FFT of a Zero-Padded Sinusoid Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


FFT of a Zero-Padded Sinusoid

Looking back at Fig.8.2c, we see there are no negative dB values. Could this be right? Could the spectral magnitude at all frequencies be 1 or greater? The answer is no. To better see the true spectrum, let's use zero padding in the time domain to give ideal interpolation in the frequency domain:

zpf = 8;            % zero-padding factor
x = [cos(2*pi*n*f*T),zeros(1,(zpf-1)*N)]; % zero-padded 
X = fft(x);         % interpolated spectrum
magX = abs(X);      % magnitude spectrum
...                 % waveform plot as before
nfft = zpf*N;       % FFT size = new frequency grid size
fni = [0:1.0/nfft:1-1.0/nfft]; % normalized freq axis
subplot(3,1,2);
% with interpolation, we can use solid lines '-':
plot(fni,magX,'-k'); grid on; 
...
spec = 20*log10(magX); % spectral magnitude in dB
% clip below at -40 dB:
spec = max(spec,-40*ones(1,length(spec))); 
...                 % plot as before

Figure 8.4: Zero-padded sinusoid at frequency $ f=0.25+0.5/N$ cycles/sample. a) Time waveform. b) Magnitude spectrum. c) DB magnitude spectrum.
\includegraphics[width=\textwidth]{eps/example3}

Figure 8.4 shows the zero-padded data (top) and corresponding interpolated spectrum on linear and dB scales (middle and bottom, respectively). We now see that the spectrum has a regular sidelobe structure. On the dB scale in Fig.8.4c, negative values are now visible. In fact, it was desirable to clip them at $ -40$ dB to prevent deep nulls from dominating the display by pushing the negative vertical axis limit to $ -300$ dB or more, as in Fig.8.1c (p. [*]). This example shows the importance of using zero padding to interpolate spectral displays so that the untrained eye will ``fill in'' properly between the spectral samples.


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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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