Minimum Feedback Rates for Multi-Carrier Transmission With Correlated Frequency Selective Fading

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1 Minimum Feedback Rates for Multi-Carrier Transmission With Correlated Frequency Selective Fading Yakun Sun and Michael L. Honig Department of ECE orthwestern University Evanston, IL Abstract We consider multi-carrier transmission through a frequency-selective fading channel with limited feedback. An on-off power allocation activates the set of sub-channels with gains above a threshold. We model the sequence of sub-channel gains as a Markov process, and give a lower bound on the feedback rate in bits per sub-channel needed to specify the sequence of activated sub-channels as a function of the activation threshold. Optimizing the threshold gives the same asymptotic growth in capacity as optimal water-filling as the number of subchannels goes to infinity. If the ratio of coherence bandwidth to the total available bandwidth is fixed, then the ratio between minimum feedback rates with correlated and i.i.d. sub-channels, respectively, converges to zero with. For a sequence of Rayleigh fading sub-channels, which are modeled( as a first-order autoregressive process, the ratio goes to zero as O with log the optimized threshold. We also consider finite-precision rate control on each sub-channel, and show that the feedback rate required to specify the sequence of assigned rate levels across sub-channels gives the same asymptotic increase in achievable rate with as the (infinite-precision on-off power allocation. I. ITRODUCTIO Feedback can substantially increase the achievable rate for multi-carrier transmission through a frequency-selective fading channel. In [1], the achievable rate for multi-carrier transmission over a frequency-selective fading channel is analyzed with limited feedback. In that work, the sub-channels are i.i.d. and time-invariant, and the asymptotic growth in capacity as a function of the number of sub-channels is specified for different power allocation schemes. For a general class of fading distributions, it is shown that the asymptotic capacity with optimal water-filling grows at the same rate as the capacity corresponding to a uniform power distribution over an optimized set of sub-channels. We refer to the latter scheme as the optimal on-off power allocation. For example, with Rayleigh fading sub-channels, the capacity grows as O(log. In practice, the sequence of sub-channels is likely to be correlated. Since the channel capacity depends only on the first-order distribution of channel gains, this correlation does 1 This work was supported by ARO under grant DAAD not affect the growth in capacity with. However, correlations among sub-channels can be exploited to reduce the minimum amount of feedback, which is needed to achieve this growth. An on-off power allocation can be specified by a binary sequence, corresponding to active and inactive subchannels. The set of active sub-channels have gains, which exceed some chosen threshold. Here we model the sequence of sub-channel gains as a Markov process, and give a lower bound on the feedback rate, in bits per sub-channel, as a function of the threshold. By choosing the optimal threshold, the minimum feedback required to achieve the asymptotic capacity can be obtained. For an on-off power allocation the minimum feedback rate is the entropy rate of the binary on-off sequence. We compute the entropy rate in terms of the channel parameters, and show how it converges to zero as. The ratio between the entropy rates for correlated and i.i.d. sub-channels also converges to zero, and we specify the corresponding convergence rate with. As an example, we consider a first-order autoregressive sequence of complex Gaussian sub-channels. In order to model a wireless system, which occupies a fixed total bandwidth, we constrain the number of coherence bands, which are contained within the sub-channels. In that case, when the activation threshold grows as O(log, which allows the achievable rate to increase as O(log, the savings in feedback relative to ( i.i.d. sub-channels is O log. umerical examples are provided, which show that the asymptotic analysis accurately predicts the average amount of feedback needed in finite-size systems for on-off power control. With finite-precision rate control, additional bits per active sub-channel are needed to choose from a discrete set of rates. We show that adding these bits does not increase the order of the feedback rate with. Furthermore, even when the rate for each active sub-channel is fixed, and optimized to maximize the total rate, we show that the growth in achievable rate with matches the optimal growth in capacity. However, as shown in [1], the absolute difference between the achievable rate and capacity converges to a Gaussian random variable, where the mean decreases with the number of possible rate levels. GLOBECOM /03/$ IEEE

2 Related work on the performance of different transmission schemes with limited feedback is presented in [2], [3], [4], [5], [6], [7], [8]. Most of that work is on characterizing the effect of imperfect channel estimates on achievable rates. A finite number of feedback bits per dimension is also assumed in [2], [5], [7], although the specific feedback schemes and analysis differ from that presented here. II. ASYMPTOTIC CHAEL CAPACITY We assume that the received vector corresponding to the i th transmitted multi-carrier symbol is given by r(i =Hs(i+n(i (1 where s(i is the vector of transmitted symbols across subchannels at time i, and H is the diagonal channel matrix, which contains the channel coefficients, h i, i = 1,,. The sub-channel gains are µ i = h i 2, i =1,,, and are identically distributed random variables. Let f µ ( and F µ ( denote the pdf and cdf of µ i, respectively, and F µ (x = 1 F µ (x. The noise, n(i, is white Gaussian with unity covariance matrix. We assume a total power constraint, i.e., trace{e{ss }} P. Conditioned on the sub-channel gains, the channel capacity is given by [9] C = log(1 + P i µ i (2 where P i is the power on the i th sub-channel, and P i P. With the optimal water-filling power allocation the channel capacity is C (wf = log (1+(λ 1µi + µ i where the water level λ is determined by P = (λ 1 + (4 µ i To reduce the amount of feedback for power and rate optimization, we consider on-off feedback, in which the transmitter allocates equal power P across a subset of sub-channels with gains that exceed a threshold µ 0. The power constraint then becomes P 1 µi µ 0 P (5 Optimizing the threshold gives the corresponding on-off capacity for finite, C (on-off = max µ 0 (3 log ( 1+ Pµ i 1µi µ 0 (6 where P depends on µ 0. The optimal threshold that maximizes C (on-off is denoted as µ 0. Data Rate (bit In what follows, we say that two sequences {x n } and {y n } are asymptotically equivalent if lim n x n /y n =1, and write x n y n. The following two theorems are restated from [1]. Theorem 1: If E[µ µ >x] x is finite for all x, then C (wf C(on-off Pµ 0 (7 where µ 0 is the optimal threshold and satisfies P F µ (µ 0 E2 [µ µ >µ 0]=2(E[µ µ >µ 0] µ 0 (8 For Rayleigh fading sub-channels, C (wf C(on-off P log (9 Suppose now that the rate R i for the i th sub-channel is chosen from the discrete set R = {0, R 1,, R n }. That is, the range of channel gains is divided into intervals, each of which is associated with a particular rate. In what follows, we assume that these intervals are selected to maximize the achievable rate [1]. The corresponding total finite-precision rate is R (fp = R i. Theorem 2: The loss in the total achievable data rate, relative to the on-off capacity, due to finite-precision rate control is a random variable, which satisfies C (on-off R (fp D ( mn,σn 2 (10 where m n and σ n are constants. Figure 1 shows plots of mean data rate vs. SR for multicarrier transmission with 512 Rayleigh sub-channels with unity variance. Achievable rates are shown with water-filling (C (wf, the optimal on-off power allocation with infinite precision rate control (C (on-off, and on-off power allocation with finite-precision rate control with n =1, 2 and 4. The figure shows that C (on-off is very close to C (wf.thegap between the achievable finite-precision data rate and C (wf increases with power, and decreases as n increases Optimal On Off Equal rate, On Off 2 level rate control 4 level rate control Water Filling 512 Rayleigh sub channels SR (db Fig. 1. Mean channel capacity vs. SR for water-filling and on-off power allocations with infinite- and finite-precision rate control. GLOBECOM /03/$ IEEE

3 III. FEEDBACK RATE WITH CORRELATED FADIG For a large class of fading distributions, an achievable rate, which is asymptotically equivalent to the capacity, can be obtained by feeding back an optimized subset of active subchannels. Given a threshold µ 0,let { 1 if µi µ χ i = 0 0 if µ i <µ 0 where 1 i, and χ =(χ 1,χ 2,,χ be the - bit on-off sequence. Define the minimum feedback rate as B, measured in bits per sub-channel. If the sequence of subchannels is a stationary process, then we have lim B = H(X (11 where H(X is the entropy rate of the sequence χ.ifthe sub-channels are i.i.d., then the corresponding feedback rate γlog γ (1 γ log(1 γ, where γ = F µ (µ 0. If µ 0 with, then γ log γ (1 γ log(1 γ as γ 0, and we have (µ 0 γ log 1 γ (12 which goes to zero with. Therefore (µ 0 is a lower bound on the feedback needed to achieve the asymptotic capacity. ow suppose that the sequence of sub-channel gains {µ 1,,µ } is a Markov process with joint second-order pdf g(µ i,µ i 1. Clearly, {χ 1,χ 2,,χ } is a two-state Markov chain, as shown in Fig. 2, with transition probabilities q = Pr{χ i =1 χ i 1 =1} = 1 γ µ 0 µ 0 g(x, y dx dy (13 p = 1 (2 qγ Pr{χ i =0 χ i 1 =0} = 1 γ (14 The corresponding entropy rate is H(X = p 1-p q Fig. 2. Two-state Markov chain corresponding to the sequence χ. (1 ph(q+(1 qh(p 2 p q = γh(q + (1 γh(p [9]. In what follows, we will assume that the ratio of coherence bandwidth to the total available bandwidth is fixed. In that case as increases, the correlation between neighboring sub-channels increases, i.e., lim q =1.LetB(mc denote the asymptotic on-off feedback rate for the preceding Markov chain model, and let Pr{µ i µ 0 } θ(µ 0 = Pr{µ i <µ 0 µ i 1 µ 0 } = γ (15 1 q q Theorem 3: If the threshold µ 0 with, then lim (µ 0 (µ 0 =0 Furthermore, if θ(µ 0 0 with, then (µ 0 (1 qγ log 1 (16 γ Given the conditions in Theorem 3, we therefore have (µ 0 (µ 0 1 q (17 where q, defined in (13, is a function of µ 0. The condition θ(µ 0 0 is satisfied in many situations of interest. In that case (17 implies that for large, the minimum feedback with correlated sub-channels is much less than that with i.i.d sub-channels. ow suppose that the data rate on each sub-channel is chosen from one of n possible rates. According to Theorem 2, the maximum achievable rate is asymptotically equivalent to the capacity, although there is an absolute (random loss in rate, which on average increases with the transmitted power. Given a set of n thresholds, the gain of each sub-channel is quantized as χ i, where χ i {0, 1,,n} indicates the rate assigned to sub-channel i. That is, R i = R χi for 0 χ i n. In this case, µ 0 is the smallest (activation threshold, i.e., if µ i <µ 0, then χ i =0and R i = R 0 =0. The sequence χ = (χ 1,,χ is an (n +1-state Markov chain, as shown in Fig 3. Let B (fp,n (µ 0 denote the p 00 p 01 p 0n n p 10 Fig. 3. (n +1-state Markov chain corresponding to feedback with finiteprecision rate control. minimum feedback rate with n-level finite-precision rate control. As, B (fp,n n (µ 0 converges to π i p ij log 1 p n0 i,j=0 p ij, where p ij is the transition probability from state i to state j, and {π i } is the steady-state distribution. Let q n = Pr{χ i 0 χ i 1 0} = Pr{µ i µ 0 µ i 1 µ 0 } = 1 g(x, ydxdy. γ µ 0 µ 0 That is, q n generalizes q in (13 to the (n +1-state Markov chain. Theorem 4: As,ifθ(µ 0 0, then for a fixed number of data rates n, B (fp,n (µ 0 (1 q n γ log 1 γ (18 GLOBECOM /03/$ IEEE

4 Comparing the preceding feedback rate with that for i.i.d. subchannels gives B (fp,n (µ 0 (µ 0 1 q n (19 We therefore conclude that given the same activation threshold, finite-precision rate control with a finite number of rate levels n requires the same order of feedback as onoff feedback, which does not depend on n and the set of thresholds. That is, for large the additional overhead needed to specify one of n data rate levels is negligible compared with the feedback needed to specify the binary on-off sequence. Furthermore, by choosing the optimal threshold set, the corresponding achievable rate is asymptotically equivalent to the capacity. IV. RAYLEIGH FADIG We now assume that each sub-channel coefficient, h i,isa complex Gaussian random variable, so that the sub-channel gains have a Rayleigh distribution with unity variance, i.e., f µ (x =e x. To model the correlation among sub-channels, we assume that the sequence {h i } is generated from a firstorder autoregressive model, h i = αh i 1 + ξ i (20 where 0 α 1, and ξ i is a complex Gaussian random variable, which is independent of h i, and has variance 1 α 2, so that E[ h i 2 ] = 1. The parameter α determines the correlations between the sub-channels. In what follows, we assume that the total bandwidth is fixed so that the width of the sub-channels tends to zero as. Hence the correlation between sub-channels increases, so that α 1. It is straightforward to show that (20 implies that {µ 1,,µ } is a Markov process, and g(x, y = 1 ( x+y 2α xy 1 α 2 e 1 α 2 I 0 1 α 2 (21 where I 0 ( is the modified Bessel function of the first kind and zero-order. Assuming the threshold µ 0 as, the transition probability q can be approximated as ( 1 α q (1 + αq 1+α 2µ 0 (22 where Q(x = 1 x 2π e x2 2 dx. Substituting (22 into (16 and (18 gives the respective asymptotic on-off and finiteprecision feedback rates. For example, if lim α =1, then choosing the threshold µ 0 = 1 1 α gives the relative feedback gain /B(iid If(1 αµ 0 0, then (22 can be further approximated as 2(1 αµ0 q 1 (23 π and the ratio of minimum feedback rates( for (1 correlated and i.i.d. sub-channels converges to zero as O αµ0. The reduction in feedback obtained by exploiting the subchannel correlations clearly depends on the rate at which α 1 with. Here we consider a special case. Let W be the total bandwidth of the channel and W c be the coherence bandwidth. Specifically, we define the coherence bandwidth W c = K f, where f is the width of the sub-channels, and K = max {k : cov(µ i,µ i+k ρ} (24 where ρ is the correlation between sub-channels separated by W c, and 0 <ρ<1. The number of coherence bands spanned by the channel is assumed to be fixed, i.e., Wc W = K/ = β where β is a constant. Since cov(µ i,µ i+k =α 2k, we can write α = ρ 1 2β = e log 1 ρ 2β (25 which increases to 1 exponentially with. µ Corollary 1: If lim 0 1 1, then the on-off 2 log feedback rate with i.i.d. sub-channels is e µ0 µ 0. (26 Furthermore, the on-off and finite-precision feedback rates for the autoregressive sequence of sub-channels satisfy B(fp,n β 1 e µ0 µ (27 log where β 1 = 1 ρ πβ. Comparing the feedback rates for the autoregressive and i.i.d cases gives B(fp,n β 1 µ0 (28 Specifically, if µ 0 b log, where b 1/2 is a constant, then B(fp,n bβ 1 log (29 The condition b 1/2 is needed so that (23 is valid. Given the total bandwidth W and coherence bandwidth W c, the minimum feedback rate for on-off and finite-precision rate control can be estimated from (27. For example, choosing the optimal threshold µ 0 for Rayleigh fading (i.e., µ 0 log ( log 3 [1] gives a minimum feedback rate of O for i.i.d. sub-channels. For the autoregressive model, the minimum feedback ( rate, relative to the i.i.d. case, is reduced by the factor O. In both cases, the maximum achievable rate is log asymptotically equivalent to the capacity, which is O(log. ote that the data ( rate normalized by the feedback ( rate increases from O for i.i.d. subchannels to O 1 log 2 log 5/2 for correlated subchannels, where the units are transmitted bits per feedback bit per channel use. GLOBECOM /03/$ IEEE

5 Feedback Rate (bits/sub channel V. UMERICAL RESULTS Here we give a numerical example motivated by a cellular system. The channel bandwidth is 5 MHz, and the coherence bandwidth with ρ = 0.5 is 146 khz [10]. We take the threshold µ 0 = 1 2 log. (Taking µ 0 > 1 2 log complicates the generation of simulated results, since for finite, active sub-channels occur relatively infrequently. Figure 4 shows the feedback rate vs. computed from Corollary 1, and by generating sample sequences of sub-channel gains according to (20, and encoding the on-off sequence with an arithmetic code. Results are shown for both i.i.d. and correlated subchannels with the same threshold µ 0. We observe that the asymptotic bounds for both i.i.d. and correlated sub-channels decrease logarithmically with, i.e., log decreases as log and log decreases as 1 2 log.as increases, correlated sub-channels greatly reduce the required feedback, relative toi.i.d. sub-channels, as indicated in (28. The variance of the simulated results with correlated channels increases with, but the asymptotic results predict the mean behavior iid, theoretical bound iid, arithmetic encoding correlated, theoretical bound correlated, arithmetic encoding model, the relative reduction in feedback rates can be explicitly computed. amely, the minimum feedback rate for correlated sub-channels, ( relative to i.i.d. sub-channels, tends to zero as O. In both cases, the achievable rate with feedback log is asymptotically equivalent to the capacity. This is also true with finite-precision rate control, where in that case, the absolute loss in achievable data rate, relative to capacity, is a Gaussian random variable with finite variance. REFERECES [1] Y. Sun and M. L. Honig, Asymptotic Capacity of Multi-Carrier Transmission Over a Fading Channel with Feedback, IEEE International Symposium on Information Theory, Japan, [2] A. arula, M. J. Lopez, M. D. Trott and G. W. Wornell, Efficient Use of Side Information in Multiple-Antenna Data Transmission over Fading Channels, IEEE Transactions on Communications, 16(8: , Oct., [3] G. Jöongren, M. Skoglund, and B. Ottersten, Combining Beamforming and Orthogonal Space-Time Block Coding, IEEE Transactions on Information Theory, 48(3: , March, [4] S. Bhashyam, A. Sabharwal and B. Aazhang, Feedback Gain in Multiple Antenna Systems IEEE Transactions on Communications, 50(5: , May, [5] K. Mukkavilli, A Sabharwal, E. Erkip, and B. Aazhang, On Beamforming With Finite Rate Feedback In Multiple Antenna Systems, submitted to IEEE Transactions on Information Theory, [6] E. Visotsky and U. Madhow, Space-Time Precoding with Imperfect Feedback, IEEE Transactions on Information Theory, 47(6: , Sep, [7] W. Santipach and M. L. Honig, Signature Optimization for DS-CDMA with Limited Feedback, Int. Symposium on Spread Spectrum Systems and Applications, Prague, Czech Republic, September 2002 [8] V.K.V. Lau, Y. Liu, and T.-A. Chen, The Role of Transmit Diversity on Wireless Communications Reverse Link Analysis With Partial Feedback, IEEE Transactions on Communications, 50(12: , Dec., [9] T. M. Cover and J. A. Thomas, Elements of Information Theory, 1991, John Wiley & Sons, Inc. [10] T. S. Rappaport, Wireless Communications, 1999, Prentice-Hall, PTR umber of Sub Channels Fig. 4. Feedback rate vs. number of sub-channels. VI. COCLUSIOS We have studied the minimum feedback rate required to specify the sequence of active sub-channels for an on-off power allocation. Optimizing the activation threshold gives an achievable rate, which is asymptotically equivalent to the channel capacity. For a sequence of sub-channels generated according to a Markov process, the minimum feedback rate has been computed in terms the transition probability, and the sub-channel activation threshold µ 0.Ifθ(µ 0 0 with, where θ(µ 0 is defined by (15, then the feedback rate, measured in bits per sub-channel, converges to zero. When the number of coherence bands is fixed, the ratio of feedback rates with correlated and i.i.d. sub-channels converges to zero. Given a complex Gaussian sequence of subchannels generated according to a first-order autoregressive GLOBECOM /03/$ IEEE