FIR filter design based on total least squares error criterion

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FIR FILTER DESIGN BASED ON TOTAL LEAST SQUARES ERROR CRITERION

Soo-Chang

Pei'

Chien-Cheng Tseng2

'Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R. 0. C.

2Department of Computer and Communication Engineering, National Kaohsiung First University of Technology and Science, Kaohsiung, Taiwan, R.O.C.

ABSTRACT

In this paper, an FIR filter design based on total least squares error criterion is investigated. The filter coefficients are obtained from the elements of the eigenvector corresponding to minimum eigenvalue of a real, symmetric and positive definite matrix. This design is not only optimal in total least squares error sense, it is also easy to incorporate linear constraints in time and frequency domain. Several design examples are used t o illustrate the effectiveness of this new design approach.

1. INTRODUCTION

Conventionally, we often use the well-known McClellan- Parks-Rabiner (MPR) computer program and standard linear programming technique t o design linear phase FIR filters according to the Chebyshev criterion which minimizes the maximum error in frequency response 111. The minimax designs usually give the designers a smallest length filter for a given specification. However, MPR algorithm is difficult to incorporate both time and frequency domain constraint. And, linear programming technique needs

a

large memory space and considerable computation time. Thus, a number of researchers have considered linear phase FIR filter design using least-squares optimality criterion.

From literature survey, two well-documented least squares approaches for FIR filter designs are matrix inverse (IM) method and eigen-approach. The IM methods are based on solving a set of linear simultaneous equations by matrix inversion [2][3], and the eigen-approaches are based on the computation of an eigenvector of an appropriate real, symmetric, and positive-definite matrix [4][5]. Compared with minimax design, the advantage of least squares design is easy to add constraint and requires simple computation. The purpose of this paper is t o design linear phase

FIR

filters using total least squares

(TLS)

error criterion which has been successfully used to solve many engineering prob- lems such as acoustic radiation problem, beamformer and harmonic retrieval etc [6]-[8]. The filter coefficients based on TLS criterion are obtained from the elements of the eigenvector corresponding t o minimum eigenvalue of a real, sym- metric and positive definite matrix. Due t o this sake, the to- tal least squares filter design is referred t o

as

the new eigenfilter approach. The main difference between conventional and new eigenfilters is that conventional method needs to specify the reference point, but new approach does not re- quire this choice. Moreover, the solution of new eigenfilter is closer to the solution of the

I M

method than the one of the conventional eigenfilter.

2. FIR FILTER DESIGN BASED O N TOTAL LEAST SQUARES ERROR CRITERION 2.1 P r o b l e m Statement:

A causal N t h order FIR filter can be represented as N

H ( t ) = h(n)z-" (1)

n=O

The magnitude responses

of

FIR

filters

is

given by

M

A ( w ) = a n t r i g ( w , T I ) (2) 11=1

where trig(w, n) is an approprite trigonometrical function [l]. The coefficient an is related t o the impulse response of the filter, whereas M is a function of the filter order N . Defining the column vector

a = [a1 a2 . . . a M I t ₍₃₎ and

C(W) = [trig(w, 1) t ~ i g ( w , 2) . . . t r i g ( w , M ) ] ~ (4)

then we rewrite eq(2) as

A ( w ) = a'c(w) = c ' ( w ) a _{( 5 )}

The notation

t

denote the vector or matrix transpose. Now, the problem is that how we can find a such that the magnitude response A ( w ) in eq(5) fit the desired magnitude response D ( w ) as well as possible. Total least squares error m e s u r e s will be studied in this paper.

2.2 TLS Design Method:

The well-known linear phase filter design problem is to find a filter weight a such that the desired magnitude response

D ( w )

is equal t o the actual magnitude response atc(w) of the filter, i.e.

D ( w ) = a'c(w) (6)

for each w E [ O , T ) . In the space

R

generated by C(W) and D ( w ) , it is clear that the expression D ( w ) = a'c(w) denotes a hyperplane. For a given frequency

4,

the ( c ( $ ) , D ( 4 ) ) denotes a point in

8.

Now, the filter design problem can be restated as "we want the point (c($),D($)) to fall on the hyperplane D ( w ) = a'c(w) for all

6

E [ O , T ) " . When the point (c(4),

D ( $ ) )

does not fall on the plane D ( w ) = a'c(w). the error between them can be measured in several ways.

0-7803-5471 -0/99/$10.0001999 IEEE

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Two typical ones with geometric interpretation are shown in Fig.1. One error (type 1) is given by

4 4 )

=

P ( 4 )

-

a'c(4)l (7)

the other (type 2) is given by

The least squares filter design problem means that the optimal filter weight a is obtained by minimizing the squared errors

J(a) =

JJ,,

le(w)12dw (9) where

R

is the region 0

5

w

5

R, but excuding the transient

band. Now, two types of least squares errors will be investigated in detail. Substitute eq(7) into eq(9), we obtain

J(a) = J J E R ( D ( W )

-

atc(w)Ydw

which is same

as

the least sauares error measure of the conventional one [2].

have

J(a) =

Next, substitute eq(8) into eq(9), we

where a and Q,, are given by

a = [at - lIt

S(w) = [ C ( w y D ( W ) ] t

Qn =

S,,,

S(~)e(~)~dw

(11) Based on Rayleigh's principle, the minimum of J(a) oc- curs a t the eigenvector of the matrix Q,, corresponding t o the minimum eigenvalue. Note that the

Q,,

is also a real, symmetric and positive definite matrix. Since the solution vector a is simply the minimum eigenvector of matrix Q,,, we call this least squares design as new eigenfilter design. In the [6), the least squares error of type 2 is named as total least squares error, so we claim that the new eigenfilter is optimal in total least squares error sense. Now, we summa- rize the design procedure of new eigenfilter as follows:

Step 1 : Compute the matrix Q,,.

Step

2:

Calculate the minimum eigenvector

io

of the matrix

&,,.

Step

3: Normalize the solution vector

&,

t o the form [ad

It is clear that this new eigenfilter does not need to specify the reference point which is a basic requirement of the conventional eigenfilter.

- 11'. The final desired solution an is equal t o h .

2.3 Design Example:

In the example, we will compare the performance of three least squares methods. This example is performed with the

MATLAB Language in an IBM PC compatible computer by using the above design procedures.

Example 1: Lowpass filter design:

Consider the problem of designing a lowpass filter with the following desired amplitude response

0 I W < W p

D ( w )

=

{

0, W s L w < R (12)

don't

l' care, wp

<

w

<

ws

There are four cases of FIR filters with exactly linear phase, but only two of these could be applied to design lowpass filters, that is, case 1 and case 2 [l]. Here, we only consider case 1 filter whose elements of matrix Qt and p are given by

q t ( i , j ) =

s,

cos((2 - l)w)cos((j - 1)w)dw p ( i ) =

l c o s ( ( i

- l)w)D(w)dw 1

5

i, j

5 M

where the region

R

= [O,wp] U [ w . , ~ ] . F i g 2 shows the magnitude responses of three least squares approaches with order N = 32, w p = 0.27r, and ws = 0 . 3 ~ . The reference point W O chosen in the conventional eigenfilter is

y.

From

this result, it is clear that the specification are well satisfied for three least squares methods. However: the stopband at- tenuation of conventional eigenfilter is slightly worse. than the proposed method, because the amplitude response a t the reference frequency W O must be satisfied exactly for the

conventional eigenfilter design. Moreover, the distances between optimal solutions are listed below:

la,f

-

at1 = 0.005109 la,,

-

at1 = 0.000267

where at is the solution of IM method and a,f is the one of conventional eigenfilter. Thus, the solution of the new eigenfdter is closer t o the solution of the conventional least squares solution than the one of the conventional eigenfilter.

3. FIR FILTER DESIGN WITH LINEAR CONSTRAINT

The main advantage of the least squares approach is that various time and frequency constraints can be incorpo- rated. In the linear case, the general form of the constraints can be stated as

where E is a

L

x M matrix and

k

is a L x 1 vector. Note that L is the number of the linear constraints which is usually smaller than the number of coefficients M . Moreover. we assume E is a full rank matrix in order t o avoid redun- dant constraints. For the conventional least squares filter design, the closed-form solution can be obtained by using the well-known Lagrange multiplier method [2]. Moreover! the procedure t o find the solution of the conventional constrained eigenfilter design is slightly complicated, the detail can be found in

[4].

As

to the new eigenfilter design which

is our main focus, the design problem becomes

E a =

k

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kfQ,k Minimize

kt

k

Subject to E a = k

where kt = [at

-

11. The basic idea of solving this problem

is to rewrite the constraint Ea = k as the following form:

E

k

.

0 (14)

(3)

where E = [E

k].

Then, the problem is reduced to

&:“&,a

Minimize

i t a

Subject to E i . 0 (15)

The key step of our method is that “all the vector I sat- isfying constraint Ea = 0 can be expressed as a

=

Bw, where the columns of

B

form an orthonormal basis of the null space of matrx E”. Based on full rank assumption of E, we have that B is a

( M

+

1) x ( M + 1 - L ) matrix and w is a ( M

+

1 - L ) x 1 vector. Due to orthonormal condition, we obtain

BtB

= I, where I is a ( M

+

1

-

L ) x

( M

+

1

-

L )

identity matrix. Thus, the problem described in eq(15) can be simplified as:

w t ~ tBW ~ n Minimize

Wt W

which is an unconstrained optimization problem. Hence, the optimal solution w, of this simplified problem is only the minimum eigenvector of matrix BtQ,B. Finally, the desired optimal solution a, is equal to Bw,. Now, the remaining problem is how to find the orthonormal basis of null space of matrix E. The well-known singular value decomposition (SVD) and

QR

decomposition can help us to solve this problem.

Finally, we summarize the entire procedure of the proposed method as follows:

Step 1: Use SVD or QR decomposition to find the orthonormal basis of null space of matrix E and construct the matrix B.

Step 2: Find the minimum eigenvector w, of matrix B ~ Q , B .

Step 3: Calculate the optimal solution a, = Bw,.

Step

4:

Normalize the solution vector a, to the form

[a: -

lit.

The final desired solution is given by a,. Note that the unconstrained eigenfilter design needs t o find the minimum eigenvector of matrix Qn with size

(A4

+

1) x ( M

+

1). However, new constrained eigenfilter design only requires t o find the minimum eigenvector of matrix BtQnB with size ( M

+

1 -

L )

x ( M

+

1 - L ) .

Thus, constrained eigenfilter has less computation load in searching minimum eigenvector.

Example 2: Notch filter design:

In this example, we will use notch filter design to demon- strate the effectiveness of proposed design algorithm described in the above. The frequency response of an ideal notch- filter has unit gain for all frequency except notch frequency in which gain is zero (21. Here, the case 1 FIR filter of even order N is used to design notch filter. Thus, the re- lation of the parameter in eq(2) is

M

=

$

+

1, trzg(w, n) = cos((n - l ) w ) , a l = h ( $ ) , and

N

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2 2

Now, the optimal filter coefficient a, can be chosen such that the amplitude response A ( w ) is as close as desired response D ( w ) defined by

a , = 2h( - - n

+

1) for n = 2,3, . . .

,

-

+

1

D ( w ) = 1 for all w E [O,.rr) (17) Moreover, to obtain zero gain

at

notch frequency W n and

control the null width, the following constraints are intro-

duced:

A ( w n ) = 0

: L - I

After some maniputation, it can be shown that the linear constraints can be written as standard form Ea = 0. Now, let us see some numerical examples. Fig.J(a) shows the magnitude response of a notch filter using proposed method for

w,

=

0.5.rr,N = 32, and various L . It is clear that the frequency response is well satisfactory. When constraint number L increases, the notch width increases accordingly. For comparison, Fig.3(b) and (c) show the magnitude responses of a notch filter using the well-known Lagrange multiplier and linear programming methods for

N

= 32,wn = 0.57r, and

L

= 1. Basically, the linear programming method is a minimax design and Lagrange multiplier approach is a conventional least squares error design. From this result, it is obvious that proposed method almost has the same frequency response as Lagrange multiplier method. This is due to both are belong to least squares error approaches. Compared with equal-ripple design using linear programming method, the new eigenfilter method has better frequency response fitting for all frequency band es-

cept at the range around the notch frequency w,,.

4. CONCLUSION

In this paper, linear-phase

FIR

filter designs using total least squares error criterion have been investigated. This design method is easy to incorporate linear constraints in time and frequency domain. Several design examples have been used to illustrate the effectiveness of this new design approach. However, only linear phase FIR filters are considered here. Thus, it is interesting to study IIR filter designs based on total least squares error criterion. This topic will be investigated in the future.

REFERENCES

[l]

A.

V. Oppenheim and

R. W.

Shafer, Discrete-time Signal Processing, Prentice-Hall Englewood Cliffs, XJ, 1989.

[2]

M. H.

Er,

”Designing notch filter with controlled null width,” Signal Processing, ~01.24, pp.319-329, Sep. 1991.

[3] M. Okuda, M. Ikehara and S . Takahashi, ”Fast and sta- ble least-squares approach for the design of linear phase FIR filters,” IEEE Trans. Signal Processing, Vo1.46. pp.1485-1493, June, 1998.

(41 T. Q. Nguyen, ”The design of arbitrary FIR digital filters using the eigenfilter method,” IEEE Tkans. Signal Processing, vo1.41, pp.1128-1139, Mar. 1993.

[5]

S. C.

Pei,

C. C.

Tseng and W.

S.

Yang, “FIR filter designs with linear constraints using eigenfilter approach

,”

IEEE Trans. Circuits Syst. 11, vo1.45, pp.232-237. Feb. 1998.

[6]

S.

V. Huffel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis. SIAM, Philadelphia 1991

[7]

S.

V.

Huffel, Recent Advances in Total Least Squares Techniques and Errors-in-variables Modeling. SIAM, Philadelphia 1997.

[8]

D.

Z. Feng, Z Bao and L.

C.

Jiao, ”Total least mean squares algorithm,” IEEE Trans. Signal Processing, vo1.46, pp.2122-2130, Aug. 1998.

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Fig.1 Geometric interpretation of two error measures at frequency w = 4 (a) type 1 error (b) type 2 error.

1 U

2od

-.-_. I. 0.3 0.4 0.5 0.1 0.2 -1201 normalized lrequency

Fig.2 The magnitude response of a lowpass filter. The dashed line and dotted line almost overlap, conventional least squares design (dashed line), conventional eigenfilter design(so1id line), total least squares design (dotted line).

Fig.3(a) The magnitude response of a notch filter using

TL-

S method. L=l(solid line), L=3(dashed line), L=S(dotted line).

l 5 y -

.

ooL

0.1 normi 0.2

0.3 0.4 0.5

I lrequency

Fig.3(b) The magnitude response of a notch filter using La- grange multiplier method for L=l.