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The most common applications of the Fourier transform are the analysis of linear time-invariant systems and spectral analysis.
The following equation defines the two-sided Fourier transform.
The following equation defines the two-sided inverse Fourier transform.
Two-sided means that the mathematical implementation of the forward and inverse Fourier transform considers all negative and positive frequencies and time of the signal. Single-sided means that the mathematical implementation of the transforms considers only the positive frequencies and time history of the signal.
A Fourier transform pair consists of the signal representation in both the time and frequency domain. The following relationship commonly denotes a Fourier transform pair.