324
Practical MATLAB
® Applications for Engineers
The Fourier coeffi cients a n , b n , and F n represent the degree of similarity between f(t)
and each of the frequency components nw 0 .
R.4.12 The relation between the exponential coeffi cients F n and the trigonometric coeffi cients a n s and b n s are as follows:
F
a
0
0
2
ϭ
F
a
jb
n
n
n
ϭ
1
2
(
)
Ϫ
F
a
j b
n
n
Ϫ ϭ
ϩ
1
2
(
)
a 0 = F n + F −n
b n = j(F n − F −n )
c
a
b
F
n
n
n
n
ϭ
ϩ
ϭ
2
2
2
(
)
R.4.13 Recall that the average power of a periodic function f(t) may be evaluated in the
time domain by
P
T
f t dt
T
ave ϭ
1
0
2
( )
∫
R.4.14 The time function f(t) may be viewed as a current or a voltage that acts on a resistor
of 1 Ω (normalized).
Recall that
Power
W
ϭ
ϭ
v t
R
v t
( )
( )
2
2
or Power = i
2
(t)R = i
2
(t)W (assuming R = 1 Ω without any loss of generality) and the
average power is therefore given by
P
T
i t dt
T
v t dt
T
T
ave ϭ
ϭ
1
1
2
0
2
0
( )
( )
∫
∫
R.4.15 Parseval’s relation (also known as Parseval’s theorem) states that if f(t) is a real and
periodic function, then the average power denoted by P ave may be conveniently
evaluated in the frequency domain by
P
T
f t dt
F
T
n
n
ave ϭ
ϭ
ϭ
1
2
0
2
( )
∫
∑
Ϫϱ
ϱ
Note that the evaluation of P ave in the frequency domain is much easier than in the
time domain, since integration is substituted by the summation. Observe that P ave
can be easily evaluated if the coeffi cients F n are known by using MATLAB in the
following way:
Let
F = [F −n F −n+1 … F −1 F 0 F 1 … F n ]
Then P ave = F * F′.
CRC_47760_CH004.indd 324
CRC_47760_CH004.indd 324
7/28/2008 12:25:50 PM
7/28/2008 12:25:50 PM
Practical MATLAB
® Applications for Engineers
The Fourier coeffi cients a n , b n , and F n represent the degree of similarity between f(t)
and each of the frequency components nw 0 .
R.4.12 The relation between the exponential coeffi cients F n and the trigonometric coeffi cients a n s and b n s are as follows:
F
a
0
0
2
ϭ
F
a
jb
n
n
n
ϭ
1
2
(
)
Ϫ
F
a
j b
n
n
Ϫ ϭ
ϩ
1
2
(
)
a 0 = F n + F −n
b n = j(F n − F −n )
c
a
b
F
n
n
n
n
ϭ
ϩ
ϭ
2
2
2
(
)
R.4.13 Recall that the average power of a periodic function f(t) may be evaluated in the
time domain by
P
T
f t dt
T
ave ϭ
1
0
2
( )
∫
R.4.14 The time function f(t) may be viewed as a current or a voltage that acts on a resistor
of 1 Ω (normalized).
Recall that
Power
W
ϭ
ϭ
v t
R
v t
( )
( )
2
2
or Power = i
2
(t)R = i
2
(t)W (assuming R = 1 Ω without any loss of generality) and the
average power is therefore given by
P
T
i t dt
T
v t dt
T
T
ave ϭ
ϭ
1
1
2
0
2
0
( )
( )
∫
∫
R.4.15 Parseval’s relation (also known as Parseval’s theorem) states that if f(t) is a real and
periodic function, then the average power denoted by P ave may be conveniently
evaluated in the frequency domain by
P
T
f t dt
F
T
n
n
ave ϭ
ϭ
ϭ
1
2
0
2
( )
∫
∑
Ϫϱ
ϱ
Note that the evaluation of P ave in the frequency domain is much easier than in the
time domain, since integration is substituted by the summation. Observe that P ave
can be easily evaluated if the coeffi cients F n are known by using MATLAB in the
following way:
Let
F = [F −n F −n+1 … F −1 F 0 F 1 … F n ]
Then P ave = F * F′.
CRC_47760_CH004.indd 324
CRC_47760_CH004.indd 324
7/28/2008 12:25:50 PM
7/28/2008 12:25:50 PM