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Null constrained adaptive beamformer—Discrete white noise source analysis
Computers Elect. EngngVol. 18, No. 3/4, pp. 331-340, 1992 Printed in Great Britain. All rights reserved
NULL CONSTRAINED DISCRETE WHITE
0045-7906/92 $5.00+ 0.00 Copyright © 1992 Pergamon Press Ltd
ADAPTIVE BEAMFORMER-NOISE SOURCE ANALYSIS
CHAIM SHENHAV a n d SALLY L. WOOD Department of Electrical Engineering and Computer Science, Santa Clara University, Santa Clara, CA 95053, U.S.A.
(Received 15 December 1990; accepted in final revisedform 15 August 1991) Abstract--A null constrained beamformer is a minimum variance beamformer in which each weight is either forced to remain zero or allowed to participate in the adaptive interference nulling process. Based on source delay modeling, a null constraint set is determined by sensor position tolerance, look-direction tolerance and signal correlation characteristics. These constraints make the desired primary input signal statistically orthogonal to the reference input. This beamformer addresses the problem of reducing broadband and narrowband sidelobe interference without rejecting the mainlobe broadband signal. Wiener solution analysis guarantees superior output signal to interference plus noise ratio as compared to a conventional beamformer. Preservation of a broadband signal is proven to be insensitive to sensor-element gain. A spatial separation of discrete white noise sources is analyzed, using the inversion lemma. This easily implemented processor is well suited to wideband low power signals.
1. I N T R O D U C T I O N In a linearly constrained minimum power linear array, the look direction constraint [1] is only effective at a single point in the directional pattern for the perfect array signal case. If, due to array signal imperfections, the desired signal is not exactly in the constrained look-direction, it will be treated as if it were an interference and partial or complete cancellation will result. The hypersensitivity of the adaptive beamformer (ABF) to imperfections [2] is reduced by additional mainlobe constraints at the expense of reduced ability to cancel mainlobe interference. Since Frost's pioneering work [1] there have been many attempts to gain better control over the mainbeam response. The most popular class of linear constraints reported in the literature was the spatial derivative constraints. These constraints were proposed by Jablon [2] for the narrow-band case, by Er and Cantoni [3] in the direct form implementation [1] and by Griffiths and Jim [4] in the generalized sidelobe canceller (GSC) form [4,5] for the broad-band case. The zero derivative constraint approach is based on the minimum variance distortionless response (MVDR) principle [6], which prohibits signal presence in the minimization process. This paper presents a new set of linear constraints [7,8], forming the null constrained-adaptive beamformer (NC-ABF), named by [2] and introduced by Widrow and McCool [9]. This robust solution to imperfection sensitivity is achieved through the use of adaptive noise cancelling techniques [10,11], resulting in a system output which is the best least mean square estimate of the signal. The adaptive noise canceller is based on a new generalized MVDR concept. The desired primary input signal is made statistically orthogonal (uncorrelated) to the reference signal which is preprocessed or linearly transformed by the blocking matrix to prevent participation of the desired signal in the adaptive nulling process. The results presented here are derived under the following assumptions prevalent in the literature. The desired signal and the beamformer are broad-band. The signal sources are in the far field having planar wavefronts. All directional signal sources and nondirectional receiver noise are zero mean, wide-sense stationary (WSS) and statistically independent of one another. The structure of the minimum constraint set systematically developed here, which to the best of the authors' knowledge has not been approached in the literature, is based on source delay modeling, and determined by the expected deviation range of sensor position, look-direction tolerance and signal correlation time. The NC-GSC is a flexible, simple and easy to implement processor. The rows of the blocking matrix are an orthogonal subset of identity matrix, and the constraint matrix rows are the complementary rows. In contrast to mainbeam zero derivative constraint methods, the null constraint set is insensitive to sensor element amplitude (gain). 331
332
CHAIM Sn~NBAVand SALLY L. WOOD
Preservation of a signal of known correlation time can be insured via the appropriate nonsensitive blocking matrix operation ( NC set), provided that array signal imperfections are of known nature and extent. Wiener output SINR superiority of this method over the conventional-BF is guaranteed under the prevailing statistical orthogonality. Using a three tap filter, an exact scalar expression for the performance improvement due to adaptation (PIA) for white noise sources is derived. This very important performance index, PIA, indicates improvement in output SINR as compared to the conventional-BF.
2. N U L L C O N S T R A I N E D
ADAPTIVE
BEAMFORMER
STRUCTURE
2.1. Direct form implementation The block diagram of the direct form implementation of the null constrained-adaptive beamformer (NC-ABF), as shown in Fig. l, is essentially a broad-band adaptive noise canceller with time-space sampling. The signal sources are sampled in space by the K-element array and steered to the desired direction. Each sensor element is connected to a tapped delay line (TDL). The central tap snapshot, is routed to the conventional sum beamformer to create the primary input to the adaptive noise canceller (ANC). All samples or stacked snapshots, are filtered by the constrained adaptive transversal filter (ATF) to create the reference input. The input to the K-element array is composed of the directional sources: the desired signal, s0(t, 00), coming from the 00 direction and the interference signal, st(t, 0~), coming from the 0t direction. Since the null constraint approach requires estimates of the uncertainties in the look direction, g0, and in the sensor position, an explicit expression for the steering delay errors will be developed. The actual propagation delay, for the qth signal source sq(t, Oq) to the ith sensor element, placed at (Yi, zi) relative to the origin, (Y0, z0) is:
Zq, = (zr/v)sin(Oq) - (yr/v)cos(Oq),
q = 0, 1, i = 0, 1. . . . . K - 1
(1)
where v is the propagation velocity. The array imperfections, represented by local element misplacement from the estimated ith element position of a uniformly spaced linear array, (f~,,~i)=(O, iD), can be expressed as: A z ~ = z ~ - ~ = 6 z , D, and Ayi=y~-fii=6yiD. The corresponding ith steering error delay is given by:
AZq, = Zq, - ~o,
(2)
where z0, = (iD/v)sin(Oo) is the ith estimated steering delay. The sources sampled in space as well as in time generate the ANC inputs. Following [12] the K-dimensional steered source vector at the first tap of the filter memory is given by: Sqo(t ) =
no
o0I
I
I I I I
I I I
[Sq(t "+"A'r:qo), Sq(t
-'1- m~'ql ) . . . . .
_•
Sq(t + A'rqx _)]r.
(3)
d +
+ r
÷
)
k- 10-
Steering delays
Tapped delay
Adaptive noise
line
canceller
Fig. 1. Block diagram of the direct form implementation of the null constrained adaptive beamformer (NC-ABF).
O
333
Null constrained adaptive beamformer
The TDL [13] which is composed of 2L + 1 taps, sampled at the time interval T, holds a total of N = K ( 2 L + 1) samples. Thus the space-time sampled reference input to the ANC is an N-dimensional stacked snapshot vector, containing 2L + 1 delayed snapshot vectors: I
xr(t)___ ~ Srq(t) + nr(t) = [x0r(t) " xr(t) " . . . " XrL(t)l
(4)
q=0
where: xs(t) = Xo(t - JT), for J = 0, 1. . . . . 2L. The central tap snapshot, xL(t) is the input to a sum conventional beamformer forming the primary input, d(t), to the ANC: d(t) = UrxL(t) = UrXo(t -- L T )
(5)
where the summation vector u = 1r consists of K l's. The reference input, x(t), is filtered via the constraint-ATF w~o, to create y(t). The imposed constraint set is represented by a n / - r a n k null-constraint matrix, C r, which is a n / - r o w subset of the identity matrix, I~. The selected orthonormal rows for I constraints are defined by the set of l indices: V = {ni}~= i 2 ~, such that the constrained weight subset is wcZ~[w,~, w~2. . . . . w,t] r = C rw = 0t. The optimization process requires the constrained minimization of H ( w ) - E{e2(t)} -- P0, hence the optimal weight vector Wconis defined as: min H(w)
subject to: Crw = f = 01.
(6)
w
2.2. Null constraint-generalized sidelobe canceller Since the optimal weights wi, i e V are zero, the constrained-ATF can be replaced with an equivalent unconstrained-ATF, the null constraint-generalized sidelobe canceller (NC-GSC). This is done by concurrently applying a linear transformation, represented by the blocking matrix, B r to both, the reference input vector, x(t) and the weight vector, w. This N - 1 rank matrix is composed of the N - l rows of IN complementary to the constraint matrix, defined by the following set of indices: U = [/--
{mj}j=l,
(7)
2. . . . . N - I .
The blocking matrix, B r, of dimension/" x N, w h e r e / = N - 1, can be written as:
Br=
(8)
L e ,r J where, emj = [0" • • 010" • • 0] r is a N-rank unit vector having 1 in the mjth plane and zero elsewhere• The preprocessed reference input, x'(t), and the resulting unconstrained-ATF can be described as follows: Wu~--[Wm,, Win:. . . . . Wm~_,IT= Brw (9) x'(t) = Brx(t)
(10)
y ( t ) = wr.x(t) = WurX'(t) = wrABx(t)
(11)
where: AB = BB r is an N x N binary diagonal matrix and Woo nr = wrAB. The optimization process simply requires a minimization over the unconstrained weight vector wu.
3. S T A T I S T I C A L
ORTHOGONALITY
CONCEPT
The objective in setting constraints is to prevent any rejection of the desired signal in the adaptive filtering process. Signal rejection will not occur if the signal component of d(t), denoted by d0(t), is statistically orthogonal (zero-mean, uncorrelated) to the signal component of y(t), denoted by yo(t). Hence, constraints will be structured to guarantee Rayo~-E{do(t)yo(t)}
O.
(12)
334
CHAIM SHENHAV and
SALLY L. WOOD
3.1. Adaptive noise cancelling approach The primary input of the adaptive noise canceller (see Fig. 1) is composed of three components: the signal, do(t), the interference, d~ (t), and the receiver noise, d~(t). The ATF output, y(t) can be treated likewise and the output power components can be written as:
Poo = Pco + E[y~(t)] - 2Rdyo
(13a)
Po, = Pc. + E[y](t)] - 2Rdy"
(13b)
Po~ = Pc~ + E[y~(t)] - 2Ray ~
(13c)
where Pco ~- E[ d2 (t)], Pc. ~ E[d2(t)], and Pc, ~- E[d~ (t)] are the conventional-BF power of the signal, the noise and, the interference components respectively. Thus, from equation (13a) if Rdyo = O, signal cancellation would not take place. Note that the more restrictive MVDR demand that yo(t) = O, also satisfies this more general statistical orthogonality condition. Since E{d,(t)yn(t)} = 0, the following equivalence relationship can be established: min E[e2(t)] ~- MMSE ~ min E{[d, (t) - y(t)] 2} ¢> min E{[e(t) - [d0(t) + d, (t)]2}.
(14)
Therefore [10,11], the outcome of the minimization process is a filter output y(t), the best least mean square estimate of the interference, and an error e(t), the best least mean square estimate of the signal (plus nondirectional noise). Using the explicit expressions for y(t) and d(t), the sufficient orthogonality condition can be written as: Rdyo = w.E{soSoL r , r u} = 0 => R~ 0 = 0N-t×l and R~s0, = 0N-I× K (15) ,
~.
t
T
•
•
•
where"• Rs,0L - E { s o S o L }. This shows that the signal component of d(t) is statistically orthogonal to the signal component of y(t), when the reference signal preprocessed by the blocking matrix is made statistically orthogonal to the desired primary input signal. The orthogonality condition Rdyo = 0 can also be fulfilled by applying the constraint matrix, C T to the weighting vector: L
W,ro.Rss0L
~ M=-L
W conM T +L RsoM = 0K r
(16)
where the terms of the summation are the 2L + 1 partitioned snapshots. This implies that a nonzero input power signal with for M = O, Rso,~ ~ OK×K must have W¢o.~= Ok. This can be indicated by defining the set of indices corresponding to the Ith delayed snapshot, as ~¢~- {IK - IK + (K - 1)}, and requiring that d L e V. Hence, ~¢L is the minimal constraint set for a perfect array when the signal is uncorrelated from tap to tap. This set will be augmented to accommodate imperfections.
3.2. Array signal imperfection null constraint set The constraint set is determined by the array imperfections, and the correlation time of the desired signal, To, such that E{so(t)So(t + At)} = 0 for IAtl > To. Using equation (2) the qth signal steering deviation difference for sensors i,j is defined a s The bounds of the normalized steering error are defined by a global maximum sensor misplacement error, y and the misdirection A sin(0q) = sin(0q) - sin(g0), where m ~ i - j , and A'rq ~-max~j 6Zqm. The normalized steering error can be explicitly expressed from equations (1) and (2):
m'~qijAArqi--A'~qj.
tSzqm~-
'
<~[m[lAsin(Oq)lm,x+y<~(g-1)lAsin(Oq)lmax+7~-Azq.
(17)
Since so(t) has a finite correlation time, To, the domain of the nonzero autocorrelation can be found: IMI ~
Null constrained adaptive beamformer
335
block-correlation time, Mq, is the largest integer satisfying Mq <~Gq. Similarly, m0, for the desired signal is defined as the smallest integer, Im L, satisfying: M0 ~< [~t I t a l i a sin(00) Im~x]+ [0ty + fl0].
(19)
If the constraint set contains a sub-block, it will be defined as
~t~AT°~-(IK+i},
ieimo
where: i~o = {0, 1. . . . . K - 1 - m0, m0, m0 + 1. . . . . K - 1}. Finally, the minimum null constraint set can be defined as:
"S/L,
M0 = 0
M0 - 1
~-¢4L +M,
Mo ¢ O
A
M= - M o + I
V :
U ~L+Mo mO U ~m°L- M0 I A s i n ( 0 o ) [ max ~ 0
A
(20)
mo > [_K/2_J
Mo ~/L + M,
otherwise.
M= -M 0
Thus increased signal correlation time increases the number of snapshots which can not participate in the minimization process. Random sensor position errors can appear similar in effect to increased signal correlation. This constitutes a minimum null constraint set, needed to satisfy the orthogonality condition of equation (12) to prevent signal cancellation, based on the imperfection bounds. 4. D I S C R E T E
WHITE
NOISE
SOURCE
PERFORMANCE
ANALYSIS
Let SNRi=a0/a,,22 SIRi=ao/a12 2 be the input signal to noise ratio and the input signal to interference ratio, where a 2, a,2 and a~ are the input powers of the signal, noise and interference respectively. For the conventional-BF output d(t) signal to interference plus noise ratio is SINRc = Pco/(ec, + Pc,). The performance quality is evaluated by three different measures: SINR, PIA and SRPq. The SINR is the resultant observed signal to interference plus noise ratio: SINRo = Poo/(Po, + Po~). The performance improvement due to adaptation, PIA, measures the SINR improvement due to the adaptive noise canceller scheme: PIA ~-SINRo/SINRc. The q th signal alone rejected power part, SRPq, measures the sidelobe interference/mainlobe signal power cancellation capability: SRPq~-Rdyq/Pcq 1~2=,~=0, where O # q. In order to evaluate the optimal performance, the convergent Wiener solution is examined: T , -- w * T ~ W/ T , tT -Wuop, RxxWuop, - -opt-xx-opt = E[y2opt(t)] = Wuop, Rdx = wovtRa, - E[yopt(t)d(t)].
(21)
Here R'~x~-Rx,~, = BrRx~B is the transformed autocorrelation matrix and R~---Ra~, = BrRd~ is the transformed crosscorrelation vector. Given w .... , y(t) can be computed along with the performance measures above. Applying the statistical orth~ogonality concept and using optimum values, the optimum performance is given by:
Pco + E[y2o(t)] SINR0 = p,. + p,~ _ {E[y2(t) ] + E[y~(t)]} PIA =
1 + E[y~(t)]/Pco i> 1 1 - {E[y2(t)] + E[yg(t)]}/[Pc. + Pc~]
0 ~< SRPq = E[y~(t)]/Pc~ 1,2=,~=0 ~< 1.
(22)
(23) (24)
It clearly follows from the orthogonality condition that SRP0 = 0 and output SINR is superior to conventional-BF SINR.
336
CHAXMSHENHAVand SALLYL. WOOD
These results will be demonstrated for a special case, where only discrete white noise sources are involved, i.e. Tq = 0, q = 0, 1. For this case the correlation function is nonzero if and only if (iff) Atq = 0 or: M T = Arq,j For simplicity, perfect (uniform) sensor array placement will be assumed or only slight misdirection will be allowed• This leads to the requirement for minimal constraint set, or: M0 = 0 which is equivalent to 0 t ( K - 1)1A sin(00)l < 1, thus defining the mainlobe as the angle domain satisfying:
1/~
IA sin(00)l < K---S]-. On the other hand, interference angles satisfying Atl = 0, for M # 0 are considered• It follows that the equation below constitutes the set of unprotected sidelobe directions: IMI = ot Im[I A sin(01)l
(25)
for 1 ~< Im[~< K - 1. Note that for a fixed [M[ there is a unique [rnl or a unique corresponding IAsin(01)[ satisfying equation (25), and that M = 0 implies m = 0 or perfect array signal A sin(0q) = 0. For full converage of all possible interference angles and for the full range of m, the extreme for which IA sin(01 )1 = 2 will be included. Therefore, the maximum IM [ still satisfying the above equation, defined and chosen as L0 is simply: L0 __42~ (K - 1).
(26)
This is a natural choice because IMI > L0 won't contribute to the set of correlated sidelobe directions• On the other extreme, for M = 1 the above set includes the smallest possible angle:
1/~
K----Z-~_ 1 ~< [A sin(01)l ~<2
(27)
for 1 ~< [M [ ~ K, which narrows the range of the possible interference angles to: 2 4 K----~-i- ~< IA sin(01 )1 ~<~..
(28)
This very limited case is chosen because of the block-diagonal structure of the correlation matrix which allows analytical matrix inversion, using algebraic methods. The lengthy algebraic derivations needed for these rather limited cases are presented in the Appendix. Final results, developed in the Appendix for both the perfect array signal and the misdirection case, will be presented• As explained in the mentioned Appendix full rejection (nulling) capability, SRPI = 1 is achieved for a very specific [ml[= K/2 for an even number of elements. This specific value [ml[ corresponds to a direction which can be interpreted as the direction defining the sidelobe region, the mainlobe width or the resolution: 4 IA sin(01 )l = ~..
(29)
Hence, only this special case will be considered. Note that the resolution under this definition is inversely proportional to the number of sensor elements, K. For the case of the look direction error the PIA is given by: l+g 2 alAu = 1 - g 2 [ 1 + 2a2o/(tr~ + try)] (30) 2 2 where, gN = trl/(a0 + a~2 + a2). Correspondingly, for the perfect array signal:
1 + (0.5~4)/[(K + g,)2cr~] PIA, = 1 - {[(tr4)l(g~a4)][1 -
(0.5K2)I(K + g/)2]}
(31)
Null constrained adaptive beamformer
337
(dB) SNR
J-
,~. sia (aB)
Fig. 2. PIA as a function of the input S N R and the input SIR for a non-perfect array with white noise sources when L = I. S N R is varied from 0 to 30 dB, SIR is varied from - 30 to 0 dB and the PIA ranges from 0.9 to 58.0 dB.
where:g1 = ( a 2 + a ,2) l a o 2 = I/SINR~. These equations indicate that the PIA in the non-perfect array signal is only a function of the relative input power of the various signals while in the perfect array it is also a function of the number of elements, K. It is interesting to find the asymptotic behavior of the PIA for which the nondirectional noise is very small in comparison with the other signals. This asymptotic behavior demonstrates the influence of the SIR,. on the potential degree of rejection of an interference. Under the assumptions that tr0~ >>tr2 and try>> 2 a~, 2 the PIA for the non-perfect array signal approaches: '/O"2~2
_ O'~
(32) Correspondingly, for the perfect array: PIA,
= 2
1 ÷ -2--)
+ O.5
.
(33)
These observations are illustrated in the following 3-D plots. Figure 2 shows the PIA as a function of the input SNR and the input SIR for a non-perfect array. SNR is varied from 0 to 30 dB, SIR is varied from - 3 0 to 0 dB and the PIA ranges from 0.9 to 58.0 dB. Figure 3 demonstrates the PIA behavior for the perfect array with K = 8, when the SNR is varied from 0 to 30 dB, and the SIR is varied from - 30 to 0 dB. For this case, the PIA ranges from 0.5 to 46.0 dB. Figure 4 presents the PIA as a function of the number of elements, K = 2, 4, 6 . . . . . 32, for the perfect array, when the input I/SIR is varied from 0 to 30 dB for SNR = 30 dB, and the PIA ranges from 3.2 to 29 dB. It is shown that the PIA is monotonically increasing with SNR~ or 1/SlR~. Wiener simulations for
P I A (~B)
Fig. 3. PIA as a function o f the input S N R and the input SIR for a perfect array with white noise sources when L = 1. S N R is varied from 0 to 30 dB, SIR is varied from , 30 to 0 dB for K = 8, where the PIA ranges from 0.5 to 46.0 dB.
338
CHAIMSt-mNgAVand SALLYL. WOOD
% Fig. 4. PIA as a function of the number of elements K = 2, 4, 6,..., 32, for a perfect array with white noise sources and with L = 1, when the input 1/SIR is varied from 0 to 30 dB for SNR = 30 dB, where the PIA ranges from 3.2 to 29 dB. L > 1 have shown that for the non-perfect array there is no further improvement with large L. On the other hand, for the perfect array, there is performance benefit in increasing the number of taps in cases where the interference direction is close to the mainbeam for high SNRi and low SIRi. 5. C O N C L U S I O N S The NC-GSC, presented in this paper, offers an easy to implement processor with performance advantages for wideband low power signals in the presence of array imperfections. The constraint set, systematically developed here, and the consequent main results are consistent with earlier investigations, intuitively approached by Widrow and McCool [9]. Preservation of signal of known correlation time is based on a new, true by construction, signal statistical orthogonality concept (GMVDR). Under the prevailing statistical orthogonality, the output S I N R is shown to be superior to that of the conventional-BF. Thorough analysis for discrete white noise sources has been carried out. In the ideal case, full rejection (hulling) capability is achieved for a very specific direction. The performance improvement due to adaptation, PIA, in the non-perfect array signal is only a function of the relative input power of the signals while in the perfect array it is also a function of the number of elements, K. The asymptotic behavior of the PIA demonstrates the influence of the SIRi on the potential degree of rejection of an interference. Performance analysis of the steady-state Wiener solution shows that this scheme is well suited to wideband low power signals, e.g. LPI (low probability of intercept), noise-like, spread spectrum signals. Acknowledgement--This work was supported by NSF grant No. 86-98035 and Sun Microsystems, Mountain View
California. REFERENCES 1. O. L. Frost III, An algorithm for linearly constrained adaptive array processing. Proc. IEEE 60, 926-935 (1972). 2. N. K. Jablon, Adaptive beamforming with imperfect arrays. Ph.D. thesis, Stanford Univ., Calif. (1985). 3. M. H. Er and A. Cantoni, Derivative constraints for broad-band element space antenna array processors. IEEE Trans. Acoust. Speech Signal Process. 31, 1378-1393 (1983).
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4. L. J. Grittiths and C. W. Jim, An alternative approach to linearly constrained adaptive beamformer. 1EEE Trans. Antennas Propagation 30, 27-33 (1982). 5. B. D. Van Veen and K. M. Buckley, Beamforming: a versatile approach to spatial filtering. IEEE A S S P Mag. 5, 4-24 (1988). 6. N. L. Owsley, Sonar array processing. In Array Signal Processing (Edited by S. Haykin). Prentice-Hall, Englewood Cliffs, N.J. (1985). 7. C. Shenhav and S. L. Wood, A null constrained adaptive beamformer in the presence of array imperfections. Proc. 21st Asilomar Conf. Signals Systems and Computers (SSC) (1987). 8. C. Shenhav and S. L. Wood, A null constrained adaptive beamformer for broad-band signal preservation. Proc. 23rd Asilomar Conf. Signals Systems and Computers (SSC) (1989). 9. B. Widrow and J. M. McCool, A comparison of adaptive algorithms based on the method of steepest descent and random search. IEEE Trans. Antennas Propagation 24, 632-634 (1976). 10. B. Widrow and S. D. Stearns, Adaptive Signal Processing. Prentice-Hall, Englewood Cliffs, N.J. (1985) 11. J. W. R. Griffiths, Adaptive array processing, lEE Proc. 130F, 3-10 (1983). 12. D. E. Dudgeon, Fundamentals of digital array processing--a tutorial. Proc. IEEE 65, 899-904 (1977). 13. W. E. Rodgers and R. T. Compton Jr, Adaptive array bandwidth with tapped delay line processing. IEEE AES-15, 21-28 (1979). APPENDIX PIA for Discrete White Noise Sources This Appendix provides proofs to equations (30) and (31). For L = 1, M 0 = 0 and M I = 1, the submatrix R 2 = 0x× x and the Wiener solution or the optimum weight vector is given by: Wuop,= R~2I R}x =
I- :lu ....
(34)
LR~'R,I The ATF output power can be written as: (35)
E[yZop,(t)lwr,R'a.~ = ur[R,R~aRr + R r R ( ' R, lu.
The signal power component of the ATF output can also be expressed in general: E[y2op,(t)]=w Uopl r R 'sso w Uopt _- -
U
rfl;, t ~ X l A~X-0I r~,X s 0 0 *l;,-llar.a_RrlR~lRsooRffiRi]u" l 0 sx I l
(36)
The following relationships simplify computations: F0,,,×~ R , = R,,, =
.
IL Ix-re'
where in general, % and zero elsewhere.
= [0.
•
0.
.,, .
.
.
.
. . . . .
a ~ [ e , , , ! . . . i e x _ 10x,,.,..
]
.
(37)
or'×x-"'
=a][Ox+m'ie°i""
ex-'-m'lr
• 010.. • 0] r is the K-dimensional unit vector having 1 in the m t h place (m = 0, 1. . . . . K - l)
A. I. Signal misdirection case
For this very simple case, where the look-direction tolerance is known, the following applies: R,oo=a~l x, Ro=(ag+~r~+~r])l x, and the symmetric R~-~= (a02+ cry+ cr2)-II x. In this case R~00, R0 and R~~ take the general form of F1 = f l x . Taking into consideration the fact that in equations (35) and (36) u (u r) are operating on a matrix from the right (left) results in: E[y2(t)] = 2a~gN(K - ml )
(38)
Ety2o(t)] = 2a~g2(K - m, )
(39)
1 + SRP,g~v l+g~ I PlAN = 1 - SRP, g:u[I + 2a~/(a~ + a,2)] - 1 - g ~ [ 1 q- 2az/~(a~ + a~)] IsRP,= ~
(40)
where, gN = a ~/(a ~ + a ~ + cr2) and SRP l = 2(1 - m I/K) = 11m, =X/2.X.... for complete rejection of a sidelobe interference coming alone (or having much more power than the signal and the noise) from the 0t direction corresponding to m~. A.2. Perfect array signal case
In this perfect array signal case, the following relationships prevail: Rso 0 = ~ 2 u u T
Ro = a~(gllK + uu r)
where: gl = (a~ + g.~)t~o~ = 1/SINRv Applying the inversion Lemma [A + bbr] I = A i
A -IbbrA -t 1 + brA - ~b
(41)
340
CHAIM SHENHAV and SALLY L. WOOD
to equation (41) results in a symmetric matrix composed of two terms, the identity terms 1r and the summation term uu r as follows: SINR,( 1 r) R°l= a~ It-UU . (42) K +gl It is observed that in this case R~00, R 0 and R o ~ take the general form of: F = f ( a l x + buu r) = F I + Fs .
Substituting Rff j into equation (35), the summation term F s results in:
(urRlu) (urRru)
+
(urR ru) (urRlu) =
2a 4 (K -- m I)2.
Hence, the following result is achieved:
ml)~r4{ _ K - ml~" g~o [ I K + g,J
2(K -
E[y2(t)]
(43)
Now, the expression for E[y2(t)], is similarly derived from equation (36): 2K - m I )2a2 E[y~(t)] = ~ + ~ .
(44)
Combining the expressions for E[y2(t)] and E[y~(t)] results in the following formula for the PIA: PIA t =
1 + (0.5SRP~a4)/[(K + g,)2ag] 1 -- {[(SRP, a~)/(g2ia4)] [1 - ( 0. SSRP, K2)/(K + g,)2]}
1 + (0.5a4)/[K + gl)Za~] SRl', = I -- { [ ( t r ~ : i ~ - . 5 K ~ + g,)Z]} =,
(45)
where, SRPt = 2(I - m t / K ) = 11m, ~ r/2. ~ov.n.
AUTHORS'
BIOGRAPHIES
Chaim Sheahav--Chaim Shenhav was born in Rishon-Le-Zion, Israel, on 3 November 1950. He received the B.S. and M.S. degrees in electrical engineering from the Technion, Israel Institute of Technology, Haifa, Israel, in 1975, and 1982 respectively. From 1977 to 1985 he was a Research Engineer at the Armament Development Authority, Haifa, Israel. Since 1986 he has been pursuing the Ph.D. degree, completed in 1991, in electrical engineering at Santa Clara University. He is currently with Applied Imaging Corp., Santa Clara, California.
Sally Wood---Sally Wood was born in Massachusetts in 1947. She received the B.S. degree in electrical engineering from Columbia University in 1969 and the M.S. and Ph.D. degrees in electrical engineering from Stanford University in 1975 and 1978 respectively. Since 1985 she has been an Associate Professor in the Electrical Engineering Department at Santa Clara University where her research has focussed on multidimensional signal processing. She is a member of Tau Beta Pi and Eta Kappa Nu.
NULL CONSTRAINED DISCRETE WHITE
0045-7906/92 $5.00+ 0.00 Copyright © 1992 Pergamon Press Ltd
ADAPTIVE BEAMFORMER-NOISE SOURCE ANALYSIS
CHAIM SHENHAV a n d SALLY L. WOOD Department of Electrical Engineering and Computer Science, Santa Clara University, Santa Clara, CA 95053, U.S.A.
(Received 15 December 1990; accepted in final revisedform 15 August 1991) Abstract--A null constrained beamformer is a minimum variance beamformer in which each weight is either forced to remain zero or allowed to participate in the adaptive interference nulling process. Based on source delay modeling, a null constraint set is determined by sensor position tolerance, look-direction tolerance and signal correlation characteristics. These constraints make the desired primary input signal statistically orthogonal to the reference input. This beamformer addresses the problem of reducing broadband and narrowband sidelobe interference without rejecting the mainlobe broadband signal. Wiener solution analysis guarantees superior output signal to interference plus noise ratio as compared to a conventional beamformer. Preservation of a broadband signal is proven to be insensitive to sensor-element gain. A spatial separation of discrete white noise sources is analyzed, using the inversion lemma. This easily implemented processor is well suited to wideband low power signals.
1. I N T R O D U C T I O N In a linearly constrained minimum power linear array, the look direction constraint [1] is only effective at a single point in the directional pattern for the perfect array signal case. If, due to array signal imperfections, the desired signal is not exactly in the constrained look-direction, it will be treated as if it were an interference and partial or complete cancellation will result. The hypersensitivity of the adaptive beamformer (ABF) to imperfections [2] is reduced by additional mainlobe constraints at the expense of reduced ability to cancel mainlobe interference. Since Frost's pioneering work [1] there have been many attempts to gain better control over the mainbeam response. The most popular class of linear constraints reported in the literature was the spatial derivative constraints. These constraints were proposed by Jablon [2] for the narrow-band case, by Er and Cantoni [3] in the direct form implementation [1] and by Griffiths and Jim [4] in the generalized sidelobe canceller (GSC) form [4,5] for the broad-band case. The zero derivative constraint approach is based on the minimum variance distortionless response (MVDR) principle [6], which prohibits signal presence in the minimization process. This paper presents a new set of linear constraints [7,8], forming the null constrained-adaptive beamformer (NC-ABF), named by [2] and introduced by Widrow and McCool [9]. This robust solution to imperfection sensitivity is achieved through the use of adaptive noise cancelling techniques [10,11], resulting in a system output which is the best least mean square estimate of the signal. The adaptive noise canceller is based on a new generalized MVDR concept. The desired primary input signal is made statistically orthogonal (uncorrelated) to the reference signal which is preprocessed or linearly transformed by the blocking matrix to prevent participation of the desired signal in the adaptive nulling process. The results presented here are derived under the following assumptions prevalent in the literature. The desired signal and the beamformer are broad-band. The signal sources are in the far field having planar wavefronts. All directional signal sources and nondirectional receiver noise are zero mean, wide-sense stationary (WSS) and statistically independent of one another. The structure of the minimum constraint set systematically developed here, which to the best of the authors' knowledge has not been approached in the literature, is based on source delay modeling, and determined by the expected deviation range of sensor position, look-direction tolerance and signal correlation time. The NC-GSC is a flexible, simple and easy to implement processor. The rows of the blocking matrix are an orthogonal subset of identity matrix, and the constraint matrix rows are the complementary rows. In contrast to mainbeam zero derivative constraint methods, the null constraint set is insensitive to sensor element amplitude (gain). 331
332
CHAIM Sn~NBAVand SALLY L. WOOD
Preservation of a signal of known correlation time can be insured via the appropriate nonsensitive blocking matrix operation ( NC set), provided that array signal imperfections are of known nature and extent. Wiener output SINR superiority of this method over the conventional-BF is guaranteed under the prevailing statistical orthogonality. Using a three tap filter, an exact scalar expression for the performance improvement due to adaptation (PIA) for white noise sources is derived. This very important performance index, PIA, indicates improvement in output SINR as compared to the conventional-BF.
2. N U L L C O N S T R A I N E D
ADAPTIVE
BEAMFORMER
STRUCTURE
2.1. Direct form implementation The block diagram of the direct form implementation of the null constrained-adaptive beamformer (NC-ABF), as shown in Fig. l, is essentially a broad-band adaptive noise canceller with time-space sampling. The signal sources are sampled in space by the K-element array and steered to the desired direction. Each sensor element is connected to a tapped delay line (TDL). The central tap snapshot, is routed to the conventional sum beamformer to create the primary input to the adaptive noise canceller (ANC). All samples or stacked snapshots, are filtered by the constrained adaptive transversal filter (ATF) to create the reference input. The input to the K-element array is composed of the directional sources: the desired signal, s0(t, 00), coming from the 00 direction and the interference signal, st(t, 0~), coming from the 0t direction. Since the null constraint approach requires estimates of the uncertainties in the look direction, g0, and in the sensor position, an explicit expression for the steering delay errors will be developed. The actual propagation delay, for the qth signal source sq(t, Oq) to the ith sensor element, placed at (Yi, zi) relative to the origin, (Y0, z0) is:
Zq, = (zr/v)sin(Oq) - (yr/v)cos(Oq),
q = 0, 1, i = 0, 1. . . . . K - 1
(1)
where v is the propagation velocity. The array imperfections, represented by local element misplacement from the estimated ith element position of a uniformly spaced linear array, (f~,,~i)=(O, iD), can be expressed as: A z ~ = z ~ - ~ = 6 z , D, and Ayi=y~-fii=6yiD. The corresponding ith steering error delay is given by:
AZq, = Zq, - ~o,
(2)
where z0, = (iD/v)sin(Oo) is the ith estimated steering delay. The sources sampled in space as well as in time generate the ANC inputs. Following [12] the K-dimensional steered source vector at the first tap of the filter memory is given by: Sqo(t ) =
no
o0I
I
I I I I
I I I
[Sq(t "+"A'r:qo), Sq(t
-'1- m~'ql ) . . . . .
_•
Sq(t + A'rqx _)]r.
(3)
d +
+ r
÷
)
k- 10-
Steering delays
Tapped delay
Adaptive noise
line
canceller
Fig. 1. Block diagram of the direct form implementation of the null constrained adaptive beamformer (NC-ABF).
O
333
Null constrained adaptive beamformer
The TDL [13] which is composed of 2L + 1 taps, sampled at the time interval T, holds a total of N = K ( 2 L + 1) samples. Thus the space-time sampled reference input to the ANC is an N-dimensional stacked snapshot vector, containing 2L + 1 delayed snapshot vectors: I
xr(t)___ ~ Srq(t) + nr(t) = [x0r(t) " xr(t) " . . . " XrL(t)l
(4)
q=0
where: xs(t) = Xo(t - JT), for J = 0, 1. . . . . 2L. The central tap snapshot, xL(t) is the input to a sum conventional beamformer forming the primary input, d(t), to the ANC: d(t) = UrxL(t) = UrXo(t -- L T )
(5)
where the summation vector u = 1r consists of K l's. The reference input, x(t), is filtered via the constraint-ATF w~o, to create y(t). The imposed constraint set is represented by a n / - r a n k null-constraint matrix, C r, which is a n / - r o w subset of the identity matrix, I~. The selected orthonormal rows for I constraints are defined by the set of l indices: V = {ni}~= i 2 ~, such that the constrained weight subset is wcZ~[w,~, w~2. . . . . w,t] r = C rw = 0t. The optimization process requires the constrained minimization of H ( w ) - E{e2(t)} -- P0, hence the optimal weight vector Wconis defined as: min H(w)
subject to: Crw = f = 01.
(6)
w
2.2. Null constraint-generalized sidelobe canceller Since the optimal weights wi, i e V are zero, the constrained-ATF can be replaced with an equivalent unconstrained-ATF, the null constraint-generalized sidelobe canceller (NC-GSC). This is done by concurrently applying a linear transformation, represented by the blocking matrix, B r to both, the reference input vector, x(t) and the weight vector, w. This N - 1 rank matrix is composed of the N - l rows of IN complementary to the constraint matrix, defined by the following set of indices: U = [/--
{mj}j=l,
(7)
2. . . . . N - I .
The blocking matrix, B r, of dimension/" x N, w h e r e / = N - 1, can be written as:
Br=
(8)
L e ,r J where, emj = [0" • • 010" • • 0] r is a N-rank unit vector having 1 in the mjth plane and zero elsewhere• The preprocessed reference input, x'(t), and the resulting unconstrained-ATF can be described as follows: Wu~--[Wm,, Win:. . . . . Wm~_,IT= Brw (9) x'(t) = Brx(t)
(10)
y ( t ) = wr.x(t) = WurX'(t) = wrABx(t)
(11)
where: AB = BB r is an N x N binary diagonal matrix and Woo nr = wrAB. The optimization process simply requires a minimization over the unconstrained weight vector wu.
3. S T A T I S T I C A L
ORTHOGONALITY
CONCEPT
The objective in setting constraints is to prevent any rejection of the desired signal in the adaptive filtering process. Signal rejection will not occur if the signal component of d(t), denoted by d0(t), is statistically orthogonal (zero-mean, uncorrelated) to the signal component of y(t), denoted by yo(t). Hence, constraints will be structured to guarantee Rayo~-E{do(t)yo(t)}
O.
(12)
334
CHAIM SHENHAV and
SALLY L. WOOD
3.1. Adaptive noise cancelling approach The primary input of the adaptive noise canceller (see Fig. 1) is composed of three components: the signal, do(t), the interference, d~ (t), and the receiver noise, d~(t). The ATF output, y(t) can be treated likewise and the output power components can be written as:
Poo = Pco + E[y~(t)] - 2Rdyo
(13a)
Po, = Pc. + E[y](t)] - 2Rdy"
(13b)
Po~ = Pc~ + E[y~(t)] - 2Ray ~
(13c)
where Pco ~- E[ d2 (t)], Pc. ~ E[d2(t)], and Pc, ~- E[d~ (t)] are the conventional-BF power of the signal, the noise and, the interference components respectively. Thus, from equation (13a) if Rdyo = O, signal cancellation would not take place. Note that the more restrictive MVDR demand that yo(t) = O, also satisfies this more general statistical orthogonality condition. Since E{d,(t)yn(t)} = 0, the following equivalence relationship can be established: min E[e2(t)] ~- MMSE ~ min E{[d, (t) - y(t)] 2} ¢> min E{[e(t) - [d0(t) + d, (t)]2}.
(14)
Therefore [10,11], the outcome of the minimization process is a filter output y(t), the best least mean square estimate of the interference, and an error e(t), the best least mean square estimate of the signal (plus nondirectional noise). Using the explicit expressions for y(t) and d(t), the sufficient orthogonality condition can be written as: Rdyo = w.E{soSoL r , r u} = 0 => R~ 0 = 0N-t×l and R~s0, = 0N-I× K (15) ,
~.
t
T
•
•
•
where"• Rs,0L - E { s o S o L }. This shows that the signal component of d(t) is statistically orthogonal to the signal component of y(t), when the reference signal preprocessed by the blocking matrix is made statistically orthogonal to the desired primary input signal. The orthogonality condition Rdyo = 0 can also be fulfilled by applying the constraint matrix, C T to the weighting vector: L
W,ro.Rss0L
~ M=-L
W conM T +L RsoM = 0K r
(16)
where the terms of the summation are the 2L + 1 partitioned snapshots. This implies that a nonzero input power signal with for M = O, Rso,~ ~ OK×K must have W¢o.~= Ok. This can be indicated by defining the set of indices corresponding to the Ith delayed snapshot, as ~¢~- {IK - IK + (K - 1)}, and requiring that d L e V. Hence, ~¢L is the minimal constraint set for a perfect array when the signal is uncorrelated from tap to tap. This set will be augmented to accommodate imperfections.
3.2. Array signal imperfection null constraint set The constraint set is determined by the array imperfections, and the correlation time of the desired signal, To, such that E{so(t)So(t + At)} = 0 for IAtl > To. Using equation (2) the qth signal steering deviation difference for sensors i,j is defined a s The bounds of the normalized steering error are defined by a global maximum sensor misplacement error, y and the misdirection A sin(0q) = sin(0q) - sin(g0), where m ~ i - j , and A'rq ~-max~j 6Zqm. The normalized steering error can be explicitly expressed from equations (1) and (2):
m'~qijAArqi--A'~qj.
tSzqm~-
'
<~[m[lAsin(Oq)lm,x+y<~(g-1)lAsin(Oq)lmax+7~-Azq.
(17)
Since so(t) has a finite correlation time, To, the domain of the nonzero autocorrelation can be found: IMI ~
Null constrained adaptive beamformer
335
block-correlation time, Mq, is the largest integer satisfying Mq <~Gq. Similarly, m0, for the desired signal is defined as the smallest integer, Im L, satisfying: M0 ~< [~t I t a l i a sin(00) Im~x]+ [0ty + fl0].
(19)
If the constraint set contains a sub-block, it will be defined as
~t~AT°~-(IK+i},
ieimo
where: i~o = {0, 1. . . . . K - 1 - m0, m0, m0 + 1. . . . . K - 1}. Finally, the minimum null constraint set can be defined as:
"S/L,
M0 = 0
M0 - 1
~-¢4L +M,
Mo ¢ O
A
M= - M o + I
V :
U ~L+Mo mO U ~m°L- M0 I A s i n ( 0 o ) [ max ~ 0
A
(20)
mo > [_K/2_J
Mo ~/L + M,
otherwise.
M= -M 0
Thus increased signal correlation time increases the number of snapshots which can not participate in the minimization process. Random sensor position errors can appear similar in effect to increased signal correlation. This constitutes a minimum null constraint set, needed to satisfy the orthogonality condition of equation (12) to prevent signal cancellation, based on the imperfection bounds. 4. D I S C R E T E
WHITE
NOISE
SOURCE
PERFORMANCE
ANALYSIS
Let SNRi=a0/a,,22 SIRi=ao/a12 2 be the input signal to noise ratio and the input signal to interference ratio, where a 2, a,2 and a~ are the input powers of the signal, noise and interference respectively. For the conventional-BF output d(t) signal to interference plus noise ratio is SINRc = Pco/(ec, + Pc,). The performance quality is evaluated by three different measures: SINR, PIA and SRPq. The SINR is the resultant observed signal to interference plus noise ratio: SINRo = Poo/(Po, + Po~). The performance improvement due to adaptation, PIA, measures the SINR improvement due to the adaptive noise canceller scheme: PIA ~-SINRo/SINRc. The q th signal alone rejected power part, SRPq, measures the sidelobe interference/mainlobe signal power cancellation capability: SRPq~-Rdyq/Pcq 1~2=,~=0, where O # q. In order to evaluate the optimal performance, the convergent Wiener solution is examined: T , -- w * T ~ W/ T , tT -Wuop, RxxWuop, - -opt-xx-opt = E[y2opt(t)] = Wuop, Rdx = wovtRa, - E[yopt(t)d(t)].
(21)
Here R'~x~-Rx,~, = BrRx~B is the transformed autocorrelation matrix and R~---Ra~, = BrRd~ is the transformed crosscorrelation vector. Given w .... , y(t) can be computed along with the performance measures above. Applying the statistical orth~ogonality concept and using optimum values, the optimum performance is given by:
Pco + E[y2o(t)] SINR0 = p,. + p,~ _ {E[y2(t) ] + E[y~(t)]} PIA =
1 + E[y~(t)]/Pco i> 1 1 - {E[y2(t)] + E[yg(t)]}/[Pc. + Pc~]
0 ~< SRPq = E[y~(t)]/Pc~ 1,2=,~=0 ~< 1.
(22)
(23) (24)
It clearly follows from the orthogonality condition that SRP0 = 0 and output SINR is superior to conventional-BF SINR.
336
CHAXMSHENHAVand SALLYL. WOOD
These results will be demonstrated for a special case, where only discrete white noise sources are involved, i.e. Tq = 0, q = 0, 1. For this case the correlation function is nonzero if and only if (iff) Atq = 0 or: M T = Arq,j For simplicity, perfect (uniform) sensor array placement will be assumed or only slight misdirection will be allowed• This leads to the requirement for minimal constraint set, or: M0 = 0 which is equivalent to 0 t ( K - 1)1A sin(00)l < 1, thus defining the mainlobe as the angle domain satisfying:
1/~
IA sin(00)l < K---S]-. On the other hand, interference angles satisfying Atl = 0, for M # 0 are considered• It follows that the equation below constitutes the set of unprotected sidelobe directions: IMI = ot Im[I A sin(01)l
(25)
for 1 ~< Im[~< K - 1. Note that for a fixed [M[ there is a unique [rnl or a unique corresponding IAsin(01)[ satisfying equation (25), and that M = 0 implies m = 0 or perfect array signal A sin(0q) = 0. For full converage of all possible interference angles and for the full range of m, the extreme for which IA sin(01 )1 = 2 will be included. Therefore, the maximum IM [ still satisfying the above equation, defined and chosen as L0 is simply: L0 __42~ (K - 1).
(26)
This is a natural choice because IMI > L0 won't contribute to the set of correlated sidelobe directions• On the other extreme, for M = 1 the above set includes the smallest possible angle:
1/~
K----Z-~_ 1 ~< [A sin(01)l ~<2
(27)
for 1 ~< [M [ ~
(28)
This very limited case is chosen because of the block-diagonal structure of the correlation matrix which allows analytical matrix inversion, using algebraic methods. The lengthy algebraic derivations needed for these rather limited cases are presented in the Appendix. Final results, developed in the Appendix for both the perfect array signal and the misdirection case, will be presented• As explained in the mentioned Appendix full rejection (nulling) capability, SRPI = 1 is achieved for a very specific [ml[= K/2 for an even number of elements. This specific value [ml[ corresponds to a direction which can be interpreted as the direction defining the sidelobe region, the mainlobe width or the resolution: 4 IA sin(01 )l = ~..
(29)
Hence, only this special case will be considered. Note that the resolution under this definition is inversely proportional to the number of sensor elements, K. For the case of the look direction error the PIA is given by: l+g 2 alAu = 1 - g 2 [ 1 + 2a2o/(tr~ + try)] (30) 2 2 where, gN = trl/(a0 + a~2 + a2). Correspondingly, for the perfect array signal:
1 + (0.5~4)/[(K + g,)2cr~] PIA, = 1 - {[(tr4)l(g~a4)][1 -
(0.5K2)I(K + g/)2]}
(31)
Null constrained adaptive beamformer
337
(dB) SNR
J-
,~. sia (aB)
Fig. 2. PIA as a function of the input S N R and the input SIR for a non-perfect array with white noise sources when L = I. S N R is varied from 0 to 30 dB, SIR is varied from - 30 to 0 dB and the PIA ranges from 0.9 to 58.0 dB.
where:g1 = ( a 2 + a ,2) l a o 2 = I/SINR~. These equations indicate that the PIA in the non-perfect array signal is only a function of the relative input power of the various signals while in the perfect array it is also a function of the number of elements, K. It is interesting to find the asymptotic behavior of the PIA for which the nondirectional noise is very small in comparison with the other signals. This asymptotic behavior demonstrates the influence of the SIR,. on the potential degree of rejection of an interference. Under the assumptions that tr0~ >>tr2 and try>> 2 a~, 2 the PIA for the non-perfect array signal approaches: '/O"2~2
_ O'~
(32) Correspondingly, for the perfect array: PIA,
= 2
1 ÷ -2--)
+ O.5
.
(33)
These observations are illustrated in the following 3-D plots. Figure 2 shows the PIA as a function of the input SNR and the input SIR for a non-perfect array. SNR is varied from 0 to 30 dB, SIR is varied from - 3 0 to 0 dB and the PIA ranges from 0.9 to 58.0 dB. Figure 3 demonstrates the PIA behavior for the perfect array with K = 8, when the SNR is varied from 0 to 30 dB, and the SIR is varied from - 30 to 0 dB. For this case, the PIA ranges from 0.5 to 46.0 dB. Figure 4 presents the PIA as a function of the number of elements, K = 2, 4, 6 . . . . . 32, for the perfect array, when the input I/SIR is varied from 0 to 30 dB for SNR = 30 dB, and the PIA ranges from 3.2 to 29 dB. It is shown that the PIA is monotonically increasing with SNR~ or 1/SlR~. Wiener simulations for
P I A (~B)
Fig. 3. PIA as a function o f the input S N R and the input SIR for a perfect array with white noise sources when L = 1. S N R is varied from 0 to 30 dB, SIR is varied from , 30 to 0 dB for K = 8, where the PIA ranges from 0.5 to 46.0 dB.
338
CHAIMSt-mNgAVand SALLYL. WOOD
% Fig. 4. PIA as a function of the number of elements K = 2, 4, 6,..., 32, for a perfect array with white noise sources and with L = 1, when the input 1/SIR is varied from 0 to 30 dB for SNR = 30 dB, where the PIA ranges from 3.2 to 29 dB. L > 1 have shown that for the non-perfect array there is no further improvement with large L. On the other hand, for the perfect array, there is performance benefit in increasing the number of taps in cases where the interference direction is close to the mainbeam for high SNRi and low SIRi. 5. C O N C L U S I O N S The NC-GSC, presented in this paper, offers an easy to implement processor with performance advantages for wideband low power signals in the presence of array imperfections. The constraint set, systematically developed here, and the consequent main results are consistent with earlier investigations, intuitively approached by Widrow and McCool [9]. Preservation of signal of known correlation time is based on a new, true by construction, signal statistical orthogonality concept (GMVDR). Under the prevailing statistical orthogonality, the output S I N R is shown to be superior to that of the conventional-BF. Thorough analysis for discrete white noise sources has been carried out. In the ideal case, full rejection (hulling) capability is achieved for a very specific direction. The performance improvement due to adaptation, PIA, in the non-perfect array signal is only a function of the relative input power of the signals while in the perfect array it is also a function of the number of elements, K. The asymptotic behavior of the PIA demonstrates the influence of the SIRi on the potential degree of rejection of an interference. Performance analysis of the steady-state Wiener solution shows that this scheme is well suited to wideband low power signals, e.g. LPI (low probability of intercept), noise-like, spread spectrum signals. Acknowledgement--This work was supported by NSF grant No. 86-98035 and Sun Microsystems, Mountain View
California. REFERENCES 1. O. L. Frost III, An algorithm for linearly constrained adaptive array processing. Proc. IEEE 60, 926-935 (1972). 2. N. K. Jablon, Adaptive beamforming with imperfect arrays. Ph.D. thesis, Stanford Univ., Calif. (1985). 3. M. H. Er and A. Cantoni, Derivative constraints for broad-band element space antenna array processors. IEEE Trans. Acoust. Speech Signal Process. 31, 1378-1393 (1983).
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339
4. L. J. Grittiths and C. W. Jim, An alternative approach to linearly constrained adaptive beamformer. 1EEE Trans. Antennas Propagation 30, 27-33 (1982). 5. B. D. Van Veen and K. M. Buckley, Beamforming: a versatile approach to spatial filtering. IEEE A S S P Mag. 5, 4-24 (1988). 6. N. L. Owsley, Sonar array processing. In Array Signal Processing (Edited by S. Haykin). Prentice-Hall, Englewood Cliffs, N.J. (1985). 7. C. Shenhav and S. L. Wood, A null constrained adaptive beamformer in the presence of array imperfections. Proc. 21st Asilomar Conf. Signals Systems and Computers (SSC) (1987). 8. C. Shenhav and S. L. Wood, A null constrained adaptive beamformer for broad-band signal preservation. Proc. 23rd Asilomar Conf. Signals Systems and Computers (SSC) (1989). 9. B. Widrow and J. M. McCool, A comparison of adaptive algorithms based on the method of steepest descent and random search. IEEE Trans. Antennas Propagation 24, 632-634 (1976). 10. B. Widrow and S. D. Stearns, Adaptive Signal Processing. Prentice-Hall, Englewood Cliffs, N.J. (1985) 11. J. W. R. Griffiths, Adaptive array processing, lEE Proc. 130F, 3-10 (1983). 12. D. E. Dudgeon, Fundamentals of digital array processing--a tutorial. Proc. IEEE 65, 899-904 (1977). 13. W. E. Rodgers and R. T. Compton Jr, Adaptive array bandwidth with tapped delay line processing. IEEE AES-15, 21-28 (1979). APPENDIX PIA for Discrete White Noise Sources This Appendix provides proofs to equations (30) and (31). For L = 1, M 0 = 0 and M I = 1, the submatrix R 2 = 0x× x and the Wiener solution or the optimum weight vector is given by: Wuop,= R~2I R}x =
I- :lu ....
(34)
LR~'R,I The ATF output power can be written as: (35)
E[yZop,(t)lwr,R'a.~ = ur[R,R~aRr + R r R ( ' R, lu.
The signal power component of the ATF output can also be expressed in general: E[y2op,(t)]=w Uopl r R 'sso w Uopt _- -
U
rfl;, t ~ X l A~X-0I r~,X s 0 0 *l;,-llar.a_RrlR~lRsooRffiRi]u" l 0 sx I l
(36)
The following relationships simplify computations: F0,,,×~ R , = R,,, =
.
IL Ix-re'
where in general, % and zero elsewhere.
= [0.
•
0.
.,, .
.
.
.
. . . . .
a ~ [ e , , , ! . . . i e x _ 10x,,.,..
]
.
(37)
or'×x-"'
=a][Ox+m'ie°i""
ex-'-m'lr
• 010.. • 0] r is the K-dimensional unit vector having 1 in the m t h place (m = 0, 1. . . . . K - l)
A. I. Signal misdirection case
For this very simple case, where the look-direction tolerance is known, the following applies: R,oo=a~l x, Ro=(ag+~r~+~r])l x, and the symmetric R~-~= (a02+ cry+ cr2)-II x. In this case R~00, R0 and R~~ take the general form of F1 = f l x . Taking into consideration the fact that in equations (35) and (36) u (u r) are operating on a matrix from the right (left) results in: E[y2(t)] = 2a~gN(K - ml )
(38)
Ety2o(t)] = 2a~g2(K - m, )
(39)
1 + SRP,g~v l+g~ I PlAN = 1 - SRP, g:u[I + 2a~/(a~ + a,2)] - 1 - g ~ [ 1 q- 2az/~(a~ + a~)] IsRP,= ~
(40)
where, gN = a ~/(a ~ + a ~ + cr2) and SRP l = 2(1 - m I/K) = 11m, =X/2.X.... for complete rejection of a sidelobe interference coming alone (or having much more power than the signal and the noise) from the 0t direction corresponding to m~. A.2. Perfect array signal case
In this perfect array signal case, the following relationships prevail: Rso 0 = ~ 2 u u T
Ro = a~(gllK + uu r)
where: gl = (a~ + g.~)t~o~ = 1/SINRv Applying the inversion Lemma [A + bbr] I = A i
A -IbbrA -t 1 + brA - ~b
(41)
340
CHAIM SHENHAV and SALLY L. WOOD
to equation (41) results in a symmetric matrix composed of two terms, the identity terms 1r and the summation term uu r as follows: SINR,( 1 r) R°l= a~ It-UU . (42) K +gl It is observed that in this case R~00, R 0 and R o ~ take the general form of: F = f ( a l x + buu r) = F I + Fs .
Substituting Rff j into equation (35), the summation term F s results in:
(urRlu) (urRru)
+
(urR ru) (urRlu) =
2a 4 (K -- m I)2.
Hence, the following result is achieved:
ml)~r4{ _ K - ml~" g~o [ I K + g,J
2(K -
E[y2(t)]
(43)
Now, the expression for E[y2(t)], is similarly derived from equation (36): 2K - m I )2a2 E[y~(t)] = ~ + ~ .
(44)
Combining the expressions for E[y2(t)] and E[y~(t)] results in the following formula for the PIA: PIA t =
1 + (0.5SRP~a4)/[(K + g,)2ag] 1 -- {[(SRP, a~)/(g2ia4)] [1 - ( 0. SSRP, K2)/(K + g,)2]}
1 + (0.5a4)/[K + gl)Za~] SRl', = I -- { [ ( t r ~ : i ~ - . 5 K ~ + g,)Z]} =,
(45)
where, SRPt = 2(I - m t / K ) = 11m, ~ r/2. ~ov.n.
AUTHORS'
BIOGRAPHIES
Chaim Sheahav--Chaim Shenhav was born in Rishon-Le-Zion, Israel, on 3 November 1950. He received the B.S. and M.S. degrees in electrical engineering from the Technion, Israel Institute of Technology, Haifa, Israel, in 1975, and 1982 respectively. From 1977 to 1985 he was a Research Engineer at the Armament Development Authority, Haifa, Israel. Since 1986 he has been pursuing the Ph.D. degree, completed in 1991, in electrical engineering at Santa Clara University. He is currently with Applied Imaging Corp., Santa Clara, California.
Sally Wood---Sally Wood was born in Massachusetts in 1947. She received the B.S. degree in electrical engineering from Columbia University in 1969 and the M.S. and Ph.D. degrees in electrical engineering from Stanford University in 1975 and 1978 respectively. Since 1985 she has been an Associate Professor in the Electrical Engineering Department at Santa Clara University where her research has focussed on multidimensional signal processing. She is a member of Tau Beta Pi and Eta Kappa Nu.