**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Zero Padding

Definition:Zero paddingconsists of appending zeros to a signal. It maps a length signal to a length signal, but need not be an integer multiple of :

For example,

The above definition is natural when represents a signal starting at time and extending for samples. If, on the other hand, we are zero-padding a spectrum, or we have a time-domain signal which has nonzero samples for negative time indices, then the zero padding is normally inserted between samples and for odd (note that ), and similarly for even . I.e., for spectra, zero padding is inserted at the point ( ). Figure 8.5 illustrates this second form of zero padding. It is also used in conjunction with zero-phase FFT windows(discussed a bit further below).

Figure 8.5:Illustration of frequency-domain zero padding: a) Original spectrum plotted over the domain where (i.e., as the spectral array would normally exist in a computer array). b) . c) The same signal plotted over the domain which is more natural for interpreting negative frequencies. d) .Using Fourier theorems, we will be able to show that

zero padding in the time domaingivesbandlimited interpolation in the frequency domain. Similarly,zero padding in the frequency domaingivesbandlimited interpolation in the time domain. This is how ideal sampling rate conversion is accomplished.It is important to note that bandlimited interpolation is

idealinterpolation in digital signal processing.