Summary: Definition of the complex Fourier series.

In an earlier module, we showed that a
square wave could be expressed as a superposition of pulses. As useful as
this decomposition was in this example, it
does not generalize well to other periodic signals:
How can a
superposition of pulses equal a smooth signal like a sinusoid?
Because of the importance of sinusoids to linear systems, you
might wonder whether they could be added together to represent a
large number of periodic signals. You would be right and in good
company as well.
Euler
and
Gauss
in particular worried about this problem, and
Jean Baptiste Fourier
got the credit even though tough mathematical issues were not
settled until later. They worked on what is now known as the
Fourier series:
representing *any* periodic signal as a superposition
of sinusoids.

But the Fourier series goes well beyond being
another signal decomposition method.
Rather, the Fourier series begins our journey to appreciate how a signal
can be described in either the time-domain or the frequency-domain with
*no* compromise.
Let
*periodic* signal with period

Assuming we know the period, knowing the Fourier coefficients
is equivalent to knowing the signal.
Thus, it makes no difference if we have a time-domain or a frequency-
domain characterization of the signal.

What is the complex Fourier series for a sinusoid?

Because of Euler's relation,

To find the Fourier coefficients, we note the orthogonality property

Finding the Fourier series coefficients for the
square wave

When integrating an expression containing i ,
treat it just like any other constant.

A signal's Fourier series spectrum

If

This result follows from the integral that calculates
the

If

Given the previous property for real-valued signals, the Fourier coefficients of even signals are real-valued. A real-valued Fourier expansion amounts to an expansion in terms of only cosines, which is the simplest example of an even signal.

If

Therefore, the Fourier coefficients are purely imaginary. The square wave is a great example of an odd-symmetric signal.

The spectral coefficients for a periodic signal
delayed by

The complex Fourier series obeys Parseval's Theorem, one of the most important results in signal analysis. This general mathematical result says you can calculate a signal's power in either the time domain or the frequency domain.

Average power calculated in the time domain equals the power calculated in the frequency domain.

Let's calculate the Fourier coefficients of the periodic pulse signal shown here.

Periodic Pulse Sequence |
---|

Also note the presence of a linear phase term (the first term in

What is the value of

The phase plot shown in Figure 2
requires some explanation as it does not seem to agree with what
Equation 11 suggests. There, the phase has
a linear component, with a jump of
*without*
affecting the value of the complex spectrum. We see
that at frequency index 4 the phase is nearly

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