Zero Padding Applications

Zero padding in the time domain is used extensively in practice to compute heavily interpolated spectra by taking the DFT of the zero-padded signal. Such spectral interpolation is ideal when the original signal is time limited (nonzero only over some finite duration spanned by the orignal samples).

Note that the time-limited assumption directly contradicts our usual assumption of periodic extension. As mentioned in §6.7, the interpolation of a periodic signal's spectrum from its harmonics is always zero; that is, there is no spectral energy, in principle, between the harmonics of a periodic signal, and a periodic signal cannot be time-limited unless it is the zero signal. On the other hand, the interpolation of a time-limited signal's spectrum is nonzero almost everywhere between the original spectral samples. Thus, zero-padding is often used when analyzing data from a non-periodic signal in blocks, and each block, or frame, is treated as a finite-duration signal which can be zero-padded on either side with any number of zeros. In summary, the use of zero-padding corresponds to the time-limited assumption for the data frame, and more zero-padding yields denser interpolation of the frequency samples around the unit circle.

Sometimes people will say that zero-padding in the time domain yields higher spectral resolution in the frequency domain. However, signal processing practitioners should not say that, because ``resolution'' in signal processing refers to the ability to ``resolve'' closely spaced features in a spectrum analysis (see Book IV [67] for details). The usual way to increase spectral resolution is to take a longer DFT without zero padding--i.e., look at more data. In the field of graphics, the term resolution refers to pixel density, so the common terminology confusion is reasonable. However, remember that in signal processing, zero-padding in one domain corresponds to a higher interpolation-density in the other domain--not a higher resolution.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.