- We have made use of the DFT (or its more efficient form, the FFT) throughout this course to analyze the frequency content of signals and in most instances, we have not made any assumptions about signal periodicity.
- If we grab a random ``piece'' of an audio signal, the signal will in almost all instances not be periodic.
- Thus, the DFT must be understood to represent an approximation, given by a set of harmonically related sinusoids, to the true spectrum of a signal.
- The number of unique harmonically related sinusoids is given by
*N*/2. Thus, the larger*N*, the more sinusoids will be available for comparison and the smaller the ``cracks'' between DFT components. - At the same time, computations over larger values of
*N*``average out'' time variations in the signal. If the frequency content of a signal is changing relatively quickly in time, the only way these variations can be identified is by using small values of*N*. - This trade-off between time- and frequency-resolution is referred to as the uncertainty principle.
- In order to evaluate changes in a signal over time, the signal is segmented into ``blocks'' of length
*N*samples and an FFT is computed for each section. - This process is referred to as The Short-Time Fourier Transform (STFT).
- Note that the blocks can overlap with one another to provide better time-domain resolution.
- The STFT can be viewed as a filterbank such that each spectral ``bin'' provides information about frequency content in a signal at a specific point in time.

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