Gram-Schmidt Forward-Backward Generalized Sidelobe Canceller

  I. IN TRODUCTION Various adaptive beamformers [1) ave h been developed using the sample covariance matrix which

  Gram-Schmidt

  is the maximum likelihood estimation (MLE) of the ensemble covariance matrix for Gaussian array inputs.

  Forward-Backward Generalized

  B y no tin g the property that the covariance matrix of the input vector of a centro-symmetric antenna array

  Sidelobe Canceller

  is Hermitian persymmetric, Nitzberg proposed utilizing per MLE the Hermitian symmetric of the covariance matrix to improve the convergence rate of adaptive [2). beamforming Furthermore, when combined

  KEH-CHIARNG HUARNG

  with spatial smoothing, the Hermitian persymmetric

  Technology Re search Institute Ind ustrial

  MLE can decrease the number of t he subarrays

  Thiwan

  multi interference [12, required for suppressing path

  CHIE'" -CHUNG YEH

  15). T he Hermitian persymmetric MLE is formed

  National Taiwan University

  by averaging the sample covariance matrices of the forward inputs and the conjugated backward inputs. T he forward-backward averaging, however,

  (FB) increases the computational load of the weight

  Addressed here is the the discrete-time and the

  update of adaptive bea m for m ing sincc each input

  continuous-time re<un;ive least-squares (RLS) modified

  vector is processed twice, forward and conjugated

  Gram-Schmidt orthogonallzation (MGSO) algorithms for

  [12] backward. To remedy this problem, proposed two

  forward-backward (FB) generali7.ed sidelohe cancelling (GSC)

  computation-saving methods, onc for the direct-form

  adaptive beamformen;. The FB-GSC requires processing each

  adaptive beamformer and the other for the generalized input data twice, forward and coQjugated backward, and hence side10be canceller (GSC) [11).

  GSC

  its adaptation would require a rank-two update each time. This As well known, the is superior to the

  direct-form adaptive beamformer in the aspect

  paper first derives a discrete-time rank-two-update RLS MGSO

  that the GSC can easily implement multiple linear

  algorithm for this purpose. Then, by imposing the coQjugate

  constraints and can transform the constrained

  symmetry constraints, the backward data processing can be

  minimum variance criterion into an unconstrained

  removed without any loss and the complex-valued algorithm can

  interference cancellation problem. 'Therefore, the he further transformed inlo an equivalent real-valued algorithm. sidelohe cancelling weight vector can be updated by

  Therefore, significant computation savings can he achieved. Also

  applying the recursive least-squares (RLS) algorithms

  described here is the continuous-time RLS MGSO algorithms for

  which converge faster than the gradient-based

  the FB-GSC and Ihe coQjugate symmetric FB-GSC. Computer

  mean square (LMS) algorithms, such as the least

  simulations of Ihe FB-GSC and the forward-only GSC are

  algorithm. The FB update of the sidelohe cancelling presented.

  RLS weight vector using the algorithm has been discussed in [12]. However, it is difficult to implcment the direct update of the sidclobe cancelling weight vector in a modular structure which is desirable in _. a d a ptive signal p rocess o rs viewpoint f

  From the o fast convergence rate RLS and modular implementation, the modified

  (MGSO) a Gram-Schmidt orthogonali7.ation is preferable approach to realize the interference cancellation in GSC [4-7). the In the configuration

  GSC MGSO it MGSO of the with a processor, is the weights, instead of the GSC s i delob e cancelling

  .

  weight vector, to be a dapt e d Both the discrete-time Manuscript received July 7, revised a ua 11, 1993. 1992; J n ry and RLS MGSO continuous-time rank-one-update IEEE Log No. T-AES/30/l/13049. algorithms have been developed in [3) and [13), fo a n ^- respectively, which arc suitable for the rw rd o ly

  Authors' addresses: K. C. Huamg, Computer and Communi c ati on GSc. Research Laboratories, Industrial 'Technology Resear In realizing the FB ad a p t i ve beamforming,

  ch Institute,

  Hsinchu, Taiwan, KO.C.; C. C. Yeh, Department of Electrical

  r p a ank-twa-u d te hecomes necessary due to the

  Engineering, National Taiwan University, Thipei, Thiwan, R.O.C

  addition of the backward data. In this paper, we RLS MGSO will first derive a rank-two-update FB-GSC.

  0018-9251/94i$4.00 1994 IEEE algorithm for the discrete-time To reduce

  @

  IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.

  30. 1\'0. 1 JANUARY 151

  1994

  (10) The determination of wa(t) in the GSC is an "unconstrained" least-squares optimization problem and its update can be achieved by using the RLS algorithms.

• element antenna array with the input vector where

  With the GSC model of (4), the beamformer output is given by yet) ⁼ d(t) - wf/ (t)u(t)

  (7) where d(t) ⁼ w

  :

  x(t) u(t) ⁼ 8Hx(t).

  (8) (9)

  In the above, d(t) is the 1 x 1 quiescent beamformer output and u(t) is the N x 1 sidelobe canceDing signal vector. By the minimum variance criterion (2), the sidelobe cancelling weight vector wa(t) is determined by

  I min

  Ee-r

  Id(r) ^- wf/ (t)u(r) 12.

  W.(I)

  r=O

  B. Forward-Backward GSC If the spatial distribution of the antenna elements are centro-symmetric, e.g., an equispaced line array, the phase response vector of the array elements to the far-field narrowband signal source would be in the form of conjugate symmetry for the phase reference point chosen at the array geometric center. That makes the characteristics of the impinging signals remain unchanged in the conjugated backward input vector

  weight vector. In (6), the symbol denotes a zero matrix (or vector) with the subscript indicating the size.

  Xb(t) ⁼ [xK(t), ... ,x2(t),xj(t»)' (11) where * denotes complex conjugate and the subscript b denotes "backward". Therefore, the backward data can be incorporated with x(t) to adapt the array weight vector. In the FB configuration, the array weight vector, subject to the constraint equation (3), is determined by

  It has been shown in [2 and 12] that as compared with the forward-only method, the FB method can significantly improve the performance of adaptive beamforming.

  The GSC realization of the forward-backward beamformer (FB-GSC) is depicted in Fig.

  1. In the figure, we have defined db(t) ⁼ w

  :

  Xb(t) (13) Ub(t) ⁼

  8Hxb(t) (14) where db(t) is the 1 x 1 backward quiescent beamformer output and Ub(t) is the N x 1 backward sidelobe cancelling signal vector. It is convenient to denote the FB quiescent beamformer output vector and the FB sidelobe cancelling signal matrix as d(t) ⁼ [d(t), db (t»)'

  U(t) ⁼ [U(t),Ub(t)]

  (15) (16)

  152 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.

  30,

  (5) (6)

  x 1 adaptive sidelobe cancelling

  the computational load, we will apply the conjugate symmetry constrained FB-GSC method [12] and then transform the complex-valued rank-two-update algorithm into an equivalent real-valued algorithm. h to the continuous-time FB-GSC, we also derive a continuous-time rank-two-update RLS

  M

  MGSO algorithm and then further develop a computation-saving algorithm by using the conjugate symmetry constrained FB-GSC method.

  II. BASIC CONFIGURATIONS

  A. Forward-Only GSC Let us consider a

  K

  I denotes transpose. The output of the adaptive

  beamformer is formed by yet) ⁼ wH (t)x(t) (1) where H denotes complex conjugate transpose and wet) is a K

  x

  1 array weight vector. According to the linearly constrained minimum variance criterion, the optimal array weight vector at time t is determined by

  (2) subject to the constraint CHw(t) ⁼ f (3) where

  0 < e < 1 is the forgetting factor, C is a K x M matrix consisting of

  M linear constraint vectors, and f is an

  x

  N

  1 vector containing the constraint values.

  The linearly constrained beamforming described above can be easily implemented by the GSC. In the GSC, the array weight vector is expressed as wet) ⁼

  Wq ^-

  8wa(t) (4) where Wq is a K

  x 1 fixed quiescent beamtormer

  defined as

  8 is a

  K

  x

  N, N ⁼ K ^- M, signal blocking matrix of full column rank satisfying

  CH8 = OMxN and wa(t) is an

  NO. 1 JANUARY 1994

2. Then the

  in the sense of least squares. To be specific,

  x

  is an i

  Ui(r)

  (19) where

  � i(t)Ui(r)

  It) = u�(r) - g

  e�1 i(r

  represents a 2 x 1 error vector defined as

  ek I i

  i}

  U(r)

  U

  I, ... ,

  estimated by {u

  Uk

  N + 1, represents the residual of

  1 � i < k �

  it

  I

  ek

  3, the symbol

  2 MGSO input data matrix consisting of the first i rows of

  Ui(r) = [Ul(r), U2(r), ... , Uj(r)]' and

  }

  For the consistency of the forward and the backward notations, we denote ek\i(r It) ⁼ [ekli(r

  {ell/-I(·I t)}!=1 HUARNG Y EH: GRAM·SCHMIDT FORWARD-BACKWARD GENERALIZED SIDELOBE CANCELLER 153

  is described below. From the orthogonal property of (21), one can see that

  � i < k � N

  t) for 1

  ek\i(-I

  The MGSO algorithm generating

  (17) would be yet) = eN+IIN(t It).

  UN(t) ⁼ U (t), we find that gN +1\N(t) ⁼ wa(t) and that the FB-GSC beamformer output vector defined by

  Comparing (19) and (20) with (18) and using UN+l(t) = d(t) and

  I t),eb,k\i(r I t)]/.

  1'=0

  ItIoli(t) is the i

  (21)

  It) ⁼ Oixl.

  t L:�t-TUi(r)ek\i(r

  From the orthogonality principle, it follows that

  UN(r) ⁼ U(r).

  I t) = Uk(r) and

  eklo(r

  (20) Note that

  1 estimation weight vector satisfying

  x

  , from which we derive the desired RLS MGSO algorithm. In Fig.

  . , t

  (18) involves a rank-two update each time. The application of the RLS algorithm to the direct adaptation of

  , the side lobe cancelling weight vector wa(t) is determined by

  The adaptation of wa(t) in the FB-GSC described by

  denotes the Euclidean norm. Note that the addition of the backward data is solely for the purpose of improving the adaptation of weights. The desired beamformer output yet) is still computed from the original forward input, i.e., yet) = [1,O]y(t).

  11·11

  where

  1'=0

  d/(r) -w�(t)U(r)112 (18) w.(t)

  ll

  L: � t

  min

  t

  2 )

  Wa (t) for the FB-GSC has been discussed in [12]. However, the RLS MGSO would be more preferable due to its modularity. We derive a rank-twa-update RLS MGSO algorithm for the

  According to (1

  1 vector yet), which is given by y/(t) ⁼ d/(t) -w�/(t)U(t). (17)

  forward and the backward beamformer outputs can be written as a 2 x

  (t) is N x

  U

  where d(t) is 2 x 1 and

  MGSO processor.

  2 Configuration of GSC with

  Fig.

  ty(t) Fig. 1. FB-GSC.

• 1"

  FB-GSC and utilize the conjugate symmetry criterion to develop a computation saving technique.

  .

  U(t),

  We now consider the "batch" MGSO processing of {ul(r)"",UN+l(r); r ⁼ 0,1, .

  3 with N = 3.

  The detailed configuration of the MGSO processor is given by Fig.

  UN +1 (t) ⁼ d(t).

  and Ub(t), respectively. As to the (N + l)th MGSO input, it is the FB quiescent beamformer output vector, i.e.,

  u(t)

  Ub,j(t) are the ith elements of

  Uj(t) and

  3. Configuration of MGSO. where

  i.e., i =1,2, ... ,N Fig.

  � N, is formed by the transpose of the ith row of the FB sidelobe cancelling signal matrix

  C. Modified Gram-Schmidt Orthogonalization The configuration of the FB-GSC incorporating the MGSO processor is depicted in Fig. 2, which can produce the beamformer output yet) without the explicit computation of

  1. The ith MGSO input, 1 � i

  . ,N +

  .

  1, .

  Uj(t) for i =

  1 vector, denoted

  x

  1 inputs and each input is a 2

  The MGSO processor has N +

  Wa(t) [3-5].

• 1

  1994

  [

  x

  i, rik(t) is i xl, rjk(t) is 1

  x

  R(t) is i

  1 (32) where

  akli(t)=

  ]

  T=O and

  I i (t) is (i

  rik(t) = 2:e-TU:(T)Uk(T) (31)

  t

  T=O

  2:e-TUi(T)U;;(T) (30)

  I rik(t) =

  (29) T=O

  I Ri(t) = 2: e-TUj(T)U[i (1')

  For the convenience of derivation, we first define

  1, and ak

  x

  (28) for k ⁼ i + 1,oo.,N

  Wik(t) is an i

  NO. 1 JANUARY

  30,

  154 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.

  (36) (37)

  Wik(t) = H . ri_l)t) rik(f)

  ]

  Ri_l(t) ri_l,k(t)

  x i matrix defined as [

  14] (J'ik(t) = af{;_I(t)Wik(t)akli-l(t - 1) where

  1. Then the solution of (20), known as the Wiener solution, can be explicitly written as gklj(t) = Ril (t)rik (t)

  (35) From (33) and (34), it follows that [

  = i) is given by (J'ik(t) = �(J'ik(t -1) + e:li_1 (t I t)ek I i

  (J'jj(t) with k

  Wik(t), we first show below that thc time update of its numerator (J'ik(t) (or denominator

  A. Time Update To derive the time updatc of

  (34) With (33) and (34), we have the following derivations.

  (24) as (J'jk(t) = rik(t) - rt:..l,i(t)Ri-_\ (t)ri_l,k(t).

  (33) and the partial covariance can also be explicitly expressed by using

  1, ... ,N. In the following, we derive an algorithm based on the a priori errors to achieve the time update of weights from Wik(t -1) to Wik(t) and obtain the a posteriori errors ekli(t I t). Especially, eN+IIN(t I t) is the desired output.

  Wik(t -1) to the present inputs Ui(t). That is, ekli(t It-I) ⁼ ekl i-let It-I) -Wikel -l)ejli_l(t It -1)

  Equation (26) is applied for computing ekl i(T I t) in the MGSO as indicated by Fig. 3. Besides, instead of

  (22) (23) (24)

  }

  (t I t -1) can be formed by applying the weights

  wik(t)elll-I(T It)

  �

  Uk (1') -

  {

  1 scalars. Equation (22) can be rewritten as ekl i(T It) =

  T=O which are 1 x

  (J'1I(t) = 2:e-Tllelll_I(T I t)W (25)

  (J'lk(t) = 2:�I-Te;II_I(T I t)Uk(T) T=O

  (24), the MGSO algorithm computes (J'jk(t) by

  I

  Wlk(t) given by Wlk(t) = (J'lk (t)!(J'1I (t)

  Uk(T)-2:Wik(t)elll-I(T It) 1=1 with the weight coefficients

  (19) can be written as i ekli(T I t) ⁼

  forms the orthogonal basis for {UI (-)} i = I' Therefore,

  Assume that at the previous instant time t - 1, we have obtained the weights

  Wik(t -1) based on the data Di(T), l' =

  O,l, .. . ,t - 1. Then, at the present time t, the a priori errors ek

  I

  i

• 1 and i ⁼
• wtk(t)eili_I(T It) which is equivalent to ekli(T It) = ekli_I(T I t) - W;k(t)eili_1(1' It). (26)

• �I;(t)
• 1)

  t (J'ik(l) = 2:e-Te:li_I(T I t)ekli_I(T It).

  (27) T=O

  T he discrete-time rank-one-update RLS MGSO algorithm has been developed in [3], which is only suitable for the forward-only GSc. Following the configurations in the preceding section, we derive in this section a rank-two-update algorithm for the FB-GSC.

  III. DISCRETE-TIME RANK-lWO-UPDATE RLS MGSO ALGORITHM

  In the above descriptions, we have considered the discrete-time adaptive beamformer. A� to the continuous-time beamformer to be discussed in Section Y, all of the above equations still are applicable except that the summations should be replaced by integrations.

  III, we derive an adaptive algorithm for the real-time update of the MGSO weights.

  Wik(t). In Section

  while Wjk(t) is the well-known partial correlation coefficient. T he batch MGSO algorithm requires all of the past data for computing

  {UI(')};:!

  and Uk(-) given

  UjO

  = i. In view of (27), (J'ik(t) can be termed the partial covariance between

  (J'ik (t) with k

  and hence (J'ii (t) in (25) is a special case of

  {D1C)};:�

  24) can be verified by using the fact that eili-ICI t) is orthogonal to

  The equivalence of (27) to (

• I (t I t - 1).
• Ui(/)[V�I(t),UZ(t)].

  155

  =

  1, ekli

  x

  1

  Wtk are

  and

  O"n

  ",_Ill) - <H(tll)e:H(lltj!(J,,(l) e:+ti,(tlt) ⁼ e:+ti,(U - I)",(t) End yet) ⁼ [1, 01 eN+IN(tlt) where

  End ",U) ⁼

  J1),,(t

  widt)

  1, and OJ is 2 X 2.

  ,:V

  For k = i-I, ^{, .}

  (t I), e:li-1(tlt)<HUII

  O',,(t) ⁼ �O' ..

  For i = 1,2jo .N

  [d(l), d,(t) I' "o(t) = 1'<2

  eN+llo(tlt) ⁼ eN+llo(tlt

  , N

  1,2, ..

  i =

  is 2 X

  Then (41) and (42) follow from the substitution of (45) into (44).

  eilo(tll) ⁼ eilo(tlt

  1 H Ril(t) =

  HUARNG & YEH: GRAM-SCHMIDT FORWARD-BACKWARD GENERALIZED SIDELOBE CANCELLER

  1 vector. In this section, we utilize the conjugate symmetry constraint method

  x

  [3], the computational load of the rank-two-update algorithm in Thble I for the FB-GSC increases because each error signal is a 2

  IV. COMPUTATION SAVING TECHNIQUES Compared with the rank-one-update algorithm in

  (46) form the rank-two-update RLS MGSO algorithm as summarized in Thble I. Like the rank-one-update RLS algorithms, the proposed algorithm is also well suited for the triangular wavefront systolic array implementation [8-10].

  Equations (28), (35), (39), (41), and

  Olx(i-l) O'ii t (47)

  I-I

  ]

  c. Order Update of Conversion Matrix Although thc a posteriori error ek I i (t It) can be obtained from (41) using the a priori error ek

  (t) O(i-l)Xl

  R� l

  [

  ,N. The order update of the conversion matrix is given by (46) which can be derived from (42) with the inverse matrix Ri- \t) substituted by [3, 14]

  1,2, ...

  Oi-I(t) for i =

  (42) is not applicable in real time processing. The conversion matrix should be iteratively obtained by the update of ai(t) from

  (t I t - 1) and the conversion matrix ai (t),

  j

  I

  [Ui(t), Ub,,(t)]',

  MGSO Algorithm for FB-GSC.

• 1) ⁼
• 1) ⁼
• e;li_l (t I t)eZIJt It - l)/ITii(t).
• 1)

  Note that O'ii(t) in (39) can be time recursively obtained by (35).

  Rl.S

• I ekl,(tlt
• I) = ekl,_Jitlt
• 1)
• w:k(t - l)e'I,_dtlt -I)
• 1) + e:Hltlt)e;I,ltlt - 1)1(J'i(t)

  In the literature concerning the rank-onc-updatc RLS algorithms, a would be a scalar and

  I t - 1) = Uk (t), it follows that ao(t) ⁼ I2x2'

  I denotes the identity matrix and its subscript indicates the size. Since eklo(t It) = eklo(t

  (42) The symbol

  (41) where ai(t) is a 2 x 2 conversion matrix given by ai(t) = I2x2 -U;'''ct)Ri-1(L)Ui(t).

  The convcrsion is stated as e�li(t It) ⁼ e�li(t It -1)ai(t)

  In this subsection, we present a simple approach to derive the conversion between the a priori and a p o s t e r io r i errors.

  Errors Both the time update formulas (35) and (39) require the a posteriori errors ek I i (t It).

  B. Conversion Between A Posteriori and A Priori

  (40) Then (39) follows from dividing cvcry term in (40) by O';;(t).

  1 - a is usually called the likelihood varinble.

  t It-I).

  O'ik(t)-Wik(/-l)O'ii(t) = e;li_l(t I t)ekl;(

  (28) and (35) to obtain

  (39) To prove (39), we use

  Wikel) = Wik(/-l)

  Wik(t), it is given by

  (38) Then substituting (38) into (36) yields (35). As to the time update of

  The matrix given by (37) can be decomposed as CJ!ik(t) = �CJ!ik(t -1)

  TABLE

  I Discrete-Time Rank-1Wo-Update

  To prove the stated conversion, we define the following (i + 1) x (i

• 2) matrix

  Ri(t) ViCt)

• ^_ ( )aili-l(t)aili_l(t).
• 1)
• hX2

  O i

  (45)

  [ Vi (t),02x2] .

  X I(i+2)x(i+2)

  }

  ] H

  2

  X

  { [

  ]

  (43) (44)

  .pik(t)= �rt£(t-l) u�(t)

  Viet) ]

  �Ri(t

  [

  We now decompose .pik(t) as follows:

  rik(t) u�(t) Then, using ( 19) and (33), we can obtain

  H .

  .pik(t) =

  [

  'ik(t)

  I t) and hence each error vector is conjugate symmetric, i.e.,

  (57) where

  [ai,l1 (t) a; 21 (t)] ai(t) = ' ai,21 (t) ai,l1 (t)

  one may see that the conversion matrix described by (46) is not only Hermitian but also persymmetric. That is, the conversion matrix can be expressed as

  ao(t) = 12x2,

  Furthermore, with (56) and

  (28), (35), (39), and (41) can be saved.

  Since every backward error is the complex conjugate of the associated forward error, thc computations involving the backward errors in

  eb,kli(T It) = eZ1i(T

  represents the (p,q)th element of aj(t). Note that

  is conjugate symmetric, it follows from (19) that

  Ui(T)

  is real and

  gkli(t)

  (55) Furthermore, since

  given by (33). Accordingly, it follows from (34) that lTik(t) is real-valued and so is the MGSO weight, i.e.,

  gkli(t)

  defined by (29)-(31) are all real-valued, and so is the weight vector

  aj,pq(t)

  ai.l1(t)

• Re{

  Re{

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  (61) IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.

  we can transform the complex-valued rank-twa-update algorithm into the real-valued rank-two-update

  I,

  By substituting (58) and (59) into Thble

  ai,21 (t)} which is also real.

  and

  is real since

  =

  aj,21 (t)} ai(t)

  (aj,l1 (t)

  (59), the transformed conversion matrix can be shown to be _{_}

  (57) into

  (58) (59) where the overbars represent the transformed quantities. We show below that the above transformations convert the complex-valued parameters into real values. Using (56), the transformed error given by (58) can be shown to be where Re and 1m denote the real and the imaginary parts, respectively. Substituting

  B. Complex-to-Real Transformation Following the above conjugate symmetry formulations, we propose the following transformations to further simplify the computations

  ai(t) is Hermitian.

  Im{ai,21 (t)} Im{ai,21(t)} ] aj,l1 (t) ^-

  R;(t), rik(t),

  to eliminate the backward data computations in the FB-GSC. After that, we utilize a complex-to-real transformation to further reduce the computational load of the rank-two-update RLS MGSO algorithm.

  given by (5) is also conjugate symmetric. That is, .IC* = C

  "

  [:

  =

  K matrix defined as J

  x

  where J is a K

  JB' =B Jw; =Wq

  Wq

  Using J, we can rewrite (11) as Xb(t) = .lx' (t).

  with columns being conjugate symmetric. Since r is usually chosen to be real-valued,

  B

  to be adapted, the conjugate symmetry constraint method still is applicable. Considering a centro-symmetric antenna array, all of the array phase response vectors and the associated derivative vectors will be complex conjugate symmetric if the phase reference point is chosen at the array geometric center. Under this circumstance, the columns of C are all conjugate symmetric if the multiple linear constraints are composed of the look-direction unit-gain constraint, the look-direction derivative constraints, and/or the known-jammer null constraints. Therefore, as described in [12], we can design a signal blocking matrix

  Wik(t)

  of the FB-GSC. Although with the MGSO processor, it is

  wa(t)

  [12], the conjugate symmetry constraint method was proposed to simplify the RLS direct update of

  A. Conjugate Symmetry Constrained FB-GSC In

  J

  (48) (49) (50) (51)

  must be real-valued. Therefore,

  ,N + 1.

  uW)u;;(t)

  From the conjugate symmetric property of (54), we find that

  N) multiplications.

  given by (13) and (14) can be avoided, which amount to K (1 +

  Ub(t)

  and

  db(/)

  (54) Since the backward MGSO inputs are the complex conjugates of the forward MGSO inputs, the computations of the backward data

  .

  With (51) substituted into (13) and (14), it follows from (48)-(50) that

  i ^{= 1,2, . .}

  Ui(t) ⁼ [Ui(t),ui(t»)',

  and each MGSO input vector would be conjugate symmetric, i.e.,

  Ub,j(t) = u;(t)

  Therefore,

  (52) (53)

  = u· (t).

  db(t) = d' (t) Ub(t)

  1994

• I) ⁼
• 1)

• - I) = ;;'1;_1(111 - 1) -w,.(1 -1);;'I;_I(t,t

• 1)
10<
• e:I,_l(lll)ihl;(tlt -1)/u,;(I) Eno

• llN real multiplications and 2
• 33N
• gkli(f)Ui(T)

  real multiplications and 4N real divisions. As to the real-valued rank-twa-update algorithm given in Table II, it is suitable for the conjugate symmetric FB-GSC and requires 2 N2 +

  12N real multiplications and 2N real divisions. Therefore, Table II can save significant amount of computations and its computational load is even less than that of the rank-one-update algorithm in [3].

  Concluding this section, we remark that the FB processing required by the FB-GSC can be replaced without any loss by the forward-only processing with conjugate symmetry constraints imposed on the FB-GSC. Besides, the conjugate symmetry constraint method, when applicable, can incorporate with the complex-to-real transformation to achieve significant computation savings for the FB-GSC with a RLS

  MGSO processor. v. CONTINUO US-TIME ALGORITHMS

  As to the continuous-time RLS algo r i t h

  ms

  [16], there has been a rank-one-update MGSO algorithm developed in [13]. That algorithm, however, is o n l y suitable for the forward-only GSc. In the continuous-time FB-GSC, the array inputs must be processed continuously together with the backward array inputs. In this section, we handle this problem and derive two continuous-time RLS

  MGSO algorithms, one for the FB-GSC and the other for the conjugate symmetric FB-GSC.

  All of the configurations in Section II are applicable here except that the discrete-time weighted summation

  L:�=o�t-r should be replaced by the

  continuous-timc weighted integration

  f� drr&gt;.(t-r),

  a posteriori errors.

  where 0 &lt; A is the exponential forgetting factor. In the continuous-time domain, there is no difference between the a priori and the

  Thereforc, (28) should be rewritten as ekli(t I t) ⁼ ekli-l(t I t)- wik(t)eili-l(t I t) (63) where

  Wiket) is to be adapted. To derive the time update, we rccall that Wik(t) ⁼ Uik(t)/Uii(t) and then we have

  Wik(t) ⁼ {O'ik(t) - O'ii(t) Wik (t)}/O"ii (t) (64) where the dot indicates d

  / d t. From (27), the continuous-time definition of O"ik(t) is given by

  Similar to (21), we have the following orthogonal property

  lot

  e-&gt;.(t-r)Ui(r)ekli(T I t)dT ⁼ Oi&gt;&lt;l. To derive

  O'ik(t), we first note from (19) that dekl;(T It) ^.

  H

  dt ⁼

  (65) (66)

  (67)

  8N2

  N real div isi ons. The complex-valued rank-two-update algorithm given in Thble I which is suitable for the FB-GSC requires

  4N2

  u;;(l) = (U,,(t - I) + ;;:h_1(tlt);;'H(tlt

  TABLE Il

  

Ttansformed Discrete-Time Rank-'lwo-Update RLS MGSO Algorithm for Conjugate Symmetric FB-GSC (Real-Valued)

  ;;;Io(tlt) = ';'Io(tlt - 1) =

  [Re{ui(t)}. Im{ui(t)} I',

  i ⁼ 1,2, . ^.

  . ,N

  ;;N+110(tlt)

  =

  ;;"'+110(111

  [Re{d(t)}, Im{d(t)} l' iio(t) = I",

  For i = 1, 2, ... , l\/

  For k ^� i +

  Detailed comparisons between the computational loads of the algorithms given in [3], Table I, and Thblc II are given below. T he complex-valued rank-one-update algorithm given in [3] which is suitable for the forward-only GSC requires

  1 .... ^.

  N + 1 e'I,(t11

  .. (1) ⁼

  w,.(/ -1)

  iii(t) ⁼ if,_l(l) -e,li-1(llt)e:H(III)/u;;(I) ;;:+11,(111)

  =

  1':+11;(111 -1)iii(l) End y(l) =

  [1,j] ;;N+1IN(III) where 0-11 and Wik are 1 x 1, ekl' is 2 x 1,

  and (}j is 2 x 2.

  algorithm listed in Thble II where we have denoted (fu(t) ⁼

  � a';;(t).

  (62) Note that only the forward data is involved in Table II. T he transformed algorithm is completely identical to the original algorithm, except that all of the quantities are transformed into real values.

  

HUARNG &amp; YEH: GRAM-SCHMIUT FORWARD-BACKWARD GENERALIZED SIDELOBE CANCELLER 157

  

TABLE III

Continuous-Time Rank-'l\vo-Update RLS MGSO Algorithm for FB-GSC

₌

  eilo(tlt) =

  Ub •• (t) i .. . , N

  [u,(t), I', 1,2, ₌ eN+1Io(tlt)

  [d(t) dl(t)]' , .

  For i=1,2, ,N ₌ . . _{= "}

• A&lt;7,.(t) e'H(tlt)

  f.&lt;7;;(t)

• +

  For k i + 1,

  II 112

  I

  . , N + ₌ ^- e'I.(tlt) = ekl,_l(tlt) w'k(t)eil.-1(tlt) l.W,k(t) eili-1(tlt)eZli(tlt)!&lt;7;;(t)

  End End yet) = [1,OJ eN+1IN(tlt)

  Will: x x

  where CTii and are 1 1.

  1 and ekli is 2 TABLE N algorithms [13, 16]. Therefore, the continuous-time

  Continuous-Time RLS MGSO Algorithm for Conjugate-Symmetric

  RLS MGSO algorithms are much simpler than the FB-GSC. discrete-time ones.

  .

  If the conjugate symmetry constraints are imposed €N+llo(tlt) = d(t) on the FB-GSC, the adaptation of _{= ,}

  €'Io(tlt) = u,(t), i = 1,2, . , N .

  Wik(t) could be even For i l,2, N simpler. Under this condition, it has been shown in ...

Section IV that

  Wik(t) is real-valued. Furthermore, we (t) = -Aa + 1€i1,-,(tlt)I'

  1.a (t) .. .. can use the conjugate symmetric property of (56) and

  For k . , N ^{= i + 1,} 1 ..

• the definition of (62) to reduce the algorithm given in

  €'Ii(tlt) = - wik(t)eiH(tlt) Eklo-1(tlt)

  Thble III into that given in Thble N where only the

  1.W'k(t) = Re {eili_1(tlt)ekl,(tlt)} I a .. (t) forward data processing is required. Note that Thble IV is a complex-valued rank-one-update algorithm.

  End Further note that weight update in Thble IV only

  End requires computing the real part of eiji-l(t I t)ekji(t It) yet) = eN+w.(tlt) while the rank-one-update algorithm in [13] used for 1 X 1. where all of the quantities are the forward-only GSC requires computing both the real and the imaginary parts of eili_l(t I t)ekji(t It). and then by using

  (66) and (67), we have (T It) de

  ;

  j

  e )d - ^•

  VI. SjMULATIONS ekji T t T - ^,

  o d t

  ;" 11 -),(1-7') ( I i &lt; k S; m.

  Computer simulation results are presented in this (68) section. We assumed a five-element equispaced line

  Using (68) and applying the formula array with a look-direction constraint only. The GSC d d¢(t T) we adopted possesses a uniform-weight quiescent dT

  ¢(t,T)dT = ¢(t,t)

• ;jf-
• dt beamformer ₌

  Wq [1,1,1,1,1]'

  we can evaluate the differentiation of O'ik(t) given by

  (65) and obtain ₌ and a conjugate symmetric signal blocking matrix

i-I (69) O',dt) e;ji-l (t I t)ek j (t It) - ,\(J'ik (t).

• j 0 0 -1.5 By substituting (69) into

  (64), we obtain the time

  1 0 -j -1 update

  2

  B = 1 ₌ Wikel) e:ji_l(t I t)ekji(t I t)/O'ii(t) (70)

  1 j -1 where ekji(t I t) is given by (63) and O'ii(t) can be

  5 a

  1

• ^- j ^.

  generated by (69). The resulting continuous-time rank-two-update RLS MGSO algorithm is summarized The look-direction desired signal source was located in Thble III. It is of interest to note that the conversion at the far-field broadside and was a baseband signal

  a a

  between source modulated by a complex sinusoid carrier of priori and posteriori errors and its order update, which are essential in the discrete-time frequency /0. The array element spacing was equal to one-half carrier wavelength. The base band signal algorithms, are absent from the continuous-time

  IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. NO. 1 JANUARY

  158 30, 1994

  FB�GSC FB·GSC 600 S[X) normalized time normalized time

  5. Output SINR of GSC with discrete-time RLS MGSO alg r i For ( a rd -u n l ) GSC, rank-one-update a lg o ri th algorithm. Same scenario as Fig.

  Fig. Output continuous-time 4. SINR of GSC with RLS MGSO Fig.

  4 i th Nyquist sampling rate. F or

  o thm. fo rw y m w

  

given in [13] is loye d . As to FB-GSC, o njug at e symmetry (forward-only) GSC, rank-one-update algorithm gi v n in [3) is

  e mp c e c on st r a ints are imposed and algorithm given in Thble

  IV is (",played. As to FB-GSC with conjugate symmetry constraints,

em pl ed. ,\ = l or th gi ven l e II employed. � =

a m in Ta b is cxp( -0.0005 /2).

  oy i 0.0005 fb. g source was modeled as having flat spectrum over algorithms for the FB-GSC, which has a faster the bandwidth extending from

• 10 to Ib where ₌₌

  convergence ratc than the forward-only GSc. To Ib fo/20. The desired signal measured at each reduce the computational load of the rank-twa-update, array element is 20 d8 over white noise. We assume we eliminated the backward data processing by that before the GSC with MGSO was at the t == 0, imposing the conjugate symmetry constraints on the interfcrence-absent steady state. At time an t == 0,

  FB-GSC. We further reduced the discrete-time and the equal-power uncorrelated interferer having the same continuous-time complex-valued rank-two-update RLS frequency spectrum as the desired signal appears at MGSO algorithms into a real-valued rank-twa-update the angle

  30° relative to array broadside. Then the algorithm and a complex-valued rank-one-update RLS MGSO processor begins to adapt the weights to algorithm, respectively. 80th reduced algorithms for achieve interference cancellation. the conjugate symmetric F8-GSC require computations

  To simulate thc continuous-time RLS MGSO even less than those of the corresponding algorithms algorithms, we utilized thc Euler method because for the forward-only GSc. only the first order derivatives arc involved in the differential equations. 80th the forward-only

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  4. T he sampling

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  VII. CONCLUSIONS

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  rop rt s ptiv algorithms. a da e

  We have derived the discrete-time and the

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