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