Digital Signal Processing Reference
In-Depth Information
evaluations for the next iteration [following (3.4)]. In case the channel impulse
response coefficients are desired, they may be obtained easily. We illustrate here
for the case of a three-tap channel.
The noise-free channel output symbols ideally satisfy the linear relation
2
4
3
5
2
4
3
5
H
(
S
0
)
111
H
(
S
1
)
111
H
(
S
2
)
1
11
2
3
h
0
H
(
S
3
)
1
11
4
5
¼
h
1
:
H
(
S
4
)
11
1
h
2
|
{z
}
h
H
(
S
5
)
11
1
H
(
S
6
)
1
1
1
H
(
S
7
)
1
1
1
|
{z
}
H
|
{z
}
F
The estimated values of the means from (3.7), collected into a vector
H
, may not
satisfy exactly this relation for any
h
. To compensate for possible errors in
H
,
a least-squares estimate of
h
may be computed, using the formula
h
LS
¼
(
F
T
F
)
1
F
T
H:
We observe here that
F
T
F ¼
8
I
(with
I
the identity matrix). More generally,
using a channel with degree
L
(having thus
Lþ
1 coefficients), we find that
F
T
F ¼
2
Lþ
1
I
, so that the least-squares solution for
h
simplifies to
h
LS
¼
2
(
Lþ
1)
F
T
H
whose calculation requires only sums, differences, and a scale factor.
B
EXAMPLE 3.11
Simulation Example for Estimating Channel Impulse Response.
Figure 3.13
shows the channel impulse response estimates from a single run of the blind
turbo equalizer applied to the same Proakis A channel setting as used in
Example 3.6, with a rate-adjusted SNR of
E
b
/
s
2
¼
6 dB, and an exponential dis-
tribution for the initial channel coefficient values, so that the various sums and
differences of the channel coefficients (which give the means used in the first
channel likelihood evaluations) are uniformly distributed. The estimated channel
coefficient values are observed to converge acceptably close to their true values,
indicated by the horizontal dashed lines. For this particular run, the blind turbo
equalizer successfully restores the transmitted information sequence.
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