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IEEE The use of the Wiener–Khinchin theorem in the reverberation chamber reveals the relationships between a number of important parameters—the coherence bandwidth and the Q-factor measured in the time domain, the coherence time and the Q-factor measured in the frequency domain, the K-factor and the Doppler spectrum, and the K-factor and the total scattering cross section. The lower bound of the average K-factor is also given. Different physical quantities, which share similar mathematical insights, are unified. Analytical derivations are given, and results are validated by measurements.

Considering that fractional correlation function and the fractional power spectral density, for $\alpha$-stationary random signals, form a fractional Fourier transform pair. We present an interpolation formula to estimate a random signal from a temporal random series, based on the fractional sampling theorem for $\alpha$ -bandlimited random signals. Furthermore, by establishing the relationship between the sampling theorem and the von Neumann ergodic theorem, the estimation of the power spectral density of a random signal from one sample signal becomes a suitable approach. Thus, the validity of the sampling theorem for random signals is closely linked to an ergodic hypothesis in the mean sense.

We extend Szego's eigenvalue distribution theorem to block Toeplitz matrices. A sequence of such matrices arises from the autocorrelation of a 2-D discrete random process. We show that the eigenvalues of the matrix sequence are asymptotically distributed like the samples of the random process' 2-D power spectrum.

We generalize the concept of the autocorrelation function and give the generalization of 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 characteristic function but the characteristic function is defined in terms of an arbitrary basis set. Both the deterministic and random cases are considered.

Random processes with stationary nth differences serve as models for oscillator phase noise. The theorem proved here allows one to obtain the structure function (covariances of the nth differences) of such a process in terms of the differences of a single function of one time variable. In turn, this function can easily be obtained from the spectral density of the process. The theorem is used for computing the variance of two estimators of frequency stability.

It is shown that for a random function of two or three dimensions and the case where statistical properties are isotropic in space, the Wiener-Khintchine theorem takes on the form of a Fourier-Bessel transform instead of a one-dimensional Fourier transform.

For a random process x(t) with autocorrelation function R(s, t) satisfying ∫ I R(t, t) dt 2 functions on I converges to x(t) in an integral mean-square sense on I. This generalizes a result due to Young, on the use of orthonormalized exponentials in representing random processes.

Szasz's theorem is extended to the exponential representation of random signals. It is shown that for the class of random processes with square integrable autocorrelation functions R(t, τ), the set {eskt} is complete if and only if this set satisfies Szasz's condition. The result also holds for the class of random processes having absolutely integrable R(t, t).