A Proof of the Strong Converse Theorem for Gaussian Multiple Access Channels

TL;DR

This paper proves the strong converse theorem for Gaussian multiple access channels, showing that any achievable rate must lie within the capacity region, using techniques like expurgation, quantization, and hypothesis testing bounds.

Contribution

It establishes the strong converse for Gaussian MACs with a novel proof combining expurgation, quantization, and hypothesis testing methods.

Findings

Any rate outside the capacity region leads to error probability approaching one.

The proof techniques can be extended to Gaussian interference channels under strong interference.

The bounds on sum-rates are derived using type-II error of hypothesis tests.

Abstract

We prove the strong converse for the $N$-source Gaussian multiple access channel (MAC). In particular, we show that any rate tuple that can be supported by a sequence of codes with asymptotic average error probability less than one must lie in the Cover-Wyner capacity region. Our proof consists of the following. First, we perform an expurgation step to convert any given sequence of codes with asymptotic average error probability less than one to codes with asymptotic maximal error probability less than one. Second, we quantize the input alphabets with an appropriately chosen resolution. Upon quantization, we apply the wringing technique (by Ahlswede) on the quantized inputs to obtain further subcodes from the subcodes obtained in the expurgation step so that the resultant correlations among the symbols transmitted by the different sources vanish as the blocklength grows. Finally, we derive upper bounds on achievable sum-rates of the subcodes in terms of the type-II error of a binary hypothesis test. These upper bounds are then simplified through judicious choices of auxiliary output distributions. Our strong converse result carries over to the Gaussian interference channel under strong interference as long as the sum of the two asymptotic average error probabilities less than one.