Mathematics for electricians - Laplace, Fourier and Z-transforms

Symbolic calculus makes it possible to determine the response of a linear transmission system to an input signal using purely algebraic transformations, thus avoiding the need to solve differential systems or partial differential equations.

From this point of view, the cissoidal transformation described in the "Complex numbers" section is a first approach to determining the steady state of a linear system subjected to sinusoidal loading.

The usual Laplace transform (TL) provides a tool for determining steady-state and transient conditions for a wide range of analog signals at the system input. Its use is generally limited to causal functions, i.e. zero for t < 0.

The Fourier transform (TF), which is very similar to the TL in its definition, may not be suitable for the analysis of common signals, but it is a very powerful tool that provides a spectral representation of the signal, and is particularly useful when using input signals that require the use of distribution theory.

With the emergence and development of digital techniques, TL had to be adapted to the processing of discrete signals: this is the purpose of the Z-transform (TZ).