3.6 Fourier Analysis MATLAB Laboratory Experiment
Purpose: This experiment demonstrates approximations of periodic signals by truncated
Fourier series as defined in formula (3.4). Using MATLAB students will plot the actual
approximate signals and observe, for large values of
๎
, the Gibbs phenomenon at the
jump discontinuity points. In addition, students will use MATLAB to plot the system
frequency spectra, and to find the system response due to periodic inputs.
Part 1. Find the trigonometric form of the Fourier series for the periodic signal
presented in Figure 3.18. Take
๎๎๎๎
๎๎๎๎๎๎๎๎
๎
and use MATLAB to calculate the
coefficients of the Fourier series for
๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎
๎
. Plot the approximations
๎๎๎๎๎๎๎๎๎
as
defined in (3.4) for
๎
๎ ๎๎๎๎๎!๎๎๎๎๎"๎๎๎$#๎๎๎๎&%'๎๎๎(๎๎๎
. Observe the Gibbs phenomenon and for
๎
๎๎๎๎๎
estimate the relative magnitude of ripples at the jump discontinuity points.
t0
. . . . . .
E
T-T TT/2 /4 /4 /2-T -T
x(t)
FIGURE 3.18: A square wave signal
Part 2. Use MATLAB to plot the amplitude and phase line spectra of the periodic
signal from Part 1.
Part 3. Plot the magnitude and phase spectra of the system defined in Example 3.19
by using the MATLAB function freqs(num,den), where the vectors num and den
contain the coefficients of the transfer function numerator and denominator in descending
order.
Part 4. For the system defined in Example 3.19, and
)๎*๎+-,๎.
determined in Part 1
with
/1032๎4656780:9<;(=๎>๎?A@
, calculate the Fourier series coefficients of the output signal
for
BC0:D๎4๎9๎4(E๎4(2
. Print the values for the magnitudes of the Fourier series coefficients
of the output signal. Plot the approximations
)๎*๎+$,๎.
and observe the convergence of
the output signal as
F
increases. Take
FG0H9๎4(E๎4(2
and
,JILK D๎4NM๎O๎P
. Comment on the
frequency of the output signal and check its value from the plot obtained.
Submit all plots and comment on the results obtained.
SUPPLEMENT:
)*+$,๎.Q0
9
RTSU7WV
*
X
Y'Z\[
KS
Y^](_
@<+`B๎5 7,๎. Vba
Y
@dc๎e8+`B๎5 7,๎.$P
(3.4)
The system in Example 3.19 is defined by
f"g!h
+$,๎.
f
,
g
V
R
fUh
+๎,๎.
f
,
VE
h
+$,๎.i0j)k+๎,๎.
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