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Design of strong symmetric cipher systems requires that the underlying cryptographic Boolean function meet specific security requirements. Some of the required security criteria can be measured with the help of the Autocorrelation function as a tool, while other criteria can be measured using the Walsh transform as a tool. The connection between the Walsh transform and the Autocorrelation function is given by the well known Wiener-Khintchine theorem. In this paper, we present an analysis of the Autocorrelation function from the Haar spectral domain. We start by presenting a brief review on Boolean functions and the Autocorrelation function. Then we exploit the analogy between the Haar and Walsh in deriving the Haar general representation of the Autocorrelation function. The derivations are carried out in two ways namely, in terms of individual spectral coefficients, and based on zones within the spectrum. The main contribution of the paper is the establishment of the link between the Haar transform and the Wiener-Khintchine theorem. This is done by deducing the connection between the Haar transform, the Autocorrelation, and the Walsh power spectrum for an arbitrary Boolean function. In the process we show that, the same characteristics of the Wiener-Khintchine theorem holds locally within the Haar spectral zones, instead of globally as with the Walsh domain. The Haar general representations of Autocorrelation function are given for arbitrary Boolean functions in general and Bent Boolean functions in particular. Finally, we present a conclusion of the work with a summary of findings and future work.

We extend the recurrence time theorem, which is originally derived by Wyner and Ziv, to individual Martin-Lof random sequences. We also present a simple, alternative proof of the algorithmic Shannon-McMillan-Breiman theorem that was first proved by V'yugin.

An autocorrelation measure of a single-frame which is not derived directly from the data is defined. A lattice method is first used to compute the frame-reflection coefficients. Then the predictor coefficients are computed, and the normal equations result in a new set of autocorrelation lags named the lattice autocorrelation measure (LAM). LAM is shown to be more accurate than the conventional Bartlett autocorrelation measure, especially in case of few samples and low SNR. The application of LAM in eigenvalues spectral analysis gives better estimates than conventional autocorrelation, mainly in the central frequency range. The use of the LAM in overdetermined autocorrelation matrices analyzed by singular value decomposition leads to an improvement in the frequency estimation of very close harmonics in the cases of short signal frames. >

A generalized bandpass signal direct sampling theorem has been derived based on the detailed study on the spectrum arrangement of the sampled signal. The theorem considers two cases of the arrangement of the shifted replicas of the original spectrum: the normal placement and the inverse placement of the spectrum. The Shannon sampling theorem is involved in this theorem as a special case in the case of normal spectrum placement.

We generalize the Wiener-Khinchin theorem. A full generalization is presented where both the autocorrelation function and power spectral density are defined in terms of a general basis set. In addition, we present a partial generalization where the density is the Fourier transform of the autocorrelation function but the autocorrelation function is defined in terms of an arbitrary basis set. Both the deterministic and random cases are considered.

The spectral characterization of second-order (or wide-sense) cyclostationarity gives rise to a generalization of the Wiener relation between the power spectral density and the autocorrelation associated with second-order stationary time-series. This generalization, called the cyclic Wiener relation, is a Fourier transform relation between the spectral autocorrelation function and the cyclic temporal autocorrelation function, both defined in terms of time averages on a single time-series. The spectral characterization is generalized from second-order cyclostationarity to n-th order cyclostationarity for n=2,3,4,5,. . ., and some basic properties of the generalised spectral characterization are presented. These include a further generalization of the Wiener relation, called the n-th order cyclic Wiener relation, which relates the n-th order joint cyclic temporal moment function to the n-th order joint spectral moment function. >