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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 window3.3.1) 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 $ N$ '' for a length $ N$ FFT. As a result, negative times $ -n$ map to $ N-n$ in the FFT input buffer.

Figure 2.6: (a) Blackman window overlaid with windowed data. b) Zero-padded windowed data loaded into the FFT input buffer.
\includegraphics[width=\twidth]{eps/zpblackmanT}

Figure 2.7a shows the result of performing an FFT on the data of Fig.2.6b. Since frequency indices are also modulo $ N$ , 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 $ -f_s/2$ to $ f_s/2$ . If $ k$ is the bin number, then the frequency in Hz is given by $ k
f_s/N$ , where $ f_s$ denotes the sampling rate and $ N$ is the FFT size.

Figure 2.7: (a) FFT magnitude data, as returned by the FFT. (b) FFT magnitude spectrum ``rotated'' to a more ``physical'' frequency axis in bin numbers.
\includegraphics[width=\twidth]{eps/zpblackmanF}

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



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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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