The Fourier Transform - DSP Using MATLAB and Wavelets - page 214

land on the actual frequencies present. This means that the actual information
\leaks" into other DFT output bins; it shows up as other frequencies. Windowing
reduces the sidelobes of the sinc function (of the Continuous Fourier Transform, or
CFT for short), which in turn decreases the eect of leakage, since the DFT is a
sampled version of the continuous Fourier transform [11].
In other words, looking at the CFT of a signal component (i.e., a single sinusoid),
we see the sinc function. Looking at the CFT of a windowed version of the same
signal, we see a sinc function with lower sidelobe levels (though a wider main lobe).
If we sample the CFT, we might be lucky and sample it right at the point between
sidelobes, so we have many zeros and a single spike. Realistically, our samples
will land where we have one or two spikes, and several nonzero values, due to the
sidelobes. The lower these sidelobes, the better our DFT indicates the frequencies
that are actually present.

6.9 Harmonics and Fourier Transform

Harmonics and the Fourier transform are closely linked. Harmonics refers to the
use of sinusoids related by a fundamental frequency f 0 , that is, adding sinusoids of
frequencies f 0 , 2f 0 , 3f 0 , etc.
The following program demonstrates harmonics. Its objective is to approximate
a triangle wave with a sum of sinusoids. When we run it, we see that the approxi-
mation gets better and better as more sinusoids are added. We use a fundamental
frequency of 1 Hz. After creating signal x as a triangle wave, the program nds
the DFT of it using the fft command in MATLAB. Next, it nds the magnitudes
and phases for all of the sinusoids corresponding to the DFT results. Finally, the
program shows the sum of the sinusoids, pausing briey between iterations of the
loop to show the progress. Essentially, this performs the inverse DFT. Figure 6.10
shows what the progress looks like about one-sixth of the way through (the solid line
is the original signal, and the sum of sinusoids is shown as a dash-dot line). Running
the program to completion shows that the nal approximation appears directly over
top of the original signal.

%
% Show how the DFT function can represent a triangle wave
%

% first, make a triangle wave
for i=1:20
x(i) = i;