2.9a Fourier Analysis

Fourier Analysis

Fourier analysis is a mathematical technique that decomposes complex functions or signals into a sum of simpler trigonometric functions, specifically sines and cosines, or complex exponentials. In the realm of linear algebra, Fourier analysis can be interpreted as a simple change of basis that compresses a signal to remove its most irrelevant components.

Any Fourier analysis method contains a synthesis and an analysis equation:

1. The synthesis equation (inverse Fourier) aims to decompose a signal $x$ into a linear combination of its Fourier coefficients in the frequency domain, $X$.
2. The analysis equation aims to decompose a signal’s Fourier coefficients $X$ into a linear combination of its signal values.

The following chart outlines all four Fourier analysis methods:

| | D-T | C-T |
| :-: | :—: | :—: |
| D-F | DTFS | CTFS |
| C-F | DTFT | CTFT |
where the rows signify whether the Fourier analysis is discrete or continuous in the frequency domain, and the columns signify whether it is discrete or continuous in the time domain.

Note (Periodicity) Fourier series works with periodic signals in the time domain and produces discrete frequency coefficients, the Fourier transform deals with aperiodic signals in the time domain and produces a continuous frequency spectrum. Additionally, the DTFT has a periodic frequency response over $2\pi$, whereas the CTFT produces a non-periodic continuous spectrum.

While the

Note (Infinite Signals) $p$). For signals of finite length, the DFT (discrete Fourier transform) and the CFT (continuous Fourier transform) are methods of Fourier analysis that can be applied. Their formulations are very close to the DTFS and CTFS.

All of the above methods deal with signals of assumed infinite length (with the Fourier series dealing with periodic signals, therefore limiting itself to a finite interval