4 research outputs found
Estimation in Gaussian Noise: Properties of the Minimum Mean-Square Error
Consider the minimum mean-square error (MMSE) of estimating an arbitrary
random variable from its observation contaminated by Gaussian noise. The MMSE
can be regarded as a function of the signal-to-noise ratio (SNR) as well as a
functional of the input distribution (of the random variable to be estimated).
It is shown that the MMSE is concave in the input distribution at any given
SNR. For a given input distribution, the MMSE is found to be infinitely
differentiable at all positive SNR, and in fact a real analytic function in SNR
under mild conditions. The key to these regularity results is that the
posterior distribution conditioned on the observation through Gaussian channels
always decays at least as quickly as some Gaussian density. Furthermore, simple
expressions for the first three derivatives of the MMSE with respect to the SNR
are obtained. It is also shown that, as functions of the SNR, the curves for
the MMSE of a Gaussian input and that of a non-Gaussian input cross at most
once over all SNRs. These properties lead to simple proofs of the facts that
Gaussian inputs achieve both the secrecy capacity of scalar Gaussian wiretap
channels and the capacity of scalar Gaussian broadcast channels, as well as a
simple proof of the entropy power inequality in the special case where one of
the variables is Gaussian.Comment: Submitted to IEEE Transactions on Information Theory, revised
MMSE of "Bad" Codes
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.Comment: 8 pages, 2 figures, submitted to the IEEE Transactions on Information
Theor
A Survey on MIMO Transmission with Discrete Input Signals: Technical Challenges, Advances, and Future Trends
Multiple antennas have been exploited for spatial multiplexing and diversity
transmission in a wide range of communication applications. However, most of
the advances in the design of high speed wireless multiple-input multiple
output (MIMO) systems are based on information-theoretic principles that
demonstrate how to efficiently transmit signals conforming to Gaussian
distribution. Although the Gaussian signal is capacity-achieving, signals
conforming to discrete constellations are transmitted in practical
communication systems. As a result, this paper is motivated to provide a
comprehensive overview on MIMO transmission design with discrete input signals.
We first summarize the existing fundamental results for MIMO systems with
discrete input signals. Then, focusing on the basic point-to-point MIMO
systems, we examine transmission schemes based on three most important criteria
for communication systems: the mutual information driven designs, the mean
square error driven designs, and the diversity driven designs. Particularly, a
unified framework which designs low complexity transmission schemes applicable
to massive MIMO systems in upcoming 5G wireless networks is provided in the
first time. Moreover, adaptive transmission designs which switch among these
criteria based on the channel conditions to formulate the best transmission
strategy are discussed. Then, we provide a survey of the transmission designs
with discrete input signals for multiuser MIMO scenarios, including MIMO uplink
transmission, MIMO downlink transmission, MIMO interference channel, and MIMO
wiretap channel. Additionally, we discuss the transmission designs with
discrete input signals for other systems using MIMO technology. Finally,
technical challenges which remain unresolved at the time of writing are
summarized and the future trends of transmission designs with discrete input
signals are addressed.Comment: 110 pages, 512 references, submit to Proceedings of the IEE
On MMSE Properties and I-MMSE Implications in Parallel MIMO Gaussian Channels
Abstract—This paper extends the “single crossing point ” property of the scalar MMSE function, derived by Guo, Shamai and Verdú (first presented in ISIT 2008), to the parallel degraded MIMO scenario. It is shown that the matrix Q(t), which is the difference between the MMSE assuming a Gaussian input and the MMSE assuming an arbitrary input, has, at most, a single crossing point for each of its eigenvalues. Together with the I-MMSE relationship, a fundamental connection between Information Theory and Estimation Theory, this new property is employed to derive results in Information Theory. As a simple application of this property we provide an alternative converse proof for the broadcast channel (BC) capacity region under covariance constraint in this specific setting. I