ON THE REALIZATION OF 2D LATTICE-LADDER DISCRETE FILTERS · ON THE REALIZATION OF 2D LATTICE-LADDER...
Transcript of ON THE REALIZATION OF 2D LATTICE-LADDER DISCRETE FILTERS · ON THE REALIZATION OF 2D LATTICE-LADDER...
Journal of Circuits, Systems, and ComputersVol. 13, No. 5 (2004) 1–5c© World Scientific Publishing Company
ON THE REALIZATION OF 2D
LATTICE-LADDER DISCRETE FILTERS
GEORGE E. ANTONIOU
Department of Computer Science, Montclair State University,
Upper Montclair, New Jersey 07043, USA
Received 17 January 2003Revised 16 September 2003
In this paper, the one-dimensional Gray–Markel lattice-ladder discrete filter structure isextended to two dimensions (2D). The proposed 2D circuit implementation has minimalnumber of unit delays. Based on this circuit implementation the corresponding 2D statespace realization is derived. The matrices A, b, c′ and the scalar d of the 2D state spacemodel are presented in generalized closed form, having minimal dimension.
1. Introduction
The area of multidimensional signal processing and systems has attracted re-
searchers from academia and industry for a few decades. This is because of the
challenging theoretical problems and the promising applications in the areas of
image processing, computer tomography, geophysics etc.1
An interesting and important problem is the circuit implementation and state
space realization for two-dimensional (2D) systems, described by a transfer function,
with minimal number of delays. The need to provide minimal realization arises not
only out of hardware requirements but also because sometimes nonminimal realiza-
tions often cause theoretical or computational difficulties. It is known that it is not
always possible to find minimal delay or state space realizations for an arbitrary
2D system in contra-distinction to one-dimensional (1D) case.1 Minimal realiza-
tions can be derived only for particular categories of 2D systems, i.e., continued
fraction expandable systems, all-pole and all-zero filters, product factorable trans-
fer functions, discrete time lossless bounded real functions, separable and factorable
systems, first order all-pass and lattice filters.2–5
In this paper, the circuit realization of the 1D Gray–Markel discrete-time lattice-
ladder filter,6 was extended to two dimensions. The proposed circuit realization
has minimun number of delay elements. Using the presented circuit realization the
corresponding state space realization, having minimal dimension, is derived.
1
2 G. E. Antoniou
2. Realization
In this section, the circuit implementation and the minimal state space realization
for the 2D discrete-time lattice-ladder filters are presented. The 2D state space
model that is used, is of the Roesser type with cyclic state space vector structure7,8:
x(i, j) = Ax(i, j) + bu(i, j) ,
y(i, j) = c′x(i, j) + du(i, j) ,(1)
where
x(i, j) =
xh1 (i, j)
xv1(i, j)
xh2 (i, j)
xv2(i, j)
· · ·
xhn(i, j)
xvn(i, j)
, x(i, j) =
xh1 (i + 1, j)
xv1(i, j + 1)
xh2 (i + 1, j)
xv2(i, j + 1)
· · ·
xhn(i + 1, j)
xvn(i, j + 1)
,
and where the dimensions of the matrices A, b, c′ are 2n × 2n, 2n × 1, 1 × 2n,
respectively. The required transformations for converting the cyclic 2D state space
model to classical state space Roesser model7 and vice versa are given in Ref. 8.
Applying the 2D Z transform to Eq. (1), its corresponding 2D transfer function
takes the following form:
H(z1, z2) = c′[Z −A]−1b + d , (2)
where
Z = diag [z1, z2, z1, z2, . . . , z1, z2] .
3. Circuit and State Space Realization
Extending the results of Gray–Markel6 for lattice-ladder discrete 1D filters to 2D,
the corresponding 2D ladder-lattice circuit realization is depicted in Fig. 1. The
new 2D section has minimal number of two delay elements, namely z−1
1 and z−1
2 .
It is noted that the cascaded circuit implementation has minimal number of delay
elements (2n).
In order to derive the state space matrices A, b, c′ and the scalar d for the state
space model (Eq. (1)), from the circuit representation given in Fig. 1, it is assumed
that the outputs of the delay elements z−11 , z−1
2 correspond to the states of the
model (Eq. (1)). Moreover, by writing one state equation for every delay element,
and after some algebraic manipulations, we can conclude that the matrices A, b,
c′ and the scalar d, of the state space model (Eq. (1)) are derived by inspection,
On the Realization of 2D Lattice-Ladder Discrete Filters 3
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(1). Moreover, by writing one state equation for every delay element, and after
some algebraic manipulations, we can conclude that the matrices A,b, c′ and the
scalar d, of the state space model (1) are derived by inspection, having the following
structure:
Fig 1: Block diagram of the lattice–ladder discrete 2D filter
A =
−∆1∆2 1−∆22 · · · 0 0 0
−∆1∆3 −∆2∆3 1−∆23 · · · 0 0
· · · · · ·. . .
. . . · · · 0
−∆1∆2n−1 · · · · · · −∆2n−2∆2n−1
. . .
−∆1∆2n −∆2∆2n · · · −∆2n−2∆2n −∆2n−1∆2n 1−∆22n
−∆1 −∆2 · · · −∆2n−2 −∆2n−1 −∆2n
b =
∆2
∆3
· · ·∆2n−1
∆2n
1
c′ =[
c1 c2 · · · c2n−1 c2n
]
,
where
cj = (1−∆2j )Vj −
2n−1∑
i=j
∆j∆i+1Vi+1 −∆jV2n+1, j = 1, · · · , 2n− 2.
c2n−1 = (1−∆22n−1)V2n−1 −∆2n−1∆2nV2n −∆2n−1V2n+1
c2n = (1−∆22n)V2n −∆2nV2n+1
and
d =
2n∑
i=1
∆iVi + V2n+1
The dimensions of the matrices A,b and c′ are 2n× 2n, 2n× 1, 1× 2n, respec-
tively. It is noted that the state space matrix A has minimal dimension 2n × 2n,
resulting from the corresponding minimal circuit realization. ∆i, i = 1, · · · , 2n are
reflection coefficients. Stability conditions require that the reflection coefficients
|∆i| < 1 [9].
4. Example
For simplicity consider the first order 2D lattice filter, with n = 1. In this the
case the corresponding state space realization takes on the form,
Fig. 1. Block diagram of the lattice-ladder discrete 2D filter.
having the following structure:
A =
−∆1∆2 1 − ∆22 · · · 0 0 0
−∆1∆3 −∆2∆3 1 − ∆23 · · · 0 0
· · · · · ·. . .
. . . · · · 0
−∆1∆2n−1 · · · · · · −∆2n−2∆2n−1
. . .
−∆1∆2n −∆2∆2n · · · −∆2n−2∆2n −∆2n−1∆2n 1 − ∆22n
−∆1 −∆2 · · · −∆2n−2 −∆2n−1 −∆2n
,
b =
∆2
∆3
· · ·
∆2n−1
∆2n
1
,
c′ =[
c1 c2 · · · c2n−1 c2n
]
,
where
cj = (1 − ∆2j )Vj −
2n−1∑
i=j
∆j∆i+1Vi+1 − ∆jV2n+1, j = 1, . . . , 2n− 2 ,
c2n−1 = (1 − ∆22n−1)V2n−1 − ∆2n−1∆2nV2n − ∆2n−1V2n+1 ,
c2n = (1 − ∆22n)V2n − ∆2nV2n+1 ,
4 G. E. Antoniou
and
d =
2n∑
i=1
∆iVi + V2n+1 .
The dimensions of the matrices A, b and c′ are 2n× 2n, 2n× 1, 1× 2n, respec-
tively. It is noted that the state space matrix A has minimal dimension 2n × 2n,
resulting from the corresponding minimal circuit realization. ∆i, i = 1, . . . , 2n are
reflection coefficients. Stability conditions require that the reflection coefficients
|∆i| < 1.9
4. Example
For simplicity consider the first order 2D lattice filter, with n = 1. In this case, the
corresponding state space realization takes on the form,
x(i, j) = Ax(i, j) + bu(i, j) ,
y(i, j) = c′x(i, j) + du(i, j) ,(3)
where
x(i, j) =
[
xh1 (i, j)
xv1(i, j)
]
,
x(i, j) =
[
xh1 (i + 1, j)
xv1(i, j + 1)
]
,
and
A =
[
−∆1∆2 1− ∆22
−∆1 −∆2
]
,
b =
[
∆2
1
]
,
c′ =
[
(1 − ∆21)V1 − ∆1∆2V2 − ∆1V3
(1 − ∆22)V2 − ∆2V3
]
,
d = ∆1V1 + ∆2V2 + V3 .
The dimensions of the state space matrix A is minimal (2 × 2).
Using Eq. (2), the 2D transfer function of the state space model (Eq. (3)) is
H(z1, z2) =V1 + (∆1∆2V1 + V2)z1 + ∆2V1z2 + (∆1V1 + ∆2V2 + V3)z1z2
∆1 + ∆2z1 + ∆1∆2z2 + z1z2
. (4)
For V1 = 1 and V2 = V3 = 0, the above transfer function (Eq. (4)) takes the
form,
H(z1, z2)′ =
1 + ∆1∆2z1 + ∆2z2 + ∆1z1z2
∆1 + ∆2z1 + ∆1∆2z2 + z1z2
. (5)
1st ReadingDecember 28, 2004 14:48 WSPC/123-JCSC 00177
On the Realization of 2D Lattice-Ladder Discrete Filters 5
It is obvious that the above transfer function (Eq. (5)) is characterized by the
all-pass property as in Ref. 9.
5. Conclusion
The 1D Gray–Markel ladder-lattice discrete filter circuit realization was extended
to 2D. The proposed circuit implementation is of minimal dimension with respect
to the required delay elements. The matrix vectors A, b, c′ of the 2D state space
model are of minimal dimension and were derived from the corresponding circuit
implementation. The results presented in this paper can be extended to three or
higher dimensions.
References
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