The Fourier Transform

The harmonic analysis of a deterministic signal serves as a fundamental tool in signal theory and processing. Fourier transformation achieves this analysis, providing a spectral representation of signals. It reveals the frequency distribution of amplitude and phase, describing the energy or power of a signal. Various formulations of this transformation exist:

Table 1: Types of Fourier Transforms.

  • The classical fourier transform is applicable to analytical expressions, functioning as a symbolic analysis tool. It is typically calculated manually."

  • The discrete fourier transform is designed for digital sequences, making it a digital tool usually calculated by software. It transforms a sequence x(n) of N samples into a sequence X(k) of N samples.

  • The fourier transform in discrete time is used for digital sequences, enabling the expression of the 'transfer function' of a convolution. It characterises a filter as a sequence of exponentials. While it can be performed manually for small sequences, software is often used for larger sequences.