Wyner has shown in his seminal paper on (discrete memoryless) wiretap channels that if the channel between the sender and an eavesdropper is a degraded version of the channel between the sender and the legitimate receiver, then the sender can reliably and securely transmit a message to the receiver, while the eavesdropper obtains absolutely no information about the message. Later, Leung-Yan-Cheong and Hellman extended Wyner's result to the case where the noise is white Gaussian. In this paper we extend the white Gaussian wiretap channel to the colored Gaussian case and show the finite block length secrecy capacity of colored Gaussian wiretap channels. We also show the asymptotic secrecy capacity of a specific colored Gaussian wiretap channel for which optimal power allocation can be found by a water-filling procedure.

High-resolution spectrum estimation techniques have been extensively studied in recent publications. Knowledge of the noise variance is vital for spectrum estimation from noise-corrupted observations. This paper presents the use of noise compensation and data extrapolation for spectrum estimation. We assume that the observed data sequence can be represented by a set of autoregressive parameters. A recently proposed iterative algorithm is then used for noise variance estimation while autoregressive parameters are used for data extrapolation. We also present analytical results to show the exponential decay characteristics of the extrapolated samples and the frequency domain smoothing effect of data extrapolation. Some statistical results are also derived. The proposed noise-compensated data extrapolation approach is applied to both the autoregressive and FFT-based spectrum estimation methods. Finally, simulation results show the superiority of the method in terms of bias reduction and resolution improvement for sinusoids buried in noise.

The processing of noise-corrupted signals is a common problem in signal processing applications. In most of the cases, it is assumed that the additive noise is white Gaussian and that the constant noise variance is either available or can be easily measured. However, this may not be the case in practical situations. We present a new approach to additive white Gaussian noise variance estimation. The observations are assumed to be from an autoregressive process. The method presented here is iterative, and uses low-order Yule-Walker equations (LOYWEs). The noise variance is obtained by minimizing the difference in the second norms of the noisy Yule-Walker solution and the estimated noise-free Yule-Walker solution. The noise-free solution is constrained to match the observed autocorrelation sequence. In the iterative noise variance estimation method, a variable step-size update scheme for the noise variance parameter is utilized. Simulation results are given to confirm the effectiveness of the proposed method.