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 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 dB to prevent deep nulls from dominating the display by pushing the negative vertical axis limit to 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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Order a Hardcopy of Mathematics of the DFT

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.