Bayes’ rule, as one of the fundamental concepts of statistical signal processing, provides a way to update our belief about an event based on the arrival of new pieces of evidence. Uncertainty is traditionally modeled by a probability distribution. Prior belief is thus expressed by a prior probability distribution, while the update involves the likelihood function, a probabilistic expression of how likely it is to observe the evidence.

Inference tasks in signal processing are often characterized by the availability of reliable statistical modeling with some missing instance-specific parameters. One conventional approach uses data to estimate these missing parameters and then infers based on the estimated model. Alternatively, data can also be leveraged to directly learn the inference mapping end to end. These approaches for combining partially known statistical models and data in inference are related to the notions of generative and discriminative models used in the machine learning literature [1] , [2] , typically considered in the context of classifiers.

Linear regression models have a wide range of applications in statistics, signal processing, and machine learning. In this Lecture Notes column we will examine the performance of the least-squares (LS) estimator with a focus on the case when there are more parameters than training samples, which is often overlooked in textbooks on estimation.

Fourier theory is the backbone of signal processing (SP) and communication engineering. It has been widely used in almost all branches of science and engineering in numerous applications since its inception. However, Fourier representations such as Fourier series (FS) and Fourier transform (FT) may not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). 

Linear time-invariant (LTI) systems play a fundamental role in signal processing. Continuity is an important property of LTI systems, without which many conclusions about LTI systems, such as convolution formula and commutative law, are not true in general. However, this concept does not receive as much attention as it should in the literature of signal processing.

In Part 1 of this "Lecture Notes" article [1], we introduced a modern perspective on the standard tools for power system analysis-the Clarke and related transforms-through the lens of data analytics. We also indicated their limitations when dealing with unbalanced power system conditions.

Functions of complex variables arise frequently in the formulation of sig - nal processing problems. The basic calculus rules on differentiation and integration for functions of complex variables resemble, but are not identical to, the rules of their real variable counterparts. 

The sliding discrete Fourier transform (SDFT) is an efficient method for computing the N-point DFT of a given signal starting at a given sample from the N-point DFT of the same signal starting at the previous sample [1]. However, the SDFT does not allow the use of a window function, generally incorporated in the computation of the DFT to reduce spectral leakage, as it would break its sliding property.

After decades of advances in signal processing, this article goes back to square one, when the word signal was defined. Here we investigate if everything is all right with this stepping stone of defining a signal.