,$,,, $,,, and .,$,, Block 48 n(t) is narrowband thermal noise which is uncorrelated
then determines a phase error for the antenna element signal between samples.
based on the power measurements made by block 46. Block The narrowband noise is:
50 then determines an amplitude error for the antenna
element signal based on the power measurements made by
block 46. Blocks 44 and 46 can be repeated as indicated by n(t)=n,(t)cos wt-n,(t)sin wt
1s
the dotted line to integrate multiple measurements of
where n,(t) and n,(t) are the inphase and quadrature
received power and improve the signal-to-noise ratio of the
components, respectively. These components are indepen-
measurement.
dent and identically distributed Gaussian processes having
Decision block 52 then determines whether each of the zero mean and variance o2=N,B with N,/2 the noise power
antenna elements have had their phases sequenced through 20 density and 2B the bandwidth of the filter.
four orthogonal phase states. If not, then the process repeats Introducing a phase of 0 on the kth element yields:
with block 44 sequencing the phase of a different antenna
edlieffmereenntt saignnteanl nsao ethleamt tehnet pshigasnea la ncadn a mbep lditeutderem eirnroedrs. for the r(t) = MC u,cos(wt + 6,) + akcos(wt + 8 + 6k) + n(t)= (2)
m-l
After the phase and amplitude errors for all of the antenna 2s m#k
element signals have been determined, block 54 adjusts the 1
M
phase of each of the antenna element signals by the corre- C a,cos6, + akcos(8 + 6k) + q(t) coswt -
sponding phase error. Block 56 then adjusts the amplitude of m-l
each of the antenna element signals by the corresponding m#k
30
amplitude error. The above procedure may be repeated until
the phase and amplitude calibration errors converge within sinwt
an acceptable level.
Referring now to FIG. 4, a measurement system 60 for
measuring power of a calibration signal 62 received by a
3s at the input to power detector 68. The output from power
receiving antenna terminal 64 is shown. Array antenna 10,
detector 68 is the square of the envelope of its input:
which is on a satellite in the example shown, transmits
calibration signal 62 to terminal 64 for calibration. Note that
pointing a beam at a fixed station (terminal 64) assumes that
~=(A,+v,+n,)Z+(A,+v,+n,)Z (3)
dependence of calibration on direction is negligible. If 40
where,
parameters are sensitive to pointing direction, then an alter-
native such as multiple receiving stations must be imple-
M M
mented.
A,= C a,cos6,,A,= C a,sin6,,
m-l m-l
As described with reference to FIG. 3, calibration signal
4s m#k m#k
62 includes a sequence of phase transitions $ ,, ,$,,, $,,, and
,,$, with array antenna signal power measurements Po, P,,,, v, = a,cos(8 + 6d, and v, = a,sin(8 + i&)
P,,, and P,,,, performed in each state. Measurement system
The output of power detector 68 is sampled at a time
60 consists of terminal 64, and a narrowband filter 66
interval T,>>l/B so that the samples are uncorrelated. The
followed by a power detector 68. Power detector 68 is
50 sampled output of power detector 68 is:
preferably a quadratic detector. The input to power detector
68 is an RF signal having an RF power. The output from
power detector 68 is a voltage proportional to the RF power.
4*=(A,+v,+n,3Z+(A,+v,+n,3Z (4)
An analog-to-digital (AD) converter 70 follows power where,
detector 68. A/D converter 70 converts the output analog 5s nct and nSta re Gaussian variables as described previously.
voltage from power detector 68 into a digital signal for For each antenna element, the statistic qt is a non-central
receipt by a calibration processor 72. Calibration processor chi-squared random variable with two degrees of freedom
72 processes the digital signal to determine the phase and and density:
amplitude error and correction.
(7)
Calibration processor 72 determines the correction data 60 (5)
according to the following derivations. It is assumed that all p(g)= (2d)-'exp[- (41 + h)/2oZYo -
of the antenna elements of array antenna 10 are driven
approximately equally. I,(.) in Equation (5) denotes the modified Bessel function of
the first kind of zero order. The non-central parameter (h)i s:
The received voltage at the input to power detector 68 65
when all of antenna elements 12 of array antenna 10 have
been set to their reference phase values is:
5,861,843
7 8
The mean @) and variance (a,') of the statistic qt are: and
-
,&E{ q,}=h+Zd (7) qo-ql,,-4a,P,cosSk (18)
and Hence, the estimates of the phase 6, and amplitude 2,
variations become:
oqZ=Var{q z}=4oZh+4d [ ( - -
(8) 4270 - 490 ] )
ik= tan-'
Assume that Lsamples of the output of the power detector 40 - 4180
are averaged to form the statistic:
and
- 1 L (9) (20)
q=- C 41
L I-1 1s a~ k = (4270 -790)'4 .4+, (40 -7180)2
w- ith the samples qt of q being independent. The statistic
q is a non-central chi-squared random variable having 2L The deviations of these estimates are readily derived from
degrees of freedom with non-central parameter: first order differentials:
-h = -1 CL [( Ac + v ~+ ()A3~ + v ,)z] = h,
L z=1
a density:
Since the elements are driven approximately equally,
am=ak for all m and A,==(M-l)a,. Using approximation
j&{q}=/L=E{q}=h+2d, (12)
As=O gives the errors:
and a variance:
3s
(23)
(13)
and
The statistic 4 is an unbiased estimate of since
and it is asymptotically efficient. Since the chi-squared where,
distribution is approximately Gaussian about the mean for 4s
large degrees of freedom, the intuitive tendency is to chose
maximum likelihood estimates for the phase variation 6, and P,=akz/2 denotes the power of the kzhe lement.
the amplitude variation ak. One may solve the gradient of the The deviation of the phase estimate from (23) is
likelihood function (11) for maxima. However, these esti- plotted in FIG, and indicates that an accuracy of
mates evolve naturally from consideration of the differences so requires approximately twelve iterations at a signal-to-nois20e
q270-q90 and qo-qls0 which are unbiased estimates:
power ratio of approximately 13 dB per element.
Because the residual phases of all elements other than the
kth element were disregarded in (17) and (18) and the
E{qz7,-q,,}=4ak (A, sins,-A, coss,) (ls)
subsequent analysis, the estimates of 6, and a, are relative to
and ss the aggregate of the other elements. Note that this reference
varies depending on which element is being tested. Hence,
caution must be exercised to update the element corrections
E{q,-ql,,}=4ak (Ac cosS,+A, sins,). (I6) only after calibration of the entire array.
The derivation of the phase and amplitude estimators in
Note that the element index k is understood for the
q, perfect amp1itude and phase
statistics and the array antenna signal power is measured 6o (19)a nd (20)
for each phase setting of each element, Since only the phase Of the The inphase and quadrature
of the kth element is varying, the sum of the other element Of this were denoted by vc(e> and v~(e>
ing (3).Actua1 phase shifters are to give exact phase
voltages forms the reference, i,e,, ASJO (assuming 6, is
settings of O", 90°, BO", and 270°, and real attenuators may
small so that Ac>>As), which gives:
65 not permit exact control of the amplitude ak. However, errors
in the settings are deterministic and may be measured.
-
qZ7,-q,,-4a,PcsinS, (17) Denote the phase settings of the kth element by 8,=mx/2,
5,861,843
9 10
m=0,1,2,3 with corresponding signal components vc=a,,
cos(8,+E,,,+6,) and vs=a,, sin(O,+E,,,+6,) having ampli-
tudes a,, and phase offsets E,,, which contain imperfections
and amplitude errors. Following the same rationale which
led to (17) and (18) gives: S
where the amplitudes a, and phase offsets E,, are from
measurements. Solution of the linear equations following
10 (27) for the amplitude estimates gives:
6, = (29)
[~22(Acco~S1+8 A0 ,sinSls0) + C21(A,sinSl,o - A,cosSl,o)l/
1s
2[(A,cosSo + A,sinSo) (A,sinSls0 - A,cosSls0) -
(A,sinSo - A,cosSo) (A,cosS180 + A,sinSls0)1,
'180 =
M
A, = C al,osin(&o + SJ. 20 -[C,,(A,COS~~+ A,sinSo) + C2,(A,sinS0 - A,cosSo)Y
1=1
kk 2[(A,cosSo + A,sinSo) (A,sinSls0 - A,cosSls0) -
(A,sinSo - A,cosSo) (A,cosS180 + A,sinSls0)1,
Evaluation of equation (24) at 8,=270" and 8,=90" yields:
- - 2s a,, =
4270 - 490 = - + (25)
[c11(Acc0s~270 + AxsinS270) + C12(AcsinS270 - Axc0s~270)Y
[%(a270si~S270 + awsinS90) -
2[(A,~inS-~ ~ (A,co~S270+ A,sinS270) -
2Ax(a270cos~27+0 a90cos~90)1cossk+
(A,CO~+S A~,~si nS,d (A,~i~S27-0 A ,cosS,dI,
30
[2Ac(a270c0s~270+ a90c0s~90)+
and
%(a270sinS270 + awsinS90)lsinsk
'270 =
and similarly for 8,=0" and 8,=180"
-[C11(A,cosS90 + A,sinS90) - C12(A,sinS90 - A,CO~S~~)Y
3s
~~
40 - q150 = a"2 - 48" + (26) 2[(A,~inS,~- A,cosS90) (A,co~S270+ A,sinS270) -
[2Asa"cosSo + ~18"COS~180+) (A,CO~+S A~,~si nS90) (A,sinS270 -
ZA,(a,sin~, + a180sin~l,o)]cos~k-
It must be emphasized that the estimators (28) and (29) for
[~,(aosinSo+ a180sinSl,o) - 40 the phase and amplitude variations are not closed form
expressions because the coefficients C,,, C,,, C,,, C,,, A,,
%(aocosSo + ~180co~S180)1~i~sl;.a nd As depend on these variations. Hence, the estimates
The subscript k indicating the element has been omitted must be solved by an iterative procedure which is described
on the amplitude and phase variations and on the power below. Further, observe that because there are array antenna
4 4s 4
measurements for simplicity in (25) and (26) because this signal power measurements at four phase settings for each
dependence is understood. These expressions may be writ- element, there are 4M data measurements. Because the
ten: estimators 6, and ,,2 constitute a set of 5M variables, the
estimator equations given by (28) and (29) are dependent.
- so This problem is circumvented by use of equations (20) for
~~270~a27~~~~~9c00s~sk+ac192 02~~cll initial amplitude estimates. Equation (19) can be used for
initial phase error estimates with equations (27) and (28)
(~o-ao2)-(q180-a1802)=~2C1O S~,+C,, sins, (27)
used for iteration of the phase error.
with To corroborate the results in (27) through (29), these
5s generalizations should reduce to the previous expressions
(19) and (20) under assumptions of small or negligible
errors. Simplification of the expression in (24) as in the
previous section obtains:
60
~q ,-q,-a,,2-a,2+~, [a,, cos(e,+~,)cos8,-a,, sin(e,+
S,)sinS,-a, cos(e,+~,)cosi&+a,, sin(e,+~,)sin8,] (30)
with the assumption that 4-0. Writing the amplitude varia-
Cz2=-~,(a0S inSo+als0 SinSl,o)+%(ao cosSo+a180c osSls0).
65 tions with phase as a,,-akn=cmn, noting 8,=8,+x, and
The equations in (27) are easily solved for 6, to obtain the ignoring terms higher than first order, i.e., E', ccosE,, csinE,,
estimate: etc., obtains: