Biomedical Engineering Reference
In-Depth Information
11.5.3 Exponential Fourier Series
The main result from the Fourier series analysis is that an arbitrary periodic signal can
approximate by summing individual cosine terms with specified amplitudes and phases.
This result serves as much of the conceptual and theoretical framework for the field of
signal analysis. In practice, the Fourier series is a useful tool for modeling various types
of quasi-periodic signals.
An alternative and somewhat more convenient form of this result is obtained by noting
that complex exponential functions are directly related to sinusoids and cosines through
Euler's identities: cos
p
.By
applying Euler's identity to the compact trigonometric Fourier series, an arbitrary periodic
signal can be expressed as a sum of complex exponential functions:
=
=
Þ¼
e
j
y
þ
e
j
y
Þ¼
e
j
y
e
j
y
ð
y
2 and sin
ð
y
2
j
, where
j
¼
þ1
c
m
e
jk
o
o
t
x
ð
t
Þ¼
ð
11
:
5a
Þ
m
¼1
This equation represents the exponential Fourier series of a periodic signal. The coefficients
c
m
are complex numbers that are related to the trigonometric Fourier coefficients
c
m
¼
a
m
jb
m
2
¼
A
2
e
j
f
m
ð
11
:
5b
Þ
The proof for this result is beyond the scope of this text, but it is important to realize that
the trigonometric and exponential Fourier series are intimately related, as can be seen by
comparing their coefficients. The exponential coefficients can also be obtained directly by
integrating
x
(
t
),
ð
T
x
ð
t
Þ
e
jm
o
o
t
dt
1
T
c
m
¼
ð
11
:
5c
Þ
over one cycle of the periodic signal. As for the trigonometric Fourier series, the exponential
form allows us to approximate a periodic signal to any degree of accuracy by adding a suf-
ficient number of complex exponential functions. A distinct advantage of the exponential
Fourier series, however, is that it requires only a single integral (Eq. (11.5c)), compared to
the trigonometric form, which requires three separate integrations.
EXAMPLE PROBLEM 11.8
Find the exponential Fourier series for the square wave of Figure 11.7a and imple-
ment in MATLAB for the first ten terms. Plot the time waveform and the Fourier series
coefficients.
Solution
Like Example Problem 11.6, the Fourier coefficients are obtained by integrating from
1to1.
Because a single cycle of the square wave signal has nonzero values between
1/2 and
þ
1/2,
the integral can be simplified by evaluating it between these limits:
