Global Positioning System Reference
In-Depth Information
Changing the variable by letting
t
−
τ
=
u
,then
∞
x(τ)
∞
−∞
h(u)e
−
j
2
πf u
du
e
−
2
πf τ
dτ
Y(f)
=
−∞
H(f)
∞
−∞
x(τ)e
−
j
2
πf τ
dτ
=
=
H(f)X(f)
(7.4)
In order to find the output in the time domain, an inverse Fourier transform on
Y
(
f
) is required. The result can be written as
y(t)
=
x(t)
∗
h(t)
=
F
−
1
[
X(f )H(f )
]
(
7
.
5
)
where the * represents convolution and
F
−
1
represents inverse Fourier transform.
A similar relation can be found that a convolution in the frequency domain is
equivalent to the multiplication in the time domain. These two relationships can
be written as
x(t)
∗
h(t)
↔
X(f)H(f)
X(f )
∗
H(f)
↔
x(t)h(t)
(7.6)
This is often referred to as the duality of convolution in Fourier transform.
This concept can be applied in discrete time; however, the meaning is dif-
ferent from the continuous time domain expression. The response
y
(
n
) can be
expressed as
N
−
1
y(n)
=
x(m)h(n
−
m)
(
7
.
7
)
m
=
0
where
x
(
m
) is an input signal and
h
(
n
m
) is system response in discrete time
domain. It should be noted that in this equation the time shift in
h
(
n
−
m
)is
circular because the discrete operation is periodic. By taking the DFT of the
above equation the result is
−
N
−
1
N
−
1
m)e
(
−
j
2
πkn)/N
Y(k)
=
x(m)h(n
−
n
=
0
m
=
0
x(m)
N
−
1
h(n
−
m)e
(
−
j
2
π(n
−
m)k)/N
e
(
−
j
2
πmk)/N
N
−
1
=
m
=
0
n
=
0
N
−
1
x(m)e
(
−
j
2
πmk)/N
=
H(k)
=
X(k)H(k)
(7.8)
m
=
0
Equations (7.7) and (7.8) are often referred to as the periodic convolution (or
circular convolution). It does not produce the expected result of a linear convo-
lution. A simple argument can illustrate this point. If the input signal and the
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