Lattice filters based on the delta operator

R. Vijayan H.V. Poor

Indexing term: Lattice filters, Rapidly-sampled signals, Delta operators

Abstract: The delta operator defined by 6 = (q - l)/A has been proposed as an alternative to the conventionall shift operator q for implementing discrete-time estimation and control algorithms for processing data obtained by sampling continuous-time processes, when the sampling rate UA is rapid relative to the process dynamics. The authors present lattice and ladder filter structures based on the delta operator. As the sampling interval goes to zero, these filters converge to classical continuous-time lattice and ladder structures.

I

1 lntroductiion

In conventionaJ discrete-time control, estimation and signal processing procedures, the dynamics of processes of interest are represented using the shift operator, the action of which on a process {X,} is described by qXk = Xk+l. Time series are commonly represented using linear models based on the shift operator, an example of this type of model being the autoregressive (AR) model given by Xk + a,Xk-] = E,, where {E,<} is a white process.

In several situations of practical interest in digital control, image processing and wideband communica- tions, it is necessary to design discrete-time procedures that operate on data obtained by sampling continuous- time processes at rates that are rapid relative to the process dynamics [l]. Algorithms based on the shift operator tend to become numerically ill-conditioned at such rapid sampling rates, since the limiting form of the shift operator, as the sampling interval goes to zero, is the identity operator (assuming that the contin- uous-time process is sufficiently smooth). This phenom- enon has been 'observed in the context of Kalman-Bucy filtering [2], polle placement [3 ] and autoregressive mod- elling of time series [4, 51. The problem of ill-condition- ing has been solved in [2-41 and in [5] by using an alternative discrete-time formulation based on an incre- mental difference operator (or delta operator) defined by 6 = (q - l)/A, where A is the sampling interval.

Since the 6 operator is a discrete approximation to

0 IEE, 1997 IEE Proceedings online no. 19971043 Paper first received 6th February and in revised form 18th November 1996 R. Vijayan is with Qualcomm, Inc., 6455 Lusk Blvd., San Diego, CA 92121, USA H.V. Poor is with tlie Electrical Engineering Department, firinceton Uni- versity, Princeton, NJ 08544, USA

the derivative, &based equations and their solutions converge to continuous-time counterparts based on dif- ferential models, as the sampling interval goes to zero. As a result, at fast sampling rates, S-based algorithms exhibit better numerical conditioning than their shift- operator-based counterparts, although the two approaches are theoretically equivalent and offer the same flexibility in modelling. An additional benefit of using the 6 operator is that the connection between continuous and discrete time models is more easily seen than with conventional formulations.

In [4], a 6-based model was introduced as a means of alleviating the numerical problems associated with the conventional autoregressive model in the fast sampling regime. An algorithm for estimating the 6 model parameters was derived that has the same order of complexity as the Levinson-Durbin algorithm for computing the conventional AR parameters. Further algorithms were derived in [6] for estimating the model parameters by forming a triangular factorisation of the covariance matrix of differenced data. On-line versions of these and related &based algorithms are found in [7,

In this paper, we consider the derivation of lattice fil- ter structures based on the 6 model. The conventional digital lattice filter structure is suggested by the Levin- son-Durbin recursions [ 101. Lattice filters are preferred over other implementations in several applications since they possess several advantageous properties such as numerical stability and a structure consisting of identical sections in cascade [Note 11.

81.

2 Shift-operator-based lattice filter

In this Section, we briefly review the conventional lat- tice filter. Given a wide-sense stationary zero mean ran- dom process {&}, we define the mth-order forward and backward prediction errors by

m m

j = O j=0

and m m

j=0 j=0

respectively, where the coefficients {am,/} that minimise the mean-squared prediction error are obtained by solving the Yule-Walker equations using, for example, the Levinson-Durbin algorithm [l 11. The Levinson- Durbin algorithm then yields the following recursions Note 1: A conventional lattice filter for determining the parameters of a 6-AR model is described in [9]. However, this previous work does not consider the use of lattice stmctures for the model itself.

IEE Proc.-Vis. Image Signal Process., Vol. 144, No. 3, June 1997 125

4

xk section 1

Fig.2 mth lattice section

.........

section n

-----+n -~ section 2 .........

3 Delta lattice structures

The model used in [4, 61 to represent rapidly sampled data is of the form

JnXk $- Pn,l&n-lXk + ' * * Pn,nXk = l/k (5) The parameter vector f i n = [l, /3n,l, .... /3,,,lT that mini- mises the mean-squared modelling error E($) in eqn. 5 satisfies

Q /3 = [ ? i n ) O , . . .,0IT (6) n-n

where Q, is the covariance matrix of [6"Xk, P I X k , .... X,IT and j& = E($). Although the matrix Q,, unlike the corresponding covariance matrices encountered in the conventional autoregressive modelling problem, is not Toeplitz, its structure yields O(n2) algorithms for solving eqn. 6, as shown in [4] and [6].

The delta-based lattice structures we present here are derived using an algorithm given in [6] for computing the triangular factorisation of Q,, which we recapitu- late here.

The algorithm recursively obtains the vectors zm = [zr, .... z,q and y" = br, .... y z ] , m = 0, .... n, that, respectively, solve the equations

zmQ(m) = [O, . . ' 7 0711

Y"Q(,) = [1,0,. . . ,Ol

(7)

( 8 )

and

where Q(m) is the (m + 1) x (m + 1) leading principal submatrix of Q,, i.e. Q(m) = [(Q,)L,J]zJ = (Note that {-z/mizfl}?j' are the coefficients of the linear minimum-mean-squared-error (MMSE) prediction of

{-fl/yr}y=l are the coefficients of the linear MMSE prediction of SnX, based on

based on SnXk, Bn-lXk, .... P m + l X k , and that

.... P m X k . )

126

An extended matrix Q, is formed by adding n col- umns to the right of Q, as follows:

(Qn)z,j = ( Q n ) z , j ,

(Qn)n,j = - ~ ( Q n ) n , j - l )

0 5 2 7 3 5 n 3 = n + 1, . . . , 2n

and ( Q n ) z , I ) = -(Qn)2+1,3-1 - A(Qn)2,I)-l

i = O , 1 , . . . , n-1 ,

respectively, are defined by

j = n + l , . . . ) 2n The matrices S and R with elements {s,"} and {$},

and

. . . . . . . . . . rE+l . . . ryn

where the blank spaces in the matrices represent zero elements.

We have the following recursions for the elements of the above matrices (throughout this paper, we fol- low the convention that prediction coefficients are assumed to be zero outside their valid range. For example, zj"= 0 if j < 0 or j > m):

and

where

The prediction coefficients are computed using the recursions

and

(C+l q - TCI) /Km (12)

(13)

(14)

(15)

Sm+l = I)

Km = C+lSZ+l - c + z

y,"+l = (y," - r:+lz:'L+l)

r:'L+1 == 7-T - T;+1s3'L+1 The algorithm is initialised by setting

and

~9 = ~ 3 0 = ( Q n ) o , j / ( Q ) n ) o , o , (17) Eqns. 11-17 represent the algorithm for computing the delta model parameters by forming a triangular factor- isation of the associated covariance matrix.

We will now derive lattice filter structures that repre- sent the delta modelling equations. The delta-based 'forward' and 'backward' prediction errors are defined by

j = 0 , . . . . 2n

and

IEE Proc.-Vis. Image Signal Process., Vol. 144, No. 3, June 1997

As noted above, fp is the error in predicting P m X k lin- early from { S n X k , Sn-lXk, ..., P m + l X k } , whereas bp is the error in predicting SnXk linearly from (Sn-'Xk, ...., P m X k } . The forward prediction coefficients are not the same as the backward prediction coefficients, unlike in the shift-operator-based model where the backward prediction coefficients are just the forward prediction coefficients in reverse order.

From eqn. 11, we have m i l

1 - - ~- m+l [yO"S-lbP + A Y P F + T,m+l43 icm]

z,+1 Km

~- [Yo"6,1bP + T,m+1z"fr] zm"++:Krn

1 - -

(20 1 where the backward delta operator Sb defined by Sb = (1 - q-l)/A satisfies S$ = 6-I + A. The operator 62 is a discrete-time integrator given by S t X k = AEJk=-,4, and is shown in Fig. 3 . As the sampling interval converges to zero, its limiting form is the continuous-time integral operator Sf,X(s)ds.

U

Fig.3 Operator 6,:

From eqn. 14, we have d m+l

Eqns. 20 and 21 suggest a cascaded realisation of the modelling filter similar to that given in Fig. 1 for the conventional lattice. This filter is shown in Fig. 4. The individual cascaded sections are as shown in Fig. 5. Thus, this fi1te:r is seen to have a ladder structure, which is similar to discrete-time ladders based on the shift operator.

Fig. 4

IEE Proc.-Vis. Image Signal Process., Vol. 144, No. 3, June 1997

Cascaded delta modelling filter

p p + l - m+l m+l I

Fig. 5 Ladder representation of cascaded section

By substituting from eqn. 20 into eqn. 21, we obtain

Eqns. 20 and 22 give a lattice representation for the cascaded sections, as shown in Fig. 6.

Fig. 6 Lattice representation of cascaded section

Thus, we have shown that the delta modelling equa- tions yield lattice and ladder filter structures where the basic operator is a discrete-time integrator. It has been demonstrated in [12] that the algorithm given by eqns. 11-17 is much less sensitive to numerical errors than the corresponding algorithms that solve for the conven- tional AR parameters. This indicates that the delta- based lattice filters are also relatively less sensitive to coefficient errors.

As mentioned earlier, the limiting form of the Si' operator is the integral operator. Thus, as the sampling interval goes to zero, the filters developed here con- verge to continuous-time ladder or lattice filters, where the integrators represent reactances. This limiting behaviour is different from that associated with con- ventional discrete-time lattice filters. The continuous- time limiting case of such filters has been shown to be the distributed parameter transmission line; the limit is obtained by letting the number of stages in the lattice go to infinity. In contrast, for the delta-based lattice, we obtain the limit by keeping the number of stages fixed, so that the limiting case is a lumped parameter continuous-time filter. Thus, the delta-based structures can be considered to be more natural discrete versions of classical continuous-time lattices and ladders.

4 Conclusions

A new class of lattice filters has been presented. Being based on difference operators, they have the property of converging to classical continuous-time lattices as the sampling interval goes to zero.

There are some important distinctions between the delta-based lattice and its shift-operator-based counter- part. As we see from Fig. 4, the input to the filter is an&. Thus, the highest order is fixed and we add sec-

127

tions ‘downward’, in a manner of speaking. Hence, to increase the order of the filter, all the sections have to be changed. This is in contrast to the lattice of Fig. I, where the order can be increased by adding new sec- tions. The errorsfl and bp are linear combinations of the m + 1 highest-order differences PX,, ..., Pm&. Thus, the recursion eqns. 20 and 21 are valid only when the highest order, IZ, is fixed. To have a lattice that can be built up using independent sections, we need the mth-order forward and backward errors to be linear combinations of the m + 1 lowest-order differ- ences Xk, ..., PIXk. The derivation of such filters, which would be based on factorisation of related covariance matrices for which suitable algorithms are not availa- ble, is a topic for further study. Other areas to be investigated include the numerical properties of the new structures, and the derivation of algorithms for adapt- ing the filter parameters.

5 Acknowledgment

This work was supported by the US Office of Naval Research under grant NOOO14-94-1-0115.

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