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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