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

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 $n_{1}^{2}$ at the legitimate receiver is not the same as the noise power $n_{2}^{2}$ 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 $n_{1} < n_{2}$ and $n_{1} > n_{2}$, respectively. We partially justify the choice of these signalings by showing that the optimal input distribution that achieves $R_{s}^{*}:= \displaystyle \sup _{P_{X}:\mathrm {E}[X^{2}]\leq P}I(X;Y_{1}) -I(X;Y_{2})$, where $X$ is the channel input with power constraint of $P$, and $Y_{1}$ and $Y_{2}$ are the channel outputs at the legitimate receiver and the eavesdropper, respectively, should satisfy some symmetric and asymmetric properties for the cases of $n_{1} < n_{2}$ and $n_{1} > n_{2}$, respectively. Moreover, for $n_{1} < n_{2}$ and sufficiently large $P$, it is shown that a BPSK using power smaller than $P$ achieves $R_{s}^{*}$.