576 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 2, FEBRUARY 2013 Maximum Expected Rates of Block-Fading Channels with Entropy-Constrained Channel State Feedback V´ ıctor M. Elizondo and Milan S. Derpich Abstract—We obtain the maximum average data rates achiev- able over block-fading channels when the receiver has perfect channel state information (CSI), and only an entropy-constrained quantized approximation of this CSI is available at the trans- mitter. We assume channel gains in consecutive blocks are independent and identically distributed and consider a short term power constraint. Our analysis is valid for a wide variety of channel fading statistics, including Rician and Nakagami-m fad- ing. For this situation, the problem translates into designing an optimal entropy-constrained quantizer to convey approximated CSI to the transmitter and to define a rate-adaptation policy for the latter so as to maximize average downlink data rate. A numerical procedure is presented which yields the thresholds and reconstruction points of the optimal quantizer, together with the associated maximum average downlink rates, by finding the roots of a small set of scalar functions of two scalar arguments. Utilizing this procedure, it is found that achieving the maximum downlink average capacity C requires, in some cases, time sharing between two regimes. In addition, it is found that, for an uplink entropy constraint ¯ H< log 2 (L), a quantizer with more than L cells provides only a small capacity increase, especially at high SNRs. Index Terms—Channel state information feedback, Informa- tion rates, fading channels, quantization, radio communication. I. I NTRODUCTION I T is well known that the achievable data rates for reliable communication over a fading wireless channel depend on the availability of channel state information (CSI) at the transmitter and receiving end [1], [2]. For single-input single- output (SISO) flat fading channels, the CSI consists of channel gain and phase. If perfect CSI is available at the transmitter (perfect CSIT) and at the receiver (perfect CSIR), the channel is slowly fading and the transmission is subject to a long-term average power constraint, then the average capacity is achieved by adapting rate and power to the channel gain in a time water- filling fashion [3], [4]. By contrast, if an instantaneous (per block) maximum power constraint is imposed, the fades are ergodic and the transmission blocks are long enough so that the fade statistics over each block converge to their ensemble Manuscript received August 14, 2011; revised January 23 and May 19, 2012. The associate editor coordinating the review of this letter and approving it for publication was O. Simeone. V. M. Elizondo and M. S. Derpich are with the Department of Electronic Engineering, Universidad T´ ecnica Federico Santa Mar´ ıa, Valpara´ ıso, Chile (e-mail: victor.elizondo.v@gmail.com, milan.derpich@usm.cl). This work was supported by CONICYT grant ACT-53 and FONDECYT grants 3100109 and 1120468. Digital Object Identifier 10.1109/TCOMM.2012.12.110537 statistics, then the ergodic channel capacity is achievable without CSIT [3], [5]. Else, if the fading is so slow that channel gain can be regarded as constant within each block (which corresponds to a block-fading scenario) then CSIT is beneficial. In this case, with perfect CSIT and per-block power constraint, the capacity is achieved by transmitting at maximum power, with only the data rate being adapted to the channel gain in each transmission block [3]. If perfect CSIR is available and the receiver feeds back this CSI via an uplink with limited information throughput, then only imperfect CSI will be available at the transmitter. In a block-fading situation, the uncertainty at the transmitter about the true channel gain in each block implies a trade-off between throughput and reliability: the larger the data-rate chosen by the transmitter, the higher the probability of exceeding the channel capacity during the transmission block [6]. This poses the problem of encoding the CSI at the receiver and decoding it at the transmitter (i.e., choosing rate and power) in a rate-distortion optimal fashion, where the distortion is some measure of the decrease in downlink throughput, as in [3], [7] or the increase in error probability, as in [8]. The capacity of memory-less block-fading SISO channels with long-term power constrained downlink transmission and fixed-rate constrained CSI feedback was studied in [9]. A similar situation was considered in [10], assuming a multi- layer downlink coding scheme in which data-blocks are de- coded perfectly or totally lost if transmission data-rate is, respectively, below or above the channel capacity during the block. The idea in [10] was to design a quantizer with a fixed number of quantization cells so as to maximize the expected downlink rate, i.e., the expected number (or long term average) of successfully decoded bits. Also for a con- straint in the number of CSI quantization cells, [11] studied the maximization of downlink throughput considering a noisy feedback channel. There exist also numerous results related to downlink throughput maximization problems for multiple- input multiple-output (MIMO) wireless channels (see, e.g. [2], [12]–[14] and the references therein). Although not directly related to the SISO problem (which is the focus of this work), it is worth mentioning that, in all the MIMO results in [2], [12]–[14] and the references therein, the only constraint on the quantizer (where there is a quantizer) is its cardinality. In [15], the maximum SISO downlink average throughput under a long-term power constraint and for a fixed number of quantization cells is analyzed. The performance of zero- 0090-6778/13$31.00 c  2013 IEEE