ON THE REALIZATION OF 2D LATTICE-LADDER DISCRETE FILTERS

Transcription

1 Journal of Circuits Systems and Computers Vol. 3 No. 5 (2004) 5 c 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 USA Received 7 January 2003 Revised 6 September 2003 In this paper the one-dimensional Gray Markel lattice-ladder discrete filter structure is extended to two dimensions (2D). The proposed 2D circuit implementation has minimal number of unit delays. Based on this circuit implementation the corresponding 2D state space realization is derived. The matrices A b c and the scalar d of the 2D state space model are presented in generalized closed form having minimal dimension.. Introduction The area of multidimensional signal processing and systems has attracted researchers from academia and industry for afew decades. This is because of the challenging theoretical problems and the promising applications in the areas of image processing computer tomography geophysics etc. 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 realizations 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 (D) case. Minimal realizations can be derived only for particular categories of 2D systems i.e. continued fraction expandable systems all-pole and all-zero filters product factorable transfer 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 D Gray Markel discrete-time latticeladder 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.

2 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 structure 78 : ẋ(i j) = Ax(i j) + bu(i j) y(i j) = c x(i j) + du(i j) () x h (i j) x h (i + j) x v (i j) x v (i j + ) x h 2 (i j) x h 2 (i + j) x(i j) = x v 2 (i j) ẋ(i j) = x v 2 (i j + ) x h n(i j) x h n(i + j) x v n(i j) x v n(i j + ) and the dimensions of the matrices A b c are 2n 2n 2n 2n respectively. The required transformations for converting the cyclic 2D state space model to classical state space Roesser model 7 and vice versa are given in Ref. 8. Applying the 2D Z transform to Eq. () its corresponding 2D transfer function takes the following form: H(z z 2 ) = c [Z A b + d (2) Z = diag [z z 2 z z 2... z z Circuit and State Space Realization Extending the results of Gray Markel 6 for lattice-ladder discrete D filters to 2D the corresponding 2D ladder-lattice circuit realization is depicted in Fig.. The new 2D section has minimal number of two delay elements namely z and z 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. ()) from the circuit representation given in Fig. it is assumed that the outputs of the delay elements z z 2 correspond to the states of the model (Eq. ()). 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. ()) are derived by inspection

3 On the Realization of 2D Lattice-Ladder Discrete Filters 3 Instructions for Typesetting Manuscripts (). 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 () are derived by inspection having the following structure: Fig.. Block diagram of the lattice-ladder discrete 2D filter. having the following structure: A =. 2n 2n 2.. 2n 2n 2 2n 2n 2 2n 2n 2n 2 2n 2 2n 2 2n 2n 2 3 b = 2n 2n c = [ c c 2 c 2n c 2n 2n c j = ( 2 j)v j j i+ V i+ j V 2n+ j =... 2n 2 i=j c 2n = ( 2 2n )V 2n 2n 2n V 2n 2n V 2n+ c 2n = ( 2 2n )V 2n 2n V 2n+

4 4 G. E. Antoniou and d = 2n i= i V i + V 2n+. The dimensions of the matrices A b and c are 2n 2n 2n 2n respectively. It is noted that the state space matrix A has minimal dimension 2n 2n resulting from the corresponding minimal circuit realization. i i =... 2n are reflection coefficients. Stability conditions require that the reflection coefficients i < Example For simplicity consider the first order 2D lattice filter with n =. In this case the corresponding state space realization takes on the form and A = ẋ(i j) = Ax(i j) + bu(i j) y(i j) = c x(i j) + du(i j) [ x h (i j) x(i j) = x v (i j) [ x h (i + j) ẋ(i j) = x v (i j + ) [ [ 2 2 b = [ ( 2 c )V 2 V 2 V 3 = ( 2 2)V 2 2 V 3 d = V + 2 V 2 + V 3. 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(z z 2 ) = V + ( 2 V + V 2 )z + 2 V z 2 + ( V + 2 V 2 + V 3 )z z z + 2 z 2 + z z 2. (4) For V = and V 2 = V 3 = 0 the above transfer function (Eq. (4)) takes the form H(z z 2 ) = + 2 z + 2 z 2 + z z z + 2 z 2 + z z 2. (5) (3)

5 December :48 WSPC/23-JCSC 0077 st Reading 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 Conclusion The D 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. J. E. Lim Two-Dimensional Signal and Image Processing (Prentice-Hall Englewood Cliffs NJ 990). 2. G. E. Antoniou 2D lattice discrete filters minimal delay and state space realization IEEE Signal Processing Lett. SPL-8() (200) G. E. Antoniou Generalized one-multiplier lattice discrete 2D filters: Minimal circuit and state space realization IEEE Trans. Circuits Syst. I CAS-48(2) (200) G. E. Antoniou S. J. Varoufakis and P. N. Paraskevopoulos Minimal state space realization of factorable 2D systems IEEE Trans. Circuits Syst. CAS-35(8) (988) G. E. Antoniou S. J. Varoufakis and P. N. Paraskevopoulos State space realization of 2D systems via continued fraction expansion IEEE Trans. Circuits Syst. CAS-33(9) (986) A. H. Gray and J. D. Markel Digital lattice and ladder filter synthesis IEEE Trans. Audio Electroacoustics AU-2(6) (973) P. R. Roesser A discrete state space model for linear image processing IEEE Trans. Autom. Control AC-20() (975) P. N. Paraskevopoulos and G. E. Antoniou Minimal realization of 2D systems via a cyclic model Int. J. Electron. 74(4) (993) N. Matsumoto B. D. O. Anderson and M. Mansour Sufficient condition for state space representation of ND discrete time lossless bounded real matrix and ND stability of Mansour matrix IEEE Trans. Circuits Syst. CAS-37(9) (990)