SIGNAL
PROCESSING EISEVIER
Signal Processing 44 (1995) 285297
Wideband forwardbackward beamforming and its efficient computation methods WenLiang Hsue, ChienChung Department of Electrical Engineering, National Taiwan Universiry
Yeh* Taipei, Taiwan, ROC
Received 21 January 1994; revised 23 September 1994
Abstract
The forwardbackward processing and narrowband adaptive beamforming was investigated recently. It was shown that the narrowband forwardbackward beamforming converges faster and requires less computations than the forwardonly beamforming. In this paper, we study the wideband forwardbackward processings of the Frost beamformer and its corresponding GSC, which converge faster than the forwardonly ones. We also describe methods to reduce the computations of the wideband forwardbackward processing. For the Frost beamformer, we observe that the constraint matrix possesses the property termed general centrosymmetry. With that property, the forwardbackward weight vector of the Frost beamformer can be transformed by a unitary matrix to achieve computation savings. As to the GSC, we show that a signal blocking matrix with general centrosymmetry may be used for reducing the computations required for calculating the weight vector. Besides, we describe a method to generate the generalcentrosymmetric signal blocking matrix. Computer simulation results are included to compare the performances of the forwardbackward and the forwardonly beamformers. Zusammenfassung
Die VorwPrtsRiickwartsVerarbeitung bei schmalbandiger adaptiver Keulenformung wurde vor kurzem untersucht. Es wurde gezeigt, dab das schmalbandige VorwlrtsRiickwiirtsVerfahren zur Keulenformung schneller konvergiert und weniger Rechenaufwand erfordert als VorwPrtsKeulenformung allein. In diesem Aufsatz untersuchen wir die breitbandige VorwlrtsRiickwartsVerarbeitung des Frost’schen Keulenformers und des entsprechenden GSCs, die schneller konvergieren als die nur vorwarts arbeitenden Verfahren. Wir beschreiben such Methodem, die den Rechenaufwand der breitbandigen VorwartsRfickwlrtsVerfahren verringern. Fur den Frost’schen Keulenformer bemerken wir, dal3 die Matrix der Nebenbedingungen die sogenannte allgemeine Zentralsymmetrie aufweist. Mit dieser Eigenschaft kann der VorwktsRiickwartsGewichtsvektor des Frost’schen Keulenformers mit einer unitaren Matrix transfomiert werden, urn Einsparungen bei der Berechnung zu erzielen. Hinsichtlich des GSC zeigen wir, daR eine SignalblockMatrix mit allgemeiner Zentralsymmetrie zur Verringerung des Aufwands bei der Bereclmung des Gewichtsvektors benutzt werden kann. AuBerdem beschreiben wir eine Methode zur Erzeugung einer SignalblockMatrix mit allgemeiner Zentralsymmetrie. Computersimulationsergebnisse dienen dem Leistungsvergleich des VorwartsRilckwiirtsKeulenformers und des nur vorwarts arbeitenden Verfahrens.
*Corresponding
author.
01651684/95/%9.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 01651684(95)000305
286
W.L. Hsue. C.C. Yeh / Signal Processing
44 (1995) 285297
Le traitement progressifrttrograde du formattage de voie adaptatif g bande ktroite a ttC Ctudii:rkcemment. 11a tt6 montrt: que le formattage de voie progressifretrograde g bande etroite converge plus rapidement et nkcessite moins de calculs que le formattage de voie progressif seul. Dans cet article, nous Ctudions les traitements progressifsretrogrades large bande du formatteur de bande de Frost, et ses GSC correspondants, qui convergent plus rapidement que ceux progressifs uniquement. Nous d&ivons tgalement des methodes pour r&duire les calculs sur le traitement progressifretrograde large bande. Pour le formatteur de voie de Frost, nous observons que la matrice de contrainte possBde la propriM appell& centrosym&ie g&&ale. Avec cette proprittt, la vecteur de pondtration progressifrktrograde du formatteur de voie de Frost peut &tre transformt en une matrice uniti pour faire des 6conomies de calcul. En ce qui concerne le GSC, nous montrons qu’une matrice de blocage de signal avec centrosym&ie g&&ale peut itre utilisk pour rtduire les calculs nCcessaires au calcul du vecteur de pondkration. En outre, nous dCcrivons une mCthode pour gCntrer la matrice de blocage de signal centrosymktrique g&&ale. Les resultats de simulations informatiques sont inclus pour comparer les performances des formatteurs de voie progressif et progressifrktrograde. Keywords: Beamforming; centrosymmetric
Forwardbackward
beamforming;
1. Introduction
The forward sample correlation matrix of an antenna array is the maximum likelihood estimator (MLE) [S] of the ensemble correlation matrix and is usually used for adaptive beamforming [1,4,6,7,1619,131. When the array elements are symmetrically distributed, the ensemble correlation matrix of the array input vector is Hermitian persymmetric [12,3]. Based on that property, Nitzberg proposed using the forwardbackward sample correlation matrix, which is the Hermitian persymmetric MLE of the ensemble correlation matrix, to achieve faster convergence rate [ 151. The application of the forwardbackward processing to the GSC was described by Huarng and Yeh [9] recently. They also considered the reduction of the computational load of the forwardbackward beamforming in which each array input vector is processed forwardly and conjugate backwardly. It was proposed to transform the complex forwardbackward sample correlation matrix into a real matrix by using a unitary matrix so that the weight vector can be calculated with real computations. As to the GSC, it was shown that when the signal blocking matrix has conjugate symmetric columns, the optimal forwardbackward weight vector in the sidelobe canceller branch is a real vector. Consequently, the weight vector can be calculated with much less computations as compared
Frost beamformer;
GSC; Centrosymmetric;
General
to a complex weight vector. The forwardbackward processing technique was also applied to the GSC with a GramSchmidt orthogonalization processor, which has the merit of modular structure [14]. The previous researches on the forwardbackward processing were focused on the narrowband processing. In this paper, we investigate the wideband forwardbackward adaptive beamformers with the Frost and its corresponding GSC structures [4,7]. The complex signals are used for discussions, which can be formed by passing the real signals through quadrature hybrids [13,18,14,2]. The ensemble correlation matrix of a wideband Frost beamformer with symmetrically distributed elements can be shown to be Hermitian persymmetric. Based on that we propose the forwardbackward beamforming of the Frost and the equivalent GSC structures with the direct matrix inversion and gradientbased algorithms. In addition, we develop computation saving methods for both beamformers. We observe that the constraint matrix of the Frost beamformer possesses the property termed general centrosymmetry. Using the property, the complex forwardbackward weight vector of the Frost beamformer can be transformed by a unitary matrix so that the transformed weight vector can be calculated by computations of almost all real. As to the GSC, we show that when the signal blocking matrix is generalcentrosymmetric, the forwardbackward sample correlation matrix
W.L. Hsue, C.C. Yeh / Signal Processing 44 (1995) 285297
287
of the sidelobe canceller branch is Hermitian persymmetric and unitary transformations can be applied to achieve computation savings. We also describe a systematic method to generate a generalcentrosymmetric signal blocking matrix. This paper is organized as follows. Sections 2 and 3 discuss the wideband forwardbackward beamforming using the Frost structure and the GSC structure, respectively. Section 4 concludes this work.
where sl(t) and si(t), i = 2, . . . , N, are the complex waveforms of the desired signal and the interferers, respectively, A,i represents the delay of the ith source at the kth element, and nk(t) is the noise. The desired signal, interferers and noises are assumed to be uncorrelated. Since the array elements are symmetrically distributed, with the time reference point chosen at the array geometric center, we have
2. Wideband forwardbackward frost beamforming
Aki =  AUK+ I k)i.
We develop in this section the wideband forwardbackward adaptive beamforming with the Frost array structure. Let us consider a K element symmetrically distributed linear array with J  1 tapped delays behind each sensor as shown in Fig. 1. The array is illuminated by a wideband desired signal and N  1 interferers. The input to the kth array element can be expressed as
In Fig. toward steered size JK
Xk(t) = f
qCt

dki)
+
nk(t)~
1, it is assumed that the array is presteered the desired signal without showing the predelays. The complex array input vector of x 1 can be written as
z(t) =
(1)
i=l
Fig. 1. The wideband
Frost
(2)
beamformer.
(3)
288
W.L. Hsue, C.C. Yeh / Signal Processing
where
40 = CXl(O, *..,XKWIT.
(4)
The complex weight vector of the array can be expressed as
44 (1995) 285297
To the desired signal, the array behaves like a transversal filter of order J with the equivalent weights specified by f& k = 1, . . . , J, i.e., fk determines the frequency response of the array to the desired signal. The solution to (8) and (9) is given by C41
w,,= R‘C(C”R‘C)‘f: (5)
where w(j) is of size K x 1 containing the weights of the jth level. T
w(j) = CW(jI)K+Iv**=,WjKl .
(6)
The array output is given by y(t) = U”
[email protected]).
With the Frost beamformer, the optimal weight vector is determined by minimizing the array output power min W”RW 02
It can be shown that the ensemble correlation matrix R, which is defined by (lo), is Hermitian persymmetric, i.e., R=R” and
(9)
where R is the ensemble correlation array input vector R = E[z(t)z(t)“],
matrix of the
(10)
C is the JK x J constraint matrix,
rb oK . oKi c=
L 1 OK 1K
.. . .
OK
*.*
.**
..
:
. OK
OK
(11)
C.fi~...,fJIT.
=
1
1
1
0
Since R is Hermitian persymmetric, the forwardbackward processing can be applied to the Frost beamformer. Practically, the ensemble correlation matrix is not available and a sample correlation matrix is used instead. The wideband forward average and forwardbackward average sample correlation matrices, denoted by f and R”, respectively, can be formed as
R = ; .i z(i)z”(i),
1K
(17)
rl
with lK and OK being the K x 1 vectors whose elements are unity and zero, respectively, and f is the J x 1 constraint vector f =
. (1 _I
0
JJK
subject to the constraint
(15)
where H and * denote conjugate transpose and conjugate, respectively, and JJK is the reflection matrix of size JK x JK,
(8)
C”w = f,
(14)
JJKR*JJK = R, (7)
(13)
(12)
where z(i) is the array sample data vector given by (3), and i = ;(i + JJKit *JJK).
(18)
W.L. Hsue, C.C. Yeh / Signal Processing 44 (1995) 285297
289
Similar to the case of narrowband beamforming [9], the forwardbackward average correlation matrix is Hermitian persymmetric, but the forward average correlation matrix is not. In addition, R”is a better estimator of R in the sense of Euclidean distance, i.e.,
formed as
lIR”RII G IlkWI,
where p is a chosen real constant, JJK is the reflection matrix,
(19)
where )I . )I is defined as II.4/I2 = tr(AHA).
G(O) = F, &(n + 1) = P{hqn)  p[:[z(n)z(n)”
1
+
JJx~*(n)rcT(n)J~~ll,(~,>+ F, (21)
F = C(C”C)+f
(20)
The proof of (19) is similar to that presented in [9] for the narrowband correlation matrices. Therefore, the forwardbackward processing is expected to improve the performance of wideband adaptive beamforming. In the direct matrix inversion processing, the weight vector of the forwardbackward Frost beamformer is computed by substituting R” into (13) for R. Other existing forward beamforming algorithms can also be adapted to perform the forwardbackward beamforming. For example, the forwardbackward Frost algorithm [4] can be
(22)
and P = IJK  C(CHC)‘CH.
(23)
Computer simulation results are presented in Figs. 2 and 3 to demonstrate the performance of the wideband forwardbackward beamforming method. Simulations are performed for both the forwardbackward method and the forwardonly method. We first use a fiveelement equally spaced array with 3 taps behind each element for simulation. The array element spacing is iA, and the equivalent tap spacing is +A,, where 1, denotes the wavelength with respect to the center frequency fc of the desired signal. The array is illuminated by
Fig. 2. Output SINR with 3 taps and 25% bandwidth: (a) forwardbackward
method; (b) forwardonly
method.
290
W.L. Hsue, C.C. Yeh / Signal Processing 44 (1995) 285297
15!m
2m
2500
sample numbu Fig. 3. Output SINR with 3 taps and 50% bandwidth of the adaptive Frost algorithm: (a) forwardbackward method.
the desired signal and an interferer, coming from 0” to 30”, respectively. The wideband sources are generated by passing white noises through bandpass filters. The fractional bandwidth of the incoming sources is assumed to be 25% relative tofc, and the sampling frequency is 4fc. The input SNR and INR are 20 and 30 dB, respectively. The signal, interferer and noise are mutually uncorrelated and have flat power spectral densities. The constraint vector f is chosen to be [0 1 0] ‘. The transient output SINRs of both the forwardbackward and the forwardonly methods based on the direct matrix inversion are plotted in Fig. 2 with the number of samples up to 4000. Fig. 2 shows that the transient output SINR for the forwardbackward method is higher than that of the forwardonly method. Then we increase the fractional bandwidth to 50%. The transient output SINRs averaged from 30 independent trials are plotted in Fig. 3, which also shows the forwardbackward method outperforms the forwardonly method. We discuss below the reduction of the computational load of the wideband forwardbackward beamforming. We begin by examining some useful properties of the constraint matrix C. It can be easily
method;(b) forwardonly
shown that C given by (11) satisfies the relation JJKC* JJ = C,
(24)
where JJK and JJ are the reflection matrices. Since a square matrix A is said to be centrosymmetric if JA* J = A, we term the property specified by (24) as general centrosymmetry. For a generalcentrosymmetric matrix, we have the following theorem. Theorem 1. For a generalcentrosymmetric matrix A of size p x q, A = l&AU+! is a real matrix, where
U, is the unitary matrix of size k x k [9, lo]:
(25)
for odd k, with 1i being the identity matrix of size i x i.
W.L. Hsue, C.C. Yeh / Signal Processing
matrix C as shown in the Appendix A. which is
Proof. The complex conjugate of 2 is A* = U*A*UT P 4
291
44 (199.5) 285297
C
r
(26)
.
for J even, 0
1K
Using JJ = Z, ZJJ = U*, and (24), (26) becomes A* = U,*J,J,A*J,J,U,T
1K g1
= U,AlJ,H
1K = A.
(27)
Therefore 2 is real.
0
0
c=. With this theorem, the transformed matrix defined by C = U&U,”
lK_
for J odd and K even,
<
0'
(28)
is a real matrix, where ZJ,, and U, are the unitary matrices defined by (25). As to the forwardbackward sample correlation matrix R”, it can also be transformed into a real matrix,
1X, for J odd and K odd,
L

R = U,,RU,,
(31)
constraint
H
where gl and g2 are K x 1 vectors defined by = Re(U,,~U~H).
(29)
Therefore, the weight vector computed from the forwardbackward sample correlation matrix. can be transformed as uJ,,
t
(30)
Since 2 and Care real matrices, the only complex term in (30) is U,f, which is a vector of size J x 1. Therefore, most of the computations involved in (30) are real instead of complex. Furthermore, the transformations of R” and C into 2 and C require very few computations since the unitary matrix given by (25) has a very simple form. It is interesting that the transformed constraint matrix C has a simple form similar to the original constraint
I
O...OIT
(32)
1 O...O]T.
(33)
and g,=[Jz...$
= UJ~k‘C(CH&‘C)‘f
= ~‘~(p~~~)q&J
g, = c $...fi L V KI2
Y V I)/2
I
Therefore, computing the transformed weight vector by (30) and then taking the inverse transformation to obtain the original weight vector & = U,H,(UJK&)
(34)
can save significant amount of computations. If the constraint vector f is conjugate symmetric, i.e., the frequency response of the array to the desired signal is linear phase as many applications desired, U’f, the only complex term in (30), also reduces to real. As a consequence, the transformed weight vector UJKG becomes a real vector and more computations can be saved.
292
W.L. Hsue, C.C. Yeh / Signal Processing
With the constraint matrix f being conjugate symmetric, it can be shown that the forwardbackward Frost algorithm shown in (21) can be transformed into a real algorithm
forming. Let us consider the GSC depicted in Fig. 4. For the constraint matrix C shown in (ll), the signal blocking matrix b is of size K x (K  1) and is orthogonal to the K x 1 all one vector, i.e.,
W(0) = F,
1,Hb= 0.
w(n + 1) = P{&(n)  C1[Yi(“)&(“) +
(35)
1 The transformed real terms in (35) are defined by G(n) = U,,,(n),
(36) 0J
P = UJKPUJHK= IJK  C(C”c) l C”
3
A(n) = ReCu(n)l, j2b)
=
u(n) = B”rc(n),
(37)
where s(n)
(38)
JK x J(K  l),
(39)
ImCYbOl~
(43)
The weight vector of the tapped delay line combiner with J taps, denoted by Mu,,, is of size J(K  1) x 1. The equivalent input vector weighted by ur, can be expressed by
Yz~4GN) + F.
F = UJ,F = @“C)’
44 (1995) 285297
(40)
&W = RWk441,
(41)
z2b4 = ImCkWl,
(42)
B=
(44)
is given by (3) and B is of size
‘b
0
...
0’
0 ..
b ..
.+ .. .
;
0
...
0 b.
(45)
0
The optimal adaptive weight vector minimizing the array output power E[ Iyo(n)12] is given by [8]
where y(n) is the array output shown in (7).
io,
3. Wideband forwardbackward GSC beamforming
=
R;‘r,
(46)
where R, and r are defined by R, = E[u(n)u”(n)] and r = E[u(n)yz(n)], respectively, with y,(n) = M$z(~) being the output of the quiescent beamformer branch.
In this section, we use the GSC structure to implement the forwardbackward Frost beamW
X1 . .
tapped delay line combiner
. xK
y,(n)
y&l
y,(n) u(n) . . .
signalblocking matrix b
:
tapped delay line combiner
Fig. 4. The wideband
GSC beamformer.
293
W.L. Hsue. C.C. Yeh 1 Signal Processing 44 (1995) 285297
From (46), the forwardbackward weight vector W, can be obtained by
adaptive
i defined by (18) is Hermitian persymmetric, i.e.,
[email protected])
t;, = &‘F,
(47)
where & is the forwardbackward tion matrix of u(n),
sample correla
*
JJK = R.
(53)
From (50), we have JJCKI,(BH)*JJK = BH.
(54)
Using (50), (53) and (54), (52) reduces to (51) and 0 therefore g,, is Hermitian persymmetric.
R”, = A ,$ [u(i)u(i)H + ub(i)ub(i)H] 1l =
In the next theorem, we describe a sufficient condition for B being generalcentrosymmetric.
BH+(i + J&.*JJK)B
= BHi&
(48)
and ? is the forwardbackward sample cross correlation vector of u(n) and y&r), F = k i$1 Cu(i)_M)* +
zMh(i)*l
Proof. With b being generalcentrosymmetric, holds
= BH +(i + J,,i* JJK)uq = BHi&,
JKb*JCKmI, = b. (49)
with Qi) and ycb(i) being the backward versions of u(i) and yC(i), i.e., u,,(i) = BHJJKz*(i) and y&i) = ufJJKz*(i). It can be shown that the wideband forwardbackward GSC beamforming is equivalent to the wideband forwardbackward Frost beamforming. We discuss below the computation reduction of the wideband forwardbackward GSC beamformer. We first derive a sufficient condition for i?” being Hermitian persymmetric. Theorem 2. ku in (48) is Hermitian persymmetric if the matrix B is generalcentrosymmetric, i.e.,
(50)
JJKB+JJW 1)= 4
Theorem 3. The matrix B is generalcentrosymmetric if the signal blocking matrix b is generalcentrosymmetric.
(55)
Using (45) and (55), we have
Jd*Jwc
1)
=[ IK :’
:I[
x[ J(Kyl, :.
1
. . . ::]
“:r:;l
Jd* Jw 1)
=
0
=
[
0
Jd*Jv
B.
where JJK and JJtK _ 1j are rejection matrices.
it
The proof is completed.
1
1)
0
Proof. We show below that
Using JJ = I, the lefthand side of (51) can be written as
From Theorems 2 and 3, we can conclude that i?” is Hermitian persymmetric if b is generalcentrosymmetric. With this property, G2, in (47) can be transformed by a unitary matrix
JJW I,(B~W*JJ~KI)
Q1$.
JJCK_l,(BH*B)*JJcKl)
= BHRB.
(51)
= u,,K1,K’F) = (u,,K l,R l GIdw?uK
= J,,,,,(B”,*J,,J,,(R”)*J,KJJK(B*)JJ~K~,. (52)
=R,‘UJ(K_l)F,
$1
(56)
294
W.L. Hsue, C.C. Yeh / Signal Processing
where
44 (1995) 285297
where
R, = &,K &J&
1).
(57)
Since i,, is Hermitian persymmetric, & is a real matrix. Though UJcK_i)r”in general is a complex vector, computation reduction can still be achieved by using (56) because performing inversion of the correlation matrix requires most of the computations. Computations can be further reduced if r”is conjugate symmetric. In that case, UJcK_1,i; is real, and so is V’,,_ 1,C;a. We describe in Theorem 4 the conditions for r”being conjugate symmetric. Theorem 4. r”is conjugate symmetric when B is generalcentrosymmetric and f is conjugate symmetric.
= JJ~K~,(B~)*JJKJJK~*J~~J~~~~. (58) With B being generalcentrosymmetric, duces to
(58) re
JJfK_ l,F* = BHRJJK&.
(59) we have
JJKuz = JJK[C(CHC)If]*
= C(C”C)‘J,f
*
*.
(60)
i = :(BHRm, + B”J& = ;(BHb,
= tDq.
(61)
0
(62)
It is worth pointing out the following comments. When the signal blocking matrix b is generalcentrosymmetric, & can be written as
(66)
with ye(n) being the output of the quiescent beamformer. Eq. (65) indicates that F can be calculated with the forwardbackward processing of E. Eqs. (63) and (65) are easier to implement than (48) and (49). In addition to the direct matrix inversion processing described above, other adaptive algorithms can also be applied to the wideband forwardbackward GSC beamformer. For example, when b is generalcentrosymmetric and f is conjugate symmetric, the forwardbackward LMS algorithm [20] can be formed as w&i + 1) = w,(n) + Cc[u(n)yZ(n) JJ(K
I,b44y,*(n))*l.
(67)
In (67), Cu(n)yo*(n)+ JJ(K &4n)y,*(n))*l is conjugate symmetric, and, therefore, the adaptive weight vector w,(n) can remain conjugate symmetric when the initial value is conjugate symmetric. Eq. (67) can be transformed into a real algorithm &An + 1) = w,(n) + 2CLReCU,,,,,(u(n)y,*(n))l
= &(4 + &4&(n)Y0~(n) + G(n)h(n)l,
(68)
where G,(n) = UJcK_l)w,(n) is real,
+ JJcK l)(BH)*k*B* JJcK 1))
= :(& + JJV &JJCK I)),
(65)
where
1, = +(BHi?B + BHJJ,i+ JJKB) = :(B”kB
+ JJcK_ l,(BH)*k*~z)
+
Substituting (61) into (59) and using (49) yields JJ(K_l$* = L
JJKu,,)
= :(; + JJ(K I$*),
Since f is conjugate symmetric, (60) becomes JJKuq* = C(CHC)‘f
(64)
Therefore, R”, can be obtained by the forwardbackward processing of & instead of ji? in (48). When b is generalcentrosymmetric and f is conjugate symmetric, from (49), r”can be written as
I 1
JJo_ 1)‘“*= JJtK_ l,(BHku,)*
= J~KJJKCJ,(J,CHJ,,J,,CJ$tf
= i ,$ u(i)u”(i). rl
i = BHk’w, = i ,$ u(i)y,*(i),
Proof. From (49), we have
Using (24) and wq = C(C”C)‘f,
& = B”k3
(63)
UJ(k i&r)
= ii(n) + j&(n)
(69)
W.L. Hsue, C.C. Yeh 1 Signal Processing
and Y&)
= Fb~(n)
(70)
+ j&(n).
Eq. (68) requires less computations as compared to (67). We want to point out that the signal blocking matrix b described above for computation saving is generalcentrosymmetric, while that proposed in [9] for narrowband beamforming consists of conjugate symmetric columns. Here we describe a systematic method to construct a generalcentrosymmetric signal blocking matrix b which satisfies (43). We first find a K x 1 real vector C = t&l,, where UK is the K x K unitary matrix defined in (25). Using E, we can construct a fullcolumnrank K x (K  1) real matrix b which satisfies CT6 = 0. Let us define b = U~iiU&
1,.
(71)
Since b is real, the matrix b is generalcentrosymmetric. Furthermore, we have
=ETbtr W 1) = 0.
(72)
Therefore, the matrix b constructed above can be used as the desired signal blocking matrix. For example, for a fourelement antenna array, a generalcentrosymmetric signal blocking matrix b constructed by the above method is
b=
0.1464
0
0.3536
 0.7071j
0.3536
0.3536
0.7071j
0.3536
L  0.8536
0
0.1464.
295
44 (199.5) 285297
gence rate because each input sample is processed forwardly and conjugated backwardly. We also developed methods to reduce the computations required for calculating the forwardbackward weight vectors. For the Frost beamformer, we used a unitary transformation matrix to transform the complex weight vector so that it can be calculated by real computations mostly. As to the GSC beamformer, we constructed a signal blocking matrix with the property of generalcentrosymmetry for computation reduction. Therefore, the forwardbackward Frost and GSC beamformers possess the merits of computation saving as well as fast convergence.
Appendix A In this appendix, we derive the general form of c shown by (31), which is the transformed version of the constraint matrix of the Frost beamformer. From (28) c is defined by _ C = uJ,CuJ”, (A.l) where C is of size JK x J defined by (1 l), and LJJKand U, are the JK x JK and J x J unitary matrices defined by (25), respectively. To evaluate (A.l), we first examine the product term lJ,KC by considering three cases: (a) J is even, (b) J is odd and K is even, and (c) J is odd and K is odd. Case (a). J is even. JK
ZK
 0.8536’ JK
IK
j&
(73)
4. Conclusions In this paper, we have described the wideband forwardbackward adaptive beamforming with the Frost and its equivalent GSC structures. The forwardbackward beamforming has a faster conver
j&i
:L 1 fK
OK
OK
1K
x.. .
OK
.
.”
OK
.”
“.
..
!
.
OK
OK
1,
W.L. Hsue, C.C. Yeh / Signal Processing 44 (1995) 285297
296
where UKis the K x K unitary matrix defined in (25) and g3
= (fiuK)
’ 1X
=[2...2
j fK
jlK
(A.3
o...oy.
(A4)
Case (c). J is odd and K is odd. UJKC ‘ZK
where ZKand JK are the identity and the reflection matrices, respectively, of size K x K and 1K is a K x 1 vector with elements equal to unity. Case (b). J is odd and K is even.
JK
. JK
ZK (fiuK)
j&
UJKC
‘ZK
jzK
JK
‘1K
(&UK)
jh
L
jzK
j&
JK
ZK
.
jzK
j&
X
OK
OK
1K
..
..
.0 K
jzK
OK
I
“*
‘. i .. . OK
‘.. .
1K
OK
‘1K
1K
1K
lK g4
jlK
jlK
. ‘1K
jlK
1K
j
1K A
(A.5) lK
=
;2
where g3
jlK
g4 = (J%K)*~K
jlK
=[2...2
Jz
O...O]T.
(A.6)
jlK
(A.3)
Substituting (A.2), (A.3) and (A.5) into (A.l), (31) can be obtained.
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585589.
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