Maximum Ratio Combining of Correlated Diversity Branches with Imperfect Channel State Information and Colored Noise

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1 Maximum Ratio Combining of Correated Diversity Branches with Imperfect Channe State Information and Coored Noise Lars Schmitt, Thomas Grunder, Christoph Schreyoegg, Ingo Viering, and Heinrich Meyr Institute for Integrated Signa Processing Systems ISS, RWTH Aachen University, Germany Emai: {Schmitt, Siemens mobie, Germany {Thomas.Grunder, Nomor Research, Germany Abstract Recenty, an anaytica expression for the mean bit error probabiity of a binary moduated signa has been derived for a receiver performing maximum ratio combining of correated diversity branches based on imperfect channe state information. In this work, these resuts are extended to the genera case of coored additive noise. Furthermore, an aternative soution is derived which yieds numericay stabe resuts aso in the case of the respective channe and interference covariance matrices having cosey-spaced or muti-fod eigenvaues. Finay, the resuts are appied to a spatio-tempora W-CDMA Rake receiver, where the effects of mutipe-access-interference (MAI) and interpath interference (IPI), due to mutipath propagation, on the mean bit error probabiity are investigated. I. INTRODUCTION Maximum ratio combining (MRC) is a specia form of genera diversity combining, by which mutipe repicas of the same information-bearing signa received over different diversity branches are combined so as to maximize the instantaneous SNR at the combiner output. It is a cassica and powerfu technique to mitigate the effects of severe fading, which occurs particuary in wireess communication systems. However, in many appications ike in CDMA systems, the channe fading coefficients are in genera mutuay correated and have different second order statistics affecting the performance of MRC. Exampes for this are cosey spaced antenna eements at the receiver and a mutipath intensity profie with unequa channe tap powers, respectivey. In addition, the channe coefficients are not known at the receiver and have to be estimated, either by the use of known piot symbos or in a bind manner. As it is we known this has a significant effect on the performance of MRC, 9. The significance of imperfect channe knowedge increases with increasing channe dynamics, since the quaity of the channe estimation worsens for increasing channe dynamics 3. There are many contributions deaing with the performance of maximum ratio combining, see e.g. 4-9, but neither of them covers the genera case of non-identicay mutuay correated fading diversity branches with ony partia channe state information being avaiabe at the receiver. In the anaysis has been carried out for the specia case of the downink of a BPSK-based W-CDMA system. Recenty, Dietrich and Utschick derived a genera anaytica expression of the mean bit error probabiity for BPSK signaing. In this contribution we extend the resut of to aso account for the genera case that the additive noise of the respective diversity branches is aso mutuay correated. Coored interference arises for exampe in case of a muti antenna eement receiver operating in a spatiay correated scenario with mutipath propagation or mutipe interfering users. In these cases there is a mutua correation between the noise sampes corresponding to the respective antenna eement outputs, i.e. the spatia diversity branches. If the eigenvaues of the channe covariance matrix are cosey spaced, the evauation of the expression for the bit error probabiity presented in becomes numericay unstabe. This happens for exampe in the case of a spatia Rake receiver empoying mutipe antenna eements where the spatia correation between the fading coefficients at the antenna eements is very ow. Therefore, we present an aternative numericay stabe expression for the mean bit error probabiity, which impies the numerica evauation of a one-dimensiona integra with a rea vaued integrand. This paper is organized as foows. After introducing the system mode in Section II, the derivation of the different expressions for the bit error probabiity is shown in Section III. Finay, in Section IV, the resuts are appied to a spatio-tempora W-CDMA Rake receiver, where the effect of mutipe-access-interference (MAI) and interpath interference (IPI), due to mutipath propagation, on the mean bit error probabiity is investigated. Section V concudes the paper. A. Information-Bearing Signa II. SIGNAL MODEL Assume that BPSK moduated data symbos a (d) n {+σ d, σ d are transmitted with a n (d) = σd and n denoting the time index. Then, the vector x n = x,n,..., x L,n T comprising the L spatio-tempora diversity branches, which are used for maximum ratio combining at the receiver, may

2 be generay modeed as x n = a (d) n h + v n, () where h = h,..., h L T and v n = v,n,..., v L,n T denote the effective channe vector and the additive noise vector, respectivey. According to the assumption of Rayeigh fading, the effective channe vector h n and the noise vector v n are modeed as zero-mean compex Gaussian random vectors with covariance matrix K h and K v, respectivey, i.e. h N C (, K h ) () v n N C (, σ vk v ), (3) where σv denotes the noise power. Note, that without oss of generaity, it can be assumed that the noise and channe covariance matrices are normaized in the way, that it is diag{k v = and diag{k h =, where diag{ is a vector containing the diagona eements. Obviousy, x n is aso zero-mean compex Gaussian distributed with covariance matrix K xn = E{x n x H n = σ dk h + σ vk v, (4) where it has been assumed, that the noise sampes and the effective channe coefficients are independent. In order to stress the generaity of the mode in (), note that x n may be the resut of a inear processing of the R dimensiona received signa vector z n, i.e. x n = W z n with W being a (L R)-matrix. B. Piot Signa As in, we assume that the receiver does not have perfect channe state information, but that a noisy channe estimate ĥ = ĥ,..., ĥl T is avaiabe. The channe estimate is obtained by maximum ikeihood channe estimation using a bock of M piot symbos a (p) m {+σ p, σ p, with a m (p) = σp, and can be expressed as ĥ = h + M Mσp a (p) m vm. (5) m= The covariance matrix of the channe estimation vector is given by Kĥ = E{ĥĥH = K h + Mσ p K v, (6) Now, assuming that the additive noise in () and the channe estimation noise in (5) are independent, the cross-covariance matrix of x n and ĥ is given by K xn ĥ = E{x nĥh = σ d K h, (7) where without oss of generaity a (d) n to be the transmitted symbo. = +σ d has been assumed C. MRC Decision Variabe The decision variabe d n, which is obtained by maximum ratio combining of the diversity branches 4 is given by d n = Re {ĥh xn. (8) III. BIT ERROR PROBABILITY In this section, the bit error probabiity P B, i.e. the probabiity that it is sign(d n ) sign(a n ), is cacuated by foowing the derivation in. By defining ĥn r n = (9) and A = x n I L, () it is d n = ) (ĥh n x n + ĥt n x n = r H n Ar n () a hermitian quadratic form. Hence, according to, the characteristic function of d n is given by Φ d (ω) = L = ( jωλ ), () where the λ are the eigenvaues of AK r and K r = E{r n r H n is the (L L) covariance matrix of r n. If a eigenvaues λ, =,..., L are distinct, the pdf of d n can be cacuated via partia fraction expansion and inverse Fourier transform of Φ d (ω) in (). Subsequenty, the bit error probabiity is obtained from the pdf via integration (see ). For distinct eigenvaues, one obtains P B = L = λ < L k= k λ λ λ k. (3) However, if the eigenvaues are not distinct, but mutifod eigenvaues exist, it is difficut to evauate a genera cosed form soution of the coefficients of the partia fraction expansion. Aso, if the eigenvaues get very cose to each other, the cacuation of the coefficients of the partia fraction expansion, and hence the evauation of (3), becomes numericay unstabe. An aternative method is to cacuate the bit error probabiity directy via the characteristic function, which resuts in a numericay stabe soution. According to the emma of Gi- Peaez (see e.g. ), the cumuative density function (cdf) F X (x) of any random variabe X can be cacuated via the characteristic function Φ X (ω) of X as foows F X (x) = π Im { Φ X (ω)e jxω dω. (4) ω Noting that the bit error probabiity is obtained from the cdf of d n as foows P B = F dn (d = a n = +σ d ) (5) the bit error probabiity can be expressed in terms of a singe integra P B = { Im π ω L = ( jωλ dω, (6) )

3 which can be evauated numericay. Note that the integrand is rea vaued and decays very fast for increasing ω. The imit of the integrand for ω exists and can easiy be cacuated as foows im Im ω { ω L = ( jωλ ) = L = λ. (7) It remains to give an expression for AK r. Obviousy, using (4), (6) and (7) it is AK r = σ d K h σ d K h + σvk v, (8) K h + Mσ K p v σ d K h which can be rewritten as AK r = ( ) σd I L K h + ( σ d K h K h + σ d K v Mσ p K v K h I L ). (9) Now, noting that the eigenvaue decomposition is commutative with respect to matrix mutipication and that it is ( ) ( ) σd I L I L = σ d I L () σ d and, finay, that scaing a eigenvaues by the same factor does not effect the bit error probabiity, as can bee seen in (3), it can be stated that the eigenvaues λ in (3) and (6) can be determined by the eigenvaue decomposition of K h K h + ρ d K v K h + γρ d K v K h. () Note that the matrix in () depends on the key system parameters ρ d = σ d σ v, γ =M σ p σd, () denoting the data bit-energy-to-interference ratio and the effective piot-energy-to-data-energy ratio, respectivey, the channe covariance matrix K h and the interference covariance matrix K v. IV. APPLICATION TO W-CDMA In this Section we consider the appication of the genera resuts from the previous section to the upink of a W- CDMA system 6 empoying mutipe antenna eements at the basestation and examine the effect of spatia coored interference. The receiver structure is iustrated in Figure. A simpe fixed beamforming technique is appied 3. A imited number of Q static beams are steered in different directions to cover a sector as it is iustrated in Figure for a 4 eement uniform inear array (ULA) with λ/ inter-eement spacing. The number of beams is assumed to equa the number of antenna eements and the beam directions are assumed to be equi-spaced resuting in the set of steering directions { 45, 5, 5, 45. The beamforming operation can be expressed in terms of a Q H W f Fig.., N, Q, Q,N Q channe estimation Fixed Beamformer Receiver Structure. MRC (Q Q) inear transformation of the antenna outputs, where the coumns of W f are the respective fixed beamforming vectors. For the reevant channe tap deays the best beam outputs are seected and used for maximum ratio combining. A. W-CDMA Spatio-Tempora Signa Mode The foowing spatio-tempora signa mode is appied. In the deay domain, the channe of each of the K users is modeed as a tapped deay ine with L k, k =,..., K, independent Rayeigh fading channe taps and normaized tota channe power, i.e. L k = σ,k =, where σ,k is the mean channe tap power of the -th channe tap of user k. Since the focus is on the effects of the spatia correations and for the ease of notation, the mutipath deays τ,k are assumed to be integer mutipes of the chip period T c, i.e. τ,k = d,k T c with d,k integer. Hence, the T c -samped version of the received signa after puse-matched fitering at the q th antenna eement can be written as z (q) n = z (q) (nt c ) = K L k k= =,k b n d,k,k + v (q) n, (3) where n is the time index and v n (q) is AWGN with variance σv. The quantity b n d,k,k denotes the k-th user s transmitted QPSK moduated sequence 5 deayed by d,k chips consisting of the scrambed superposition of the binary piot signa dedicated physica contro channe (DPCCH) and the data signa dedicated physica data channe (DPDCH) b n,k = c scram,n,k (c data,n,k c k + jβc piot,n,k )/ + β, (4) where β is the piot-to-data ampitude ratio, c scram,n,k denotes the compex scrambing sequence, c data,n,k and c piot,n,k denote the binary data and piot spreading sequences, and c k is the k-th user s transmitted data symbo. Note that it is b n,k =. Due to the Rayeigh fading assumption, the channe coefficients are competey characterized by their second order statistics E{,k h(q ),k = σ,k K S,,k (q, q )δ(, )δ(k, k ), (5) where δ( ) denotes the Kronecker deta and K S,,k (q, q ) denotes the eement of the spatia correation matrix K S,,k

4 corresponding to the q-th row and q -th coumn and the -th channe path of user k K S,,k = a(θ)a(θ) H f θ,k (θ) dθ. (6) The quantity f θ,k (θ) denotes the anguar power density function accounting for an anguar spreading of the signa energy. For a uniform inear array with inter eement spacing of = / waveengths, the array response vector is given by a(θ) =, e jπ sin θ,..., e jπ(q ) sin θ. (7) Note that the diagona eements of K S,,k are unity. B. Bit Error Probabiity In order to derive an expression of the signa vector prior to maximum ratio combining in Figure we note that the beamforming and ing operations can be interchanged. Let the desired user correspond to k =, then the output after ing at the q-th antenna eement corresponding to the -th channe tap is given by y (q) = n= = +β, c L = c data,n d,,c scram,n d,,z (q) n, b n d,,c n d,, n= K L k k= = n=,k b n d,k,kc n d,, n= v (q) n c n d,,, (8) where is the data spreading factor and the c n d,, = c data,n d,,c scram,n d,, denote the effective chips used for ing. Note that the first term in (8) is the desired component, the second term is the interpath interference (IPI) component, the third term characterizes the mutipe access interference (MAI) and the ast term is AWGN. Since the scrambing code in the upink is a fraction of a very ong God sequence (ength 4 ) 5, the imperfect autocorreation properties of the pseudo noise sequences can be statisticay described by making use of the foowing common approximation 4:,k b n d,k,kc n d,, (9) n= is approximatey N C (, σ,k ) distributed for k > or. By defining y = y (),..., y (Q) T, the signa vector of the desired user corresponding to the -th channe tap after ing is given by y = c +β h + ṽ, (3) with h the desired component and ṽ the interference component in (8) being zero-mean compex Gaussian distributed. The respective covariance matrices are given by K h = K S,, (3) L K L k Kṽ = σ,k S,, + σ,kk S,,k + σvi Q. (3) = k= = Now, accounting aso for the fixed beamforming operation we define x = J T W H f y = c +β J T W H f h + J T W H f ṽ = c +β h + v, (33) where J is a seection matrix with ony one eement being in each row whie the other eements are a. The matrix J accounts for the fact that except from a spatiay uncorreated scenario ony a subset of beams is used for maximum ratio combining with respect to the -th channe tap. These are usuay the beams steering into the direction of arriva of the channe taps. Due to the inear transformation with J T W H f, h and v are aso zero-mean compex Gaussian distributed and characterized by the respective covariance matrices. = J T W H f K S,, W f J (34) K v = J T W H f Kṽ W f J. (35) The tota input diversity vector used for maximum ratio combining is given by stacking the vectors x,..., x L of a channe taps on top of each other x = x T,..., x T L T = c +β h + v, (36) with h = h T,..., h T L T and v = v T,..., v T L T, respectivey. Note that the covariance matrices and K v of h and v are bock-diagona, since the channe taps have been assumed to be mutuay uncorreated. The derivation for the ML-channe estimate is anaogous. Finay, if a bock of N p piot chips is used for the channe estimation, then the bit error probabiity is given by (6) with the λ being the eigenvaues of + +β K v + +β β N p K v where (37) is in accordance with (). C. Exampe, (37) In the foowing a user scenario is considered in order to demonstrate the effect of coored interference. The antenna configuration and the choice of the fixed beam matrix W f is as iustrated in Figure. A tap fading channe with channe tap powers {, 3 db, channe tap deays {, T c, direction of arriva {5, 5 and an uniform anguar spread of 5

5 interfering user 3 desired user 33 no IPI, no MAI no MAI fuy correated interferer uncorreated interferer antenna case Fig.. Considered user scenario: desired user at 5, interfering user at 8. Antenna configuration: Q = 4 antenna eements with beam steering directions { 45, 5, 5, 45. mean BEP 3 is considered for the desired user. According to the 3GPP standard 6 the data spreading factor is set to = 64 and the piot-to-data ratio is set to β = /5, according to a typica speech user. The ML-channe estimation is performed by coherenty accumuating over 6 piot symbos, i.e. N p = 536 chips. A fat fading channe has been assumed for the interfering user. By evauating (6), the mean bit error probabiity is potted versus the bit-energy-to noise ratio (E b /N = /( + β )/σv) in Figure 3. It is assumed that ony the fixed beam pointing into the direction of the desired user is used for maximum ratio combining. The interferer is assumed to have the same E b /N as the desired user. The dotted curve shows the resut if no IPI and no MAI are considered. The dashed curve aso considers ony the desired user but accounts for IPI. As expected, for high E b /N a suturation effect can be observed. If the fading coefficients of the interferer are spatiay fuy correated at the antenna eements (the markers), the saturation effect worsens significanty. On the other hand, if the interferer fading coefficients are spatiay uncorreated (the markers) ony a sight degradation can be observed. This is due to the fact that the fixed beam performs some kind of interference suppression by rejecting the interference power coming from directions other than that of the desired user. However, in case of a spatiay correated interferer a rejection of the interference power is not possibe, since the DOA of the interferer is too cose to the DOA of the desired user. However, even in the case of a spatiay correated interferer, a performance gain of neary 6 db is obtained compared to the one antenna case. This is due to the fact that the DOA of the desired user matches the steering direction of a fixed beam and hence, neary the fu antenna array gain can be expoited. V. CONCLUSION An anaytica cosed form expression for the mean bit error probabiity of a binary moduated signa has been presented for the genera case of maximum ratio combining with imperfect channe state information accounting aso for coored additive interference. Furthermore, an aternative soution has been derived, which aows a numericay stabe evauation of the bit error probabiity aso for the case of cosey-spaced eigenvaues of the respective channe and interference covariance E b /N in db Fig. 3. BER versus E b /N for a -tap mutipath fading channe and one fat fading interfering user. matrices. The effect of coored interference has been demonstrated by appying the resuts to a muti antenna eement receiver in the upink of a W-CDMA system, where the signa reception is disturbed by interpath and mutipe access interference. REFERENCES F.A. Dietrich and W. Utschick. Maximum Ratio Combining of Correated Rayeigh Fading Channes with Imperfect Channe Knowedge. IEEE Trans. Communications Letters, Vo. 7, No. 9, 49-4, Sept. 3. D. Brennan. Linear diversity combining techniques, Proc. IRE, vo. 47, pp. 75-, June H. Meyr, M. Moenecaey and S. Fechte. Digita Communication Receivers: Synchronization, Channe Estimation and Signa Processing, John Wiey and Sons, New York, J.G. Proakis. Digita Communications, McGraw-Hi, J.G. Proakis. Probabiities of Error for Adaptive Receiption of M-Phase Signas, IEEE Trans. Commun. Techno., vo. 6, pp. 68-7, Feb M. Z. Win and J. H. Winters. Virtua Branch Anaysis of Symbo Error Probabiity for Hybrid Seection/Maxima-Ratio Combining in Rayeigh Fading, IEEE Trans. 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