Signals & Systems II: Second Order Statistics

This lecture covers the analysis of second order statistics in signal processing, focusing on complex random vectors and the duality of energies in Fourier transformation. It explains the modeling of physical noises with Gaussian processes, the concept of stationarity in random signals, and the distinction between ergodic and non-ergodic processes. The instructor discusses the properties of stationary signals, the importance of separate notations, and the relationship between time averages and statistical means. Examples are provided to illustrate the concepts of stationarity and ergodicity.

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In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

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In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit.

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In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.

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