Interference-plus-noise covariance matrix estimation for adaptive space-time processing of DS/CDMA signals *. Ioannis N. Psaromiligkos and Stella N. Batalama.
Interference-plus-noise covariance matrix estimation for adaptive space-time processing of DS/CDMA signals * Ioannis N.Psaromiligkos and Stella N.Batalama Department of Electrical Engineering State University of New York at Buffalo Buffalo, NY 14260 E-Mail: { ip2, batalama}@eng.buffalo.edu
Abstract The presence of the desired signal during the estimation of the minimum-variance-distortionless-response ( M V D R ) or auxilia y-vector ( A V ) filter under limited data records leads t o significant signal-to-interferenceplus-noise ratio ( S I N R ) performance degradation. W e quantify this observation in the context of DS/CDMA communications by deriving two new close approximations for the probability density functions (under both desired-signal- Npresent ” and desired-signal“absent” conditions) of the output SI” and bit-errorrate (BER) of the sample-matrix-inversion (SMI) MVDR receiver. To avoid such performance degradation we propose a DS/CDMA receiver that utilizes a sample palot-assisted algorithm that estimates and then subtracts the desired signal component from the received signal prior to filter estimation. Then, to accommodate decision directed operation we develop two recursive algorithms for the on-line estimation of the M V D R and A V filter and we study their conuergence properties. Finally, simulation studies illustrate the 3 E R performance of the overall receiver structures.
1. Introduction The ideal minimum - variance - distortionless - response (MVDR) filter evaluated using the perfectly known covariance matrix of the desired-signal-free input vector can be shown to be equivalent to the MVDR filter evaluated using the perfectly known signal-present covariance matrix. However, their estimated filter counterparts are not equivalent in terms ‘This work was supported in part by the NSF under Grant ECS-0073660 and in part by the AFOSR under Grant F4962099-1-0035.
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of statistical performance measures of interest. In this paper we derive a close approximation of the probability density function (pdf) of the output signal-tointerference-plus-noise ratio (SINR) and the bit-errorrate (BER) of DS/CDMA receivers that utilize the following sample-matrix-inversion (SMI) MVDR filter estimates: The first estimate is calculated using desiredsignal-free received vectors while the second estimate is calculated using desired-signal-present received vectors. The newly developed SINR and BER approximate pdf expressions prove and quantify the need for filter estimation (training) under desired-signal-free conditions (this need was also observed long ago by array radar practitioners). In particular, comparing the two pdfs we will reason that the use of the desiredsignal-present covariance matrix estimate can lead to significant SINR and BER performance degradation when the estimate is based 0n.a limited record of input observations. To avoid the requirement for silent periods of the user of interest that requires significant coordination among the DS/CDMA users, we propose to proceed by estimating and then subtracting the desired transmission from the received vectors prior to the sample-average estimation of the covariance matrix. This way we obtain an estimate of the interferenceplus-noise covariance matrix which is then used to evaluate the MVDR or the auxiliary-vector (AV) [l],[2] filter estimates of interest. To this end we propose to use initially a simple pilot-assisted (supervised) algorithm [3] and then switch to decision-directed mode. The latter operational mode, however, requires the use of on-line recursive algorithms for the estimation of the AV and MVDR filter. Recursive AV filter estimators have not been reported in the literature so far, while the LMS and RLS recursions qualify as candidates for the recursive on-line estimation of the MVDR filter. In this paper we develop a new recursive algorithm for
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the on-line estimation of the AV filter and a modified LMS-type algorithm for the estimation of the MVDR filter. These algorithms represent low-complexity alternatives to their batch counterparts. While their development was motivated by decision directed operation needs, they can be viewed as useful stand-alone tools, as well. Theoretical results included in this work establish formally the convergence of the proposed recursive algorithms. Finally, simulation studies illustrate the performance levels achieved by the overall proposed receiver structure that operates under limited pilot signaling followed by decision-directed mode. The paper is organized as follows. In Section 2 we introduce the signal model and we review the MVDR and AV filtering principles. Close approximations of the output SINR and BER pdfs of the joint spacetime (ST) SMI MVDR receiver are developed in Section 3. Section 4 describes and analyzes a supervised interference-plus-noise covariance matrix estimator. Recursive online implementations of the MVDR and AV receiver are developed in Section 5. Finally, in Section 6 simulations studies demonstrate the performance of the proposed structures.
2. System Model and Background We consider K DS/CDMA users that transmit over a multipath Rayleigh fading additive white Gaussian noise (AWGN) channel. The multipath channel is modeled as a tapped-delay line (TDL). The k-th user baseband transmitted signal is given by u k ( t ) = Cbk(i)&~k(t-iT), k = 0,. . . ,K-1, (1)
to be identical across all antenna elements (no antenna diversity is considered) and 6'k,n identifies the angle of arrival all with respect to the n-th path of the k-th user. Tk is the relative transmission delay of user k with respect to user 0 (70 = 0) and with ~ ~ (bant ) dlimited to B = l/Tc the TDL has taps spaced at the chip interval T,. In (2), nm(t) represents additive sensor noise modeled as temporally and spatially complex white Gaussian (WG) with variance 02.The received signals ro(t),. . . , r ~ - ~ (can t ) be grouped to form the vector
where a(&) = A
(2) where N, is the number of resolvable paths, assumed to be the same for all users. In (2), Ck,n is the effective complex Gaussian channel coefficient which is assumed
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[1,e-Jksin6h,.
. . ,e - j ( M - l ) ~ s i n 6 k 1T, k =
0,. . . ,K - 1, is the steering vector associated with the n k-th user, and n(t) = [no(t),. . . ,~ M - I(t)IT. Chip-matched filtering of r ( t ) and sampling at the chip rate, l/T,, over the symbol time interval (L+N,1 chip periods) prepares the data for one-shot detection of the information bit of interest bo. By stacking the vector samples r(O), . . . , r((L N p - 1)T,) one below the other we obtain the space-time received data vector
+
~M(L+Np-I,,l~[r(o)Tr(T,)T. . . r T ( ( L +Np
- 1)TC)lT.
(4) The cornerstone for any form of joint space-time filtering is the space-time signature which, for user 0, is A defined as Vo = Cr',; CO,~ @ a(&,) where
SF'
-
" p ' q,.". .,o, dO(O),. .., d o ( L - 1),0,. ..,O]T
i
where b k ( i ) E {-1, tl} is the i-th data (information) bit, T is the information bit period and EI,denotes the transmitted energy. The normalized signature waveform s k ( t ) is given by s k ( t ) = E;";' dk(Z)$(t - IT,) where d k ( Z ) E {-1, +1} is the I-th bit of the spreading sequence of the k-th user, $(t) is the chip waveform, T, is the chip period, and L = TIT, is the system spreading gain. The received signa1 is collected by a uniform linear antenna array consisting of A4 elements, spaced halfthe-wavelength apart. The baseband received signal a t the m-th antenna element (& = 0,. . . ,M - 1) is given by
T
A
r(t) = [ro(t),T1 ( t ) , . .,TM-1 ( t ) ]
(5)
N,-n
n
and €3 denotes the Kronecker product. We assume (without loss of generality) that llVoll = 1. A linear joint S-T receiver with tap weight vector w E C'(L+Np-l) detects the transmitted bit of the user of interest as
&, = sgn(Re{wH F}) where sgn(.) identifies the sign operation, and Re{.} extracts the real part of a complex number. In this work we consider two types of linear receivers. The first type is the minimum-variance-distortionless-response (MVDR) linear receiver whose tap-weight vector is designed to minimize the variance at the filter output E { IwH F 12} while maintaining unity response in the vector direction VO.The MVDR-receiver tap weight vector is given by [4]:
(7)
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e
In (7), R E { Z H }is the covariance matrix of the received vector f . The second type is the AuxiliaryVector (AV) linear receiver [l], [2] whose tap weight vector is a member of a sequence of vectors that converges to the MVDR solution. The AV filter sequence can be obtained as follows:
+
where the index I n is used to distinguish the interference-plus-noise (desired-signal-free) input covariance matrix R I + ~= R - EoVOVCfrom the desired-signal-present input covariance matrix R. We are interested in obtaining the pdf of the output SINR of the MVDR filter estimators G ~ S M =I R&vo
A
= VO f o r p = 1,2,3, ...
(8)
wAV(0)
G,
=
PP
=
- VfRWAV(p-l)VO
RWAV(p-1)
(9)
GfRWAV(p-l)
GFRG, 0
a= 1
The auxiliary vector generation procedure may stop when Gp+l = 0. In that case W A V ( ~ )is exactly equal t o WMVDR. Formal theoretical analysis of the sequence of auxiliary-vector filters WAV(O), wAV(1), . . ., was pursued in [2] where it was shown that The MVDR and AV-type algorithms outlined above, require knowledge of the covariance matrix R which is unknown in practice and it is usually estimated by sample averaging over a finite set of joint S-T data Fj, j = 0,. . . ,N - 1. The resulting estimator R is given by N-1 R=f&. (13) h
A
N
j=o
Using 6 in (7) and (8)-(11)we obtain the MVDR and AV filter estimates f s ~ rand G S A V ( ~ ) ,respectively, where the subscript SMI stands for “samplematrix-inversion.” As illustrated in [2] for a fixed finite data-record-size N , the sequence { S A V l(p~pro) vides filter estimators with varying bias versus covariance characteristics that converge to % S M I . For short data records N , the early, non-asymptotic, elements of the generated sequence of AV estimators offer favorable bias/covariance balance and are seen t o outperform significantly in mean-square estimation error the S S Mestimator. I
v R-lv ~ ~ L and- WSMI,l+n ~ ~ o = v,HR;;nv, using N-point sample-average estimates of the desired-signal-present and desired-signal-free covariance matrix, respectively. Evaiuation 0f- these pdfs requires knowledge of the pdf of R and R I + ~ . We make the following simplifying assumption: We assume that the received vectors are identically distributed according to a multivariate Gaussian distribution N ( 0 ,R). Then, the estimator 6, is distributed according to a Wishart distribution with N degrees of freedom, WM(L+N,-~)(N-~R; N) [5]. Similarly, we assume that &l+nis distributed according to a Wishart distribution with N degrees of freedom, WM(L+N,-l)(N-’Rr+n; NI. The following theorem provides close approximations of the pdf of the output SINR of the estimated SMI MVDR filter for the case in which filter estimation is performed in the presence of the desired signal as well as the case in which filter estimation is performed in the absence of the desired signal. The proof is omitted due to lack of space.
Theorem 1 (i) The p d f of S ( @ S M Ican ) be approximated by N(1-t So)So[(N- M ( L N p - 1) 2)So+ fs(x)= 2&[Nz(So - z) (1 + S 0 ) ] 3 / 2 + z ( M ( L N , - I)(& 2) - N - 2So - 411
+
+
(ii) The pdf
+
+
o f S ( B s ~ r , ~ +can ~ )
be approximated by
3. SINR and BER pdfs of the joint S-T SMI MVDR receiver Let w be a linear S-T receiver that is distortionless in the VOdirection i.e., wHVo = 1. Then the SINR at the filter output is given by S(W) =
-
b0
’
wHR~+~w
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A
where so = ~ ( W M V D R = ) S(WMVDR,I+n) aS the Output SINR Of the i d d filters WMVDR and WMVDR,Ifn, N is the data record size, M is the number of antenna elements, L is the system processing gain, and Np is U the number of resolvable paths.
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In Fig. la and l b we plot the actual and the approximated pdf of S(G.SMI) and S(GsMl,l+n)with the range of the m i a b l e 2 in (15) and (16) normalized to [0,1]. The data-record-size N is equal to 200 and 300, respectively. An antenna array consisting of M = 5 elements is assumed. The system processing gain is L = 15 and the number of paths is N p = 3. The actual pdfs are obtained by simulating a 4user DS/CDMA system. The SNR ofthe user of interest is fixed at 8dB while the SNRs of the interferers are fixed at 13, 14, and 15dB. The delays, angles of arrival and path coefficients are chosen randomly and kept constant for the duration of the pdf evaluation. We see that equations (15) and (16) offer a very good approximation of the actual pdf. Moreover, we see that for given N, M , L and N p the SINR performance of S S M I , I + n is significantly more likely to lie near the optimal performance S(wSMI.r+n) = s W S M I = 1) than the SINR S ( L M Y D L point ( S ( W M " D R . l + J performance of G S M I . In the following, we examine how the results of Theorem 1which are based on the filter output SINR translate into BER terms. Under the assumption that the received vector is Gaussian distributed the BER P e (w) at the output of a sign detector that follows an arbitrary linear filter w (distortionless in the Vo direction) can be expressed as follows:
(m)
(17) pe(4= Q where S(.)is defined by (14). The accuracy of the BER expression in (17) is examined in Fig. 2 where we compare the approximate BER performance evaluated using (17) with the exact BER of the ideal and the estimated MVDR receiver. The BER curves of Fig. 2 are plotted as a function of the SNR of user 0. A DS/CDMA system with K = 4 users is assumed. The SNRs of the interfering users are fixed at 13, 14, and 15dB. The receiver's antenna array consists of M = 5 elements while the system processing gain L is equal to 15. The number of resolvable paths is N p = 3. The results presented are averages over 1,000 independent experiments. The delays, angles of arrival and coefficients of the paths are chosen randomly and kept constant for 10 experiments. We see that (17) offers a very accurate, yet simple, alternative tool for evaluating the BER performance of linear receivers. The following proposition provides a close approximation of the pdf of the BER of the estimated MVDR receiver for the desired-signal-present and desiredsignal-absent case. The proof is omitted due to lack of space.
Proposition 1 (i) The pdf of P,(G.sMI)can be approximated b y
x E [Pe(WMVDR), 1/21 (22)
The Pdf ofPe(GsMI,I+n) can be
by
~ P ~ , I + ~ ( Z ) = ~ ~(~)fs,r+~([Q-'(2)]~)e[~-'(~)1~/~ GQ-' 2 E [Pe(WMVDR,I+n),1/21 (19) where fs(z)and f s , ~ + ~are ( z given ) by (15) and (16), respectively. 0 In Fig. 3a and 3b we plot the actual and the approximated pdf of loglo P e (GsMl,l+n) and loglo P e ( B s ~ r ) . The latter pdfs are given by ln10zfp,,I+n(102) and In lozfP,(lo"), Z E [loglo pe(WMVDR),1OglO 1/21>respectively. The simulation setup is identical to the setup of Fig. 1. Comparing the two pdfs, we see that the BER performance of G S M ~ , I + is,significantly , more likely to lie near the performance of the ideal filter (Pe(WMVDR,l+n)= Pe(WMVDR))-
4. Interference-plus-noise covariance ma-
trix estimation The theoretical developments of the previous section reveal the advantages of evaluating the MVDR filter using an estirf;late of the interference-plus-noise covariance matrix RI+,, in place of the estimated covariance matrix 6. Although the estimate can be obtained easily during the silent periods of the user of interest, such an approach requires significant amount of coordination among users. Alternatively, we propose to estimate the interference-plus-noise component by subtracting an estimate of the desired transmission from the received samples. The proposed algorithm is based on the observation that E{bo(j)F j } = ~ V [3]. O Thus, given a known transmitted bit sequence { b o ( j ) } j we may estimate the product PO ~ V by O
e
$0
l
=-
N
N
b o ( j ) ij
(20)
j=1
and the interference-plus-noise component of the vectors F j by -rj =rj- -bo(j)PO. (21) The interference-plus-noise covariance matrix estimate is then given by A
A recursive implementation of the desired-signal-free estimator of the input covariance matrix RI+,, is summarized below:
fp, (2)= 2~Q-1(s)f~([Q-1(~)]2)e[Q-'(Z)1Z/2
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(18)
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= 0
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G1 can be expressed in the form [2]: G I = (I - VoV,H)RI+nVo.
f o r j = 1 , 2 , ...
(31)
Expression (31) motivates an equivalent evaluation of G1,i.e. as the unique solution of the equation 2; (I?) = 0 where
z;(r)S r - (I - V ~ V ~ ) R , + ~ V (32) ~. while the inverse of 6,yJn that is used by the desiredsignal-free SMI-MVDR filter estimator ( G s M I , I + n ) can be evaluated recursively by
On the other hand, the steering scalar p1 minimizes the mean square error between VfY and GFF and is given by (cf. (10))
(33) Expression (33) motivates an equivalent evaluation of as the unique solution of the equation Z:(v) = 0 with p1, i.e.
The following theorem presents the properties of the desired-signal-free recursive estimator kyinin (27) of the input covariance matrix. The proof is omitted due to lack of space.
Theorem 2 The estimator @?n in (2'7) is an asymptotically unbiased estimator of R I + ~ ,Moreover, for @ed j , RP2n is a biased estimator with bias 1 E gyJn}= ~ $ ( + j 1) + T] RI+n (29) where $(.) is the Digamma function and y is the Euler const ant. 0
Z:(v) v(GFRI+nGl) - GyRI+,Vo. (34) If we define the functions
