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 $N$ '' 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.

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.

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