330
Practical MATLAB
® Applications for Engineers
component, it is possible to predict the output of the composite input by adding all
the individual outputs (superposition principle).
Recall that the superposition principle states that the sum of the inputs applied to
a linear system returns as its output, the sum of the individual outputs.
R.4.33 Practical considerations dictate that the FS representation of an arbitrary function
f(t) cannot have an infi nite number of terms and must be truncated at some point,
and will, therefore, consist of a fi nite number of terms. This truncation process
creates an error that can be best evaluated by what is referred to as the mean square
error (MSE), defi ned as follows:
MSE
error
ϭ
1
2
0
T
t dt
T
[
( )]
∫
where
error( )
( )
t
f t
Fe
m
n
n m
n m
jw nt
for a finite
ϭ
ϭ
ϭ
Ϫ
Ϫ
Ϫ
∑
0
then
MSE ϭ
ϭ
ϭ
1
0
2
0
T
f t
F e
dt
n
n m
n m
jw nt
T
( ) Ϫ
Ϫ
Ϫ
∑
∫
R.4.34 The process of truncation, or approximation of f(t) by a partial sum is equivalent to
passing f(t) through an ideal low pass fi lter, and eliminating the high frequencies. If
f(t) presents discontinuities, then high frequencies are present with a considerable
amount of power, and the elimination of these frequencies would create distortion
and oscillations.
The resulting truncated waveform can be improved if a window (see Chapter 1) is
used with the objective of gradually reducing the high-frequency components.
R.4.35 Observe the existence of negative frequency components in the exponential FS
expansion. These frequencies have no physical meaning at this point, and represent a convenient mathematical model of representing the series. The physical
implications will become evident when analyzing, for example, the process of
modulation.
R.4.36 Let f(t) be now a nonperiodic function of t. Then by using the FT equations given
below, f(t) can be transformed from the time domain to the frequency domain, and
vice versa. These equations are also referred as the inverse and direct FT, given by
f t
F e dt
jwt
( )
( )
ϭ
ϩ
1
2
Ϫ∞
∞
∫
and
F w
f t e dt
jwt
( )
( )
ϭ
ϩ
Ϫ
Ϫ∞
∞
∫
CRC_47760_CH004.indd 330
CRC_47760_CH004.indd 330
7/28/2008 12:25:52 PM
7/28/2008 12:25:52 PM
Practical MATLAB
® Applications for Engineers
component, it is possible to predict the output of the composite input by adding all
the individual outputs (superposition principle).
Recall that the superposition principle states that the sum of the inputs applied to
a linear system returns as its output, the sum of the individual outputs.
R.4.33 Practical considerations dictate that the FS representation of an arbitrary function
f(t) cannot have an infi nite number of terms and must be truncated at some point,
and will, therefore, consist of a fi nite number of terms. This truncation process
creates an error that can be best evaluated by what is referred to as the mean square
error (MSE), defi ned as follows:
MSE
error
ϭ
1
2
0
T
t dt
T
[
( )]
∫
where
error( )
( )
t
f t
Fe
m
n
n m
n m
jw nt
for a finite
ϭ
ϭ
ϭ
Ϫ
Ϫ
Ϫ
∑
0
then
MSE ϭ
ϭ
ϭ
1
0
2
0
T
f t
F e
dt
n
n m
n m
jw nt
T
( ) Ϫ
Ϫ
Ϫ
∑
∫
R.4.34 The process of truncation, or approximation of f(t) by a partial sum is equivalent to
passing f(t) through an ideal low pass fi lter, and eliminating the high frequencies. If
f(t) presents discontinuities, then high frequencies are present with a considerable
amount of power, and the elimination of these frequencies would create distortion
and oscillations.
The resulting truncated waveform can be improved if a window (see Chapter 1) is
used with the objective of gradually reducing the high-frequency components.
R.4.35 Observe the existence of negative frequency components in the exponential FS
expansion. These frequencies have no physical meaning at this point, and represent a convenient mathematical model of representing the series. The physical
implications will become evident when analyzing, for example, the process of
modulation.
R.4.36 Let f(t) be now a nonperiodic function of t. Then by using the FT equations given
below, f(t) can be transformed from the time domain to the frequency domain, and
vice versa. These equations are also referred as the inverse and direct FT, given by
f t
F e dt
jwt
( )
( )
ϭ
ϩ
1
2
Ϫ∞
∞
∫
and
F w
f t e dt
jwt
( )
( )
ϭ
ϩ
Ϫ
Ϫ∞
∞
∫
CRC_47760_CH004.indd 330
CRC_47760_CH004.indd 330
7/28/2008 12:25:52 PM
7/28/2008 12:25:52 PM