AD A129 868 A SUMMARY OF RESULTS IN DATA ADAPTIVE DETECTION AND I1ESTIMATION(U) RHODE ISLAND UNIV KINGSTON DEPT OF

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00OA Summary of Results(0) in Data Adaptive DetectionC4 and Estimation

Steven Key

'Department of Electrical EngineeringUniversity of Rhode Island

Kingston, Rhode Island 02881

April 1983

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A Summary of Results in Data Adaptive Detectionand Estimation

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Steven Kay N00014-81-0144A

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This report describes a number of results which have been obtained inthe Data Adaptive Detection and Estimation project. The various paperssummarized deal with properties of autoregressive representations asthey relate to detection and estimation in partially known signal and/ornoise environments. Complete copies of the works can be obtained fromthe author.

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SECUITy CLAsSIFICATIoNl or THOS PACE (Wheng be gEeeoe.

A Summary of Results

in Data Adaptive Detection

and Estimation

Steven KayDepartment of Electrical Engineering

University of Rhode Island

Abstract

This report describes a number of results which have been obtained in

the Data Adaptive Detection and Estimation project. The various papers

summarized deal with properties of autoregressive representations as they

relate to detection and estimation in partially known signal and/or noise

environments. Complete copies of the works can be obtained from the author.

Aooession 1o--ins ' -DTIC TABUnannounced 0justification

Distributionl/Availability Codes

.Avail and/or .

Dist Special

SUMMARY

1) "on the Statistics of the Estimated Reflection Coefficients of an

Autoregressive Process" (with John Makhoul of Bolt, Beranek, and Newman)--

submitted to IEEE Trans. on Acoustics, Speech, and Signal Processing.

This paper derives a recursive means of computing the Cramer-Rao (CR)

lower bound for the estimated reflection coefficients of an autoregressive

(AR) process. The exact algorithm which computes the bound is given in

Figure 1. Since the CR bound is attained asymptotically (for large data

records) for a Maximum Likelihood Estimate of the reflection coefficients,

one can also use the algorithm of Figure I to describe the statistics of

the MLE. It is also shown in this paper that all currently available

estimators for the reflection coefficients are MLE's for large data recc.ds.

Thus, the results can be used to statistically characterize these estima-

tors, i.e., the Burg algorithm I as an example. An example of the actual

covariance matrix (obtained via computer simulation) and the CR bound is

shown in Figure 2. The data length was 1000 points.

2) "Simple Proofs of the Minimum Phase Property of the Prediction Error

Filter" (with Louis Pakula of U.R.I.)--IEEE Trans. on Acoustics, Speech, and

Signal Processing, April 1983.

Presented are two simple proofs which assert the well known property

that all the zeros of the optimal prediction error filter lie on or within

the unit circle of the Z plane. The first proof is somewhat incomplete in

that it only shows that the zeros lie on or within the unit circle. It

gives no indication as to when the zeros must be totally within the unit

circle. However, it is extremely simple. The second proof which is slightly

more complex also determines conditions under which the zeros may be on the

ultt circle.

3) "Recursive Maximum Likelihood Estimation of Autoregressive Processes"--

IEEE Trans. on Acoustics, Speech, and Signal Processing, February 1982.

An estimation algorithm is derived which is a closer approximation to

the true MLE of the parameters of an AR process than currently existing

procedures. Specifically, the algorithm does not make the standard assump-

tion that the determinant of the filter covariance matrix can be neglected

in the maximization of the likelihood function. This common assumption is

not valid for short data records and/or highly peaked spectra. By incorpo-

rating the determinant and maximizing the likelihood function recursively

(in model order) more accurate parameter and spectral estimates are obtained.

The algorithm is summarized in Figure 3. Also, Figure 4 summarizes some

simulation results which verify the improved performance of the algorithm

for short data records. (RMLE - Recursive MLE, FB - Forward-Backward,

YW = Yule-Walker).

4) "More Accurate Autoregressive Parameter and Spectral Estimates for

Short Data Records"--IEEE Workshop on Spectral Estimation, Hamilton, Ontario,

August 1981.

This paper is a shortened version of 3.

5) "Some Results in Linear Interpolation Theory"--to be published in IEEE

Trans. on Acoustics, Speech, and Signal Processing.

The optimal finite length interpolation filter for a wide sense stationary

process is derived. It is shown that for an AR process one can significantly

reduce the power out of an FIR filter by performing interpolation rather than

the more common one sided prediction. Also, the optimal interpolation filter

for an AR process, i.e., the one which minimizes the interpolation error

power, is found to be the one which attempts to interpolate at the midpoint

of a data record. This result although intuitive is shown not to hold for

2

more general processes. Finally, some well known results in infinite

length FIR filter interpolation are simply derived using AR approximations

of infinite order.

6) "Accurate Frequency Estimation at Low Signal-to-Noise Ratios"--sub-

mitted to IEEE Trans. on Acoustics, Speech, and Signal Processing.

An iterative algorithm for frequency estimation of sinusoids in white

noise is described and analyzed. The algorithm iteratively computes AR

parameter estimates after the original data has been filtered by an all

pole filter. The algorithm is shown to be related to the Steiglitz-McBride

algorithm 2 . It is summarized in Figure 5, where it has been termed the

Iterative Filtering Algorithm (IFA). For two sinusoids closely spaced in

frequency the algorithm performance is shown in Figure 6. Also, shown is

the CR bound (for unbiased estimators) and the principal component (PC)

3approach of Tufts and Kumaresan . The IFA outperforms the PC approach

at low SNR's. The PC approach had been claimed to offer the best per-

formance other than a direct MLE, at low SNR's. At higher SNR's a bias

is present which tends to degrade the performance of the IFA. In Figure 7

a comparison is made with a direct MLE, which amounts to a nonlinear

least squares estimator. Again at low SNR the IFA appears to perform

better.

7) "Asymptotically Optimal Detection in Unknown Colored Noise via

Autoregressive Modelin "--to be published in IEEE Trans. on Acoustics,

Speech, and Signal Processing.

The problem of detecting a known signal in Gaussian noise of unknown

covariance is addressed. The noise is assumed to be an AR process of known

order but unknown coefficients. Thus, the parameterization of the covari-

ance leads to a problem in composite hypothesis testing. Since in general

no uniformly most powerful test exists, the Generalized Likelihood Ratio

3

Test (GLRT) is applied. The GLRT detector which results is shown in Figure

8. As expected adaptive prewhiteners are included which assume either the

signal is present (upper channel) or absent (lower channel). The detector

is not a single estimated prewhitener and matched filter as might be ex-

pected. The performance of the detector is shown in Figures 9 and 10 for

an AR process of order p-I and parameter a--0.9 and for various data record

lengths N-20,100. The optimal curve is the performance assuming a known

covariance function, i.e., a prewhitener and matched filter. It is seen

that even for short data records the performance is nearly optimal. Finally,

it is proven that for large data records the GLRT performance is equal to that

of a prewhitener (assuming a known covariance) and matched filter and hence

is optimal.

8) "Detection for Active Sonars via Autoregressive Modeling,"--Workshop

on Maximum Entropy and Bayesian Methods in Applied Statistics, University

of Wyoming, Laramie, Wyoming, August 1982.

This paper is a shortened version of 7.

4

1) For n=1a. = a(l) - K1

2 1

22

- " (1-K ) I

2) n - n) a2 (1K2 ) 2 )

N 1 2)i n, n+1 2 2 2n D_ _ ~_-n-l i-2 A (I-K J) 1

1-K

C' n) -g I [c,,2 D -1 T 1LT 1_K2

n

3) If n-p, let = C(n) and exit.

4) 0~~£I n--n 2J

r

(1 [(Kn-11) an.2]An-i-

K n-1

Bn-i l n

B =

0 1n Po T n

S) Go to 2

Note that 0T is lx(n-l).

Figure 1. Summary of Cramer-Rao Bound Computationfor Reflection Coefficients.

5

TABLE 2

TRUE VALUES ESTIMATED VALUES

K K2 K3 0.570 0.570 0.570 0.569 0.568 0.564

0.541 - 0.490 0.000 0.523 - 0.479 - 0.006

NCA - 0.490 0.675 0.000 - 0.479 0.689 0.028K

0.000 0.000 0.675 - 0.006 0.028 0.665

TRUE VALUES ESTIMATED VALUES

K1 K2 K 3 0.570 -0.570 0.570 0.557 -0.563 0.563

7.210 -1.790 0.000 6.988 -1.858 -0.151

NCA -1.790 0.675 0.000 -1.858 0.716 0.016

0.000 0.000 0.675 - 0.151 0.016 0.692

Figure 2. True and estimated values of the reflectioncoefficients and their covariances for two AR(3)processes. The two processes differ by the signof K2.

6

60 -Soo

fornl:C,$=So, d, =S,

to find KI solve

N-2C, E.0 +Nd N C,

N- I d, (N-I)dl N- I d,

and choose toot within [- !1.'(' )

. g l

I, -So0 + 2KISO, + KISI,

2 ( l) 6 1

Fornm2,3,- .p:

I Sn-2 Pn-tJ

= b fT Sn- n-l bT

to find Kn solve (for d. > 0)*

K -2n K n.n-, + Ndn N C,,

N-n d, (N-n)d, N-n d,

and choose root within 1- i, I1.

a().fa(n-i .n n-i)

K() .' +n, n- , i = n,2(nI

rbn l - :.I+Cnd I

i= an-i + 2CnKn + dngKn

02(n) =I IV 1

where(n-i) (n-i) (,-I) T

a.Il-I a3 a "a,,-(.-i) (n-I) ("-i) 7bn-l [an-, a.- '. . at

N-i-l I = 0, 1.

{SMn1=S#= / x(.ir1

rTSOO qn-I SO"I X lt n- I I Ix

Sn qT-1 S$-, P'I-I\t I) I i -- ) x (of - II I

,p0, iN,,

"or do -. 0 see Sclin III for dis cusin .t 'roots.

Figure 3. SumMary of Recursive ILE Algorithm.

7

TABI.E ICOMPARIS,s oF AR PARAMFTIR FSPN.,ITIw.N MITI('

N I I Data Point%

a 2 a 1 04

True AR Parameters -2.76070 3.81060 -2.65350 0 92388;Mean-YW -1.01527 061949 0.04466 0 05735Mean-FB -2.5S333 3.42185 2.273o3 0.77262Mean- RMLE -2.6^975 351086 -2.37156 '.81412Variance - YW 3.13780 10.29o27 7,33884 0.7584tVariance 11B 0.11598 0.52574 0 4930 1 ( 08462Variance RMLE 0.10960 033371 0.26227 0,04148

N = 50 Data Points

a, a'2 0"1 d4

True AR Parameters 2.76271 3.8 1 60 2.65 Is() 0I7 2,

Mean YW 1.30552 0.92378 0.0515 (1, 7 19Mean -FB 2.7236H 3.71024 -2.54984 11.87_6Mean -RMLI-. 2.71515 3.69876 -2.54345 h.h?,765Variance-YW 2.22544 8.56804 6.90835 ().745:6)Variance- 1.B 0.00758 0.14138 0.04219 0.I ,,s 4Variance RMLI* 0.01076 0.0448. 0.04069 7.:)4 1

Figure 4. Comparison of AR Parameter Lstimation Methods

" .. ... . II I II II-- . .. . .. .

'0

boC:

4-3

cj'%

/0 LoG C-1. ~iA~ 7/

60-

S-01CRAPS-AA0

-30 2-LZA~A4

/0

0 /0 Zo

Figure 6. Comparison of Frequency Estimators,S77~t .r. 0

10

/0e 001

SA/

Figue 7.Comarisn /o Frequency j Estim ars, 0

0 w

Mi i0 0c C,A Vc-

+

z

ac

zl Zi IL

0

LUI

N zj< 0)

<0. I-I-- +4

wd 4)

o

W (A

zL

- -12

z00

0mw

Ho Nz z 44U, 0

00

:.

000w- w

L N40d

00

0 .. . ......

C;l

loll

00

U-4

LLI w

w In

~>1

0ow

144

REFERENCES

1) J. Burg, "Maximum Entropy Spectral Analysis," Ph.D. Dissertation,Stanford University, Stanford, CA, 1975.

2) K. Steiglitz, L.E. McBride, "A Technique for Identification of

Linear Systems," IEEE Trans. on Aut. Control, Vol. AC-10, pp. 461-

464, October 1965.

3) D. Tufts, R. Kumaresan, "Estimation of Frequencies of Multiple

Sinusoids: Making Linear Prediction Perform Like Maximum Likeli-

hood," IEEE Proc., Vol. 70, pp. 975-989, September 1982.

1

15!

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