Digital Signal Processing Reference
In-Depth Information
suppose we create a new channel
L
L
z
∗
m
−
z
−
1
c
new
(
n
)
z
−n
=
c
(0)
z
−
1
z
k
)
C
new
(
z
)=
(1
−
×
z
−
1
z
m
1
−
n
=0
k
=1
This is an FIR channel with the
m
th zero
z
m
replaced by 1
/z
∗
m
.Sincethe
factor
z
−
1
1
− z
−
1
z
m
is an allpass filter, the magnitude response is unchanged:
z
∗
m
−
C
new
(
e
jω
)
C
(
e
jω
)
|
|
=
|
|
.
Only the phase response of the channel is changed by this.
Since the
autocorrelation
r
(
k
) is the inverse Fourier transform of
|
2
, it follows
that
c
(
n
)and
c
new
(
n
) have the same autocorrelation. Thus, even though
the full banded Toeplitz matrix
A
is different for
C
(
z
)and
C
new
(
z
), the
matrix
A
†
A
is identical for both channels.
|
C
(
e
jω
)
2.
Zero locations of channel, and noise gain
.Since
A
†
A
isthesamefor
C
(
z
)
and
C
new
(
z
), it follows that
(which depends only on
A
†
A
)isalso
unchanged. Since the reconstruction error at the receiver (due to channel
noise) has the amplification factor
A
#
A
#
2
/M
(Eq. (8.10)) it then follows
that the channel noise amplification is insensitive to whether the zeros of
the channel are inside or outside the unit circle. This is a surprising result
and follows from the fact that the receiver uses all
P
noisy samples in
every block for the identification of the transmitted symbols. By contrast
if the receiver had used only
M
of the received samples, then the equaliza-
tion would be tantamount to inverting a square lower triangular Toeplitz
matrix, like
A
3
in Eq. (8.19). In this case, zeros of
C
(
z
) outside the unit
circle can create a large noise gain as demonstrated in Ex. 8.1 (where
C
(
z
)
has zeros with magnitudes 2.2611, 2.2611, 2.0529, and 0.0953).
3.
Channel with unit circle zeros
. If an FIR channel has unit circle zeros, then
the inverse 1
/C
(
z
) is unstable (even if we are willing to accept noncausal
inverses). Thus there is no stable equalizer at all (if there is no redundancy
like zero padding), and the channel noise is amplified in an unbounded
mannerby1
/C
(
z
)
.
But in a zero-padded system, the equalization works
perfectly well: the full banded matrix
A
still has full rank, so
σ
i
>
0for
A
#
2
/M
is finite.
all
i,
and the noise gain
4.
Norm of A
. Even though only the norm of the inverse
A
#
is involved in
the discussion of error, we would like to point out an interesting property
satisfied by the norm of
A
itself. We have
A
2
=Tr(
A
†
A
). Since the
diagonal elements of
A
†
A
are all equal to
r
(0), it follows that
A
2
=Tr(
A
†
A
)=
Mr
(0)
,
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