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.4.2a). 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.^5.4

Figure 4.4a shows a windowed segment of some sinusoidal data, with the window also shown as an envelope. Figure 4.4b 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 4.5a shows the result of performing an FFT on the data of Fig.4.4b. Since frequency indices are also modulo , the negative-frequency bins appear in the right half of the buffer. Figure 4.4b 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 4.4 and 4.5 is listed in in §F.1.1.

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Subsections

• Matlab/Octave fftshift utility
  • Index Ranges for Zero-Phase Zero-Padding
  • Summary

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Order a Hardcopy of Spectral Audio Signal Processing

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.