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 known as the Fourier Transform. 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:
Real-world data is typically continuous and aperiodic. To process this data on a computer, we must first convert it into a discrete format by sampling it at regular intervals. Our goal is to ensure that the sampled, discrete data accurately retains the frequency characteristics of the original continuous signal. This process is crucial for tasks like signal processing, where the frequency content holds vital information. Our discussion of the next Fourier transforms will illustrate how the frequency content of a continuous signal is altered when it is sampled to produce a discrete signal.