Digital Signal Processing Reference
In-Depth Information
On the other hand, the second-order modes are invariant under similarity transformation
because of the following relationship
KW
=
T
−
1
(
KW
)
T
.
(7)
Hence
it
follows
that
the
Gramians
depend
on
realizations
of
the
system,
while
the
second-order modes depend only on the transfer function.
In the literature on synthesis of filter structures [13-25], it is shown that the two Gramians and
the second-order modes play central roles in analysis and optimization of filter performance
such as the roundoff noise and the coefficient sensitivity. In other words, given the transfer
function of a digital filter, we can formulate some cost functions with respect to the
aforementioned filter performance in terms of the two Gramians
(
K
,
W
)
, and a filter structure
of high performance can be obtained by constructing the two Gramians appropriately in such
a manner that they optimize or sub-optimize the corresponding cost functions.
An example of high-performance digital filter structures is the balanced form [15, 16, 18, 23,
25]. This form consists of the two Gramians given by
K
=
W
=
Θ
(8)
where
Θ
is the diagonal matrix consisting of the second-order modes, i.e.
Θ
=
diag
(
θ
1
,
θ
2
,
· · ·
,
θ
N
)
.
(9)
Another example is the minimum roundoff noise structure [13, 14, 16, 17], which consists of
the two Gramians that satisfy the following relationships
!
2
N
I
=
1
θ
I
1
N
W
=
K
K
II
=
1
(10)
where
K
II
denotes the
I
-th diagonal entry of
K
.
Finally, we address the significance of the second-order modes from two practical aspects.
First, it is known in the literature that the second-order modes describe the optimal values
of the aforementioned cost functions. Therefore, it follows that the optimal performance
is determined by the second-order modes of a given transfer function. Another important
feature of the second-order modes can be seen in the field of the balanced model reduction
[26-28], where it is shown that the second-order modes provide the upper bound of the
approximation error between the reduced-order system and the original system.
