Secrecy Capacity of a Gaussian Wiretap Channel With ADCs is Always Positive

We consider a complex Gaussian wiretap channel with finite-resolution analog-to-digital converters (ADCs) at both the legitimate receiver and the eavesdropper. For this channel, we show that a positive secrecy rate is always achievable as long as the channel gains at the legitimate receiver and at the eavesdropper are different, regardless of the quantization levels of the ADCs. For the achievability, we first consider the case of one-bit ADCs at the legitimate receiver and apply a binary input distribution where the two input points have the same phase when the channel gain at the legitimate receiver is less than that at the eavesdropper, and otherwise the opposite phase. Then the result is generalized for the case of arbitrary finite-resolution ADCs at the legitimate receiver by translating the input distribution appropriately. For the special case of the real Gaussian wiretap channel with one-bit ADCs at both the legitimate receiver and the eavesdropper, we show that our choice of input distribution satisfies a necessary condition of optimal distributions for Wyner codes.