Digital Signal Processing Reference
In-Depth Information
s
(
n
)
m
R
(
z
)
0
,m
R
(
z
)
1
,m
C
(
z
)
m
P
P
z
−1
z
channel
P
ignore these
L
samples
P
z
−1
z
P
z
−1
s
(
n
)
k
E
(
z
)
k,L
P
P
z
z
R
(
z
)
P
− 1
,m
E
(
z
)
k,L+
1
P
z
E
(
z
)
k,P-
1
P
Figure P7.9
.
7.10.
In Problem 7.8 we showed how ISI can be eliminated in a multiuser system
by zero padding when the filter lengths are restricted to be
P
,where
P
=
M
+
L
and
L
is the order of the FIR channel
C
m
(
z
)
.
We now show how
these restricted filters can be designed such that
multiuser interference
is
also eliminated. Let
A
m
be the
P
≤
×
M
banded Toeplitz matrix given in
Problem 7.8.
1. For any nonzero number
ρ
k
, show that
1
ρ
−
1
k
k
... ρ
−
(
M−
1
k
,
where
C
m
(
z
)=
n
=0
c
m
(
n
)
z
−n
.
In other words, any exponential row
vector is like a left-eigenvector of
A
m
(except that the vectors on the
two sides have different sizes).
2. Using this, show that if the transmitting and receiving filters are
A
m
=
C
m
(
ρ
k
)
1
ρ
−
1
... ρ
−
(
P−
1)
k
F
m
(
z
)=
r
0
,m
+
r
1
,m
z
−
1
+
...
+
r
M−
1
,m
z
−
(
M−
1)
,
z
2
+
...
+
ρ
−
(
P−
1)
k
H
k
(
z
)=
a
k
(1 +
ρ
−
1
k
z
+
ρ
−
2
k
z
(
P−
1)
)
,
then the transfer function from
s
m
(
n
)to
s
k
(
n
)isgivenby
T
km
(
z
)=
a
k
C
m
(
ρ
k
)
F
m
(
ρ
k
)
.
(P7
.
10
a
)
3. Given
M
distinct numbers
ρ
k
,
0
1
,
write a set of
M
linear
equations from which we can solve for the
M
nonzero coe
cients of
F
m
(
z
) such that
≤
k
≤
M
−
F
m
(
ρ
k
)=
δ
(
k
−
m
)
,
0
≤
k
≤
M
−
1
.
This will eliminate multiuser interference by forcing
T
km
=0for
k
=
m
.
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