Digital Signal Processing Reference
In-Depth Information
the resulting output, the speed of the execution, and the throughput, and other
factors would depend not only on the finite wordlength but also on so many
other factors, including the architecture of the DSP chip, program instructions
per cycle, and dynamic range of the input signal. We will discuss these factors
in a later chapter.
6.4 ALLPASS FILTERS IN PARALLEL
Next in importance is the structure shown in Figure 6.16. The transfer function
G(z)
1
2
[
A
1
(z)
A
2
(z)
], where
A
1
(z)
and
A
2
(z)
are
the allpass filters connected in parallel. But in this figure, there is another trans-
fer function,
H(z)
=
Y(z)/X(z)
is given by
+
1
A
2
(z)
].
The structure shown in Figure 6.16 is also called the
lattice structure
or
lattice-
coupled allpass structure
by some authors. A typical allpass filter function is of
the form
=
V(z)/X(z)
, which is given by
H(z)
=
2
[
A
1
(z)
−
a
n
−
1
z
−
1
a
n
−
2
z
−
1
a
1
z
−
n
+
1
a
0
z
−
n
N(z)
D(z)
=±
a
n
+
+
+···+
+
A(z)
=
(6.32)
a
1
z
−
1
a
2
z
−
2
a
n
−
1
z
−
n
+
1
a
n
z
−
n
a
0
+
+
+···+
+
which shows that the order of the coefficients in the numerator is the reverse of
that in the denominator, when both the numerator and denominator polynomial
are expressed in descending powers of
z
. Equation (6.32) can be expressed in
another form as
z
−
n
(a
0
+
a
2
z
2
a
n
−
1
z
n
−
1
a
n
z
n
)
a
1
z
+
+···+
+
A(z)
=
a
0
+
a
1
z
−
1
+
a
2
z
−
2
+···+
a
n
−
1
z
−
n
+
1
+
a
n
z
−
n
z
−
n
D(z
−
1
)
D(z)
=
(6.33)
The zeros of the numerator polynomial
D(z
−
1
)
are the reciprocals of the zeros
of the denominator
D(z)
, and therefore the numerator polynomial
D(z
−
1
)
is the
mirror image polynomial of
D(z)
.
When the allpass filter has all its poles inside the unit circle in the
z
plane, it is
a stable function and its zeros are outside the unit circle as a result of the mirror
A
1
(z)
Σ
Y(z)
1/2
X(z)
A
1
(z)
V(z)
Σ
−
1
1/2
Figure 6.16
Two allpass filters in parallel (lattice-coupled allpass structure).
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