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Identification for Colored Gaussian Channels

Mohammad Javad Salariseddigh

Abstract

We study the identification capacity of discrete-time Gaussian channels impaired by correlated noise and inter-symbol interference (ISI). Our analysis is formulated for deterministic encoding functions subject to a peak power constraint and colored noise whose covariance matrix features a polynomially bounded singular value spectrum, i.e., $\sim [n^{-μ} , n^{μ/2}]$ where $n$ is the codeword length and $μ\in [0,1/2)$ is the spectrum rate. A central result establishes that, even when the ISI memory length grows sub-linearly with $n,$ i.e., $\sim n^κ$ where $κ\in [0,1/2)$ and $κ+ μ\in [0,1/2),$ the codebook size continues to exhibit super-exponential growth in $n$, i.e., $\sim 2^{(n \log n)R},$ with $R$ representing the associated coding rate. Moreover, by employing the well-known Mahalanobis-distance decoder induced by colored Gaussian noise statistics, we characterize bounds on the identification capacity, with the resulting bounds parameterized by $κ$ and $μ.$

Identification for Colored Gaussian Channels

Abstract

We study the identification capacity of discrete-time Gaussian channels impaired by correlated noise and inter-symbol interference (ISI). Our analysis is formulated for deterministic encoding functions subject to a peak power constraint and colored noise whose covariance matrix features a polynomially bounded singular value spectrum, i.e., where is the codeword length and is the spectrum rate. A central result establishes that, even when the ISI memory length grows sub-linearly with i.e., where and the codebook size continues to exhibit super-exponential growth in , i.e., with representing the associated coding rate. Moreover, by employing the well-known Mahalanobis-distance decoder induced by colored Gaussian noise statistics, we characterize bounds on the identification capacity, with the resulting bounds parameterized by and

Paper Structure

This paper contains 15 sections, 7 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. System Model and Coding Preliminaries
  3. Colored Gaussian Channel
  4. Identification Coding
  5. Identification Capacity of the Colored Gaussian Channel with ISI
  6. Main Results
  7. Achievability
  8. Upper Bound (Converse Proof)
  9. Conclusion
  10. Acknowledgments
  11. Anaysis of Whitening Noise Transformation
  12. Upper Bound on The Rayleigh Quotient
  13. Bounds on the Singular Values of the Inverse Matrix
  14. Proof of Lemma \ref{['Lem.Converse']}
  15. Spectrum of Covariance Matrix Power

Key Result

Theorem 1

Consider the ISI Gaussian channel, $\mathcal{G}_{\mathbf{h}},$ with CIR $\mathbf{h}$ and covariance matrix $\boldsymbol{\mathbf{\Sigma}}$ fulfilling conditions C1-C3 and assume that the the number of ISI channel taps grows sub-linearly with the codeword length, i.e., $K(n,\kappa) = n^{\kappa},$ wher $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Definition 1: Colored Gaussian identification code
  • Definition 2: Colored Gaussian identification capacity
  • Theorem 1
  • proof
  • Lemma 1: minimum distance of the convoluted codebook
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 11 more