In this paper, the k-th order autoregressive moving average (ARMA(k)) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of the signals in both forward and feedback channels. It is shown that a variant of the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the feedback Gaussian channel, achieves the same maximum rate for the same channel with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the 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. As the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
Index TermsSecrecy Capacity, Feedback, Colored Gaussian, Schalkwijk-Kailath Scheme
I. INTRODUCTIONIt has been more than a half century since the information theorists started to investigate the capacity of feedback Gaussian channels. As the pioneering studies on this topic, Shannon's 1956 paper [2] showed that feedback does not increase the capacity of the memoryless AWGN channel, and Elias [3] [4] proposed some simple corresponding feedback coding schemes. Schalkwijk and Kailath [5] [6] then developed a notable linear feedback coding scheme to achieve the capacity of the feedback AWGN channel. Thereafter, the problem of finding the feedback capacity and the capacity-achieving codes for the memory Gaussian channels (e.g. ARMA(k)) has been extensively studied. Butman [7] [8], Wolfowitz [9] and Ozarow [10] [11] extended Schalkwijk's scheme to ARMA(k) Gaussian channels, leading Chong Li is with Qualcomm Research, Bridgewater, NJ, 08807, USA (chongl@qti.qualcomm.com). Yingbin Liang is with the