Global Positioning System Reference
In-Depth Information
impulse response of the linear system both have
N
data points, from a linear
convolution, the output should be 2
N
1 points. However, using Equation (7.8)
one can easily see that the outputs have only
N
points. This is from the periodic
nature of the DFT.
The acquisition algorithm does not use convolution; it uses correlation, which
is different from convolution. A correlation between
x
(
n
)and
h
(
n
) can be
written as
−
N
−
1
z(n)
=
x(m)h(n
+
m)
(
7
.
9
)
m
=
0
The only difference between this equation and Equation (7.7) is the sign before
index
m
in
h
(
n
m
). The
h
(
n
) is not the impulse response of a linear system
but another signal. If the DFT is performed on
z
(
n
) the result is
+
N
−
1
N
−
1
m)e
(
−
j
2
πkn)/N
Z(k)
=
x(m)h(n
+
n
=
0
m
=
0
x(m)
N
−
1
m)e
(
−
j
2
π(n
+
m)k)/N
e
(j
2
πmk)/N
N
−
1
=
h(n
+
m
=
0
n
=
0
N
−
1
x(m)e
(j
2
πmk)/N
H(k)X
−
1
(k)
=
H(k)
=
(7.10)
m
=
0
where
X
−
1
(k)
represents the inverse DFT. The above equation can also be
written as
N
−
1
N
−
1
m)h(m)e
(
−
j
2
πkn)/N
H
−
1
(k)X(k)
Z(k)
=
x(n
+
=
(
7
.
11
)
n
=
0
m
=
0
If the
x
(
n
) is real,
x(n)
∗
=
is the complex conjugate. Using this
relation, the magnitude of
Z
(
k
) can be written as
x(n)
where
∗
H
∗
(k)X(k)
H(k)X
∗
(k)
|
Z(k)
|=|
|=|
|
(
7
.
12
)
This relationship can be used to find the correlation of the input signal and
the locally generated signal. As discussed before, the above equation provides a
periodic (or circular) correlation and this is the desired procedure.
7.8 ACQUISITION BY CIRCULAR CORRELATION
(
2
)
The circular correlation method can be used for acquisition and the method
is suitable for a software receiver approach. The basic idea is similar to the
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