Identification of Continuous Systems from Noisy Sampled Input-Output Data

Copyright © IFAC Identification and System Parameter Estimation. Budapest. Hungary 1991

CONTINUOUS-TIME SYSTEMS

IDENTIFICATION OF CONTINUOUS SYSTEMS FROM NOISY SAMPLED INPUT-OUTPUT DATA S. Sagara, Zi-Jiang Yang and K. Wada Department of Electrical Engineering . Faculty of Engineering. Kyushu University. Hakozaki Fukuoka 812. Japan

Abstract. The problem of identification of continuous systems is consid ered when both the disc rete input a nd output measurements are cont aminated by whi te noises. Usin g a pre-designed digi t al filter, a discrete-time estimation model is cons tructed easily with out direct approxim ations of syst em signal deriva tives. If th e pass-band of the filt er is designed so t hat it includes t he main fr equencies of both the syst em input and output signals in some range, the noise effec ts a re sufficiently reduced , accurate estimates can be obtained by least squares (LS) algorithm in the presence of low measurement noises. Two classes of filt ers(infinite impulse response(IIR) filt er and finit e impulse response(FIR) filt er) are employed. The former requires less computational burden and memory than the latter while the latter is suitable for the bias compensated leas t squ ares(BCLS) method , which comp ensates the bias of the LS estimate by the estimates of the input-out put noise variances and thus yi elds unbi ased es timates in th e presence of high noises.

Key words. Cont inuous syst ems; id entificati on; input-out put meas urement noises; di gital fil ter; leas t-squares estimation.

found th at in th e prese nce of in put meas urement noise, it is not appropriate t o let the pass-band of the filt ers mat ch t hat of the continu ous sys tem und er study as sugges ted in some previous works. Our simula tion res ul ts will show that in this case the pass-band of the digit al low-pass filt ers should be chose n such that it includ es t.he main frequ encies of both t.he syste m input and output signals in some range. Since most physical sys tems are low- pass sys tems, we emphasize t hat the selection of th e pass-band of th e pre-filters should be based on the main fr equ encies of the input signals which excite the system modes.

INT ROD UCTI ON Recently, t he d irect a pproaches to continu ous system paramete r es timati on purely usin g di git al co mputers have received much attention due to the rapid development of digit al computers. A maj or difficulty of identificati on of continuous-time models is that the deri vatives of the system input-output signals are not meas ured directly and t he differentiations may accentu ate the noise effects. T herefore a n important problem is how to ha ndl e the ti me deriva ti ves (Unbehau en and Rao, 199 0). From the point of vi ew of using digital co mputers, a unified approach usin g the di gital filtering t echniques to dir ec t recursive ident ification of continu ous systems has been discussed by the authors(Sagara, Yang and Wad a, 1990, 199 1a 199 1b; Ya ng, 1989). T he approach includes the foll owing steps:

T wo cl asses of fil ters(FIR filter and IIR fil te r) are employed . T he IIR filt ers require less comput.ational burden and memory th a n the FIR filt ers whil e the FIR fil ters are suit able for the BCLS method which yields unbi ased es timates in the prese nce of high inpu t-output meas urement noises(Sagara and Wad a, 1977; Wada a nd Eguchi , 1986 ).

1: Find a low-p ass digit al filt er to prefilt er the noise accent uating signal derivatives. 2: Construct a disc rete- time es tim at.i on model wi t h continuous system para meters. 3: Use a rec ursive identifi cation algorit hm to estim ate t he sys tem par ameters from sampl ed in put-output d ata.

STATE MENT OF TH E P ROBLEM Consider th e foll owin g SISO continu ous syste m desc ribed by the ordin ary differential eq uation

In t he case where only th e out put is corrupted by a measurement noise, it is well-kn own th at t he pass-band of t he filters should be chosen such th at it matches t hat of the system un der s tudy as closely(Youn g and Jakeman, 1980) . And wh en onl y th e out put measurement is corrupted by a high measurement noise, the boots trap method gives excell ent results. However , in mos t prac tical situ ations , it may not be possibl e t o avoid th e noise wh en meas uring the input signal. So far , although some works have discussed the id entification of disc rete-time systems in the presence of input noise( Fern a ndo and Nicholson 1985; Sooe rd trom 198 1; Wada and Eguchi 1986), we have not found such works for identificati on of cont.inuous systems.

A(p)x(l )

B (p)u(l)

A(p)

L a,pn-i

n

(ao = 1)

i=O

(1 )

n

B( p)

L bipn- , i= l

where p is differenti al ope ra tor, u(l ) and x(l) are the real in put and the real output. Our goal is to identify th e sys tem para meters from the noisy sampled input -out put da t a

y(k ) w(k)

In thi s work , t he problem of identifi cation of continuous systems is co nsidered when bot h t he discrete inpu t an d out put measurements are cont amin ated by whit.e noises. It will be

x(k) + e(k) u(k) + v(k)

(2)

wh ere k denotes t he sam pling time instant s I = kT for co nveni ence of nota tion, and T is the sampling peri od. v(k) and

603

e(k) are white noises such that E[e(k)] = 0, E[e(k)2] = a; E[v(k)] = 0, E[v(k)2] = E[e(k)v(k)] = 0, E[u(k)v(k)] = 0, E[u(k)e(k)]

a;

of knowledge of the continuous-time or analog filters. When a continuous filter is designed , we can transform it in some manner such as the bilinear transformation to obtain the HR digi tal fil ter.

(3)

=0

In this report , we choose an mth(m ~ n) order Butterworth filter FJ(p):

Since differential operations may accentuate the noise effects, it is necessary here to introduce a digital low-pass filter which would reduce the noise effects sufficiently. Then we can obtain a discrete-time estimation model with continuous system parameters.

FJ(p) = (/)m I p Wc + Cl (/)m P Wc

~ C2 (/ P Wc )m

2 + ... + Cm (10) where c.(i = 1,2"" , m) are the coefficients which let the Butterworth filter has the basic form for the 'amplitudesquared function' as

DISCRETE-TIME ESTIMATION MODELS In this section, we describe the design techniques of the two classes of digital filters and the discrete-time estimation models derived by the pre-designed filters(Sagara, Yang and Wada, 1991b) .

(11) and

Wc

is the cut-off frequency for which

1W

I~

(12)

Wc

FIR filtering approach Multiplying both sides of the system (1) by the pre-designed FJ(p), we have

It is known that the differential operator in (1) can be replaced by the bilinear transformation:

n

FJ(p)pnx(t)

+ I: a.FJ(p)pn-'x(t)

n

=

i=l

I: b.FJ(p)pn-'u(t) i=1

(13) Discretizing it by the bilinear transformation and using (2), we obtain

Introduce a low-pass digital filter F(Z-l) as

F(Z-l) = QF(z-I)(~)" where QF(Z-l) is a kind of FIR filter.

1W I::; Wdc otherwise

rJ(k) =

(6)

I: qmz-m

I: a'~F'y(k) = I: b'~F'w(k) + rdk)

(7)

(8)

QF(Z-I)(f),(l + z-I)'(I- z-l)n-'f(k) 2M+n

I:

fjz-J f(k)

j=O

n

i=O

i=1

I:ab.e(k) - I:b'~F'v(k) 2M+n

I:

crjz-je(k) -

j=O crj =

i=O

I:

j=O

n

I: fja. ,

2M+n

/3J

=

n

i=O

i=1

z-lt-'f(k)

Both types of digital filters are useful. The HR filters have simple design methods using the bilinear transformation. And the HR filters can produce desired amplitude response with significantly few coefficients than nonrecursive FIR filters. Therefore, usually, the HR filters are more convenient for the LS or the bootstrap algorithms. However, as shown later, the FIR filters are suitable for the so-called BCLS method which yields unbiased estimates in the presence of high input-output measurement noises. It has been shown by the authors that some distinct methods such as the integral equation(Whitfield and Messali , 1989) , state variable fiIters(Young and Jakeman, 1980) and the integral operations over finite interval(Sagara and Zhao, 1990j Schoukens, 1990) can be viewed as either the HR or the FIR filtering approach. Detailed discussions are given in our previous works(Sagara, Yang and Wada, 1990, 1991a 1991b j Yang, 1989).

where

n

n

I:a.6'e(k) - I:b.6'v(k )

I:c. (__

Multiplying both sides of (4) by the pre-designed F(Z-l) and using (2), we have

i:;::)

(14)

T (_)m-n(1 + z-l)m-n Q (Z-l) _ 2 J -1- z l m I_ZI . T . ( _ _ )m+ )m-'(_)'(1+z-I), Wc 1=1 Wc 2 ( 15)

m:;O

i=O

1=1

2

2M

n

i=O

~Iif(k) = QJ(Z-I)(~)'(1 + z-I)'(1 -

And QF(Z-l) can be obtained by the Hamming window technique with an appropriate length 2M(Roberts and Mullis 1987j Sagara, Yang and Wada 1991b):

n

n

where

Many types of FIR digital filters can be applied to QF(z-I). For simplicity, we may choose QF( Z-l) as a desired ideal lowpass filter which has the specification:

QF(Z-l) =

n

I: a'~Iiy(k) = I: b'~J'w(k) + rJ(k)

(5)

(9)

LS METHOD

/3jz- j v(k)

When the digital low-pass filters have been designed, we have the discrete-time estimation model of (8) for the FIR filtering approach , or the model of (14) for the HR filtering approach. Both can be written in vector form:

n

I: fib. i=l

zT(k)l1

+ r(k)

HR filtering approach

[-~ly(k),·

There are many design methods for HR digital filters. One of the most popular formulations is to use the large body

[aJ, " ', an, 6)1 "', bn ]

."

-~ny(k) , 6w(k),·

."

~nw(k)]

(16)

604

where

For the HR filtering approach, it is difficult to express the bias explicitly by and since usually expressions of the correlations of the outputs of the HR filters are not so simple. However, it should be noted that if we approximate the HR filter QJ(Z-l) by an FIR filter with a sufficiently large length, very similar results can be obtained for the HR fil tering approach. In this report, we restrict our discussions on the FIR filtering approach.

0';

= ~F;y(k),~;w(k) = ~F;w(k), r(k) = rF(k) (FIR filter) ~;y(k) = 6;y(k) , ~;w(k) = ~J;w(k),r(k) = rJ(k) (HR filter)

~;y(k)

(17) We can estimate the continuous system parameters by the following LS method: ~+N

{j = [

~+N

L

L

Z(k)ZT(kWl. [

k=ko+l

z(k)~oy(k)]

(18)

0';,

BCLS method

k=ko+1

From (21) and (23), we have When the noise effects can not be neglected, it is well-known that the LS estimate is asymptotically biased in general. For the case where both the input and output signals are corrupted by low measurement noises, if the pass-band of the pre-designed digital low-pass filter includes both the system input and output signals in some range, the LS method is still efficient in the presence of low measurement noises. When the discrete input-output measurements are corrupted by high white noises, we will extend the BCLS method which was first proposed by Wada and Eguchi( 1986) for discrete-time systems, to the case of continuous systems.

11 = plim B N-oo

0';

B(N) = B(N - 1) + L(N)[~FOy(N) - zT(N)B(N - 1)] L(N) _ P(N - l)z(N) - p(N) + zT(N)P(N - l)z(N) P(N) = _1_[P(N _ 1) _ P(N - l)z(N)zT(N)P(N - 1)] p(N) p(N) + zT(N)P(N - l)z(N) (28) where p( N) is forgetting factor and is chosen to be

Expression of the bias Consider the discrete model derived by the FIR filter:

p(N)

+ rF(k)

[-~Fly(k) , . .. '-~Fny(k), ~F1w(k),· .. ,~Fnw(k)] [al," " an, bl ," ', bn] = [aT, b T]

(19)

Z(k)ZT(kW l . [

L

k=ko+l

~+N

L

Z(k)~FOy(k)]

Estimation of

O.OI)p(N - 1) + 0.01

p(O)

= 0.95

(29)

0'; and 0';.

We

0'; and 0';

The equation error TF(k) for the LS estimate B(N) is (20)

k=ko+l

(30)

It can be shown that

11

= (1 -

The BCLS method requires the estimates of will show the method to find iJ; and iJ;.

The LS estimate is ~+N

0';,

and B(N), P(N) are obtained by

In this section, we extend the BCLS method to the problem of identification of continuous systems in the presence of inputoutput measurement noises using the FIR filtering approach .

e= [

(26)

which implies that an unbiased estimate of the unknown parameters can be obtained by substracting an estimate of the bias. If we have the estimates of and the BCLS algorithm is given by

BCLS METHOD FOR THE FIR FILTERING APPROACH

zT(k)11

+ N P(N)DH(I1)

Using (19), we have 1

ko+N

N

k=ko+l

+ N P(N)[plim N-oo

L

z(kJrF(k)]

(21)

(31) From (20) and (30), we have

where

ko+N

P(N) = [plim

L

Z(k)ZT(kW l

ko+N

(22)

L

N -+00 k~ko + 1

1 plim /i N-+ oo

z(kJrF(k) = -DH(B)

(23) ko+N

k:::k o +l

L

g(N) =

where

fF(k)fdk)

k=ko+l ko+N

H(I1) = [he(a) , hV(bW he(a) = [hHa),· . " h~(a)J, hV(b) = [h~(b) , 2Af+n

hi(a)

=

L

(32)

Using (31) and (32), we have the sum of squared residuals:

ko+N

L

Z(k)TF(k) = 0

k=ko+l

With straight calculations, we have

L

k=~+l

···, h~(b)]

h;'(b)

=

L

+

L

zT(kJrF(k)(11 - B(N))

k=ko+l

(33)

2M+n

JjCtj(a) ,

ko+N

r}(k)

Since

Jjf3) (b)

1 plim /ir}(k)

(24) and the 2n x 2n matrix D is of the form

N-+oo

~~

= E[r}(k)] = L

r=O

Ct~(a)O';

+

~~

L

f3~(b)O'~

)=0

(34) and

(25) where In, On are an identity matrix and a zero matrix of order n respectively.

605

where

then the following result can be obtained: .

g(N)

N-oo

N

2M+n

2M+n

2M+n

L ;_0

p I lm--

a;(a)o-;

+

L j_O

L

f](b)o-;

-he(aBC(N - 1» [aBc{N - 1) - a(N)]

-he(a)[a - a(N)]o-; - hV(b)[b - b(N)]o-; (36)

2M+n

L

Similar to the above discussions, we define the instrumental variable estimte e(N) by ~+N

e(N) = [

L

Z(k)ZT(k - LW I

L

. [

f; (bBC(N - 1»

)=0

-hV{bBc{N - 1» [bBc{N - 1) - b(N)] 2M+n-L a;(aBc{N - 1» a;+L{aBc(N - 1))

L j_O

~+N

k-ko+1

a;(iIBc(N - 1»

;-0

Z(k)~FOy(k - L)]

(45)

-iie(aBC{N - 1) [aBc{N - 1) - a(N)]

k-ko+1

(37)

2Af+n-L

L

where L is a natural number. The recursive form is

f;(bBC(N - 1) f)+L(bBc(N - 1»

;=0

e(N) = B(N - 1) + L(N)[~FOy(N - L) - zT{N - L)B(N - 1)] P{N - l)z(N) L(N) = p(N) + zT(N - L)P(N - l)z(N) P(N) = _1_[P(N _ 1) _ P(N - l)z{N)zT(N - L)P{N - 1)] p(N) p{N) + zT(N - L)P(N - l)z{N)

-iiV(bBc{N - 1» [bBc(N - 1) - b(N)] It should be noted that the delay L should be chosen such that A and [L:Z~~~+I Z(k)ZT(k - L)] are nonsingular. To our experiences, the results are not so sensitive to L. Now we will consider the methods to calculate g(N) and f(N). It can be shown that

(38) The equation error for B(N) is defined by

kf [-~FOy(k)] [-~Foy(k), zT(k)] [ • z(k) I1(N) = kf [-~FOy(k)] [-rF(k)] = [ g(N) ] z(k) 1

]

k-ko+1

ko+N

f'F{k) = zT(k)[I1- B(N)]

k_~+1

+ rF{k)

ko+N

L

0

k-ko+1

and it can also be shown that

(40)

z(k)rF{k - L) = 0

[

-~Foy(k) ] T z(k) [-~Foy(k - L), z (k - L)]

[1] B(N)

[-~FOy(k)]

]

kf

=

= [ f(N) 0

(46)

Hence we have

Hence we can express g(N), f(N) as ko+N

L

f(N)

[-rF(k - L)]

z(k)

k-ko+l

k-ko+1

ko+N

f'F{k)i'F{k - L)

g(N) =

k-ko+1 ko+N

L

rF(k)rF(k - L)

~~oy(k) -

k=ko+l ko+N

(41)

f(N) =

k-ko+1 ko+N

L

ko+N

L

~Foy(k)zT(k)e(N)

k=k o +l

~Foy(k)~FOy(k - L)

(47)

k=k o +l

L

+

L

ko+N

- L

zT(k - L)rF{k)(l1- B{N»

k-ko+1

~Foy(k)zT(k - L)B(N)

k=ko+1

and thus Based on the above discussions , we can summarize the on-line BCLS algorithm as:

2M+n-L

L

plim f(N) N-oo N

aj{a)aj+L{a)o-;

1: by 2: 3: 4: 5:

;-0

2M+n-L

+

L

fj{b)fj+L(b)o-;

;_0

-iie{a)[a - a(N)]o-; - iiV{b)[b - b{N)]o-; (42) where

iie(a) = [h~(a), ... , h~(a)J, 2M+n-L hi{a) = L fjaj+L(a),

i(l) + alx(i) + a2x(i) = blu(i) + b2u(i) al = 3.0, a2 = 4.0, bl = 0.0, b2 = 4.0

0-; and 0-; are

Then the estimates of the unknown variances given by the solution of the following equation:

=

A

[o-~] = ~ [ g(N) o-v

N

f(N)

]

er;.

Consider a second-order system described by

j-O

( 43)

all a12] [ o-~ ] [ a21 a22 o-v

er;

ILLUSTRATIVE EXAMPLES

iiV(b) = [h~(b) , . . " h~(b)] 2M+n-L h;'{b) = L fjfj+L{b)

;-0

Calculate the LS estimate e(N) and the IV estimate B(N) (28) and (38) respectively. Calculate g(N) and f(N) by (47). and Solve (44) to have the variance estimates Compensate the bias of the LS estimate e(N) by (27). Return to 1 untill convergence

(48)

The input uti) is the output of a second-order continuoustime Butterworth input filter driven by a stationary random signal ((i):

(44)

u(l)

= L(p)((i) = (p/wy + h(p/wc ) + 1 ((i),

Wc

= 4.0 (49)

The sampling interval is taken to be T = 0.04, and in this case, o-u = 2.38,o-x = 0.7. The LS est.imates for ·the case of

606

low noises where a v = 0.24 ,a e = 0.07(N/S ratio~ 0.1) are shown in Table 1 for the FIR filters(M = 25), and Table 2 for the HR filters(m = 2). In Table 1, Wac denotes the actual cut-off frequency of the FIR filter F(Z-I) defined in equation(5) which lets 1F(w) 12::; 1/2, for 1W 12 Wac' Each of the tables includes the mean and standard deviation of the estimates obtained from Monte-Carlo simulation of 20 experiments. 10000 samples are taken for each experiment. And in each table, t.11811 = 118 - 811. The frequency responses of the system, the digital pre-filters used in Tables 1",2 and the input filter L(p) ill (49) are shown in Figs.1",2. It is clear that accurate estimates can be obtained if the pass-band of the pre-filters includes that of the low-pass input filt er L(p) in eq uation( 49) in some range. Therefore, for the case of low input-output measurement noises, if the pass-band of t.he digital low-pass filters is chosen such that it includes th e main frequencies of the input signa ls in some range which excit e the system modes, the noise effects are sufficiently reduced, and

thus the LS estimates are still acceptable. For the case where only the output is corrupted by a measurement noise, it is known that the pass-band of the filt ers should be chosen such that it matches that of the system under study as closely. This suggestion is not appropriate in the presence of input measurement noise. The LS estimates and the BCLS estimates(L = 5) for the FIR filters are shown in Table 3 and Table 4 when a v = 0.60, a. = 0.17(N/S ratio~ 0.25). In the presence of high input-output measurement noises, it is difficult to obtain accurate estimates with the LS method. However, the BCLS method is very efficient in this case. Figs.3",4 plot the LS estimates and the BCLS estimates of one experiment(when Wdc = 7.0). For short samples, it is clear that the BCLS estimates are more sensistive than th e LS estimates. However, when sufficiently long samples are taken, the BCLS estimates converge to their true values in a very accurat.e mann er. CONCLUSION In this report , the digital filtering approach to recursive identificat.ion of continuous systems from noisy sampled inputoutput data have been discussed. Using a pre-designed digitallow-pass filt er, a discrete-time estimation model with continuous system parameters is constructed easily. And it was emphasized that in the presence of input measurement noise, the pass-band of the filters should be chosen such that it includes the main frequ encies of the real system input-output signals in some range to reduce the noise effects. Two classes of filt ers(FIR filter and HR filter) have been applied. Both classes of filters have excellent noise reduc·· ing effects. Usually, the IIR filters require less computationai burden and memory than the FIR filters and are more convenient. However , it has been shown that for the discrete-time model derived by the FIR filters, the bias of the LS estimates can be exp ressed explicitly by a~ and a~. And it was shown by examples that the BCLS method combined with the FIR filtering approach yields unbiased estimates in the presence of high measurement noises. For the HR filtering approach, it should be mentioned that if we approximate the HR filter by an FIR filter with a large length , the bias of the LS estimates can also be expressed explicitly by a~ and a~ in a similar way. REFERENCES Fernando, K. V. and H. Nicholson (1985). Identification of linear systems with input and output noise: the KoopmansLevin meth od. Proe. lEE , 132, Pt.D., 30-36. Roberts, R.A. and C.T. Mullis (1987). Digital Signal Processing. Addision- Wesley Publishing Company.

607

Sagara,S. and K.Wada (1977). On-line modified least-squares parameter estimation of linear discrete dynamic systems. Int.] .Control, 25, 329-345. Sagara, S. , Z.J. Yang , and K. Wada (1990). Consistent parameter estimation for continuous systems via IIR digital filters. Trans.Soe.Instrum.Contr.Eng., 26, 39-45, in Japanese. Sagara., S., Z.J. Yang, and K. Wada (1991a) . Recursive identification algorithms for continuous systems using an adaptive procedure. Int.J.Control(to appear) . Sagara, S., Z.J. Yang, and K. Wada (1991b) . Identification of continuous systems using digital low-pass filters. Int.].Systems Sci.(to appear). Sagara, S., and Z.y. Zhao (1990). Numerical integration approach to on-line identification of continuous-time systems. Automatica, 26, 63-74. Schoukens, J. (1990). Modeling of Continuous time systems using a discrete time representat.ion . Automatica, 26, 5790-583. Siiderstriim, T. (1981). Identification of stochastic linear systems in presence of input noise. Aut.omatica, 11, 713725. Unbehauen, H. and G.P. Rao (1990). Cont.inuous-time approaches to system identification-a survey. Automatica, 26 23-35. Whitfi eld, A.H. and N. Messali (1987). Integral-equation approach to system identification. Int .J.Control, 45,14311445. Wada, K., and M. Eguchi (1986). Estimation of pulse transfer function model via bias compensated least-squares method in the presence of input and output noises . SICE 9-th Dynamical System Theory Symposium, Kumamoto, 315-318, in Japanese. Yang, Z.J. (1989). Digital Filtering Approach to Parameter Estimation of Continuous Systems, Master's thesis, Department of Electrical Engineering, Faculty of Engineering, Kyushu University, in Japanese. Young, P. and A. Jakeman (1980) . Refined instrumental variable methods of recursive time-series analysis. Part IH. Ext.ensions. JnU .ControJ, 31, 741-704. TABLE 1: LS estimates(FIR filter, N/S ratio~ 0.1)

bl ClI Cl2 Wde (wae ) (3.0) (4.0) (0.0) 14.0 2.892 4.089 0.009 (12.47) ±0.041 ±0.030 ±0.006 12.0 2.918 4.052 0.009 (10.57) ±0.037 ±0.027 ±0.006 10.0 2.930 4.020 0.010 (8.62) ±0.030 ±0.025 ±0.005 8.0 2.934 3.993 0.011 (6.69) ±0.025 ±0.023 ±0.004 7.0 2.934 3.980 0.013 (5.69) ±0.024 ±0.023 ±0.004 5.0 2.920 3.945 0.018 (3.78) ±0.024 ±0.023 ±0.005 4.0 2.9018 3.914 0.025 (3.01) ±0.026 ±0.0245 ±0.006

b2 (4 .0) 4.012 ±0.034 4.004 ±0.030 3.989 ±0.025 3.972 ±0.021 3.963 ±0.020 3.933 ±0.022 3.904 ±0.026

~1I811

0.141 0.097 0.074 0.073 0.080 0.120 0.163

TABLE 2: LS estimates(IIR filter, N/S ratio~ 0.1)

a} We

10.0 8.0 6.0 4.0 3.0 2.0

(3.0) 2.857 ±0.039 2.906 ±0.035 2.936 ±0.030 2.944 ±0.024 2.930 ±0.024 2.875 ±0.030

.12 (4 .0) 4.100 ±0.029 4.050 ±0.027 4.013 ±0.025 3.979 ±0 .023 3.950 ±0.022 3.866 ±0.024

b} (0.0) 0.010 ±0.006 0.009 ±0.005 0.009 ±0.004 0.011 ±0.004 0.015 ±0.004 0.030 ±0.005

b2 (4.0) 3.988 ±0.030 3.990 ±0.026 3.984 ±0.022 3.965 ±0.019 3.938 ±0.020 3.852 ±0.027

1. 2 ---- . digital filter ~lIell

O. 8 0.176

<:: ';;l

Cl

0.107

O. 4

0.068

8. 0

o.

~~~~~~~~:~;:20. 0

10. 0

Frequency(w) Figure 1. Frequency responses of the FIR filters

0.070 1. 2

0.108 0.238

O. 8 <:: ';;l

Cl

O. 4 TABLE 3: LS estimates(FIR filter, N/S ratio~ 0.25) Wde (Wo e )

12.0 (10.57) 10.0 (8.62) 7.0 (5.69) 5.0 (3.78) 4.0 (3.01)

a} (3.0) 2.363 ±0.072 2.477 ±0.062 2.570 ±0.051 2.536 ±0.049 2.471 ±0.051

O. .12 (4.0) 4.219 ±0 .063 4.042 ±0.057 3.840 ±0.053 3.672 ±0.053 3.545 ±0.053

b} (0.0) 0.041 ±0.0131 0.045 ±0.01l 0.056 ±0.01O 0.083 ±0.012 0.110 ±0.012

b2 (4.0) 3.703 ±0.067 3.697 ±0 .056 3.649 ±0.043 3.535 ±0.044 3.420 ±0.050

8. 0

10 . 0

20 . 0

Frequency(w) Figure 2. Frequency responses of the IIR filters

~lIell

0.737

5.

0.608 0.581

~-------------------------------------------0.2

3.

¥-;----------~~-------------------------------

0.738

1.

0.914

b2 /2

........

h} - 1.0 •c.. . . . . . . . .o. . . :-1"";:-0~O.~---;:-2-::-0~O.-. . . . -. . --:: -3"-::-0':::-0-...............,4~0 0 • TABLE 4: BCLS estimates(N /S Wde ( W oe )

12.0 (10.57) 10.0 (8.62) 7.0 (5.69) 5.0 (3.78) 4.0 (3.01)

a} (3.0) 3.076 ±0.096 3.052 ±0 .077 3.034 ±0.061 3.037 ±0.062 3.039 ±0 .070

.12 (4.0) 4.029 ±0 .072 4.024 ±0.066 4.023 ±0.070 4.030 ±0.061 4.034 ±0.066

b} (0.0) 0.001 ±0.015 0.002 ±0.012 0.001 ±0.010 -0.000 ±0.013 -0.001 ±0.016

ratio~

TIME (SEC) Figure 3. LS estimates(N /S ratio~ 0.25)

0.25)

b2 (4.0) 4.093 ±0.076 4.071 ±0.060 4.052 ±0.051 4.057 ±0.060 4.061 ±0.072

5. ~-----

~lIell

3.

a2

~

a}

/!'::.

0.124

b2 /2

1. 0.091

-1.0.

0.067

J

100.

200.

b} 300.

400.

TIME (SEC) Figure 4. BCLS estimates(N /S ratio~ 0.25)

0.074 0.080

608