FourierTransform: Continuous, Non-periodic Time Series

 

If a signal is continuous in time, but not periodic, it can be represented by a continuous sum (i.e. the integral) of sinusoidal waves of continuous frequencies. In other words, the Fourier transform of a continuous signal is also continuous in frequency. A speech waveform can be represented by a Fourier transform, although it has been argued that so many people repeat themselves, speech might best be represented by a Fourier series.

 

A continuous, non-periodic time series has the following Fourier transform: