On what basis we depend when we choose the tool to transform to frequency domain, I can't distinguish in what case we use one of these transformations ( trig. fourier series, complex fourier series, fourier transform, laplace transform, z-Transform)
2026-04-11 21:33:14.1775943194
Transformations Difference
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These transformations are generally applicable to different types of time-domain signals:
The Fourier series (real or complex, here the difference is of presentation and not substance) exists to periodic signals (which are integrable over a period). Note that in the general case periodic signals aren't integrable over the entire line (time domain).
The Fourier transofrm can be applied to integrable (or finite-energy, if you develop the theory carefully enough) signals on the entire line. These signals aren't periodic.
The Laplace transform is a method of generalizing the Fourier transform to signals which aren't integrable (or have bounded energy), but whose growth can be exponentially dampened. Note that the imaginary axis is in the region of convergence of the Laplace transform iff the signal has a Fourier transform, in which case its restriction to that axis is the function's Fourier transform.
The $Z$-transform is applicable to signals in discrete time-domain, unlike former transforms which are applicable to signals of continuous time-domain.