ASYMPTOTIC PERFORMANCE OF MIMO WIRELESS CHANNELS WITH LIMITED FEEDBACK

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1 ASYMPTTIC PERFRMANCE F MIM WIRELESS CHANNELS WITH LIMITED FEEDBACK Wiroonsak Santipach and Michael L. Honig Department of Electrical and Computer Engineering Northestern University 2145 Sheridan Road Evanston, IL USA Abstract We consider a single-user, point-to-point communication system ith transmit and receive antennas ith independent flat Rayleigh fading beteen antenna pairs. The mutual information of the multi-inputmultioutput (MIM) channel is maximized hen the transmitted symbol vector is a Gaussian random vector ith covariance matrix. The optimal depends on ho much channel state information is available at the transmitter. Namely, in the absence of any channel state information, the optimal is full-rank and isotropic, hereas ith perfect channel knoledge, the optimal has columns hich are the eigenvectors of the channel, and has rank at most. We assume that the receiver can feed back bits to the transmitter (per codeord). The feedback bits are used to choose the columns of from a random set of i.i.d. vectors. We compute the mutual information as a function of both and the rank of. ur results are asymptotic in the number of antennas, and sho ho much feedback is needed to achieve a rate, hich is close to the capacity ith perfect channel knoledge at the transmitter. 1. INTRDUCTIN Adding antennas to the transmitter and receiver of a single point-to-point fading link can substantially increase the achievable data rate [1, 2]. The channel capacity depends on the amount of channel state information (CSI) knon at the transmitter and receiver. Assuming perfect CSI at the receiver, and independent fading across different transmitterreceiver antenna pairs, the channel capacity has been evaluated for the cases here there is no CSI and complete CSI at the transmitter. Here e examine the capacity ith complete CSI at the receiver, but partial CSI at the transmitter. We assume This ork as supported by the U.S. Army Research ffice under grant DAAD transmit antennas, receive antennas, and that the channels across different transmitter-receiver antenna pairs are i.i.d. Rayleigh. (The capacity ith correlated fading across antenna pairs is studied in [3].) The mutual information is maximized hen the transmitted symbol vector is Gaussian ith covariance matrix, hich depends on ho much CSI is available at the transmitter. Specifically, in the absence of any CSI, the optimal is full-rank and isotropic, hereas ith perfect CSI the optimal has rank at most. We assume that the channel is stationary, and that the receiver relays bits to the transmitter (per codeord) via a reliable feedback channel (i.e., no feedback errors). The purpose of the feedback is to specify. In this paper, e analyze the performance of a Random Vector Quantization (RVQ) scheme, hich is used to specify a ith rank, here can be optimized. The RVQ scheme is motivated by related ork on signature optimization for CDMA ith limited feedback, here it is shon to be optimal [4, 5]. ther related ork on the performance of MIM channels ith limited feedback has been presented in [6 10]. Although our model is similar to that originally presented in [6], our asymptotic approach, based on RVQ, along ith the associated results, differs substantially from prior ork. We compute the sum mutual information beteen the transmitted and received symbols per receive antenna as the number of transmit antennas, the number of receive antennas, the number of feedback bits, and the rank all increase to infinity ith fixed ratios. The limiting mutual information can be evaluated by using results from extreme order statistics and random matrix theory. We sho ho the optimal rank for the covariance matrix ith RVQ decreases ith the amount of feedback, and observe that relatively little feedback is needed to achieve a rate close to the capacity ith perfect CSI hen is large. ptimizing the rank can also results in a substantial increase in the sum 1 of 6

2 d mutual information. The paper is organized as follos. In section 2, e describe the system model. In section 3, e revie the behavior of the capacity ithout any CSI at the transmitter, and ith perfect CSI. We present RVQ and the associated performance analysis in section 4 and the numerical results in section 5. Section 6 concludes the paper. 2. CHANNEL MDEL We consider a single-user, point-to-point ireless channel ith transmit antennas and receive antennas. We assume i.i.d. flat Rayleigh fading channels, so that the set of channel gains! beteen the " th transmit antenna and the # th receive antenna contains i.i.d complex Gaussian random variables ith zero mean and variance $&%' ( )',+.-0. The received vector is given by here 2:-;%<( =+ is the?>@ channel matrix, 4A-;% BC+ is the >A transmitted symbol vector, 1 -D% EF+ is the G>9 received symbol vector, and the noise 8 is complex Gaussian ith covariance matrix H I)JLK here JLK is the M> identity matrix. ur performance metric is sum mutual information beteen 4 and 1. The mutual information is maximized hen the transmitted symbols BC are complex Gaussian random variables, in hich case the sum mutual information is given by N 4QP 1SR -A$UTWVYX[Z]\_^(`=acb J K 69de2f 2hgjik (1) here the covariance matrix l-3$&% 4m4 g +, dn-3$&%po4coq+!hm I is the background Signal-to-Noise Ratio (SNR), and X[Z]\ denotes natural logarithm. We are no interested in maximizing the sum mutual information over the covariance matrix subject to a poer constraint asr=] tvu. 3. PERFECT CSI VS. N CSI Here e consider the to extreme cases in hich the transmitter has no CSI available, and the transmitter has perfect CSI. These correspond to normalized feedback values x-3, -3y and x-z, respectively Zero Feedback If there is no CSI available at the transmitter (corresponding to zero feedback bits from the receiver), then the optimal :- { J [2]. That is, each antenna transmits an independent symbol, and the transmitted poer is allocated equally across transmit antenna. As hq Rf} z ith fixed, the capacity per receive antenna is given by ~- ^(`=ax[z]\ b JLK g i }~ X[Z]\ _69d R ˆ R ^ - Š x-3y R here convergence is almost surely, and ˆ R is the asymptotic probability density function for a randomly chosen eigenvalue of 2h2 g, and is given in [11]. The integral in (2) has been evaluated in [12], and gives the folloing closed-form expression here E - - x-3y R -3X[Z]\ d EŽ Infinite Feedback d d 6 š 6 6 X[Z]\& 6 6 f d d œ œ žÿ žÿ If D- z, then the transmitter has perfect CSI, and the optimal is given by (2) (3) l-3 5g (4) here is an unitary matrix hose columns are eigenvectors of 2 g 2, and is a diagonal matrix hose elements are computed by ater-filling over the available dimensions. The rank of the optimal is at most Y, and depends on the SNR. In contrast, ith -Ay the optimal is full-rank. Let Š -z R denote the capacity per receive antenna in the limit as hq R } z. We have that -z R - X[Z]\ R Žˆ R ^ (5) here the ater level is determined by the poer constraint ( & ˆ. R ^ -ªd (6) here B -«! Fy sbš. We can verify that Š - y R u Š ~- z R, and no consider the ratio Š ~- 2 of 6

3 K Ó Ô K z R F -y R to see ho much of an increase in capacity is attainable ith feedback. We evaluate this ratio in the regimes of both large and small SNR. For large SNR, from [3] e have that X {± -Az R x-3y R - ²Z]r ³µy Hence for large SNR, there is clearly little incentive to relay CSI back to the transmitter. This is because for large SNR, the transmitter uses all available dimensions, and an equal poer allocation asymptotically gives the same capacity as ater pouring. For small SNR, e have that XY {± x-z R -ªy R - 6 ²Z]r ³µy (7) This limit is an extension of the analogous limit in [3], hich is for G-. Hence the potential gain due to feedback decreases ith. This is because for large, the receiver obtains more independent copies of a transmitted symbol and is able to decode the symbol hether the poer is allocated equally or optimally. 4. RANDM VECTR QUANTIZATIN To maximize the sum mutual information, is given by (4) here the columns of are the eigenvectors of 2 g 2. If has rank then the eigenvectors correspond to the largest eigenvalues. In hat follos, e assume that the receiver quantizes the matrix, and relays this back to the transmitter. We also assume that the transmitter distributes equal poer and rate over the eigenvectors, i.e., the optimal ater pouring distribution is not relayed back to the transmitter. (The degradation in capacity due to using a constant poer allocation ith optimized rank, rather than the optimal ater pouring poer allocation has been shon to be small [13].) Given feedback bits, e can construct a vector quantizer, hich chooses the matrix from the set, or codebook Lº =»=»=» sº ¼. That is, assuming that 2 is knon at the receiver, the receiver chooses the º that maximizes the sum mutual information, and relays the corresponding index back to the transmitter. ptimizing this codebook for finite and is quite difficult; hoever, as R } z, the eigenvectors of 2 are isotropically distributed. This suggests that the columns of the codebook entries should also be isotropically distributed. Hence in hat follos, e assume ]¾ are independent and that the matrices Lº ], # -½=»=»=» isotropic, and refer to this scheme as Random Vector Quantization (RVQ). An analogous RVQ scheme has been previously proposed for CDMA signature optimization ith limited feedback in [4, 5], and has been shon to maximize the SINR in the large system limit. We are interested in the mutual information ith RVQ in the limit as hq 7 R } z ith fixed ratios?-l, À-,, and «- t. (Note that the number of feedback bits is normalized by the number of matrix entries.) Given the rank uá, the covariance matrix is chosen from the set ], #Â- ½=»=»=»Ã ]¾, here ÄU- º =º g ²pZ]r Äu#u and º is an >Å random unitary matrix, i.e., º g º Ä- JLÆ. The matrices º for all # are chosen independently, and the transmitted poer asrl] Ž½Ž-. The receiver selects the quantized covariance matrix Ç -3 ½r\5! K - X[Z]\ ^`=a b JLK 69d)2@ Ä =2hg i Ë sè È ¼ É Hence the sum mutual information per receive antenna ith feedback bits is given by ÍYÎ,Ï K - $ ÐÑ! K Ò sè È ¼ - $ Ð X[Z]\.^(`=a b J K 69d)2 ¾ Ç n2ågsi Ò (8) here the expectation is over the channel and covariance matrix. To evaluate (8), e must kno the cumulative distribution function (cdf) for K. Since the matrices are i.i.d., the random variables K are also i.i.d.. Letting Ó K denote the corresponding cdf, e have that K ÍpÎ±Ï -3 BÓ ¾ÕÔ B R Ö K B R ^(B here Ö K B R is the probability density function (pdf) for K. In hat follos e consider the limit as hq all tend to infinity. Theorem 1 As q & R } z ith fixed - ³Áy, Ñ-~,Å, and -~, the sum mutual information per receive antenna K surely to ÍpÎ±Ï - XY K Æ ¾šØ ÍpÎ±Ï converges almost ¾ (9) 3 of 6

4 Ù ä 6 Ô 6 ä œ The inverse cdf Ó K depends on,, and. The proof relies on the asymptotic theory of extreme order statistics [14]. With appropriate shifting and scaling [14], Ó K converges in distribution to a Weibull cdf as } z. An analogous result in the context of CDMA signature optimization has been presented in [4]. Computing the exact cdf Ó K appears to be difficult. We therefore resort to a Gaussian approximation, hich is motivated by the folloing lemma. In hat follos, e drop the superscript in K to simplify the notation. here Ü and here Lemma 1 As hq R } z ith - and Ú-, K fû R }¹Ü y H R v^c Ýsasrq [Þß(a [Zà y H R is the Gaussian cdf ith zero mean and variance H, Ûh- X[Z]\ _6 dâ d á 6âX[Z]\ á R fá án- 6 H - 5XZ]\ hd 6 á d hd This lemma is an extension of a similar result in [15] ith -l, and follos from Theorem 1.1 in [16]. To compute Û and H, e rite K - K ãä=å X[Z]\ 69d here is the æ th eigenvalue of K 25ºtº g 2 g. As hq R.} z, the empirical eigenvalue distribution converges to a deterministic function, hich can be represented in terms of its Stieljes transform [16]. The asymptotic mean and variance of K can then be evaluated according to results in [16]. Although Lemma 1 states that the asymptotic distribution of K is Gaussian, this assumption leads to an approximation for the asymptotic sum mutual information because the distribution for finite is still needed to compute ÍpÎ±Ï in Theorem 1. Specifically, using the Gaussian assumption in Theorem 1 gives ÍpÎ±Ï -ªÛ76âH è âx[z]\ (10) hich approximates ÍYÎ,Ï. As } y, this approximation becomes exact. Hoever, as } z, ÍpÎ±Ï } z, hereas Š é- z R is finite, and can be computed by (5) and (6). Therefore the Gaussian assumption gives an inaccurate estimate of ÍpÎ±Ï for large. This is because K is bounded for all, hereas the Gaussian pdf assumes that K can assume arbitrarily large values. Note that ÍpÎ±Ï is a function of both the rank. For a given, e ish to select the ÍpαÏ. The optimal and depends on, and feedback, hich maximizes can be found by differentiating ÍpαÏ,, and d. 5. NUMERICAL RESULTS Figure 1 shos ÍpÎ±Ï ith optimal rank &ê ÍYÎ,Ï versus ith different values of d and ë-my íì. The dashed lines sho the ater-filling capacity Š?- z R. As expected, the performance of the optimal poer allocation, relative to equal poer allocation, decreases as SNR increases. Also, the amount of feedback needed to reach Š Á-¹z R decreases ith SNR. The dashed-dotted lines sho ÍpÎ±Ï ith fixed rank î-ï and different values of d. In this case, the rates ith the optimized rank and full-rank do not differ much. Also shon in the plot are simulation results ÍYÎ,Ï ith -ð. We can see that ÍpÎ±Ï is a good estimate even for small (e.g., 8) NM = 0.5 ρ = 6 db ρ = 3 db ρ = 0 db C (B = ) I rvq (D rvq ) I (D = 1) rvq Simulation for I (D = 1) ith M = 8 rvq BN 2 Figure 1: Sum mutual information per receive antenna for RVQ ith optimized rank and full-rank versus normalized feedback. for dif- Figure 2 shos the optimal RVQ rank &ê ÍYÎ,Ï versus ferent values of d and ;-3y íì. The rank starts at ÍpÎ±Ï ê - (full-rank) for M-y, and decreases ith. The rank con- 4 of 6

5 ñ verges to the value associated ith the optimal covariance matrix ith perfect CSI. For given, the optimal rank increases ith SNR, reflecting the fact that the ater pouring distribution is spread across more eigenvectors. In the figure, the optimal rank converges to for d-óò and 6 db, but converges to a loer number for dn-ô and 0 db. Figure 4 shos mutual information per receive antenna versus normalized rank from (10) ith «-;y, dµ- õ db, and different values of. The maximal rates are attained at é-3y for cases shon. As increases, the rate increases and the difference beteen the rate ith optimized rank and full-rank also increases NM = 0.5 ρ = 0 db ρ = 3 db ρ = 6 db ρ = 9 db NM = 0.1; ρ = 6 db D rvq I rvq BN 2 Figure 2: The optimal RVQ rank versus normalized feedback. 1 BN 2 = 1 BN 2 = 5 BN 2 = DM Figure 4: Mutual information per receive antenna versus normalized rank ith different feedback. Figure 3 shos the mutual information per receive antenna versus normalized feedback ith different values of. The ater-filling capacity is also shon. As decreases, the performance gap associated ith - y and - z idens, hich is consistent ith (7). Also, the sum mutual information per receive antenna decreases ith. I rvq ρ = 6 db 1.35 NM = 0.9 NM = 0.8 NM = BN 2 Figure 3: Sum mutual information per receive antenna versus normalized feedback ith different values of. 6. CNCLUSINS We have studied the achievable rate of a single-user MIM fading channel ith limited feedback. ur results are asymptotic in the number of antennas, and indicate the minimum feedback, hich is needed to achieve the capacity associated ith perfect CSI at the transmitter. The asymptotic mutual information as evaluated ith a random vector quantization scheme in hich the covariance matrices in the source codebook are independent and isotropic. This approach allos us to study the rank of the covariance matrix hich maximizes the mutual information. ur numerical results sho that less feedback is needed as the ratio beteen the number of receive antennas and transmit antennas increases. For the cases shon, the covariance ith the optimized rank gives a large increase in capacity over the one ith full-rank (e.g., 50-60%). REFERENCES [1] G. J. Foschini and M. J. Gans, n limits of ireless communications in a fading environment hen using multiple antennas, Wireless Personal Communications, vol. 6, no. 3, pp , Mar of 6

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