PPubMed23 May 2018Cited 8×

Rate-Distortion Function Upper Bounds for Gaussian Vectors and Their Applications in Coding AR Sources

Jesús Gutiérrez-Gutiérrez, Marta Zárraga-Rodríguez, Fernando M. Villar-Rosety, Xabier Insausti

Abstract

'Fernando M. Villar-Rosety' 'Xabier Insausti'] In this paper, we give upper bounds for the rate-distortion function (RDF) of any Gaussian vector, and we propose coding strategies to achieve such bounds. We use these strategies to reduce the computational complexity of coding Gaussian asymptotically wide sense stationary (AWSS) autoregressive (AR) sources. Furthermore, we also give sufficient conditions for AR processes to be AWSS.

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The paper derives general upper bounds on the RDF of Gaussian vectors and introduces low‑complexity DFT‑based coding schemes that efficiently compress AWSS AR sources.

What’s new

i.Upper bounds for the rate‑distortion function of any Gaussian vector
ii.Two low‑complexity DFT‑based coding strategies that achieve the bounds
iii.Application of these strategies to reduce computational complexity for coding Gaussian AWSS AR sources
iv.New sufficient conditions for AR processes to be asymptotically wide‑sense stationary

How they did it

The authors prove new results on the discrete Fourier transform of random vectors, use these to derive analytical upper bounds on the RDF, and design coding schemes based on the DFT that meet the bounds. They then apply the schemes to Gaussian asymptotically wide‑sense stationary autoregressive sources and validate the theory with a numerical example.

What they found

Upper bounds converge to the exact RDF as the data‑block size grows
DFT‑based coding achieves the bounds with O(n log n) complexity versus O(n²) for the optimal eigen‑decomposition method
Numerical example (AR(1) with a₋₁=‑½, σ²=1, D=4/9, n≤100) confirms the rates match the theoretical predictions

Caveats

✕Results are limited to Gaussian vectors and AR sources
✕Upper bounds may be loose for small block lengths
✕Coding strategies require the source to be asymptotically wide‑sense stationary
✕Implementation needs knowledge of the transformed covariance Ey_k y_kᵀ

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