On the Expected Rate of Slowly Fading Channels with Quantized Side Information

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1 On the Expected Rate of Slowly Fading Channels with Quantized Side Information Thanh Tùng Kim and Mikael Skoglund IR S3 KT 0515 c 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/repulish this material for advertising or promotional purposes or for creating new collective works for resale or redistriution to servers or lists, or to reuse any copyrighted component of this work in other works must e otained from the IEEE. ROYAL INSTITUTE OF TECHNOLOGY Department of Signals, Sensors & Systems Signal Processing S STOCKHOLM KUNGL TEKNISKA HÖGSKOLAN Institutionen för Signaler, Sensorer & System Signalehandling STOCKHOLM

2 On the Expected Rate of Slowly Fading Channels with Quantized Side Information Thanh Tùng Kim and Mikael Skoglund School of Electrical Engineering Royal Institute of Technology SE Stockholm, Sweden {kimt, Astract A multiple-layer variale-rate system employing quantized feedack to imize the expected rate over slowly fading channels is studied. The transmitter utilizes partial channel-state information, otained via an optimized resolutionconstrained feedack link, to adapt the power and to assign code layer rates, under different power constraints. We develop an iterative algorithm to optimize the system parameters, which successfully exploits results from the study of parallel roadcast channels. Unlike in the ergodic case, even coarsely quantized feedack improves the expected rate consideraly. Our results indicate that with as few as one it of feedack information, the role of multi-layer coding reduces significantly. I. INTRODUCTION Many wireless applications require a strict constraint on transmission delay. This motivates the lock fading Gaussian channel model 1], where a transmitted codeword is assumed to span a fixed and finite numer of independent fading locks. In such scenarios, capacity in the traditional sense of the term is generally not a useful performance measure. Therefore, other ways to characterize the channel, for example in terms of outage proaility 1], 2], are often considered in these cases. While outage proaility is a valid measure of the performance of a fixed-rate system over slowly fading channels, for certain applications it may e more reasonale to consider the achievale expected rate over multiple fading locks 3], 4]. Expected rate can also e seen as a measure of relialy decodale rate, from the receiver s perspective 5]. With perfect channel-state information at the receiver (CSIR), the receiver knows whether the transmission of the present lock is in outage, and it can therefore disregard unreliale locks. Hence, from the receiver s perspective, loss of data may occur while there will never e any transmission errors. Consequently, all codewords, at rates allocated y the transmitter, are either supported without errors or lost. It therefore makes sense to discuss expected rate in the sense of the average numer of relialy received its, per channel use over a large numer of transmitted codewords. Interestingly, the traditional outage approach has een shown to e suoptimal in an expectedrate sense, as higher rates can e achieved using a roadcast strategy or multi-layer coding 6], 7], 8]. This idea was first proposed in Cover s seminal work on roadcast channels 3]. The multi-layer approach is particularly appealing for, e.g., successive refinement systems, which can produce a coarse version of a source such as an image, when some information is availale and gradually improve the quality of the reproduced source as more information is received. In this paper we consider a slowly fading frequency-flat link with perfect CSIR, and partial channel-state information at the transmitter (CSIT). Our aim is to study the properties of adaptive systems optimal in an expected-rate sense suject to some particular coding strategies, and with optimized quantizers in the feedack link. Inspired y the roadcast strategy 6], 7], 8], in contrast to a fixed-rate system considered in some previous related works on finite-rate feedack, we study a multi-layer variale-rate coding scheme under different power constraints. Essentially, quantized CSIT transforms the original channel, which can e viewed as a composite one 4], into a finite numer of parallel composite channels. By exploiting the inherent connection etween our design and prolems in parallel roadcast channels 9], we develop a simple iterative algorithm to optimize the feedack and the power allocation, as well as some strategy-dependent parameters. This is, of course, vastly different from feedack design for an ergodic channel 10], 11] where there is only a single centroid, namely the power, associated with each quantization region. Furthermore, unlike in the ergodic channel 12], 11] where even perfect CSIT improves the ergodic capacity only marginally, our results show that even coarsely quantized CSIT provides a significant improvement on the expected rate. Interestingly, our results also indicate that with as little as one it of feedack information, the role of multi-layer coding decreases dramatically. II. SYSTEM MODEL Consider the discrete-time complex-aseand model of a flat-fading single-input single-output (SISO) communication system illustrated in Fig. 1, where the complex-valued channel gain is assumed to e random ut constant during one fading lock consisting of N channel uses. The received signal at time instant k within fading lock m can e written as y m (k) =h m s m (k)+w m (k), k=0,...,n 1, (1) where h m denotes the channel gain and the s m (k) s are the transmitted symols. The noise samples w m (k) s are independent and identically distriuted (i.i.d.) complex Gaussian with zero mean and unit variance. In the current work, we exclusively consider the case that one transmitted codeword

3 Encoder s(k) i n(k) y(k) h γ I(γ) Fig. 1. System model. Decoder support. The coding scheme to imize the expected rate over such a composite channel is unknown in general. We focus our attention on two specific strategies: the traditional outage approach, also referred to as single-layer coding, and the roadcast strategy or multiple-layer coding. Although single-layer coding is a special case of the more general multilayer coding, we consider this special case separately ecause the prolem is more analytically tractale and therefore, more instructive. The main challenge is to optimize the index mapper I(γ), the allocated power P(i) and some strategy-dependent parameters jointly. We will often refer to the set of all parameters to e designed as a feedack scheme. spans a single fading lock. As pointed out in 2], 13], it is reasonale to study the case N. We assume that the h m s are i.i.d. according to some distriution. For revity, the fading lock index m will e omitted in the following discussion. Define γ Δ = h 2, which will e referred to as the channel power. With a slight ause of notation, oth the random variale representing the channel power and its realization will e denoted y γ. We assume that γ is a continuous random variale. Denote the cumulative distriution function (cdf) and the proaility density function (pdf) of γ as ) and f(γ), respectively. Furthermore, assume that f(γ) takes on positive values over the entire region 0, ). The channel is assumed to e known perfectly at the receiver. Given γ, the receiver employs a deterministic index mapping I(γ) that partitions the nonnegative real line into K quantization regions I(γ) =i, if γ γ i,γ i+1),,...,k 1, (2) where the γi s denote the oundary points of the quantization regions. For convenience, we use the convention γk = and γ0 =0. Herein K is a given positive integer, i.e., we consider a resolution-constrained quantizer. The index I(γ) is sent to the transmitter via a noiseless feedack channel. Conditioned on the index fedack I(γ) = i, any transmitted sequence {s(0),...,s(n 1)} satisfies 1 N 1 N k=0 s(k) 2 P(i) where P(i) is a deterministic mapping from random index to power allocated. Denote P i = P(i), i =0,...,K 1. We consider two different types of power constraint. The short-term power constraint requires that the power allocated cannot exceed P, independent of the received index, i.e., P i P, i {0,...,K 1}. (3) Under the more relaxed long-term power constraint, the transmitter can adapt the power so that the average power over multiple locks does not exceed P, E γ P(I(γ))] P, (4) which, in the scenario considered, is equivalent to i+1 ) i )] P i P. Due to our assumption of nonzero and continuous density, the channel conditioned on an index i is still a composite channel 4], as in the case of no CSIT, however with a smaller III. SINGLE-LAYER CODING With the single-layer coding approach, given the index i, the transmitter selects a codeword from a rate-r i capacityachieving codeook where R i s are design parameters. The system is in outage if the imum instantaneous mutual information of the channel is smaller than R i 1]. A. Feedack Design Under a Short-Term Power Constraint It is clear that with a short-term power constraint, the optimal power Pi = P, i, as there is no cost incurred with increasing the power allocated to each fading lock up to the upper limit. Given the index i, the transmitter chooses an operating rate R i, which is associated with a reconstruction point γ i via the relation R i = log(1 + γ i P ). If the actual channel power γ γ i, the codeword will e successfully decoded. On the other hand, if γi γ<γ i, the system is in outage. Designing a feedack scheme optimal in the sense of expected rate is equivalent to solving the following prolem {γ i,γ i } i+1 ) i ) ] log(1 + γ i P ) s.t. γ i+1 γ i 0, γ i γ i 0. (5) By a direct investigation of the activeness of the linear constraints, we can simplify the Karush-Kuhn-Tucker (KKT) conditions for a scheme {γi,γ i } to e optimal to: γ i = γ i, i 1 (6a) i+1) = i ) +f(γ i ) 1+γ i P P log 1+γ i P 1+γi 1,i 0 (6) P with the convention γ 1 =0, γk =. It is relatively easy to solve for {γi } from (6) since one can express γ 1,...,γ, as a function of γ0. (Recall that ) is invertile due to our nonzero density assumption.) Therefore, (6) with i = K 1 can e expressed as an equation with a single unknown γ0, which can e solved numerically. Given a γ0, we can compute γ1,...,γ (in that order) using (6).

4 B. Feedack Design Under a Long-Term Power Constraint Under the more relaxed long-term power constraint, given an index i, the transmitter selects a codeword from a codeook of rate R i = log(1 + γ i P i ). The operating rate R i depends not only on the reconstruction point γ i ut also on the power allocated P i. The design prolem can thus e formulated as {γi,γi, Pi} s.t. P i+1 ) i ) ] log(1 + γ i P i ) i+1 ) i ) ] P i 0 (7) Throughout this section, we assume that a constraint qualification holds at the imizers of (7), so that the KKT conditions are necessary-optimality conditions. Let us introduce the Lagrange multiplier λ 0 associated with the power constraint. It can e shown that we can consider γi = γ i, i 1 without loss of optimality and focus on the dual prolem. A simple iterative, Lloyd-like algorithm can e developed to otain a sequence {γ (k) i,p (k) i,λ (k) }. For clarity, we omit the iteration index k whenever this does not cause any confusion. Given a set {γ i } so that γ > >γ 0, solving the dual prolem is equivalent to allocating power over a set of parallel scalar AWGN channels to imize a linear comination of the achievale rates. The solution is readily otained y the following water-filling algorithm 1 ) 0 ) 1 P 0 = 1 ) λ 1 ] + (8a) γ 0 1 P i = λ 1 ] + i =1,...,K 1 (8) γ i where x] + Δ = (x, 0) and λ is chosen such that the power constraint is active. In the next step, we fix λ and {P i } and solve the dual prolem for {γ i }. Setting the first partial derivatives to zero and simplifying leads to i+1 )=F(γ i )+f(γ i ) 1+γ ip i P i ] 1+γ i P i log + λ (P i P i 1 ), (9) 1+γ i 1 P i 1 which can e solved with the same technique used to solve (6). The two asic steps are iterated until convergence. IV. MULTIPLE-LAYER CODING It is possile to achieve a higher expected rate over a composite channel y means of superposition coding. The no- CSIT case has een considered in 6], 7], 8]. Notice that conditioned on perfect CSIT, the transmitter no longer sees a composite channel, thus the single-layer results apply. A. Feedack Design Under a Short-Term Power Constraint Consider a feedack scheme with K quantization regions, over each region, L-layer coding is employed. Over quantization region i, the code layer j, j =0,...,L 1 is characterized y a reconstruction point γ ij and a power P ij. Without loss of generality, assume γ ij < γ i(j+1), i, j. Viewing the conditionally composite channel as a L-user roadcast channel with channel with channel power γ ij, we design the code rate γ ijp ij 1+γ ij Pk>j P ik ). Code layer Δ of layer j as R ij = log (1+ j is successfully decoded and sutracted from the received signal for any channel power realization γ γ ij y successive decoding and treating the interference from all the layers k with k>jas AWGN (due to the fact that the elements of a capacity-achieving codeook are i.i.d. Gaussian 3]). The short-term power prolem is thus explicitly formulated as {γi,γij, Pij} j=0 s.t. L 1 i+1 ) ij ) ] R ij L 1 P P ij. (10) j=0 Similar to the single-layer coding case, it is necessary that γi = γi0, i > 0 for a scheme to e optimal. However, unlike the single-layer case, the necessary conditions in general cannot e solved directly. Herein we focus on a low-complexity iterative algorithm that successfully exploits results in parallel roadcast channels 9]. The procedure is summarized as follows. Given {γ i0 } and {P ij }, setting the first partial derivatives to zero results in a set of single-unknown equations and can e solved numerically for {γ ij }.Onthe other hand, given {γ ij }, optimal P ij can e found separately for each quantization regions. Over the quantization region i, the optimization prolem is equivalent to allocating a total power of P to imize a linear comination of the achievale rates of an L-user Gaussian roadcast channel, where User j has channel power γ ij and rate reward (i+1)0 ) ij ) ]. This can e solved y a simple algorithm 9]. The special structure of the feedack prolem allows us to otain some interesting results. Clearly, it is not necessary that all the coding levels are assigned nonzero power. The following proposition states a simple sufficient condition that is convenient to identify the quantization regions where singlelayer coding is optimal, independently of the numer of code layers L that the system can afford. The proof is referred to 14]. Proposition 1: Consider a quantization region γi i+1),γ. Suppose that L-layer coding is employed. For any L 2, allocating all the availale power to a single reconstruction point is optimal in an expected-rate sense if and only if arg i+1 ) ) ] γ = γi. (11) γ γ i,γ i+1) If (11) holds, the optimal reconstruction point that receives all the availale power is γ i0 = γ i. Finally, we need to find optimal {γ i0 } given {γ ij }, j>0 and {P ij }. The necessary conditions can e simplified to (i+1)0 )= i0 )+f(γ i0 ) (1 + γ i0p )(1 + γ i0 (P P i0 )) P i0 (R i0 R i 1 ) (12)

5 where R i Δ = L 1 j=0 R ij. Clearly, (6) is a special case of (12) where P i0 = P, i. Again, we have een ale to solve (12) with relative ease using non-derivative numerical techniques. B. Feedack Design Under a Long-Term Power Constraint In this section, we consider the less restrictive prolem {γi,γij, Pij} j=0 s.t. P L 1 i+1 ) ij ) ] R ij i+1 ) i ) ] L 1 P ij. (13) j=0 We can extend the algorithm in Section IV-A to take into account the Lagrange multiplier λ associated with the power constraint. As in Section III-B, it is sufficient to consider γi = γ i0, i > 0. For a fixed {γ ij }, the optimal power allocation and corresponding Lagrange multiplier λ can e found with a greedy algorithm for parallel scalar AWGN roadcast channels 9]. In our scenario, we first need to find the optimal power allocated to each quantization region ( (i+1)0 ) ij ) 1 P i = j (i+1)0 ) i0 ) λ 1 )] + (14) γ ij where λ is chosen such that the power constraint is active. The individual P ij s are found as in the short-term case. With a given set of {γ ij } for j>0, {P ij } and λ, the optimal {γ i0 } can e found y solving (i+1)0 )=F(γ i0 )+f(γ i0 ) (1 + γ i0p i )(1 + γ i0 (P i P i0 )) P i0 R i0 R i 1 + λ(p i P i 1 )], (15) which is a generalization of (9). V. NUMERICAL RESULTS In Figure 2, we plot the expected rate achieved y several feedack schemes with different numers of quantization regions and different numers of code layers over a Rayleigh channel with unit mean power, i.e., ) =1 exp( γ). The average signal-to-noise ratio is defined as SNR = Δ P since the noise variance is assumed to e unit. Significant gains can e oserved even with coarsely quantized systems. For example, to achieve a target expected rate of 2 nats per channel use, feedack schemes with 2 and 4 quantization regions requires a power of roughly 3 and 5 db less than a no-csit system does, respectively. As can e seen, most of the gain of multi-layer coding is oserved in the high-snr regime. Furthermore, the enefit of multi-layer coding appears to e more pronounced as the SNR increases. However, as the quality of partial CSIT improves, the role of multi-layer coding reduces sustantially. For instance, in a system with K =4 quantization regions, there is practically no enefit ofusing 2-layer coding over single-layer coding for any SNR smaller than 30 db. This perhaps can e attriuted to several factors. Firstly, as the resolution of the quantizer increases, the mutual information of a composite channel conditioned on a feedack index ecomes more and more deterministic. Thus transmitting a single layer ecomes less and less suoptimal. Additionally, more quantization regions satisfies the single-layer optimality condition (11) as the quality of CSIT improves. To emphasize the important role of temporal power control in systems with very limited power, we plot the expected rate in the low-snr region in Fig. 3. It should e noted that the SNR range depicted may not e relevant for some wireless communication systems. Since multi-layer coding only provides a negligile improvement over this SNR range, we only plot expected rate achieved y single-layer coding. Interestingly, even a power-control system with coarsely quantized CSIT, namely K =2or 1-it feedack, outperforms a short-term system with perfect CSIT for any SNR smaller than 5 db. REFERENCES 1] L. H. Ozarow, S. Shamai (Shitz), and A. D. Wyner, Information theoretic considerations for cellular moile radio, IEEE Trans. Veh. Technol., vol. 43, pp , May ] G. Caire, G. Taricco, and E. Biglieri, Optimum power control over fading channels, IEEE Trans. Inform. Theory, vol. 45, pp , July ] T. Cover, Broadcast channels, IEEE Trans. Inform. Theory, vol. 18, pp. 2 14, Jan ] E. Biglieri, J. 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6 7 6 L = 1. No CSIT L = 2. No CSIT Perfect CSIT. K =2. L =1. K =2. L =2. K =4. L =1. K =4. L =2. Expected Rate (nats/channel use) SNR (db) Fig. 2. Expected rate achieved with different feedack schemes over a Rayleigh channel. A short-term power constraint is assumed. K = 2. Short-term power constraint. K = 2. Long-term power constraint. K = 4. Short-term power constraint. K = 4. Long-term power constraint. Perfect CSIT. Short-term power constraint. Perfect CSIT. Long-term power constraint. Expected Rate (nats/channel use) SNR (db) Fig. 3. Performance of short-term and long-term power constrained systems over a Rayleigh channel at low SNR. Solid and dashed curves correspond to a short-term and a long-term power constraint, respectively.