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.2.3.^3.10

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.