The Discrete Fourier Transform
Neither the Fourier Series nor the Fourier Transform lends itself easily to calculation by digital computers. To overcome this hurdle, the so-called Discrete Fourier Transform, or DFT was developed. Probably the first person to conceive the DFT was Wilhelm Friederich Gauss, the famous 19th century German mathematician, although he certainly did not have a digital computer on which to implement it. The DFT operates on a sampled, or discrete, signal in the time domain, and generates from this a sampled, or discrete, spectrum in the frequency domain. The resulting spectrum is an approximation of the Fourier Series, an approximation in the sense that information between the samples of the waveform is lost. The key to the DFT is the existence of the sampled waveform, i.e., the possibility of representing the waveform by a series of numbers. To generate this series of numbers from an analog signal, a process of sampling and analog to digital conversion is required. The sampled signal is a mathematical representation of the instantaneous signal level at precisely defined time intervals. It contains no information about the signal between the actual sample times.
If the sampling rate is high enough to ensure a reasonable representation of the shape of the signal, the DFT does produce a spectrum very close to a theoretically true spectrum. This spectrum is also discrete, and there is no information between the samples, or "lines" of the spectrum. In theory, there is no limit to the number of samples that can be used, or the speed of the sampling, but there are practical limitations we must live with. Most of these limitations are the result of using a digital computer as the calculating agent.
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