Quantized vs. analog feedback for the MIMO downlink: a comparison between zero-forcing based achievable rates

Abstract-We consider a MIMO fading broadcast channel and compare the achievable ergodic rates when the channel state information at the transmitter is provided by "analog" noisy feedback or by quantized (digital) feedback. The superiority of digital feedback is shown, with perfect or imperfect CSIR, whenever the number of feedback channel uses per channel coefficient is larger than 1. Also, we show that by proper design of the digital feedback link, errors in the feedback have a minor effect even by using very simple uncoded modulation. Finally, we show that analog feedback achieves a fraction 1 − 2F of the optimal multiplexing gain even in the presence of a feedback delay, when the fading belongs to the class of "Doppler processes" with normalized maximum Doppler frequency shift 0 ≤ F < 1/2. I. MODEL SETUP AND BACKGROUND We consider a multi-input multi-output (MIMO) Gaussian broadcast channel modeling the downlink of a system where the base station (transmitter) has M antennas and K user terminals (receivers) have one antenna each. A channel use of such channel is described by where y k is the channel output at receiver k, z k ∼ CN(0, N 0 ) is the corresponding AWGN, h k ∈ C M is the vector of channel coefficients from the k-th receiver to the transmitter antenna array and x is the channel input vector. The channel input is subject to the average power constraint E[|x| 2 ] ≤ P . We assume that the channel state, given by the collection of all channel vectors M×K , varies in time according to a block fading model where H is constant over each frame of length T channel uses, and evolves from frame to frame according to an ergodic stationary jointly Gaussian process; i.i.d. block-fading channel, where the entries of H are Gaussian i.i.d. with elements ∼ CN(0, 1) is a special case of this. A. Capacity results If H is perfectly and instantaneously known to all terminals (perfect CSIT and CSIR), the capacity region of the channel (1) is obtained by MMSE-DFE beamforming and Gaussian dirty-paper coding (see (2) We consider the situation where K = M , and thus do not consider user selection. Furthermore, we are mainly interested in the high-spectral efficiency regime, where we can characterize the achievable sum rate as κ log P/N 0 + O(1), and κ is the "system multiplexing gain" or "pre-log factor" of the ergodic sum rate. Hence, it is well-known that using uniform power P k = P/M for all k = 1, . . . , M , rather than performing optimal water-filling, incurs a loss only in the O(1) term, and we shall restrict to this choice in the rest of this paper. It is well-known that, under perfect CSIT and CSIR, both the optimal "Dirty-Paper" sum-rate C and the zero-forcing sum-rate R ZF are equal to M log P/N 0 + O(1). On the contrary, under non-perfect CSIT the rate sum may behave in a radically different way; for example, if there is perfect CSIR and no CSIT when H has i.i.d. Gaussian entries, the sum rate is equal to log P/N 0 + O(1) [1] B. Channel state feedback models We consider some specific CSIT and CSIR models and derive lower-bounds to the corresponding achievable ergodic rates by analyzing a naive beamforming scheme that computes a mismatched ZF beamforming matrix V from the CSIT. In particular, we consider an "analog" CSIT feedback scheme where the transmitter observation at frame time t is given by where {W(τ )} is a spatially and spectrally white Gaussian process with elements ∼ CN(0, N 0 ) and d is the feedback delay. This models the case where the channel coefficients are explicitly transmitted on the reverse link (uplink) using unquantized quadrature-amplitude modulatio