Zero Padding in the Time Domain
Unlike time-domain interpolation [270], ideal spectral interpolation is very easy to implement in practice by means of zero padding in the time domain. That is,
Since the frequency axis (the unit circle in the![]()


Practical Zero Padding
To interpolate a uniformly sampled spectrum
,
by the factor
, we may take the length
inverse DFT, append
zeros to the time-domain data, and take
a length
DFT. If
is a power of two, then so is
and
we can use a Cooley-Tukey FFT for both steps (which is very fast):
![]() |
(3.45) |
This operation creates





In matlab, we can specify zero-padding by simply providing the optional FFT-size argument:
X = fft(x,N); % FFT size N > length(x)
Zero-Padding to the Next Higher Power of 2
Another reason we zero-pad is to be able to use a Cooley-Tukey FFT with any
window length
. When
is not a power of
, we append enough
zeros to make the FFT size
be a power of
. In Matlab and
Octave, the function nextpow2 returns the next higher power
of 2 greater than or equal to its argument:
N = 2^nextpow2(M); % smallest M-compatible FFT size
Zero-Padding for Interpolating Spectral Displays
Suppose we perform spectrum analysis on some sinusoid using a length
window. Without zero padding, the DFT length is
. We may
regard the DFT as a critically sampled DTFT (sampled in
frequency). Since the bin separation in a length-
DFT is
,
and the zero-crossing interval for Blackman-Harris side lobes is
, we see that there is one bin per side lobe in the
sampled window transform. These spectral samples are illustrated for
a Hamming window transform in Fig.2.3b. Since
in
Table 5.2, the main lobe is 4 samples wide when critically
sampled. The side lobes are one sample wide, and the samples happen
to hit near some of the side-lobe zero-crossings, which could be
misleading to the untrained eye if only the samples were shown. (Note
that the plot is clipped at -60 dB.)
![]() |
If we now zero pad the Hamming-window by a factor of 2
(append 21 zeros to the length
window and take an
point
DFT), we obtain the result shown in Fig.2.4. In this case,
the main lobe is 8 samples wide, and there are two samples per side
lobe. This is significantly better for display even though there is
no new information in the spectrum relative to Fig.2.3.3.10
Incidentally, the solid lines in Fig.2.3b and
2.4b indicating the ``true'' DTFT were computed
using a zero-padding factor of
, and they were virtually
indistinguishable visually from
. (
is not enough.)
![]() |
Zero-Padding for Interpolating Spectral Peaks
For sinusoidal peak-finding, spectral interpolation via zero-padding gets us closer to the true maximum of the main lobe when we simply take the maximum-magnitude FFT-bin as our estimate.
The examples in Fig.2.5 show how zero-padding helps in clarifying the true peak of the sampled window transform. With enough zero-padding, even very simple interpolation methods, such as quadratic polynomial interpolation, will give accurate peak estimates.
![]() |
Another illustration of zero-padding appears in Section 8.1.3 of [264].
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Zero-Phase Zero Padding
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Interpolating a DFT