Secrecy Capacity of a Gaussian Wiretap Channel with One-bit ADCs is Always Positive

Abstract

We consider the Gaussian wiretap channel with one-bit analog-to-digital converters (ADCs) at both the legitimate receiver and the eavesdropper. In this channel, we show that a positive secrecy rate is always achievable whenever the noise power n12 at the legitimate receiver is not the same as the noise power n22 at the eavesdropper. A binary phase-shift keying (BPSK) and an asymmetric BPSK are shown to achieve a positive secrecy rate for the cases of n1 < n2 and n1 > n2, respectively. We partially justify the choice of these signalings by showing that the optimal input distribution that achieves Rs∗ := supPX: E[X2]≤ PI(X;Y1)-I(X;Y2), where X is the channel input with power constraint of P, and Y1 and Y2 are the channel outputs at the legitimate receiver and the eavesdropper, respectively, should satisfy some symmetric and asymmetric properties for the cases of n1 < n2 and n1 > n2, respectively. Moreover, for n1 < n2 and sufficiently large P, it is shown that a BPSK using power smaller than P achieves Rs∗.