Waterfilling Gains O(1/SNR) at High SNR

We show that the gain for using a waterfilling power allocation instead of a flat allocation over non-singular channel components is negligible at high signal-to-noise ratios. Consider the standard additive noise communication channel with a quadratic power constraint. Although waterfilling provides the optimal input distribution for Gaussian noise channels, sub-optimal distributions are often not too bad. For example, [1] shows that transmitting at only 2 levels (either on or off) in each sub-channel loses very little. Similarly, [2], [3] show that the mutual information for an optimal waterfilling power allocation on any additive noise channel is at most 0.5 bits per real channel use higher than the mutual information for an independent, identically distributed, Gaussian input. Here we use elementary arguments to show that the gain in mutual information for the optimal waterfilling power allocation versus a flat power allocation over non-singular channel modes is at most O(SNR) where SNR denotes the signal-to-noise ratio parameter. When power is allocated across all n modes instead of only the k non-singular modes, the mutual information penalty (in bits per complex channel mode) is (k/n) log2(n/k) + O(SNR ) (which is at most e log2 e +O(SNR ) ≈ .53 +O(SNR)). In [3] this bound is tightened to show that the mutual information loss for a flat Gaussian input is at most e log2 e ≈ .53 bits per complex channel mode for any SNR. A. Problem Model and Notation We denote vectors and sequences in bold (e.g., x) with the ith element denoted as xi. Matrices are capitalized bold letters. Random variables are denoted using the sans serif font (e.g., x) while random vectors and sequences are denoted with bold sans serif (e.g., x). A variety of scenarios involve transmitting over a channel with different gains or noise powers which can be ergodic, non-ergodic, time-selective or frequency-selective. Since the role of waterfilling is essentially the same in all these cases, we focus on the following model and briefly mention how to translate the main results to other scenarios. The transmitter selects an n-vector, x, as input to the channel which produces the output y according to y = H · x + w (1) where w are independent, complex, zero mean, Gaussian random variables with an identity covariance matrix. We model a power constraint by requiring E[||x||] < SNR. Furthermore, we consider the case where the receiver always has perfect knowledge of the deterministic channel matrix H. For example, (1) may represent a multi-antenna channel. Alternatively, we can model a time or frequency selective fading channel by considering a diagonal H where the coefficients correspond to separate time-slots or frequency bands. In any case, maximizing the average mutual information between the channel input sequence and channel output sequence is often desirable for maximizing capacity or minimizing outage probability. In the following we will consider the difference between the mutual information for the optimal Gaussian waterfilling input distribution and a Gaussian input distribution with a flat power allocation either over the non-singular channel modes or all channel modes. B. Non-singular Channels Theorem 1. Consider the setting in (1) with H non-singular. If we let I(SNR) represent the mutual information for the optimal waterfilling input distribution and let Iflat(SNR) represent the mutual information for an independent, complex, circularly symmetric, zero mean, Gaussian input distribution with flat power allocation, then I(SNR)− Iflat(SNR) < O(SNR ). (2) Proof. Without loss of generality we assume that H is diagonal. If this is not the case, we can always take the Singular Value Decomposition (SVD).1 Thus we consider the equivalent channel yi = si · xi + wi (3) where the wi are zero mean, unit variance, complex, circularly symmetric, Gaussian random variables and the si are the singular values of H. The resulting mutual information for a flat power allocation is Iflat(SNR) = n