Chapter 7

Fourier Methods

Core Topics

Approximating a Square Wave by a Series of sine functions (7.2).

General (Infinite) Fourier Series (7.3).

Complex Form of the Fourier Series (7.4).

The Discrete Fourier Series and Discrete Fourier Transform (7.5).

Complex Discrete Fourier Transform (7.6).

Power (Energy) Spectrum (7.7)

Aliasing and Nyquist Frequency (7.8)

Alternative Forms of the Discrete Fourier Transform (7.9).

Use of MATLAB Built-In Functions for Calculating Fast Fourier Transform (7.10).

Leakage and Windowing (7.11).

Bandwidth and Filters (7.12).

Complementary Topics

The Fast Fourier Transform (FFT) (7.13).

7.1 BACKGROUND

Fourier methods are mathematical methods that use sinusoidal functions to represent and approximate other functions. They are widely used in applied mathematics, the basic sciences, and in many applications in engineering and medicine. These functions that are represented by simpler sinusoidal functions may be discrete or continuous, e.g. intensity of an image versus position, intensity of sound versus time, or intensity of light versus time or position. The methods are named after Jean Baptiste Joseph Fourier, a French mathematician and scientist, who lived in the late 18th to the early 19th century. Fourier was a student of Joseph-Louis Lagrange1, and was briefly at the Ecole Normale Superieur and later at the Ecole Polytechnique established by Napoleon Bonaparte2. Fourier was interested in (among other subjects) the propagation of heat and presented ...

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