Quantized vs. Analog Feedback for the MIMO Downlink: A Comparison between Zero-Forcing Based Achievable Rates

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 yk = h H kx + zk, k = 1, . . . ,K (1) where yk is the channel output at receiver k, zk ∼ CN(0, N0) is the corresponding AWGN, hk ∈ C 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 H = [h1, . . . ,hK ] ∈ CM×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.