MMSE of “Bad” Codes

Abstract

We examine codes, over the additive Gaussian noise channel, designed for reliable communication at some specific signal-to-noise ratio (SNR) and constrained by the permitted minimum mean-square error (MMSE) at lower SNRs. The maximum possible rate is below point-to-point capacity, and hence these are non-optimal codes (alternatively referred to as "bad" codes). We show that the maximum possible rate is the one attained by superposition codebooks. Moreover, the MMSE and mutual information behavior as a function of SNR, for any code attaining the maximum rate under the MMSE constraint, is known for all SNR. We also provide a lower bound on the MMSE for finite length codes, as a function of the error probability of the code.

On the Minimum Mean $p$ th Error in Gaussian Noise Channels and Its Applications

“…in [8], by using various extensions of the SCPP, to prove a special case of the vector EPI, a converse for the capacity region of the parallel degraded BC under per-antenna power constraints and under an input covariance constraint, and a converse for the compound parallel degraded BC under an input covariance constraint; and in [10] to provide a converse for communication under an MMSE disturbance constraint.…”

“…in [13], for the Gaussian BC with outputs Y snr 1 and Y snr 0 , where snr 0 ≤ snr 1 and rate pair (R 1 , R 2 ) = 1 2 log(1 + βsnr 1 ), 1 2 log 1+snr 0 1+βsnr 0 for some β ∈ [0, 1], with γ = 0; in [13], for the Gaussian wiretap channel with outputs Y snr 0 (primary) and Y snr 1 (eavesdropper) with maximum equivocation d max and rate R ≥ d max , for β = γ = 0; and in [10], for the Gaussian pointto-point channel with output Y snr 1 and an MMSE disturbance constraint at Y snr 0 measured by mmse(X, snr 0 ) ≤ β 1+βsnr 0 for some β ∈ [0, 1] with γ = β. The jump discontinuities in (76) at snr = snr 0 and snr = snr 1 are referred to as the phase transitions.…”

On Communication Through a Gaussian Channel With an MMSE Disturbance Constraint

“…The subscript n in C n (snr, snr 0 , β) emphasizes that we seek to find bounds that hold for any input length n. Even though this model is somewhat simplified, compared to the G-IC, it can serve as an important building block towards characterizing the capacity of the G-IC [12], [13].…”

“…In [12] the capacity of the channel in Fig. 1 was properly defined and it was shown to be equal to lim n→∞ C n (snr, snr 0 , β).…”

On MMSE properties of optimal codes for the Gaussian wiretap channel

“…Moreover, similar to the analysis of optimal point-to-point codes done in [11] and [12], such an understanding allows us to determine the effect/disturbance of these codes on other possible unintended receivers, which are neither eavesdroppers nor require the reliable decoding of the message. For the point-to-point Gaussian channel such an analysis has been done in [13] where the disturbance has been measured in terms of the mutual information and in [14] where the disturbance has been measured in terms of the MMSE.…”