__Problem Statement__

A periodic pulse train has a fundamental period of *T*_{0} = 8
seconds and a pulse width of 2 seconds. The fundamental frequency is *w*_{0}
= pi/4 = 0.7854 rad/sec. Three periods
of the signal are shown in the figure. The amplitude of the signal is 1 during
the pulse width and 0 elsewhere. Since the signal is periodic, it repeats itself for all time in
both directions.

Periodic Signal *x*(*t*)

Since *x*(*t*) is periodic and satisfies the Dirichlet conditions, it can be represented by
a Fourier series.
Since the signal is not a pure sinusoidal signal, the Fourier series will
consist of an infinite sum of sine and/or cosine terms. It is clear from
the graph of *x*(*t*) that the average value of the signal is positive, and
that the signal does not contain symmetry (even, odd, or half-wave odd).

__Calculating the Trig Coefficients__

All three forms of the Fourier series (trig, cosine, and exponential) will be
computed, beginning with the trig form. Because of the definition of *x*(*t*),
computation of the *a*_{0}, *a _{n}*, and

The
expression for *a*_{0}, the average value of the signal, is

The expression for the *a _{n} * coefficients is:

Thus, the *a _{n}* coefficients will be 0 for all even values of

The expression for the *b _{n}* coefficients is

The *b _{n}* coefficients will be 0 whenever cos(

__Approximating x(t)
with Partial Sums of Fourier Series Terms__

The complete Fourier series representation of a signal requires an infinite number of terms in general. When the complete series is used, the series converges to the exact value of the signal at every point in time where the signal is continuous and converges to the midpoint of the discontinuity wherever the signal is discontinuous.

The signal *x*(*t*)
can be approximated by using a truncated form of the Fourier series, that is,
stopping the summation after a finite number *N* of terms. The
approximation may be 'good' or 'bad' in a subjective sense, but it will be the
best approximation for a given *N* in terms of minimizing the mean squared
error between the approximation and the actual signal. The approximation is
given by

The
next figure shows the approximations to the *x*(*t*) in this example
for *N* = 10, 30, 50. The figure shows clearly that as the number of terms
increases, the approximation becomes better. The transitions between the two
amplitude values become steeper and the magnitudes of the oscillations become
smaller for the larger values of *N*. An interesting point to note in the
graphs is that the magnitudes of the oscillations just before and just after the
discontinuity do not decrease. In fact they increase slightly with our values of
*N* from approximately 7.3% to 8.4% to 8.6% of the pulse height for the
values of *N* = 10, 30, 50. The magnitude of the overshoot will eventually
approach a value of 9% of the pulse height. This is known as the Gibbs
phenomenon.

Approximations to *x*(*t*)

When
plots of the *a _{n}*
and

__Plotting
the Single-Sided Spectrum from the Cosine Series__

The frequency
spectrum of *x*(*t*) can be easily plotted by converting the trig
version of the Fourier series into the cosine form. This allows the magnitude
and phase of each frequency component to be readily visible. The expressions for
magnitude and phase in the cosine form are

Since
*a _{n}* takes on negative values, care must be used when computing
the phase angle in a computer or calculator using the

The single-sided magnitude and phase
spectra are shown in the next figure. The independent variable in each of the
plots is frequency in rad/sec. Since *x*(*t*) is periodic, all the
frequencies in the signal are integer multiples of the fundamental frequency *w*_{0}
= 0.7854 rad/sec. The magnitude is seen to generally decrease with increasing
frequency. This indicates that the higher frequency terms contribute less and
less to the construction of *x*(*t*). The phase follows a periodic
pattern, repeating itself after every four terms.

Single-Sided Magnitude and Phase Spectra

__Plotting the Double-Sided Spectrum from the Exponential Series__

The complex exponential form of the Fourier series is derived from the trig form by making use of the Euler relations between sine and cosine functions and the complex exponential function

The
complex exponentials provide a more convenient and compact way of expressing the
Fourier series representation of *x*(*t*) than either the trig or
cosine forms. It also allows the magnitude and phase spectra to be easily
calculated. It does, however, introduce negative frequencies into the
representation, which were not present in either of the other two forms. The
expressions for the complex exponential form are given by

These
equations show that *x*(*t*) is represented in the complex exponential
form by terms taking on all integer multiples of *w*_{0}, negative
as well as positive. The magnitude spectrum has even symmetry, with the
magnitudes in this form being one-half the magnitudes in the cosine form (except
at 0 frequency, where they are equal). The phase spectrum has odd symmetry. At
positive frequencies, the phases in the complex exponential form have the same
values as the phases in the cosine form. At negative frequencies, the phases
have the opposite signs as they do at positive frequencies. The next figure
shows the double-sided spectra for this example, and the symmetry properties of
the spectra are easily seen.

Double-Sided Magnitude and Phase Spectra

*Click the
icon to return to Dr. Beale's home page*

*Latest revision on
Thursday, May 18, 2006 10:54 PM
*