IEEE SIGNAL PROCESSING LETTERS, VOL. 5, NO. 7, JULY 1998
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Minimum Order Input–Output Equation for Linear Time-Varying Digital Filters Cishen Zhang, Song Wang, and Yu Fan Zheng
Abstract— The objective of this letter is to obtain minimum order input–output equations for a class of linear time-varying digital filters in state equation with constant dimension state vector. It is shown that the minimum order of the input–output equation may not be identical to the dimension of the state vector and there exists a nonunique solution for the minimum order input–output equation.
I. INTRODUCTION
where is the state vector of the filter with a constant and are the input and output dimension , and of the filter, are time varying matrices of the filter. Without loss of generality, we assume that the state equation (1) is reachable and observable where the reachability and observability follow from the standard definition, e.g., [1]. and an , let For each
L
INEAR time-varying digital filters, which include periodic filters as a special set, have been extensively studied and found a number of applications in digital signal processing and communications; see for example, [2]–[4] and [5]. While state and input–output equations are widely used for representing linear time-varying filters, there has been no procedure for efficiently computing a minimum order input–output equation from a given state equation. In this letter, we study this problem for a class of linear timevarying filters in state equation with constant dimension state vector to present a computational procedure for obtaining a minimum order input–output equation of the filter. We show that, considerably different from the case of linear time-invariant filters, the minimum order of the input–output equation for linear time-varying filters can be greater than the dimension of the state vector and the solution for the minimum order input–output equation can be nonunique. Section II presents the state equation for linear timevarying filters and useful properties relating to observability. Section III presents a procedure for computing a minimum order input–output equation and analysis of the minimum order equation. An example is given in Section IV to illustrate the computational procedure.
.. .
be the observability matrix of (1). It follows from [1] that the time-varying filter (1) is observable if and only if there exists such that a finite integer for all
(3)
It is noted that the observable time-varying filter (1) with the -dimensional state vector may satisfy (3) only for some but not for . This implies that the state is determined from the input and output sequences and over an interval of length . This is different from the case of linear time-invariant observable filters with constant and whose matrices state can always be determined from the input and output sequences over an interval of length . In this letter, we call the observable linear time-varying filter (1) -step observable satisfies if its observability matrix for all for some
II. STATE EQUATION FOR LINEAR TIME-VARYING FILTERS Consider a linear time-varying digital filter in the following state equation
(2)
(4)
We now present some useful properties of (1). Let (5)
(1) Thus, the output of state equation (1) satisfies Manuscript received April 7, 1998. This work was supported by the Australian Research Council. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. P. P. Vaidyanathan. C. Zhang and S. Wang are with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Vic. 3052, Australia (e-mail:
[email protected]). Y. F. Zheng is with the Institute of Systems Science, East China Normal University, Shanghai 200062, China. Publisher Item Identifier S 1070-9908(98)05138-4.
(6) where we obtain the formulations shown on the bottom of the next page. -step observable, is If (1) is is linearly represented by satisfied and the last row of
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IEEE SIGNAL PROCESSING LETTERS, VOL. 5, NO. 7, JULY 1998
the rows of exists an
. It follows that for each
there (7)
such that (8) III. MINIMUM ORDER INPUT-OUTPUT EQUATION FOR LINEAR TIME-VARYING FILTERS A linear time-varying digital filter can also be represented by an input–output equation written as (9) and are in the form (5) representing inputs where is in the form (7) and and outputs of the filter, and representing time-varying coefficients of the filter. We define the order of time-varying if its coefficients satisfy input–output equation (9) as or(and) and
for for all
and some and all
For the problem of finding an input–output equation of a linear time-varying filter from a given state equation, we define the following. Definition 1: The equation in the form (9) is an input–output equation of the time-varying state equation (1) if the output sequence of the for all and state equation (1) satisfies (9) for any initial state . input sequence Definition 2: An th order input–output equation in the form (9) is a minimum order input–output equation of the state equation (1) if it is an input–output equation of the state equation and there exists no input–output equation of the state equation with order less than . is upper-bounded by a known integer . Assume that We now present a procedure for computing a minimum order input–output equation of the reachable and observable linear time-varying filter (1) as follows. Step 1: Form the observability matrix (2) with row dimen, and determine such that (1) is sion an -step observable state equation. , compute and in (6); Step 2: For each
Step 3: For each , determine an in the form (7) in (6) to such that (8) is satisfied and use . obtain and determine an th order input–output Step 4: equation in the form (9), which is a minimum order input–output equation of the state equation (1) (see proof of Step 4 below). Proof of Step 4: Since (8) is satisfied, (9) is obtained in Step 4 by multiplying (6) from left by . It follows that the input and output sequences of the state equation satisfy (9) . Thus it is an input–output equation for any initial state of the filter. -step observable, For (1) being for all and there exists some such that . For this , it follows from with and that the last row of is not rows of . linearly dependent on the other , Therefore, for any such that there always exists a nonzero vector . It follows from (6) that the equation is . Thus, there exists no dependent on the state th order input–output equation in the form for all for the -step observable state equation. Moreover, there exists no th order input–output for the -step observable equation for any allow for state equation since the coefficients of some or all . Hence, the statement of Step 4 is established. Remarks: 1) If (1) represents a periodic filter with period such that and are satisfied for any integer , we only need to and for to obtain the compute minimum order periodic input–output equation where and are satisfied. 2) The proof of Step 4 shows that the minimum order of the input–output equation is possibly greater than the state dimension of the filter. This is because the required to observe the state of interval of length the -step observable state equation is possibly greater than the state dimension . Hence, to fully represent the information contained in (1), the minimum order of the input–output equation is .
ZHANG et al.: LINEAR TIME-VARYING DIGITAL FILTERS
3) For some , there may exist more than one such that (8) and (9) are satisfied. Thus, the solution for the minimum order input–output equation in the form (9) is possibly nonunique. IV. AN EXAMPLE We now present an example to illustrate the computational procedure for obtaining the minimum order input–output equation and show that the order of the minimum order input–output equation may be greater than the dimension of the state vector and the nonuniqueness of the input–output equation. Consider a two-periodic reachable and observable filter in and the form (1) with
Since the filter is two-periodic, we only need to determine and for to obtain the minimum order input–output equation. By Step 1, using the observability matrix (2), it is easy to for , and . verify that . Thus, , obtain and as follows: By Step 2, for
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By Step 3 and Step 4, determine and for . We obtain . These give a third-order (minimum order) input–output equation, whose order is greater . than the dimension of the state vector and Alternatively, we can determine to satisfy (8) and obtain and , which represent a different third-order (minimum order) input–output equation. V. CONCLUSION This letter presents a procedure for computing minimum order input–output equations for a class of linear time-varying digital filters in state equation with constant state dimension. The procedure covers linear periodic filters as a special case. It is shown that the minimum order of the input–output equation is determined by the length of the interval over which the state equation is observable. It is possible that the minimum order of the input–output equation is greater than the state dimension of the filter and the solution for the minimum order input–output equation is nonunique. REFERENCES [1] G. Ludyk, Time Variant Discrete Time Systems. Braunschweig, Germany: Vieweg & Sohn, 1981. [2] C. W. King and C. A. Lin, “A unified approach to scrambling filter design,” IEEE Trans. Signal Processing, vol. 43, pp. 1753–1765, 1995. [3] R. A. Meyer and C. S. Burrus, “A unified analysis of multirate and periodically time varying filters,” IEEE Trans. Circuits Syst., vol. CAS22, pp. 162–168, 1975. [4] J. S. Prater and C. M. Loeffler, “Analysis and design of periodically time varying IIR filters, with applications to transmultiplexing,” IEEE Trans. Signal Processing, vol. 40, pp. 2715–2725, 1992. [5] C. Zhang and Y. Liao, “A sequentially operated periodic FIR filter for perfect construction,” Circuits, Syst., Signal Process., vol. 16, pp. 475–486, 1997.
