Digital Signal Processing Reference
In-Depth Information
5.
Existence of inverses.
Note that the eigenvectors are polynomial functions of
z.
If
C
(
z
) is an FIR filter, then the blocked version
C
(
z
)
,
the eigenvalues
C
(
zW
k
), and the determinant of
C
(
z
P
) are all FIR functions. Except
in the trivial case where
C
(
z
) is identically zero, the determinant (D.20)
cannot be identically zero. So the inverse of
C
(
z
P
)
always
exists as a
rational function. If the determinant has no unit circle zeros this inverse is
stable
; if the determinant has all zeros inside the unit circle then there is
a
causal stable inverse
. Similar statements follow when
C
(
z
) is a rational
IIR transfer function.
C
(
z
)
C
(
z
)
1
2
(a)
P
P
P
P
z
z
−1
z
−1
z
P
P
P
P
C
(
z
)
2
C
(
z
)
1
z
−1
z
z
z
−1
z
−1
z
z
−1
z
P
P
P
P
(b)
P
P
z
−1
z
P
P
C
(
z
)
2
C
(
z
)
1
z
−1
z
(c)
z
−1
z
P
P
Figure D.2
. (a) A cascade of two LTI systems, (b) the blocked versions, and
(c) the simplified system.
6.
Cascade of pseudocirculants.
Next consider a cascade of two LTI systems
C
1
(
z
)and
C
2
(
z
) as shown in Fig. D.2(a). If we block these two systems
(Sec. 3.9), the result is as shown in part (b). But the system shown in
the gray box is the identity system (Fig. 3.17) so we obtain the simplified
version of Fig. D.2(c). This means that the blocked version of the cascade
C
2
(
z
)
C
1
(
z
) is the cascade of blocked versions
C
2
(
z
)
C
1
(
z
)
.
But since
C
1
(
z
)
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