25-08-2017, 09:32 PM
Fourier Analysis Made Easy
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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 received signal is a time domain
signal and it is hard to get much useful information from it. It is much easier to
understand signals in frequency domain. This is what Fourier analysis allows us to do.
One could say that it is a linear algorithm that can take a time domain signal into the
frequency domain and back. Fourier analysis allows a more intuitive look at an
unknown signal in frequency domain.
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.
(a) - A complicated looking wave
Easy Fourier Analysis Part 1 Complextoreal.com 1
(b) - Sine wave 1 ©- Sine wave 2 (d) - Sine wave 3
Let’s look at signal 1a in three dimensions. The signal here is given in time
domain, with time from left to right. We see the amplitude going up and down with
time. So this is what we mean by time domain, time the independent variable on the xaxis.
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 arrayed along a frequency axis. 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.
The concept of spectrum came about from the realization that any arbitrary wave is
really a summation of many different frequencies, associated amplitudes and phase.
The composite wave, let’s call it f(t) of Fig. 1 is composed of just three frequencies and
can also be depicted as in Fig. 3. The view into the z-y plane can be thought of as the
spectrum of the wave. This is called a one-sided amplitude spectrum or amplitude
response. One-sided not because anything has been left out of it, but because only
positive frequencies are represented. (So what is a negative frequency? Is there such a
thing? We will discuss this in another chapter. )