Fast-Fourier-Transform-Made-Easy

Easy Fourier Analysis Part 1 Complextoreal.com

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SIGNAL PROCESSING & SIMULATION NEWSLETTER



Fourier analysis made Easy Part 1

Jean Baptiste Joseph, Baron de Fourier, 1768 - 1830

While studying heat conduction in materials, Baron Fourier (a title given to him by
Napoleon) developed his now famous Fourier series, approximately 120 years after
Newton published the first book on calculus. It took Fourier another twenty years to
develop the Fourier transform which made the theory applicable to a variety of
disciplines such as signal processing where Fourier analysis is now an essential tool.
Fourier did little to develop the concept further and most of that work was done by
Euler, LaGrange, Laplace and others. Fourier analysis is now also used in thermal
analysis, image processing, quantum mechanics and physics.

Why do we need to do Fourier analysis – In communications, we can state the
problem at hand this way; we send an information-laced signal over a medium. The
medium and the hardware corrupt this signal. The receiver has to figure out from the
received signal which part of the corrupted received signal is the information signal and
which part the extraneous noise and distortion. The transmitted signals have well
defined spectral content, so if the receiver can do a spectral analysis of the received
signal then it can extract the information. This is what Fourier analysis allows us to do.
Fourier analysis can look at an unknown signal and do an equivalent of a chemical
analysis, identifying the various frequencies and their relative “quantities” in the signal.

Fourier noticed that you can create some really complicated looking waves by just
summing up simple sine and cosine waves. For example, the wave in Figure 1a is sum
of the just three sine waves shown in Figures 1b, 1c and 1d of assorted frequencies and
amplitudes.



Easy Fourier Analysis Part 1 Complextoreal.com

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(a) - A complicated looking wave




(b) - Sine wave 1 (c)- Sine wave 2 (d) - Sine wave 3

Figure 1 - Sine waves

Let’s look at signal 1a in three dimensions. With time progressing to the right we
see the amplitude going up and down erratically, we are looking at the signal in Time
domain. From this angle, we see the sum of the three sine waves as shown in Fig
(1b,c,d).

When we look at the same signal from the side along the z-axis, what we see are the
three sine waves of different frequencies. We also see the amplitude but only as a line
with its maximum excursion. This view of the signal from this point of view is called
the Frequency Domain. Another name for it is the Signal Spectrum.




Figure 3 - Looking at signals from two different points of view

The concept of spectrum came about from the realization that any arbitrary wave is
really a summation of many different frequencies. The spectrum of the composite wave
f(t) of Fig (1) is composed of just three frequencies and can be drawn as in Fig (3.1).


Easy Fourier Analysis Part 1 Complextoreal.com

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This is called a one-sided magnitude spectrum. One-sided not because anything
has been left out of it, but because only positive frequencies are represented. (So what
is a negat...