Digital Signal Processing Reference
In-Depth Information
function has an input signal the Fourier transform of the output signal
is given by the product of the transfer function and the Fourier transform of
the input function.
Energy and power spectral densities are defined by (5.49) and (5.53), respec-
tively.
The usefulness of the energy and power spectral density functions in the
analysis of systems and signals is discussed. An important application is the study of
power-signal transmission through a linear system. In this case, the equation for the
power spectral density of the output signal is given in terms of the system transfer
function and the power spectral density of the input signal.
Several additional applications of the Fourier transform are discussed in
Chapter 6. See Table 5.3.
H(v)
X(v),
Y(v)
TABLE 5.3
Key Equations of Chapter 5
Equation Title
Equation Number
Equation
q
f(t)e
-jvt
dt
Fourier transform
(5.1)
f
5
f(t)
6
= F(v) =
L
-
q
q
1
2p
L
F(v)e
jvt
dv = f
-1
5
Inverse Fourier transform
(5.2)
f(t) =
F(v)
6
-
q
rect(t/T)
Î
f
"
T sinc(T
v/2)
Fourier transform of
rect
function
(5.4)
1
ƒ a ƒ
v
a
f(at - t
0
)
Î
f
"
e
-jt
0
(v/a)
Time-transformation property
(5.14)
F
¢
≤
F(t)
Î
f
"
2pf(-v)
f(t)
Î
f
"
F(v)
Duality property
(5.15)
when
f
1
(t)*f
2
(t)
Î
f
"
F
1
(v)F
2
(v)
Convolution property
(5.16)
1
2p
F
1
(v)*F
2
(v)
f
1
(t)f
2
(t)
Î
f
"
Multiplication property
(5.17)
x(t)e
jv
0
t
Î
f
"
X(v - v
0
)
Frequency-shifting property
(5.18)
q
k=-
q
"
2p
q
k=-
q
C
k
e
jkw
0
t
Î
f
Fourier transform of periodic signal
(5.35)
C
k
d(v - kv
0
)
f
s
(t) = f(t)d
T
(t) = f(t)
q
n=-
q
d(t - nT
s
) =
q
n=-
q
Sampled signal
(5.40)
f(nT
s
)d(t - nT
s
)
d
T
(t) =
q
n=-q
"
v
s
q
k=-q
d(t - nT
s
)
Î
f
Ideal sampling function
(5.41)
d(v - kv
s
)
2p
F(v)*[v
s
q
k=-q
T
s
q
1
1
Fourier transform of sampled signal
(5.42)
F
s
(v) =
d(v - kv
s
)] =
F(v)*d(v - kv
s
)
k=-q
Frequency response
Y(v) = X(v)H(v),
h(t) 4H(v)
