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.