AD-AI06 753 VIRGINIA POLYTECHNIC INST AND STATE UNIV BLACKSBURG D--ETC F/6 12/1ARMA SPECTRAL ESTIMATION: AN EFFICIENT CLOSED FORM PROCEDURE.(U)AUG 81 J A CADZOW AFOSR-80-O069

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MFINAL REPORT

AR&. Spectral Estlmation: An Efficient Closed Forn Procedure

Prepared by: Dr. James A. Cadzow

Academic Rank: Professor

Department and Department of Electrical Engineering

University: Virginia Polytechnic Institute - '.

Blacksburq, VA. 24060

Prepared for: Air Force Office of Scientific Research

Date: August, 1981

Contract No. AFOSR 80-0069 V,

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ARMA SPECTRAL ESTIMATION:

AN EFFICIENT CLOSED FORM PROCEDURE

by

JAMES A. CADZOW

ABSTRACT

In this report, an effective procedure for effecting on ARMA

spectral model of a time series is described. This procedure is

predicated on minimizing as set of "basic error" terms as generated

from an ARMA model that is hypothesized as characterizing the time

series under analysis. The spectral estimation performance achieved

in using this approach has been empirically found to be generally

superior to that obtained using such contemporary methods as maximum

entropy and the periodogram.

AIR FO',cE OFF" .. "TTT~TIC IMSEA (ySe)N TICE OF T . 70 DTTCThis technic l'e . en "eview-A and isapproved for .. .e'se IAWAFR 1 .- 12,

M4.TTHEW 3. JcNRT'Ch ef, Teohnical Inforwation.DIviSiM

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T A BLE O F CO0N TE NT S

I. INTRODUCTION .. ..... ........

II. DESCRIPTION of ARMA MODELING METHOD -2,3

III. CONTRACT PUBLICATIONS. ..... ... 3

APPENDIX..... .. ...... ..... 4,8

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I. INTRODUCTION

In this report, a description of the high performance ARMA spectral

estimator, as in-part developed under the AFOSR contract 80-0069, shall

be given. To begin, the main objective of the project was that of deve-

loping an effective method for estimating the ak and bk coefficients

governing the ARMA model

p qx(n) + I akx(n - k) = 7 bkc(n - k) (1)

k=l k-O

in which the excitation time series {((n)} is taken to be white. These

coefficients are to be selected so that this ARMA model is "most consis-

tent" with a set of time series observations.

x(l),x(2),",x(N). (2)

which are available. The term "most consistent" is here being used in the

sense that the hypothesized ARMA model is most compatable with the task of

minimizing a set of "basic error terms". A brief description of this pro-

cedure shall be now given while a more detailed description is to be found

in the appendix.

k7

II. DESCRIPTION OF ARMA MODELING METHOD

The basis for the ARMA method is dependent on the so-called basic

error terms. These terms are generated by first multlplyinq each side

of relationship (1) by the delayed entity x*(n - M) to obtain

p

e(m,n) = x(n)x*(n - m) + I akx(n - k)x*(n - m) (3)

k=l

q+l <m<N

max (mp) < n < N

It can be shown that these basic error terms £(m,n) are each zero mean

when the underlying time series in an ARMA process of order less than or

equal to (q,p) and the ak coefficients in expression (3) correspond to

the exact process'. autoregressive coefficients.

With this in mind, a method for selecting the ARMA models ak

coefficients is suggested. Namely, they are chosen so as to cause the

basic error terms (3) to be as close to their mean value of zero as

possible. This objective may be realized by introducing the following

quadratic functional

f(a) = e (4)

in which e is a column vector whose elements are appropriate arrangements

of the ensemble of basic error terms (3), W is a nonegative matrix, a is

the autoregressive coefficient vector with elements ak, and, t denotes the

operation of complex conjugate transposition. It is readily shown that

the minimization of functional (4) will result in a set of consistent

2

linear equations for the optimum ARMA model autoregressive coefficients.

Once the ak coefficients have been obtained, the ARMA model's

movinq average coefficients are estimated by first generattnq the so-

called residual sequence {c(n)} according to

p

e(n) = x(n) + I akx(n - k) p + 1 < n < N (5)

k=l

The spectrum of this residual time series is theoretically given byq

I bke-jkw 12. A technique for obtaining a moving average estimate

k=Oof this residual time series is fully described in the appendix and

entails the utilization of the smoothed periodogram. Once this smoothed

periodogram has been generated, the required ARMA spectral estimate is

achieved,. Empirically derived results indicate that this report's pro-

cedure provides a superior spectral estimation performance than that

achieved by such contemporary alternatives as the maximum entropy method,

and the periodogram.

III. CONTRACT PUBLICATIONS

The following two refereed publications resulted from the sponsored

AFOSR contract.

[ll " Autoregressive Moving Average Spectral Estimation: A Model

Equation Error Procedure", IEEE Trans. on Geoscience and

Remote Sensing, vol. GE-19, No. 1, Jan. 1981, pp 24 - 28.

[2] "Two Dimensional Spectral Estimation", IEEE Trans. on Acoustics,

Speech and Signal Processing, vol. ASSP-29, No. 3, June 1981,

pp 396 - 401.

3

34 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. GE-19. NO. 1. JANUARY 1981

Autoregressive Moving Average Spectral Estimation:A Model Equation Error Procedure

JAMES A. CADZOW, SENIOR MEMBER, IEEE

Abrt-A liocaduuis Im sod for gsteeatiq a autorgrusuve I. INTRODUCTIONmoving saeme (ARMA) I aodl of a statiomary time m I N this paper, we shall be concerned with the task of esti-beid spon a alft set of time sad.' al I'm- The ARMA model'slatmleulod coefficliets a estiited by mimnumin3 a quamtlic .1mating the statistical characteristics of a stationary random

functiom of a st of basc saw terms. In eumples treated to dat, this time series {x(n)} from a finite set of observations. For manymethod has damo ated a exceptio m al ability in reacing dosely applications, knowledge of the time series' underlying autocor-,ead nrow band aW" in a low signllI-noln envlromemnt where relation sequence as formally defimed by

othe proeeduees such an U. maximum entrop method often fail. Itseffeedeuu omn ot cimamm a d ime W is w promits and a r. (n)-E {x(n + k) x*(k)}

Move Iur evaluation Is pmasently baling coaductad. With this i r()~~~~~ 1mind, he new ARMA peohdue promma to be an important semaseddon 21. conveys all the information required. In this expression, E and

* denote the expected value and complex conjugation opera-

Mmuctipt received Aprl 27, 1"a0. This work was supported in part tions, respectively. It is often advantageous to equivalentlyby the Office of Naval Research under Contract N00014-80-C-0303 characterize stationary time series in the frequency domainand the Air For= Office of Scientific Re a4 under Grant AFOSR where their intrinsic properties may be more discernible. This0-0069.The author is with de Department of Eletriclngineering, Virgini is particularly true for so-called narrow-band processes. The

Polytechnic Institute, lacks bu , VA 24061. vehicle for this characterization is the associated power spec-

0196-2892/81/0100.0024S00.75 0 1981 IEEE

I

CADZOW: ARMA SPECTRAL ESTIMATION

tral density as given by In this paper, a generalization of a recently developed closed-form ARMA method shall be presented [71 and [81. This

Sx (w) = rx(n) e'"" (2) new procedure has been empirically found to provide signifi-n--- candy better spectral estimation performance than the above

which is recognized as being the Fourier transform of the two ARMA approaches as well as the maximum entropyautocorrelation sequence. method. In what is to follow, a time domain.approach for de-

Upon examination of expressions (1) and (2), it is apparent tertmining the ARMA model's AR ak coefficients will be given.that the required time series characterization will generally en. This in turn will be followed by a frequency domain proce-tail complete knowledge of the generally "infiite" length dure for estimating the effects of the moving average bk coef-autocorrelation sequence. Here, we shall be concerned with ficients on the overall spectral estimate. Use will be made ofextracting this information from the following "finite" set of the well-known fact that the power spectral density corre-time series observations sponding to ARMA model (4) is specified by

x(l),x(2),'",- x(N). 2 (3) s.(W)= lbo +bIe jw +" +bqe W!e

Unless some constraints are imposed upon the time series' 1 +a1 eiw+. . . +a p e"t ,, ! obasic nature, it is clear that there exists a fundamental incom- C7IB()2

2(5

patability in estimating the required statistical knowledge from =A(w)this finite set of measurements. In accordance with well ac-cepted practices, this dilemma is here resolved by postulating a II. AUTOREGRESSIVE COEFFICIENT ESTIMATESfinite parameter linear model to represent the random phe- An effective method for estimating the ARMA model's ARnomenon. Specifically, the response of this model to "ran- coefficients from the set of given observations will be now pre-dom" shock excitations will be conceived of as generating the sented. This first entails multiplying each side of expressiontime series under study. (4) by the entity x*(n - m) to yield the "basic error terms"

In terms of parameter parsimony, the causal autoregressivemoving average (ARMA) model of order (p, q) as specified by p

e(m, n) x(n) x*(n - m) + E akx(n - k) x*(n - m) (6a)p q 1 k-i

x(n) + F, akx(n - k) E bke(n - k) (4)k-i k-o bke(n - k)x*(n - m), for q+1 m<N

is generally the most effective linear model. In this representa- -omax (mp)n Ntion, the unobserved excitation {e(n)} is taken to be a whitenoise time series with zero mean and variance 02. It is impor- (6b)tant to appreciate the fact that the more specialized autore- where the range on the m and n variables is dictated by thegressive (i.e., bk 0 for k * 0) linear model will generally en-tail a much higher order choice so as to achieve a comparable random variables e(n - k) and x*(n - m) be uncorelated. tfstatistical representation. Conceptually, the more efficient the time series is in fact an ARMA process of order less than orARMA model is the logical model choice when the exact na- equal to (p, q) it fows that the basic error terms e(m, n) areture of the time series is unknown. each zero mean random variables.' This is a consequence of

Due to the relatively difficult task of estimating the ARMA the causality of model (4) and the whiteness of the excitationmodel's ak and bk parameters from the given observations (3), which results in the random variables e(n - k) and x*(n - m)however, the preponderance of activity has been directed to- being uncorrelated.wards the specialized autoregressive (AR) model. This is a di- With these thoughts in mind, relationship (6) provides anrect consequence of the simpler AR parameter estimation ideal vehicle for determining a set of AR coefficients which areproblem which results. In particular. the basically equivalent consistent with the given time series observations. Specifi-maxiimum entropy, linear predictive coding, and AR methods cally, the ak coefficients will be selected so as to cause each ofhave been developed for efficiently obtaining the AR model the basic error terms (6a) to be as close to their mean value ofparameters [II. Nonetheless, the inherent superiority of an zero as possible. This goal is achieved by minimizing a squaredARMA model representation is widely recognized and a num- error criterion of the formber of procedures for estimating the ARMA model parametershave been advanced. These include the whitening filter ap- f(a) = el We A (7)proach which is iterative in nature, generally slow in conver- in which e is a column vector whose elements are appropriategence, and typically requires a relatively long data length N tobe efecive[21and 3].A mre esiablecloed-ormap-arrangements of the ensemble of basic error terms (6a). It' is abe effective [2) and [3). A more desirable closed-form ap-

proach which is based on the autocorrelation relationship 'As a side note, it may be shown that the basic error terms have iden-

governing ARMA processes has also been developed [41-[6]. tical variances given byThis approach often provides good estimates and does not re- qquire N to be excessively large. Unfortunately, its perfor- rx(O) a' F Ibkl2 .mance is not always as good as that provided by AR methods. k-0

26 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. GE-19. NO. 1. JANUARY 1961

nonnegative definite square matrix, and, I denotes the opera- where t = 1 + max (n, p) and c is a p X I vector with elementstion of complex conjugate transposition. This criterion is ci " -c10. for 1 4i 4 p. (I b)readily shown to be a function of the AR coefficients uponsubstitution of expression (6a) into criterion (7). Thus the optimum set of ARMA autoregressive coefficient

The weighting matrix W is to be selected based on statistical estimates are obtained by solving the linear system of equa-reasoning which must be tempered by an appreciation that lit. tion (10). It is possible to utilize the Hermitian structure oftie is generally known about the time series' statistics. To ius- matrix C to yield an efficient solution procedure.trate this point, let us consider the usually hypothesized condi- A. Forward and Backward Data Approachtion that the excitation {e(n)} is a Gaussian white process. In

One can more effectively use the observed data (3) to achievethis case, if" the m atrix W is se t equal to E {ee t }-z th e m in u a A R s e t l etim atebs n h t d t ' a k a d v rzation of criterion (7) will result in a maximum likelihood esti- an a spect ie bamate of the AR coefficients. Unfortunately, the generation ofthis particular matrix requires knowledge of the time series' i( 1),i(2)," ,.i(N) (12){x(n)} second-order statistics. This statistical information, in whichi(n)=x(N+ I - n). In particularone formulatesahowever, is precisely what we are seeking to estimate via theARMA model (4).2 This dilemma has arisen due to the unreal- mean-squared error criterion as given byistic desire to seek a maximum likelihood ARMA model. N-1 I N 2More modest objectives must be sought if a tractable proce- f(A) = w(m) e(m, n)dure is to be evolved. " 1 fin -I -max (,p)

A reasonable implementation of the above autoregressive Ncoefficient selection philosophy then depends on a prudent + ir(m, n) 1 (13)choice for the weighting matrix. In this paper, we will be con- n - I + max (,n,p)cerned exclusively with a choice that results in the following where i(m, n) denotes the basic error term associated with themean-squared error criterion: backward sequence (12). It is a simple matter to show that

the AR coefficient vector which minimizes this forward-fa) = E e(m, n) (8) backward criterion must satisfy the linear system of equations

M-q-l n -i+max(m.p) [(C1 =C+,F (14)

although other choices are suggested in Section V. In this ex- where C and C are each p X p nonnegative definite Hermitianpression, the w(m) are nonnegative parameters which are log matrices given by relationship (I Ia) with the forward (3) andcally selected to be inversely proportional to the variance of backward (12) data used, respectively, and, the p X I vectorsthe term which they multiply. A general expression for these c and i are given similar interpretations.variance entities, unfortunately, also reveals a dependency on It is important to note that the rank of matrix [C + C] atthe time series' unknown second-order statistics and the least equals the rank of either of its constituent nonnegativeARMA model coefficients. Empirical evidence suggests, how- defmite matrices C or C. Thus, in using this forward and back-ever, that a choice of the weighting parameters as given by ward approach to estimate the AR coefficients, one is typi-

cally able to use a higher order model (i.e., larger value of p)w(m) - [N- m] q + 1 •m 4N- 1 (9) than would be the case for the forward or backward models

only. Moreover, it has been empirically found that the for-results in satisfactory performance. An examination of otherony Mrevithsbnemrcalfudtathef-

ward and backward approach usually results in better spectralweighting parameter selections is currently being conducted. estimates.

Using standard calculus, it is readily shown that the set ofAR coefficients which render criterion (8) a minimum must 11. NUMERATOR DYNAMICSsatisfy the following linear system of equations: A variety of procedures exist for determining the numerator

dynamics of an ARMA time series once the AR coefficientsc e (10) have been estimated [4]-[9]. In this section, a procedure

where i is the p X I AR coefficient vector estimate with ele. which has been found to produce improved results will be pre-

ments ak, C is a p X p nonnegative definite Hermitian matrix sented. It makes use of the characteristic equation (4) which

with elements can be interpreted as generating the auxiliary "residual" se-quence {e(n)} according to

N-I N N p -c,, ' F .F, E iw(m)x(n-k)x(s-m) e(n)=x(n) + T akx(n-k) (I a)

mq+i no t kset(

x*(n-m)x*(s-i), for i,ka1,2,"•.,p (lla) q= bk e(n - k). (15b)

kc-oIn the special case of an AR model, however, it is possible to obtain One may straightforwardly show that the spectrum of the mov-

the maximum likelihood estimate. ing average residual time series Se(w) is given by 1B()1 2 '2

CADZOW: ARMA SPECTRAL ESTIMATION 27

This observation in conjunction with relationship (5) provides 27

the vehicle for obtaining the underlying time series spectral es-timate, that is

Sx(w)=Se()/IA(w))12. (16)

With this in mind, a method will be now given for obtaining aqth-order moving average spectral estimate of the residual se-quence. This estimate will be based on the specific residualelements

e(p+ l),e(p + 2),'.. ,e(N) (17)

which are calculated using relationship (15a) and the giventime series observations (3). The AR coefficient used in (15a)will correspond to the ak estimate obtained upon solving ex-pression (14).

The approach to be now presented in an adaption of thewell-known method of Welch for obtaining smoothed peri-odogram estimates [101. In essence, one first segments thecalculated residual elements (17) into L segments each oflength q + I as specified by

ek(n)=a(n)e(n+ I +p+kd), O n~q AO<k'L- 1 (18) 0.00 0.20 0.40 0.60 0.80 1.00

where a(n) is a data window and "d" a positive integer which Normalized Frequencyspecifies the time shift between adjacent segments. These in- Fig. 1. Fifteenth-order ARMA spectral estimate of two sinusoids at

normalized frequencies f, a 0.4 (10 dB) and f = 0.426 (0 dB) baseddividual segments will overlap for a shift selection of d <q. upon individual sequencs of length 64.Furthermore, to include only observed data, the relevant pa-rameters must be selected so that p + q + 1 + (L - 1) d < N.Finally, the periodogram of each of the L segments is takenand these are averaged to obtain the desired smoothed qth.order periodogram, that is 1

YSjq -+II a(n) e(n + p + kd)- /' | mOnO n

(19)

The data window is to be normalized according to 2 al(n) = 1.In using this segmentation, the variance of the estimate S (w)is reduced. The price paid for this reduction, however, is a loss A...

in frequency resolution, an increased bias of the estimate, and,a possible deterioration in power level estimates. It must be . . .. . .noted, however, that the overall resolution capability of thisARMA procedure is predominately influenced by the AR co-efficient selection. If one is basically interested in resolutionperformance, an examination of the ARMA model's pole loca.tions then need only be considered.

IV. NUMERICAL EXAMPLE - kAIn order to compare the effectiveness of the new ARMA

spectral estimator with the maximum entropy method, the, j A A A kclassical problem of resolving two closely spaced (in fre- 0.00 0.20 0.40 0.60 0.8o 1.00quency) sinusoids in white noise will be considered. Specifi- Normalized Frequencycally, the time series under study is specified by Fig. 2. Thirtieth-order maximum entropy AR spectral estimate of two

sinusoids at normalized frequencies f, a 0.4 (10 dB) and f2 n 0.426

x(n) a ,/2 cos (0.4frn) + r" cos (0.426frn) + w(n) (20) (0 4B) based upon individual sequences of length 64.

where {w(n)} is a white Gaussian noise process of zero mean 0.4 and 0.426 are readily calculated to have a signal-to-noiseand unity variance. The sinusaids of normalized frequencies ratio (SNR) of 10 dB and 0 dB, respectively. A sequence of

. ,1

"28 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. GE-19. NO. 1. JANUARY 19S1

apparent. In addition, a thirtieth-order AR spectral estimate8th order which arises from this 640-length sequence is shown in Fig. 4.

Clearly, the new ARMA method has outperformed the ARmaximum entropy method in both the short and long data

0.00 0. 20 0.40 0.60 0. 0 1.00 length examples here considered.

V. CONCLUSIONS

15th order A generalization of a recently developed ARMA spectral esti-mation method has been presented. This has included the in-

,. 0.0 0..6 o -, troduction of an error weighting matrix, the concept of usingforward and backward data, and, the utilization of Welch's

Fig. 3. ARMA spectral estimates of two sinusoids at normalized fr method for obtaining estimates of the spectrum's numeratorquencies m = 0.4 (10 dB) ande stma0.426 (0 dB) based upon all 640observations. dynamics. Empirical evidence suggests that this new proce-

dure has an improved spectral resolution performance whencompared to popular contemporary methods. Its full poten-tial will be realized, however, only after a number of relevantissues are resolved. These include the task of determining

0.0 0.20 0.40 0.60 0.80 L. 00 good choices for the weighting matrix W, obtaining a data de-

Fig. 4. Thirtieth-order maximum entropy AR spectral estimate of two pendent procedure for estimating the ARMA model order,inusoids at normalized frequencies f, = 0.4 (10 dB) and f2 = 0.426 and, investigating other numerator dynamics methods.

(0 dB) based upon all 640 observations. In addition to the specific weight matrix selection consid-ered in this paper, the following two choices of criterion

length 640 defined over 0 -4 n - 639 was next generated using N -1 N 2

this relationship. Furthermore, in order to provide a statistical T F je(m, n)l

basis for our comparison, this 640 length sequence was then m.q i m.m+i 1

decomposed into ten dioint sequences each of length 64 de- and N w2 ( )an Z w(n) N-I m n

fined on 0 4 n • 63, 64 • n • 127,- • •, 576 • n • 639. An +

ensemble consisting of ten subsequences each of length 64 hasthereby been generated with each subsequence having a differ- have given preliminary evidence of providing satisfactory spec-ent noise sample and a different initial phase between the two tral estimation performance. Further investigation of thissinusoids. This latter condition is useful in revealing any po- most critical weighting matrix choice is now being conductedtential sensitivity to initial phase that the new ARMA spectral and will be subsequently reported upon.estimation method may possess. REFERENCES

The spectral estimates which resulted when the new ARMAspectral estimator (forward data only) and the maximum en- [11 D. G. Childer, Modem Spectral Analysis. New York: IEEE

Press, 1978.tropy (covariance) method were applied to these ten random [21 P. R. Gutowki, E. A. Robinson, and S. Treitel, "Spectral estima-time series samples are displayed in Figs. 1 and 2, respectively. tion, fact or fiction," IEEE Trans. Geosd. Electron., voL GE-16,The ordinates were scaled from 0 to 50 dB for each of the ten pp. 8044, Apr. 1978.

(31 K. Steiglitz, "On the simultaneous estimation of poles and zerosindividual plots with each AR maximum entropy estimate be- in speech analysis," IEEE Trans. Acoust., Speech, Sgnal Process-

ing of order thirty and each ARMA estimate being of order fif- ing, vol. ASSP-25, pp. 229-234, June 1977.teen (with d 16). The spectral estimates are plotted one 141 G. Box and G. Jenkins, Time Series Analysis: Forecasting and

Control, revised edi. San Francisco, CA: Holden-Day, 1976.above the other (lower one corresponds to 0 4 n • 63) so as [51 M. Kaveh, "High resolution spectral estimation for noisy signsls"to better reveal the consistency of the estimates and to con- IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-27,serve space. It is clear that the maximum entropy method is pp. 286-287, June 1979.

(61 J. F. Kinkel, J. Perl, L. Scharf, and A. Stubberd, "A noteable to resolve the sinusoids in only five of the ten sample on covariance-Invariant digital and autoregressive moving aver-cases. On the other hand, the new ARMA method successfully age spectrum analysis," IEEE Trans. Acoust., Speech,and Signalresolves the sinusoids in each case and provides excellent fre- Processing, vol. ASSP-27, pp. 200-202, Apr. 1979.

s e(71 J. A. Cadzow, "ARMA spectral estimation: An efficient dosedquency estimates off1 I 03980 andf2 - 0.4295 obtained by form procedure," in Proc. Rome Air Development Center Work.averaging over the ten cases with associated sampled standard shop Spectral Estimation (Rome, NY, Oct. 3-5, 1979), pp. 81-deviations of o - 0.0027 and oj - 0.0 107. 97.

(8-) , "High performance spectral eimation: A new ARMATo illustrate the new ARMA method's effectiveness on long method," IEEE Trans. Acourt., Speech, Signal Processing, vol.

data length sequences, an eighth- and fifteenth-order ARMA ASSP-28, pp. 524-529, Oct. 1980.model were next obtained using the entire sequence of length 19] S. M. Kay, "A new spectral estimator," IEEE Trans. Acoust.,

Speech, Signal Processing, vol ASSP-28, pp. 585-588. Oct. 1980.640 as given above. The spectral estimates which resulted are [101 S. A. Tretter, Discrete-Time Signal Processing. New York:displayed in Fig. 3 where a sharpness in frequency resolution is Wiley, 1976.

L- A, I-

*k UNI I ' CLASSIFI' 1 I j jk THIS PA0 F ,.. I .r _rjl

REPORT D -- , A il PAGEREAD INSTRUCTIONSBEFORE COMPLETING FORM

I~~a 1 F bS .2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

4 TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

APECTRAL ESTIMA'ION: AN EFFICIENT CLOSED -4 FINALFORM PROCEDUR. 6. PERFORMING ONG. REPOR f NWI8ER

, 0RIrO'Hs) 6. CONTRACT OR GRANT NUMBER(s)

i. tames A/ Cadzow .3$ AFOSR-8g-d69

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK Ut4IT NUMBERSDepartment of Electrical Engineering A // (OR. UT E /S

Virginia Polytechnic Institute 2305-D9Blacksburg, VA. 24060

I I. CONTROLLING OFFICE NAME AND ADDRESS 410& I' ALQ&4 -

AFOSR of Scientific Research Au mR 81Building 410.UMBER OF PAGESBoiling AFB, Washington, D.C. 20332 I

14. MONITORING AGENCY NAME & ADDRESS(iI different from Controlling Office) IS. SECURITY CLASS. (of this report)

i Unclassif ied

t. )S.I.. DECLASSIFICATION DOWNGRADING- - SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

fpr~6 oar PUbl Ic releasesdistribution UzI11ted.

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, it different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side it necessary and identify by block number)

- 4~~~'1/6(9...u, ,,:.20. ABSTRACT (Continue on reverse side If necessary and identify by block number)

A procedure was devised for generating an autoregressive moving average (ARMA)spectral model of a stationary time series based upon a finite set of timeseries' observations. The ARMA model's autoregressive coeficients are estimate-by minimizing a quadratic function of a set of basic error terms. In examplestreated to date, this method has demonstrated an exceptional ability inresolving closely spaced narrow band signals in low signal-to-noise environmentwhere other procedures such as the maximum entropy method often fail. -OVER -

FORM

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SECURITY CLASSIFICATION O3'TIS PAGE (When Data Entered)

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bat. ., 9 -LASSIICAIW. HIS PAGI(slh ,.. f ne"ed)

its effectiveness on other classes of time series aloo shows promise and a moregeneral evaluation is presently beii., conducted. With this in mind, the newARMA procedure promises to be an important spectral estimation tool.

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