FFT of a Not-So-Simple Sinusoid

Now let's increase the frequency in the above example by one-half of a bin:

% Example 2 = Example 1 with frequency between bins

f = 0.25 + 0.5/N;   % Move frequency up 1/2 bin

x = cos(2*pi*n*f*T); % Signal to analyze
X = fft(x);          % Spectrum
...                  % See Example 1 for plots and such

Figure: Sinusoid at Frequency $f=0.25+0.5/N$. a) Time waveform. b) Magnitude spectrum. c) DB magnitude spectrum.

The resulting magnitude spectrum is shown in Fig. 8.2b and c. At this frequency, we get extensive ``spectral leakage'' into all the bins. To get an idea of where this is coming from, let's look at the periodic extension of the time waveform:

% Plot the periodic extension of the time-domain signal
plot([x,x],'--ok');
title('Time Waveform Repeated Once');
xlabel('Time (samples)'); ylabel('Amplitude');

The result is shown in Fig. 8.3. Note the ``glitch'' in the middle where the signal begins its forced repetition.

Figure: Time waveform repeated to show discontinuity introduced by periodic extension (see midpoint).