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Lecture Notes

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). 

A window function is a mathematical function that is zero valued outside some chosen interval [1] , [2] . For applications like filtering, detection, and estimation, the window functions take the form of limited time functions, which are in general real and even functions [3] , [4] , while for applications like beamforming and image processing, they are limited spatial functions. A spatial window can be a complex function for optimizing the beams in magnitude as well as in phase, as in the case of certain antenna arrays, where the phasor currents in the array are complex numbers [5].

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.

In many signal processing applications, filtering is accomplished through linear time-invariant (LTI) systems described by linear constant-coefficient differential and difference equations since they are conveniently implemented using either analog or digital hardware [1]. An LTI system can be completely characterized in the time domain by its impulse response or in the frequency domain by its frequency response, which is the Fourier transform of the system’s impulse response.

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