Abstract
In this paper, the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of signals in both forward and feedback channels. It is shown that the generalized Schalkwijk–Kailath scheme, a capacity-achieving coding scheme for the Gaussian feedback channel, achieves the same maximum rate for the same channel even with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the Gaussian feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. Our result implies that as the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
Highlights
It has been more than a half century since information theorists began investigating the capacity of feedback Gaussian channels
We show that the proposed coding scheme provides non-trivial positive secrecy rates and achieves the feedback capacity of the additive white Gaussian noise (AWGN) channel as the amplitude of the quantization noise vanishes to zero
We have considered the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with feedback and have shown that the feedback secrecy capacity equals the feedback capacity without the presence of an eavesdropper
Summary
It has been more than a half century since information theorists began investigating the capacity of feedback Gaussian channels. As the pioneering studies on this topic, Shannon’s 1956 paper [3] showed that feedback does not increase the capacity of the memoryless additive white Gaussian noise (AWGN) channel, and Elias in [4] and [5] proposed some simple corresponding feedback coding schemes. Schalkwijk and Kailath in [6] and [7] developed a notable linear feedback coding scheme to achieve the capacity of the AWGN channel with feedback. Thereafter, the problem of finding the feedback capacity and capacity-achieving codes for Gaussian channels with memory, e.g., finite-order autoregressive moving average (ARMA) channels, has been extensively studied.
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