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Discrete Fourier Transform

Module by: Don Johnson. E-mail the author

Summary: The Fourier transform can be computed in discrete-time despite the complications caused by a finite signal and continuous frequency.

The discrete-time Fourier transform (and the continuous-time transform as well) can be evaluated when we have an analytic expression for the signal. Suppose we just have a signal, such as the speech signal used in the previous chapter. You might be curious; how did we compute a spectrogram such as the one shown in the speech signal example? The big difference between the continuous-time and discrete-time worlds is that we can exactly calculate spectra in discrete-time. For analog-signal spectra, use must build special devices, which turn out in most cases to consist of A/D converters and discrete-time computations. Certainly discrete-time spectral analysis is more flexible than in continuous-time.

The formula for the DTFT is a sum, which conceptually can be easily computed save for two issues.

Signal duration. The sum extends over the signal's duration, which must be finite to compute the signal's spectrum. It is exceedingly difficult to store an infinite-length signal in any case, so we'll assume that the signal extends over 0 N−1 0 N1 .
Continuous frequency. Subtler than the signal duration issue is the fact that the frequency variable is continuous: It may only need to span one period, like −12 12 1 2 12 or 0 1 0 1 , but the DTFT formula as it stands requires evaluating the spectra at all frequencies within a period. Let's compute the spectrum at a few frequencies; the most obvious ones are the equally spaced ones ∀k,k∈k…K−1:f=kK k k k … K1 f kK .

We thus define the discrete Fourier transform (DFT) to be