Frequency Transformation for Linear State-Space Systems and Its Application to High-Performance Analog/Digital Filters - Digital Filters and Signal Processing - page 120

We next discuss the Gramian-preserving frequency transformation from a realization
point of view. From (27), we first see that realization of the Gramian-preserving
frequency transformation requires us to construct the structure of the all-pass filter 1/ F ( Z )
appropriately such that its state-space representation becomes a balanced form. Although
formulation of the balanced form is known to be non-unique for a given transfer function,
we presented a useful technique [31]: given an all-pass transfer function 1/ F ( Z ) , its
normalized lattice structure becomes a balanced form, which enables us to realize the
Gramian-preserving frequency transformation. This is derived from the fact that 1/ F ( Z )
is all-pass. Now, recall that the frequency transformation of digital filters means that each
delay element in a prototype filter is replaced with an all-pass filter (and delay-free loops,
if any, are eliminated after this replacement) 4 . In view of this, we can conclude that the
Gramian-preserving frequency transformation is interpreted as the replacement of each delay
element in the prototype filter with the all-pass filter that has the normalized lattice structure.
Figure 2 illustrates this scheme. Given a state-space prototype filter as in Fig. 2(a), we carry
out the aforementioned replacement and we obtain the transformed state-space filter as in
Fig. 2(b). The all-pass filter that is included in this structure consists of M lattice sections
Φ 1 , · · · , Φ M , and each section for 1 ≤ I ≤ M Φ I is given as in Fig. 2(c). The variable ξ I

1 − ξ 2 .
Finally, we provide the mathematical formulation of the Gramian-preserving frequency
transformation based on the normalized lattice structure. The normalized lattice structure of
1/ F ( Z ) can be given by the following state-space representation:

denotes the I -th lattice coefficient for 1/ F ( Z ) , and ξ I =

0
@

1
A

− ξ 1 −

ξ 1 ξ 2 − ξ 1 ξ 2 ξ 3 · · · − ξ 1 ξ 2 ξ 3 · · · ξ M − 3 ξ M − 2 − ξ 1 ξ 2 ξ 3 · · · ξ M − 2 ξ M − 1 − ξ 1 ξ 2 ξ 3 · · · ξ M − 1 ξ M

ξ 1 − ξ 1 ξ 2 − ξ 1 ξ 2 ξ 3 · · · − ξ 1 ξ 2 ξ 3 · · · ξ M− 3 ξ M− 2 − ξ 1 ξ 2 ξ 3 · · · ξ M− 2 ξ M− 1 − ξ 1 ξ 2 ξ 3 · · · ξ M− 1 ξ M

0 ξ 2

− ξ 2 ξ 3 · · · − ξ 2

ξ 3 ξ 4 · · · ξ M− 3 ξ M− 2 − ξ 2 ξ 3 ξ 4 · · · ξ M− 2 ξ M− 1 − ξ 2 ξ 3 ξ 4 · · · ξ M− 1 ξ M

α =

. . . . . .

. . .

0 0 0

· · ·

0 ξ M− 1

− ξ M − 1 ξ M

0
@

1
A

ξ 1 ξ 2 ξ 3 · · · ξ M− 1 ξ M

ξ 1 ξ 2 ξ 3 · · · ξ M − 1 ξ M

ξ 3 · · · ξ 2 ξ M− 1 ξ M .

β =

ξ M− 2 ξ M− 1 ξ M

ξ M − 1 ξ M

γ = 0 0 0 · · · 0 ± ξ M

δ = ± ξ M . (29)

Therefore, substitution of (29) into (26) carries out the Gramian-preserving frequency

representation ( A , B , C , D) given transformation. Note that the state-space in this way

To be precise, the set ( A , B , C , D) becomes sparse due to many zero entries in α and γ .

4

Note that the mathematical formulation of the Gramian-preserving frequency transformation (26) is derived after
elimination of delay-free loops. Therefore, (26) does not have the problem of delay-free loops. See [30] for the details.