Genetic Algorithm Based Mixed Spectrum Estimation for Discrimination of Sinusoids and ARMA Noise

3a-1O 5

Copyright © 1996IFAC 13th Triennial World Cong ress. San Fran(:is(:o. USA

GENETIC ALGORITHM BASED MIXED SPECTRUM ESTIMATION FOR DISCRIMINATION OF SINUSOIDS AND ARMA NOISE Y. Ashida, H. Ohmori and A. Sano Departmeflt of System Design Engineering. Keio University, 3-14-1 Hiyoshi, Kohokll-ku, Yokohama 223. Japan

Abstract: This paper is concerned with a method for estimating a mixed spectrum which is composed of multiple sinusoids and an ARMA noise with continuous spectrum. In order to aVQ,id simultaneous minimization of a prediction error criterion with respect to all unknown model parameters, we give an efficient iterative algorithm for estimating the frequencies of the sinusoids and other parameters separately. By adopting the genetic algorithm to choose initial values of the ARMA parameters in the iterative estimation, we can attain a globally optimal estimates of unknown parameters. For executing the proposed hybrid genetic algorithm. a modification of the Toeplitz approximation method is given to estimate the frequencies in [he presence of the MA noise. by using a shifted correlation matrix of thc filtered observed data. The effectiveness of the proposed algorithm is validated in numerical simulations using the beneh mark test data. Keywords: Genetic algorithm, spectral analysis, signal detection, parameter estimation. !-inusoids.

I. INTRODUCTION Estimation of mixed spec trum which is composed of line and continuous spectra plays an important role in wide area of digital signal processing and system identification. Line spec tra can be expressed by a sum of multiple sinusoids. while continuous spec trum is characterii.cd by an AR, MA or ARMA noise modeL This problem was first investigated by using the periodgram (Priestley 198 I) via a non~parametric approach. From a parametric point of view. AR. MA and ARMA spectral eSlimation methods are applicable (Kay 1988), how~ver, wc cannot discriminate the si nusoids from the continuous spectral signal hy using only the nonparametric or parametric approach. On the other hand. for estimation of line spectra only, we can apply the maximum likelihood (ML) method (Haykin 1991 , 1995) and the signal or noise subspace approaches (Haykin 1995) such as the Pisarenko's melhod (Kay 1988), Ihe TO
These problems can be fonnulatcd as a mixed spectrum eSlim'lion (Challerjcc et al. 1987 ; Kay et al. 1994; Tsuji et al. 1989; SanD er al. 1985; Yu et al. 1987). The ML approaches normally search all unknown parameters simultaneously so thal the prediction error criterion may be minimized (Chatterjee el.1. 1987; Kay el al. 1994). However. the simultaneous s~arch needs the minimi zation wilh respect to too many parameters and it suffers from [raps into local minima. Onc of the purpose of Ihis paper is to clarify a high-resolulion method forthe mixed spectrum estimation incorporating with a genelic algorithm (GA) (Goldberg 1989). If an ARMA model is given as the co ntinuous spectrum part , we can implemenl a lilter with the ARMA paramclcn; and then apply estimation techniques to the filtered output of the observed signal to estimate the frequenci<..~s of sinusoids. The Toeplitz approximation method is a noi se~suppressed frequency estimation melhod based on {he subspace approach. However, it cannot be directly applied to th<.! filtered signal, because Ihe signal contains a MA noise pan. Therefore in this paper an extension of the TA method is given to overcome this problem by using its shifted submatrix of the correlation matrix of the filtered signals. This implies that the total estimation problem can be simplified to an iterative search of only the ARMA parameters minimizing a prediction errur criterion. 'The featufC of the proposed method is that a genetic aJgorilhm

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is efficiently utilized in the choice of initial values of the ARMA parameters in the iterative prediction error method. nut in a direcl search o f the ARMA parameters adopted in usual approaches (Kristinsson et al. 1992; Hunt 1993). which can save computational burden and improve the precision of the parameter estimates. Detennination of thenumber of sinusoids and the order of ARMA noise model is also discussed. Finally the ef'fectiveness of the proposed algorithm is investigated by applying it to a bench mark test data. 2. SIGNAL MODEL WITH MIXED SPECRUM 2.1 Prublem statement It is assumed that the observed signal y(k) is composed of multiple sinusoids and autoregressive moving-averaged (ARMA) noise process as M

L am sin(2nfmt + (/Jm) + o(t)

y(t) =

(I)

,.. ..,1 I la

u(1) -

L

n,

a,v(t - i ) ~ w(1 ) +

1= 1

L

bjw(1 - j)

(2)

jal

where fm.. am, and IPm are the frequt!n cy. amplitude and phase of each sinusoid respectively, lad and fbjl are the coefficient parameters of the ARMA nui se process which is actuated by a while Gaussian noise l.L!(t ) with 7.e-ro mean and variance a,~, M is the number of sinusoids and na and nb is the ord er of the AR and MA pan respec tively. These model parameters If",) , laml, {IP", I, la, I. {bj }, (Y~, M, na and are all unknown .

n"

2.2 ParametrizatiOfI of signal rum/et

Let the polynomials A(Z-I) and defined by A(Z-I)

= 1-

Ia.z-', .

8(z-l)

be stable an d be

8(z - l )

~ 1+ Ib,z-'

!,=I

(3)

'", 1

Multiplying A(Z-I ) with the both s ides of ( I ) gives

3. SEPARABLE ESTIMATION ALGORITHM 3. J OUlline of hyhrid genetic algorithm based estima tion method

Direct parameter estimation schemes like the ML estimation are based on the signal model (4a) in which all the unknown parameters la,!, Ib, I, lemI, Idm ' and Ifm , appear explicitly. Thus the scheme requires a simultaneous optimization with respe ct LO the all many parameters and then it sometimes suffers from traps into local minima. and especially the resolution and precision of the frequency estimates tend to be degraded. An outline o f the proposed scheme is summarized as follows: First if the ARMA parameters are given a priori. we can estimate the frequencies by applying a modified Tocplitz appro ximat ion method to the liltered observed signal (4b). as will he clarified late r (the tirst prohlem). Then, secondly if the ARMA parameters and the frequencies are given a priori. we can estimate the parameters {ern I and IdIOt I of the signal model (5) by utilizing the ordinary least squares (LS ) cstimati on method, and then calculate the amplitudes and phases of si nusoids from all of Ihe given or estimated parameters (the second problem). Finally. conversely. if the frequencies and amplitudes of sinusoids are g iven a priori. the ARMA parameters can be estimated by using a proposed hybrid algorithm incorporating the ge netic algorithm with the prediction error method (lhe third problem). Thus the proposed algorithm is constructed by the three basic procedures to si mplify the simu ltaneous optimization for the total estimalion problem. 3.2 £ftimatiorl of frequencies by extension of lire Toeplirz.

approximation method The first problem is how to estimate the frequencies of sinusoid~ in the signal representation (4b). if the ARMA parameters are given correctly. We first obtain the filtered output signal z(t) whit:h is composed of the sinusoids and the MA noise. as z(t) = A(z-l)y(l)

A(z -I)y(t) ~ A(z-I) Lam sin(21(,. + ~m) + B(Z-I)W(I) rti: 1

lit

M

M

m_ I

In", 1

~ L e" sin(2nf,,, tJ+ Ldm cos(2I(ml ) +B(z-I)w(t)

J.f

= L em s in(2I(ml) + L d .. c"s(2nf~, I) +B(Z- I)w(t) (4a)

(4b)

Cm and d", depend on the unknown parameters lam I, la,!, Ifm , and Ib, I.

Unlike a while noi se ca<;e with B(Z-I ) = 1, we cannot apply the ordinary Toeplitz approximation method (Kung et af. 1983) directly to estimate the frequencies {fm J of the sinusoids in (4b) . So, we should co nsider an extension of the Toeplitz approx.imation method to a case in which the MA noise is present As the result, the algorithm is summarized as foll ows. As for the proof, see (Ashida et at. 1994).

Further, dividing the both si des of (4bl by B(z-I), we can obtain an alternative model description as

Step J: Calculate the correlation matrix R" with the overdetennined size (p + q) x (p + q) for the tiltered output z(l) as

11'1 - 1

m", 1

z(l) ~ A(Z - I)y(l) =

M

M

m=1

m. _1

Lem sin(2I(m l ) + L d m c08(21(,,,I) (4b)

+8(z-I)W(tI

where

1 1/(1) = - B I z(l) = {z - )

L cm lot

+

LM d lti'd

1

- - I sin

RI" I

B (z- )

1

r, (0)

21(m l

m - - 1 - eos

R = 2Trf,n t + wCt)

(5)

:r

B(z -)

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r,(1)

[

rCl(Q+:p-l)

r,(-l} .. r,(-q - p + l}] r,(O) :: r,(-q~p+2) . .

r,(O)

where p and q are chosen so that q > flb + p and p 2: 2M arc satisfied.

can also calculate the actual amplitudes and phase of the sinusoids as

Step 2: Pick up the lower left block mal
I)]

rz (q+1)

r,(q - P+ r,(q -p + 2)

r,(q+p-ll

r,(q)

~

R, (q ) =

,

aM =

(6)

Cm

(10)

.

fJcostpm -rsm~m

respectively. where

[

{J = 1-

t

ah

r = ta, sin(2nfm k ) h_'

cos(2nfm k ) ,

k=l

Step 3: Take a manipulation matrix 8 dclined by

8=[~o .. 1 0

3.4 Estimation of ARMA parameters

o

1 1 0

(7)

o

The final problem is how to estimate the ARMA parameters {ad and {h j } if the frequencies and amplitudes of sinusoids are given a priori. In order to execute the estimation. we use the signal representation (4a) and rewrite it by A (Z - ' )y(t) = x(1) + B(Z-I)w(t)

and calculate the product of R,(q) and 8 as R,(q)8 which becomes a Hankel matrix.

Step 4: Take the singular value decomposition of the matrix R,(q)8

as

x(t) =

Vr

Step 5: Takc out thc matrix tP, = U I LF2 from UII, V,r and then calculate a system matrix F as

1n= 1

m=J

LemHin(2nfmt} + Ld", cos(2n{mt)

(I la) (lIb)

e = (at. .... all-.,blo ... ,b,4)T and the prediction error eet,9) hy e(t,9) = yet) -

y(e IB)

where the predicted output y(1 IB) =

(12)

yU IB) of a model is given by

[1- B(zA(z -II)}(t) +_1_1 x(l) ) B(z - )

(13)

It is followed from (13) that B(z-I )y(1 IB) = [8(, - ') - A (Z - I)Iy(t) + x(l)

(9)

cP/

M

Let the ARMA parameter vector be denoted by

(8) where the singular values appear in the diagonal matrix I. The Tank should become 2M if M is the number of sinusoids. Therefore by only retaining 2M singular values in El> the matrix R,(q)S can be approximated by the matrix UtI I with rank 2M.

.,

( 14)

where is cb t deprived from its lasl row, and 4>1 T is 41, deprived from its first row.

Adding y(tI 9) and y(t) to the hoth side of (14) and rewriting it gives Ihat

Step 6: Calculate the frequency eslimales f". = (j)".J2rr from

j(t 18) ~ j(t 19) - B(Z-I)j(t 19) + [B(Z -I) - A(z - I)[y(t) +x(t) + y(t) - y(t) = (1- A(z-I ))y(t) + x(t) - (1- B (Z - I »e(t, 9) (I Sa) = rpTU,B)IJ + x (t) (l5b)

the eigenvalucs of F . which arc given hy exp(±j(.(.Im) .

The ahove algorithm makes use of the property of the MA process that the aUlocorrclation functi ons of the MA noise become 1.ero for lags q larger than nb. The ordinary Toepliu approximation algorithm can use the eigenvalue decomposj~ lion of the corre lation matrix Rl with the size p x p , while the proposed extended method needs the singular value d e~ composition for the submatrix Rz(q)S taken out of the over~ determined correlation matrix Rz with size (p + q) x (p + q), Thus, it is clarified that the frequency estimation of sinusoids can be executed if the ARMA parameters arc given a pri ori. 3.3 Esrif7Ultioll of amplitudes of sjnu.'iOid~·

The second problem is how to estimate the ampliludes ICm I and {dm } of the sinusoids in the signal representation (5) if the frequencies and the ARMA pardmelers are given correctly. TI,e least squares cSlimales for fe,It} and {d", I can be easily obtained because the ampliludes are linear parameters in (5) and the regressor signals arc the sinusoids filtered through I/B(z - '). Thus oy using the eSlimates o f \cm I and Id.. I, we

where rp(t,9) is the regressor signal defined by ",T(I,B) = (y(t -ll, ···,y(t -n,),e(t -I,B), .. ,ett - nb,9». We take aprediction error criterion for the parameter estimation as 1

J(9) = N

L {y(t) - yet I B)} N

..

(16)

t=' Applying a Newton type of non linear optimization algorithm to the minimization of (16). wc obtain the estimate of 9 in an iterative manner as

0,,·1, = 8'" - jl[J" (9"'W'r (0"')

(17)

where the derivatives of j(1 19) with respect to [he ARMA paramerers which arc nceded in the minimi zation in (17) are given hy 1 ( . -Jy"(l I B)=--yk-
4239

(18a)

...i.. yet 10) = _l_E(k ibj

j , O)

B (z)

( ISb)

Thus, instead of simultaneous minimi zation with respect to all the paramclcrs, the estimation algorithm is reduced to only the iterative search of the ARMA parameters. However. this searching scheme still suffers from local minima. Therefore , a genetic algorithm is efficiently introduced into the choice of the starting values of the ARMA parameters in the iterative search to obtain the global minimum, as will be described in the next sCL:tion. 4. HYBRID GENETIC ALGORITHM FOR ARMA MODEL ESTIMATION

4.1 Genetic Algorithm One of d istinctive feature of the proposed approach is that the iterative search of the minimum of the prediction error is executed by incorporating a genetic algorithm for the choice of the iniLial values of the ARM A paramelers into the iterative prediction error method , In this meaning. the proposed approach can he categorized to a class of hybrid genetic algo rithm, The basic clement processed by a genetic algorithm (GA) is the string (genes) formed by concatenating substri ngs, each of which is a binary coding of a parameter of (he search space. Eac h s tring represe nts a possible solution to the optimization problem. The genetic algorithm work s with a set of strings, not a singl e string. whi ch is ca1led the population, nlis population then evolves from generation to generation th rough the application of three bas ic genetic operators: reproduction, crossover and mutation (Goldgerg 1989), The outline of the genetic algorithm utili zed in the proposed scheme is given as follows: M

(a) Coding: Coding is required ~o map a finite IC!:Igth of s!ring to the initial cond itions 8(0) = (aiD), ... , a~) ,biO)•...• b~l) T of thc iterative search of the ARMA parameters. It is one of the advanlages o f the proposeu sc heme (hat the ARMA parameters them sd vcs are nm directl y coded, but only the initial conditions, therefore this coding may be rather rough. A set of strings are referred to as the population, (b) Evaluation of fitn ess Junclion: The fitness of the nth string is denoted by g(n) which is dell"cd by VJ. (O), where J .(IJ) is given in (16). In thi s paper. the scaling of V.I.(O) by the ranking is performed so that average strings and best strings get nearly th~ same number of copies in the next generation. (c) ReprodlKtion: Re production is an operation in which individual strings are copied in probabilistic ways according 10 the fitness values . Mo reover we employ {wo sc hemes: One is the stochastic remainder selection without replacement by which the strings with the fitness higher than average can have more than one o ffspring (Booker 1982). The other is the scheme. in which the best string is always copied. (d) Crossover: Reproduction directs the search toward the best copy of strings but does not produ ce new strings. This operation is divided into two steps. First [WO strings from reproduced population are mated at rand om. Then each pair

of strings are cut randomly between any two bits on the strings. The new pair of strings are then created by interchanging the tails. Thus the new offsprings are created from their parent strings. (e) Mutation: Mutation is introduced to produce a new string by flipping the state of a bit from 0 to I or vice versa. However, lhis operation should be used with small probability.

4,2 Separation Alf?ori thm Usi,lg Genetic Algorithm We will apply the genetic algorithm to the choice of inirial value of the ARMA parameters for starting the iterative search to avoid traps into local minima of the prediction error criterion. The algorithm is given below: (Step I)Set M, n o and n b ' (Step 2) Set the population size 8 and the initial population of the strings 10 w'(l), ... ,9 10 '(8)1. (Step 3) Maximize the fitness g(n ) = VJ.(O) (see (16)) for the each Ilew string with respect to (J in an iterati ve way . (3-1) Calculate the filtered HUtPUt z"'(t ) i" (4b) by using 19(;' ) and apply the ptoposed extended TA met!lOd to the output z(j)(t) to ohtain the frequency estimates If::)}. (3-2 ) Calculate the LS estimate {c~) } and td~ll on a basis o f thc model (5) to obtain the estimates of the amplitudes and phases in ( 10). (3-3) Update the estimate of ARMA parameters 19"' ) by the NewlOn method as 9(1-+11 =

e{i) -11 [J;(8(j»)]-lJ~ (8li»)

and re tu rn (3-1) un lil 6(j·H ) and J~ (9(i+li) converge to a constant 8 and J,,(8) respectively. Store the value of the fitness fun ct ion for the eaeh string. (Step 4) Scale the fitness g(n) hy ranking J,(ii). (Step 5) Operate the reproduction including the lwo schemes stated above (see (c)). (Step 6) Perfonn the crossover on the each string. (Step 7) Take mutations on the each stting. (Step S) Return Step 3 and produce the next generation until the minimum of .111 " is not more updated . 5. OPTIMAL DETECTION OF NUMBER OF SINUSOIDS AND ORDER OF ARMA NOISE In the previo us sections we presented the iterative algorithm assuming that the number (M) of sinusoids and the order (na.n o ) of the ARMA model are given a priori. Now we will discuss how to decide M, nu and no. In this paper, we first apply the X' statistical tcst to val idate the triplet (M , na ,Rb) by testing the whitenes~ of the predicted error y{t)j(t 19,,) fat the obtained hest string by the genetic algorithm. The null hypothesis H o is that the prediction error is white. Let a be the level of significance indicating the risk probability of rejecting Ho when Ho holds. Normally a is chosen between 0.0 l and 0.1. The dc(:is ion rule is summarized as follows: I) For all triplets (M,n. ,n,), first apply the x2-test for validating the whitenes ~ of the prediction error of the best string minimizing J in order to eliminate trivial cases and simplify the order decision procedure.

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2) For the eac h validated triplet (M ,n., n,). evaluate the ABIC criterion (Sano ct al. 199 1; Akaike 1980) gi ven by ABIC(K )

~ (N - p)\o{ lry !' - t, A, +~YK lut gl') +

t, IOg(~K I). +

I

I i

!

( 19)

1 5 K5p

where a = 0.01. ()j and Ai are the eigenvector and eigenvalue of the correlation matri x of the filtered observed s ign al {ry(tll given by (4) and (5). g is defined by g ~ 'PT ry where ~ is the data veclor ry : ( '~p + 1)" " IJ{N)T , and 'P is the data matrix of q(t ). p is an order of orverdetermined linear model fo r 1J{t) . We can find the number of sinusoids which minimi zes ( 19) for the p~j r (na,nb). 3) If there is only one M ' that coim;idcs with the assumed M, the number of sinusoids is decided as ir. The order (n a,nb) of the ARMA model is chosen by minimizing the Ale or MDL. 4) If there are more than lWO M, me number of sinusoids is decided as the maxim um if' amon g M's.

,..

»

" i

I~

i, ", I •

L

I

I __ ~

~_ .

~ IS

0 .1

'"

_....

" ...,

/

"'",--

i _'-

"_'-.," .'. _. .. _'I.

0:15

" .1

,,~

0 ._

( ,, ~

..... ,

Fig.2 Estimated mixed power spectrum by proposed method as.surning that the continuous part is the ARMA = :5 ). model: (M = 3, nu ::: 2,

By usi ng this criterion , the numoor and order can be determined at the same time.

nb

Table! I MOL Criterion for (if,

6. EXAMPLE USING BENCH MARK TEST DATA

n no) Q ,

(01 M= 2

We applied the proposed algorithm to the signal data given by Gardner ( 1988) for benchmark tests. The signal data was generated from the true signal mode l composed of three sinuso ids (M = 3) and noises with continuous spectrum as plotted in Fig. I. The number of data is N: 64. and the risk rate is chosen a = 0.03 in the X 2 statistical test. In the genetic algorithm giv en in Section 4, we assume that the popul ati on size is S = 30. the maximum generation is 50, the coding of the initial value 8(0) is exec uted by assuming that 8(0) will tak.e values between -2 and 2 and the quantization leve l is 1/2 10 • The crossover is processed at a s ingle pos ition with probabihty o nc.

na

,r'

."

(20)

The calcul atio n for the detection of the number and order was don e for M=1,2, .. · ,5. no =1,2, · ·· ,6, n,=1,2, .. · ,6. Some o f the triplets were rejected by the X2 test. By the steps 2) and ~) in Section 5 , we could decide the numher of sinuoids is M = 3. Then appy ing the MOL c riterion 10 the order detection for the continuous spectrum. we obtained as = 2 , lib = 5. The al1ernative MDL criterion also indicated that the optimal eastirnale o f the triplet are (M = 3, = 2, nb = 5) as summarized in Table I. The estimated power spectrum is satisfactorily close to the true o ne as shown in Fig. 2. Fig.3 illustrates the process or the search by the hybrid genetic algorithm and shows that the number of genes giving smaller value of J(S ) int.:reases and the minimum of Jno(0 ) can be attained by the best gene, a .... the generation proceeds. For comparisons. Fig.4 5ihows the estimation results obtained

.. ,

Fig. t True profile of mixed power spectrum generating benchmark test data.

Alternative criterion for sim ultaneous detection of the number and orders is given based on the MOL approach, which is described by C(M,"" n,): N logii' +(2M + p +q)(IugN)/2

~

n ~,\ ,ilt

2 3 4 5

6

------

-

2

1

132 164 190 246

3

5

4

-

161 - 190- 209 172 - 197 - 214 198 - 202· 222 255 - 244 - 254 - 252 - 2SO - 249 - 253 - 298 - 292 - 254 - 285

- -

- 209

6

- 208

-169 - 209 -182 - 249

-264 - 266

- 277

- 213 - 181

- 2SS

(b) M- 3

E~~ ~I . --,2=----,',1 2 3 4 5 6

- 1:15 - 222 - 225 - 289 -307 - 308

na'\ nit 1 2 3 4 S 6

na

-

- 163 - 243 - 242 - 291 -321 - :HO

186 22 t 255 300 322 285

-

202 243 204 272 269 303

2

1

190 263 235 306 282 312

-

-

19 1 253 249 296 298 281

3

5

4

- 214 -· 241 - 2&l -- 300 -·332 - 304

1-

4 -

239 256 244 264 280 298

6

- 236 3S4.0~ - 353. 8 - 263 - 293 _ 298 -

_ -

226

258 213 278 260 254

5

6

241 266 274 209 262 291

- 226 - 226 - 253 - 217 - 253 - 256

by the proposed algorithm, on the condition that the continuous spectrum is assumed to be given by the AR noise modeL Thc separation is su(;ccssful, but the estimated AR mode l

4241

n

of the ARMA parameters. The simulation results havc revealed that the discrimination of the mixed s pec tra can be achieved by using a small number of the o bserved data .

-",' . . I. : '

"

..

.,

REFERENCES

,~

Akaike, H .. (1980). Likelihood and the Bayes procedure, In: Baye.,iall StatistiC>' (J.M. Bemado et al ., (cd.)), pp. 141-

166.

'~ "'"' '''

Fig.3 Distribution of strings having allained J in the iteratio n at each generation wflcre the best strings are plotted by solid lins.

'"

--j

-. ' - .

!"

I'

!'

l/\

i, .,.. ..., _. 0., "'.

iT-

~ i --f,;~: ,,-- 0;;"- -;. -- 0" '........,

---•

.1

Fig.4 Estimated mixed power spectrum by proposed methud a,ss uming th at the continuous part is the AR· mode l: (M ~ 3, ri. ~ 8).

., .

'

..

"

-,

, ..

,. ,. '.

Fig.S Distribution of strings havin g attained J in the iteration a1 each generation where the best strings are plotted by solid lins in case of Fig.3. with na = 8 becomes rather peaky and Fig.S indicates that the attained minimum criterion is more degraded than Fig.3.

7. Conclusions The genetic algorithm based estimation scheme has been presented to discriminatc the line and ARMA spectra. The simultaneous minimization of the predicted error criterion with respec t to all unknow n parametcn;,was reduced to the simple iterative search or the ARMA parameters only, in which the hybrid type of genetic algorithm is applied to the optimal choice of the initial condition for the iterative search

Ashida, H. and A. Sano, (1994). Estimation of mixed spectrum incorporating with genetic algorithm, Proc, the 26th ISClE Int. Symp. Stochastic Systems Theory and Its Applications, pp. I 19-124. Dooker, BL, ( 1982). Intelligent behavior as an adaptation to task environment, Res. Report No. 243, Univ. Michigan, Ann Arbor. Chauerjcc. C., RL Kashyap and G. Boray. (1987). ESlimati on of close sinusoids in colored noise and model discrimination, IEEE Trans. ACOUS I .• Speech, Signal Processing. ASSP.35 , pp328-337. Gardner. W.A., (1988). Statistical Spectral Allalysis, pp.254350, Prenlicc Hall. Goldberg, D., (1989). Genetic Algorithms ill Search. Optimi"alioll, and Mach;,Je Learning, Addiso n-Wesley. Haykin, S .• ( 1991 . 1995). Ad..ances in Spectrum Analysis and Array Processing. Vols.2 &3, Prenticc-Hall. Hunt, K.J . , ( 1993). "Black-box and partially known systems identification with ge netic algorithms". Pmc. ECC'93, pp.2286-2290. Kay, S.M., (1988). Modem Spectral Estimation. Theory & Applications, Prentice-Hall . Kay. S .M. and V . Nagesha. (1994). Maximum likelihood estimation of signals in autoregressive noise. IEEE Trans. Acuusl. Speech. Signal Processillg. ASSP·42, pp.88. 101. Kristinsson, K. and G.A . Dumont, (1992). System identification and control usi ng genetic algorithms. lEEE Trami. SYSI. Mall alld Cybem SMC-22 . pp. 1033-1046. Kung, S.Y .• K.S . Arun and DV. Bhaskar Rao, (1983). Stalespace and singular value decomposition based approx.im af ion methods fo r the harmonic retrieval problem, Opt. Soc. Amer., 73, pp.1799-1811. Priestley. M.B .. (1981). Spectral Allalysis and Time Series, Prentice Hall. Sano. A. and H. Hashimoto. (1985). Spectral analysis of sinuso ids embedded in noise. with un known colored spectrum. TrailS. SICE, 21 . pp.lOO7-IOI2. Sano, A .. H. Ohmori and N. Kamegai. (1991). Stabilized identifi cation via GSVD optirnaized based on Baycsian infonnation theo ret ic criterion, Proc. IFAC Symp. Oil Identi{tu.ltioIJ and System Parameter Estimation, pp. 15 36- 1541. Tsuji, T. and A Sano (1989). Separable eSlimation of discrete and co ntinuous spectra of signal with mixcd spectrum, Proc. ICASSP '89. pp.22 18-222 1. Tuft D.W. and R. Kumaresan, (J 982). Estimation of frequencies of multiple sinusoids. Proc. IEEE , 70, pp.975-989. Yu, X. and Z. He. ( 1987). A modified Yu!e-Walkerequations method for harmonic analysis in unknown colored noise, Proc. ICCASP, pp.2043-2046.

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