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Zero-Padding for Interpolating Spectral Displays

Suppose we perform spectrum analysis on some sinusoid using a length $ M$ window. Without zero padding, the DFT length is $ N=M$ . We may regard the DFT as a critically sampled DTFT (sampled in frequency). Since the bin separation in a length-$ N$ DFT is $ 2\pi/N$ , and the zero-crossing interval for Blackman-Harris side lobes is $ 2\pi/M$ , we see that there is one bin per side lobe in the sampled window transform. These spectral samples are illustrated for a Hamming window transform in Fig.2.3b. Since $ K=4$ in Table 5.2, the main lobe is 4 samples wide when critically sampled. The side lobes are one sample wide, and the samples happen to hit near some of the side-lobe zero-crossings, which could be misleading to the untrained eye if only the samples were shown. (Note that the plot is clipped at -60 dB.)

Figure 2.3: (a) Hamming window. (b) Critically sampled sinusoidal peak = frequency-shifted Hamming-window transform.

If we now zero pad the Hamming-window by a factor of 2 (append 21 zeros to the length $ M=21$ window and take an $ N=42$ point DFT), we obtain the result shown in Fig.2.4. In this case, the main lobe is 8 samples wide, and there are two samples per side lobe. This is significantly better for display even though there is no new information in the spectrum relative to Fig.

Incidentally, the solid lines in Fig.2.3b and 2.4b indicating the ``true'' DTFT were computed using a zero-padding factor of $ L=N/M=1000$ , and they were virtually indistinguishable visually from $ L=100$ . ($ L=10$ is not enough.)

Figure 2.4: (a) Hamming window zero-padded by a factor of 2. (b) $ 2\times $ -oversampled sinusoidal peak = frequency-shifted Hamming-window transform.

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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University