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Discrete Fourier Transform

Module by: Don Johnson

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 0N-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 -1212 1 2 12 or 01 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

Discrete Fourier transform

∀k,k∈k…K-1:Sk=∑n=0N-1Snⅇ-ⅈ2πnkK k k k … K1 S k n 0 N 1 S n 2 n k K (1)

Here, Sk S k is shorthand for Sⅇⅈ2πkK S 2 k K .

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