### Zero-Phase Zero Padding

The previous zero-padding example used the *causal* Hamming
window, and the appended zeros all went to the *right* of the
window in the FFT input buffer (see Fig.2.4a). When using
*zero-phase* FFT windows (usually the best choice), the zero-padding
goes in the *middle* of the FFT buffer, as we now illustrate.

We look at zero-phase zero-padding using a *Blackman window*
(§3.3.1) which has good, though
suboptimal, characteristics for audio work.^{3.11}

Figure 2.6a shows a windowed segment of some sinusoidal data, with the window also shown as an envelope. Figure 2.6b shows the same data loaded into an FFT input buffer with a factor of 2 zero-phase zero padding. Note that all time is ``modulo '' for a length FFT. As a result, negative times map to in the FFT input buffer.

Figure 2.7a shows the result of performing an FFT on the data of Fig.2.6b. Since frequency indices are also modulo , the negative-frequency bins appear in the right half of the buffer. Figure 2.6b shows the same data ``rotated'' so that bin number is in order of physical frequency from to . If is the bin number, then the frequency in Hz is given by , where denotes the sampling rate and is the FFT size.

The Matlab script for creating Figures 2.6 and 2.7 is listed in in §F.1.1.

####
Matlab/Octave `fftshift` utility

Matlab and Octave have a simple utility called `fftshift` that
performs this bin rotation. Consider the following example:

octave:4> fftshift([1 2 3 4]) ans = 3 4 1 2 octave:5>If the vector

`[`1 2 3 4] is the output of a length 4 FFT, then the first element (1) is the dc term, and the third element (3) is the point at half the sampling rate ( ), which can be taken to be either plus or minus since they are the same point on the unit circle in the plane. Elements 2 and 4 are plus and minus , respectively. After

`fftshift`, element (3) is first, which indicates that both Matlab and Octave regard the spectral sample at half the sampling rate as a negative frequency. The next element is 4, corresponding to frequency , followed by dc and .

Another reasonable result would be `fftshift([1 2 3 4]) == [4 1
2 3]`, which defines half the sampling rate as a positive frequency.
However, giving
to the negative frequencies balances giving dc
to the positive frequencies, and the number of samples on both sides
is then the same. For an odd-length DFT, there is no point at
, so the result

octave:4> fftshift([1 2 3]) ans = 3 1 2 octave:5>is the only reasonable answer, corresponding to frequencies , respectively.

#### Index Ranges for Zero-Phase Zero-Padding

Having looked at zero-phase zero-padding ``pictorially'' in matlab
buffers, let's now specify the index-ranges mathematically. Denote
the window length by
(an odd integer) and the FFT length by
(a power of 2). Then the windowed data will occupy indices 0
to
(positive-time segment), and
to
(negative-time segment). Here we are assuming a 0-based indexing
scheme as used in C or C++. We add 1 to all indices for matlab
indexing to obtain `1:(M-1)/2+1` and `N-(M-1)/2+1:N`,
respectively. The zero-padding zeros go in between these ranges,
*i.e.*, from
to
.

#### Summary

To summarize, zero-padding is used for

- padding out to the next higher power of 2 so a Cooley-Tukey FFT can be used with any window length,
- improving the quality of spectral displays, and
- oversampling spectral peaks so that some simple final interpolation will be accurate.

*spectral modifications*in the short-time Fourier transform (STFT). This is because spectral modifications cause the time-domain signal to

*lengthen in time*, and without sufficient zero-padding to accommodate it, there will be

*time aliasing*in the reconstruction of the signal from the modified FFTs.

Some examples of interpolated spectral display by means of zero-padding may be seen in §3.4.

**Next Section:**

Rectangular Window Side-Lobes

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Zero Padding in the Time Domain