Time-delay estimation performance in a scattering medium

so IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 1, FEBRUARY 1985 [3] A. Gelb, Ed., Applied Opfimal Estimation. Cambridge, MA: M.I.T.

Press, 1974.

[4] E. S. Gale, "Passive sonar delay estimate improvement using a priori knowledge and increased number of sensors," M.S.E.E. thesis, Univ. Wyoming, July 1982.

[5] G. C. Carter, "Variance bounds for passively locating an acoustic source with symmetric line array," J. Acoust. Soc. Arner., vol. 62, pp. 922-926, Oct. 1977.

[6] C. H. Knapp and G. C. Carter, "The generalized correlation method for estimation of time delay," IEEE Trass. Acoust., Speech, Signal F’rocessing, vol. ASSP-24, pp. 320-327, Aug. 1976.

| R. Lynn Kirlin received the B.S. and M.S. degree from the University of Wyoming, Laramie, in 1962 and 1963, respectively, and the Ph.D. de- gree from Utah State University in 1968, all in electrical engineering.

He has also had coursework in statistics and a variety of industrial experience. He has analyzed data transmission circuitry and an EMP pulse simulation system at Martin—Marietta (Denver, CO, 1963 to 1964). At Boeing (1965 to 1966) he performed analysis and design tasks for vari- ous space communication projects. Datel (Riverton, WY, 1968 to 1969), afforded him experience in computer peripheral design and analysis. At

Floating Point Systems (Beaverton, OR, 1978 to 1979), he specified and developed a basic image processing library and a signal processing library addition for the array processor AP-12OB. There he also ana- lyzed 1/0 times for the processor with attached disk. Since 1969 he has been on the faculty in the Department of Electrical Engineering, Univer- sity of Wyoming, attaining the rank of Professor in 1978. During this time he has taught nearly 30 different courses in electrical engineering, establishing several in communication theory and signal processing. He has also published nearly 40 technical papers in control theory applica- tions, signal demodulation and detection, pseudonoise applications, and speech and signal processing. His current research and consulting at the University of Wyoming involves image, sonar, and seismic signal processing.

Ernest S. Gale was born in Casper, WY, on Jan- uary 6, 1959. He received the B.S.E.E. and M.S.E.E. degrees from the University of Wyo- ming, Laramie, in 1981 and 1982, respectively, concentrating in the areas of signal analysis and control systems.

Currently, he is employed by IBM's General Product Division, Tucson, AZ.

Time-Delay Estimation Performance in a Scatt,ering Medium

SURENDRA PRASAD, MEMBER, IEEE, M. S. NARAYANAN, AND SAMPATH R. DESAI

Abstract—The effects of a scattering medium on the performance of time-delay estimation are considered. The medium is assumed to ex- hibit angular scattering, causing angular (as well as delay) dispersion and hence loss of signal coherence across the array aperture. Both the variance (via the Cramer-Rao bound) and the bias introduced in the time-delay estimates are studied. The results have been converted to bearing and range error standard deviation and bias. It is shown that there is an optimum range of values for the separation distance between the sensors in the design of an array for time- delay estimation, for range and bearing measurements.

I. INTRODUCTION

T

S. Prasad is with the Department of Electrical Engineering, Indian In- stitute of Technology, Delhi, New Delhi 110016, India.

M. S. Narayanan is with the Defence Research and Development Or- ganization, Sena Bhavan, New Delhi 110001, India.

S. R. Desai is with the Indian Navy.

tions is particularly important for source localization [l ]. A number of workers have therefore analyzed the performance of the range and bearing estimators based on the delay estimates and the geometry of the problem, in terms of variance and bias [ 2 ] - [ 4 ] . Some studies have also reported on the effects of moving targets and/or platforms on the performance of time- delay estimators.

The purpose of this paper is to present some results on the effects of a scattering medium on the estimation performance.

For simplicity, the medium is assumed to exhibit angular scat- tering, causing angular dispersion and hence loss of signal co- herence across the array aperture. The performance in terms of variance is studied via the evaluation of the Cramer-Rao lower bound (CRLB), which is known to be achieved asymp- totically (i.e., if the processing .time is large enough) by the maximum likelihood estimator presented by Carter [5]. How- ever, there are some cases when the CRLB will not be achieved, e.g., the case studied by Scarbrough et al. [8] and Ianniello et al. [9] when the processing time is not long enough. The results of this paper are therefore applicable only when the bandwidth-observation time product is large enough. The bias

PRASAD et al.: TIME-DELAY ESTIMATION IN A SCATTERING MEDIUM 51

introduced by the TDE is also studied here. These results have been converted to the standard deviation and bias of the re- sulting bearing and range estimates. Numerical results are pre- sented for different signal-to-noise ratios (SNRs), and sensor separation distances as a function of a "spatial coherence loss coefficient" introduced in the sequel, for typical signal spectra and observation intervals.

It is found from these studies that there is an "optimum"

value for the separation distance in the design of an array used for time-delay estimation. The optimum distance depends not only on the scattering loss coefficient, but also whether we in- tend to minimize the bearing error variance or the range error variance, or the corresponding biases. Fortunately, however, the performance curves are quite flat near the minima, so that a reasonable compromise is easy to obtain. It should be noted here that the results obtained pertain to a particular scattering model. Hence the validity of these results as applied to differ- ent problems may vary in detail, although the general behavior is expected to be similar.

11. MODEL FOR THE SCATTERING MEDIUM [lo]

A. Generalized Scattering Function

In a general scattering medium, the transmission character- istics of the medium depend on space and time parameters.

We introduce the five-dimensional vector

P’i= M,tⁱ,xⁱ,yⁱ,zⁱI (1)

where fi and ti are frequency and time parameters, respectively, and (xi, yi, zi) are space parameters. Thus, in the most general case, the transfer function I?(-) between two points in the medium depends on the associated vectors Ci.

We assume here that the transfer function process is station- ary in time, frequency, and space. Defining the difference vector

= dl-$2 = [Af, At, AX, Ay, Az] (2) we can then define the space-time correlation function of the medium as

)- (3)

(4) We next define a dual, "transform domain" vector i:

q = [7,CP,u,0,WI

where T = delay rp = Doppler

and where u, u, and ware variables in the angular domain related to the direction cosines cow, cosp, and cosy as shown below:

u = — cos a = — sin 8 cos 0 X X

v = - cos p = — sin e sin <p

x h

1 1

Here X is the wavelength and the angles 0, $ are with reference to a spherical coordinate system. The generalized scattering function L(4) is then obtained as the five-dimensional Fourier transform of the space-time correlation function:

(6) where p ’ q represents the scalar product between the vectors p’ and q and d$ is the five-dimensional volume element

d; = Wf) ,aw,d(Ax),d(Ay), d(Az)l^T. (7) The inverse transform of L(;) yields

(8) The scattering function L(i) describes the distribution of the signal power with respect to channel delay, Dopplerrp, and the three angular coordinates u, u, and w.

B. Special Case of Angular Scattering

Here we consider a simple example of a hypothesized me- dium, which has only angular scattering in one plane. In other words, the space-time correlation function is assumed to depend only on the separation distance Ax, so that the vector 6 be- comes a scalar variable Ax, and the vector < contains the scalar variable u = l/h cos ar. Let 8 = 90 - ar, so that u = l/h sin d.

We then have

L(6)

= ^[

In order to get a clear picture of the above relation, it is in- structive to obtain (9) from elementary principles for the case of two sensors separated from each other by a distance Ax, as shown in Fig. l(a). The scattering model is summarized in Fig. 1(b).

Assume that the source is transmitting an impulse 6 (t). Due to angular scattering, the two sensors receive energy from an angular sector, say between (8, - 0m to 0, +8,), where Bo

is the mean direction and Om is usually small. Assume first, that the energy comes via, say N, different directions withm this sector, Le., from N different plane waves. The kth plane wave arriving from direction ek has the associated delays ?-k k Ax sin ekl2C at the two sensors, where ?^-k is the delay of the plane wave to the midpoint along the line of the two sensors, relative to an unscattered plane wave in the mean direction.

Thus the impulse response of the medium to the two sensors is given by

(t, ±=?j =^

w = — cos 7 = — cos 1

X x (5)

where ak is the strength associated with the kth plane wave.

The corresponding transfer functions are given by

(

/,± —) = £

2 / fc = i

52 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 1, FEBRUARY 1985

k ' th plane wave

Scattering medium

Scattered energy

Fig. 1. (a) The array geometry for time-delay estimation. (b) The scattering model. Arrows in the scattered energy indicate directions of plane wavefronts.

In the limiting case, when we assume a continuum of plane having uncorrelated cross sections, we have waves arriving from all directions [-e, to e,] where 8, is P

small, we have n (14)

a(6) e

giving the distribution of energy in the angular domain. This where the mean direction Bo is taken to be the broadside direc- definition is motivated by the analogous uncorrelated scattering tion, without loss of generality, and where a(6) and T(Bj now assumption of Bello for a time-varying channel [ l l ] . From represent the path strength and relative delay, respectively, (13) and (14), it follows that

associated with the plane wave arriving at an angle 8.

The space-time correlation function for this case, then be- e,

comes R^H(Ax)= (( L(d)8(6'-e)e-⁽ⁱ²«f^Ax!^2c)⁽-^sine'^+!iiae'>de

m

=Ji E[a(6)a(6')e'

"

-Or,

- J L r

which is the inverse transform of (9), as required.

(1³) C. Remarks

Assuming uncorrelated scattering, i.e., assuming that the wave- 1) The one-dimensional scattering function L(0) introduced fronts coming from different directions arise from scatterers above presents a highly simplified picture of the real medium

PRASAD et al.: TIME-DELAY ESTIMATION IN A SCATTERING MEDIUM 53

t o 9 H, [f) H^* ( f r o m C a r t e r C51)

Fig. 2. Generalized cross-correlator for time-delay estimation (from [5]).

characteristics in that it does not reflect the loss in coherence between the signals at the two sensors (or across the array aperture, as the case may be) due to other effects of scattering, viz. spreads in the delay and Doppler domains,.etc. However, the simplicity of the resulting model not only permits easy evaluation of its effect on the time-delay estimation perfor- mance, but also makes the interpretation of the results simpler.

It is, of course, of interest to generalize the results to be pre- sented in the next section for more general scattering models.

2) It has been shown above that the space-time autocorrela- tion function and the scattering function are Fourier transform pairs. Thus if the scattering surface is assumed to be associated with a Gaussian, space-time autocorrelation function, then the resulting scattering function also has a Gaussian shape [7].

Thus, if we assume

where

we then have

(16)

(17) where u is defined to be "spatial coherence loss coefficient."

The reciprocal of u is a measure of the "coherence distance."

It is this correlation function which is used in the next sec- tion for typical performance calculations.

111. PERFORMANCE OF TIME-DELAY ESTIMATORS IN INCOHERENT MEDIA

A. Cramer-Rao Bound for TDE

Carter [5] has derived an expression for the variance of the time-delay estimate in the neighborhood of the true delay, for a general cross correlation receiver with a weighting func- tion W^g(f) (see Fig. 2). This is given by

\W

(f)\

G

(f)G

(f)[l-C

(f)]f

df

U

8n2Tl\ \GXiX2(f)\Wg(f)f2df\

(18)

Gxlx, ( f ) = c r o s s spectrum of the two signals x1 (t) and x2(t) received at the two sensors (19a) Gx,xl ( f ) , G~Z~2 ( f ) = autos~ectra of x,(t) and x,(t),

respectively (19b) C12(f) = magnitude square coherency function

<?*,*,(/)

T = observation interval.

(19c) ( 1 9 4 In particular it has been shown that for the maximum likeli- hood (ML) processor, with

[GXlXl(f)\[l ~ Cl2(f)] (20)

the variance is givpn (under high SNR conditions) by var^{MLn = ,}

I

(21) which is identical with the Cramer-Rao lower bound giving the minimum obtainable variance for delay estimation. This ex- pression has been extensively evaluated in the literature for various special cases. It takes a particularly simple form, being a function only of the bandwidth, T, and the SNR, when the signals have flat spectra. For example, Quazi [3] has shown that for the ideal medium variance is given by

var

-(

\ 1

\47r

r/S]S

1 for SNR>> 1 (21a) when the signal autospectrum is flat between f, to fi Hz. It should be noted, however, that even the ML estimator attains the CRLB performance of (21) only under certain conditions.

It has been shown by Scarbrough et al. [8] that for a given bandwidth-observation time product, there is a threshold SNR below which CRLB performance will not be attained.

It is this bound, however, that has been evaluated in the sequel to compare the performance in an ideal medium with that for a nonideal or scattering medium.

54 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 1, FEBRUARY 1985

It is easy to see that, for our scattering model,

X2{f) =

S{f) , ^fj + N

(f)

(22)

(23) where S(f) is the spectrum of the desired signal, D is the delay to be estimated, and N,(f) and N2(f) are the spectra of the noises in the two sensors, assumed to be statistically indepen- dent with respect to (w.r.t.) each other as well as with the signal of interest. It follows that

(24) where G,(f) ~ E{lS(f)lz } is the autospectrum of the signal.

Thus the value of the cross-spectrum G,, x z(f) and hence of the coherency function C, 2 (f) would depend on the separa- tion between the two sensors.

A close examination of (16) and (24) clearly shows that for the scattering medium considered in the previous section; as Ax increases, Gx x (/) decreases and it follows from (21) that the variance of the delay error increases.

B. Bearin& and Range Error Variances

The bearing error variance is related to the time-delay error variance by the relation [6]

-M-V

\ A x c o s 0 /

(25) where c is the velocity of sound in the medium, u; is the variance of delay estimation error, and 19 is the true angle of the source, with respect to the perpendicular to the line join- ing the two sensors.

For an ideal medium, therefore, the bearing error variance decreases as the separation distance between the sensors is increased, even though it is very high for small angles of the source with respect to the array line.

Since in a scattering medium UD is expected to increase with separation distance due to coherence loss, it follows that there will be an optimum distance for which OQ is mini- mum. This is demonstrated in the next section, where detailed numerical results are presented for the performance.

Range can be estimated with a minimum of three sensors.

Let the distance between both adjacent pairs of sensors be equal and let the bearing of the source from the central ele- ment with respect to line perpendicular to the array be 0.

Assuming that OD as obtained from both pairs is also same, it can be shown that [6]

OR = range error variance / V2R*c V 2

\(Ax)2cos20/ °D

where R is the true range.

(26)

Once again, unlike in an ideal medium where the performance is expected to improve as the fourth power of the separation distance, a range-dependent optimum separation distance would yield the minimum value of the range error variance for a scattering medium.

C. Bias Effects

Quazi [6] has derived expressions for the bias introduced in the bearing and range calculations from time-delay measure- ments. The results are reproduced here for convenience:

OB = bearing bias = (-0; tan 0)/2 where 0 is the true angle, and

~ range bias = — R

(27)

(28) where R is the true range.

It is seen that bias in angle measurement is very high for graz- ing angles close to end fire and is negative or positive depend- ing on 8.. The range bias also becomes significant when the range variance is of the order of true range. It maybe recalled from (26) that u& a R4 so that the range variance can quickly become greater than or equal to R asR increases.

IV. TYPICAL PERFORMANCE CALCULATIONS AND NUMERICAL RESULTS

In this section we present typical performance curves for the estimation of bearing and range from time-delay measurements in scattering media. The parameters selected for study are the effects of SNR, separation distance between the sensors, and the "spatial coherence loss coefficient" introduced in Section 11.

The signal and noise spectra are assumed to be flat and band limited between 3500 and 4500 Hz. It was felt that the per- formance of the ML estimator is mainly dependent on the SNR, rather than the signal and noise spectra, hence spectrum shape was not taken as a parameter in these studies. The ob- servation time is taken to be 50 s, and has not been varied since oh is known to be proportional to 1 /T. The speed of sound has been assumed to be 1500 m/s.

In order to provide a "benchmark" of comparison, the asymp- totic performance of the maximum likelihood estimator in an ideal medium and additive Gaussian noise is first briefly summarized.

A. Performance of the TDE in Additive Noise in an Ideal Medium

Bearing Errors: Figs. 3 and 4 illustrate the important per- formance features of the bearing standard deviation (BSD) OQ in an ideal medium, with respect to SNR and sensor separa- tion distance, respectively. Although bearing bias 0 is not plotted here, its behavior is very similar to that of U$ in view of its direct dependance on the latter via (27). Following are some of the important observations from Figs. 3 and 4 and (27).

1) BSD (ae) varies inversely with the separation distance Ax, even though uD is independent of Ax in an ideal medium (Fig. 4).

PRASAD et ah: TIME-DELAY ESTIMATION IN A SCATTERING MEDIUM 55

zo

cco o

<

z

U'U

7-5

5 0

0

\

A

\ \\

-20

\ \

\ \\ \ _{\ \}

\ \

1 -15

— AX

R"i r inn ^-P

* • • ! • • ^,,J t t \ ^ t 1 1 1 U * ^ U ^ - , p .

Range SD.(R = 5000Meters , 9 = 0°)

= 150 M e t e r s

I ^**^~l

-10 -5 0 SO

- 40 _

- 3 0

- 2 0

- 10

UJ O

§

Q Z

CC<

SNR ( in DB's)

Fig. 3. SNR dependence of range and bearing standard deviations:

ideal medium.

UJ

a acc.

oz<

z

Q:

UIm 4 -

B e a r i n g 5.0.

Range SO-

J _ 18 36 5L 72 90 108 SEPERATION DISTANCE BETWEEN SENSORS ( m e t e rs )

Fig. 4. Dependence of, range ahd bearing standard deviations on sensor separation distance: ideal medium.

0.

z

e

o 85 c

UJa cco

<f

o

UJto

<CE;

12G

2) The value of u^o also depends on the true bearing, being as low as 9.6 X lob³ degrees for 0 = 0 " (broadside direction) at an SNR of -20 dB and a separation distance of 150 m (Fig. 3).

3) The bearing bias &B is zero for 0=0". For small grazing angles with respect to the line of the array, however, the bias becomes very high in view of the tan 8 dependence (27).

Range Errors: The behavior of the range standard deviation (RSD) with respect to the SNR and Ax is also summarized in Figs. 3 and 4; respectively. The range bias RB depeqds' directly

on & (28). Some of the important features of OR and RB are as follows.

1) RSD, like BSD, displays an inverse relationship with re- spect to the SNR (Fig. 3).

2) RSD decreases more rapidly than BSD as the separation distance between the sensors is increased. This isbecause of the inverse square-law relationship between UR and AX (Fig. 4).

3) The range bias R^B depends both on range and bearing.

The bias, like the RSD OR , is least for a broadside direction (e=0").

56 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. I, FEBRUARY 1985 75, , 1 200

- 60

2 45

a

Q O

cr

~r

\

\

B e a r i n g s t a n d a r d d e v i o t t o n

<r : 0 . 0 0 1 , AX : 150 m e t e r s

R a n g e s t a n d a r d d e v r o t l o n

<r = 0 001 , AX = 1 50 m e t e r s

S N R ( i n d B ' s )

Fig. 5. SNR dependence of range and bearing standard deviations:

nonided medium (u = 0.001).

180

160

1LO

4) The range bias, however, has an inverse square-law de- pendence on the SNR, an inverse fourth-law dependence on the separation distance Ax, and a direct R³ dependence on range. In order to obtain a reasonable value for RB, therefore, it is required to choose an appropriately large separation dis- tance. For example, with a separation distance of 150 m and an SNR of -20 dB, the value of the bias is only 0.316 m for a true range of 5000 m.

B. Performance of TDE in Scattering Media

For the following results, the spatial correlation function of (16) is used for calculating the cross-spectrum between the two received signals xl(t) and x²(t) via (24). The time-delay estimation error is computed via numerical evaluation of the integrals involved in (21), which in turn is used for the com- putation of bearing and range error standard deviations and biases. In order to study the effect of the medium, two repre- sentative values of u are chosen [IO], viz. u = 0.001 and u = 0.01. Figs. 5-8 demonstrate the important performance fea- tures of' range and bearing measurement in a scattering medium.

The following important observations can be made.

Bearing Standard Deviation (BSD):

1) Although the bearing standard deviation decreases with the SNR as in the ideal medium case, it can be seen that for a given SNR, BSD is considerably larger in the scattering medium.

For example, for an SNR of -20 dB, the BSD is 9.6 X 10"3

degrees in an ideal medium (Fig. 3) as against 55.3 X 10~³ de- grees in the scattering or incoherent medium (Fig. 5).

2) The behavior of the BSD in the scattering medium with respect to the sensor separation distance is illustrated in Fig. 6.

As expected, we have an optimum separation distance for the minimum value of BSD. For the two cases shown here at an SNR of 0 dB, i.e., u = 0.01 and u = 0.001, the optimum dis- tances are 8 and 75 m, respectively. The corresponding mini- mum values of BSD are 6.7 X 10~3 and 0.673 X 10~3 degrees, respectively.

It is interesting to observe, however, that for these cases the performance remains nearly constant (equal to the optimum value) when the separation distance is varied around the opti- mum value.

Range 5hndard Deviation IRSD):

1) A comparison of Figs. 3 and 5 shows a considerable loss in SNR performance in the nonideal medium. Thus, with a true range R = 5000 m and a sensor separation distance of 150 m, the range standard deviation is 11.4 m in the ideal medium (u = 0.001) at an SNR of-15 dB.

2) The behavior of RSD with respect to the sensor separa- tion distance is also similar to the corresponding behavior of BSD. Fig. 7 shows the existence of optimum values for the separation distance, the optimum values being 11 m for u = 0.01 and 110 m for u = 0.001. The corresponding values of RSD are 376 and 3.7 m, respectively. Once again, however, the curves are reasonably flat near the origin, although not as flat as for BSD.

Biases: Similar conclusions can also be drawn for bias in the range and bearing measurements. The results for range bias are summarized in Fig. 8. In fact, the behavior of the biases is very similar to the corresponding variances, except for scaling factors which depend on the true range and bearing [e.& see (27) and (28)].

PRASAD et al.: TIME-DELAY ESTIMATION IN A SCATTERING MEDIUM 57

3 0 -

o. 24 o

^ 18

<

O

O CC O 2

<

O

CC

o

<

oa:

<

en o

.

CD

12

I

3 .0

2-4

1-2

0 6

0 0

8 12 SEPARATION DISTANCE ( meters 1

(a)

SNR : 0 dB 6~ -- 0 0 0 1

16

I 20

I

. 80 SEPARATION

120.

DISTANCE ( m e t e r s )

180 200

(b)

rig. 6. Dependence of bearing standard deviation on sensor separation distance: nonideal medium. (a) u = 0.01. (b) u =0.001.

Choice of Sensor Separation Distance: A comparison of Figs. 6 through 8 clearly shows that the exact values of the separation distance are different for the minimum values of BSD, RSD, and the corresponding biases. However, for a given

value of u, these values are reasonably close. For example for a = 0.01, the minimum value of BSD is obtained for AX = 8 m, whereas the minimum value of RSD is obtained for Ax = 11 m.

This fact, coupled with the flatness of the associated curves,

58 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 1, FEBRUARY 1985

700

350

o

C£

O 2

<

I—

to UJ

CE 0

SNR = Od B

True range : 5000 meters 6- : 0:01

_ L

12 16

70

O

5

Q

cr

Q

35

S E P A R A T I O N D I S T A N C E ( m e t e r s ) (a)

SNR =0 dB

True range = 5000 meters tf~ = 0 001

80 SEPARATION

120 DISTANCE (b)

160 ( mete r s )

200 240

Fig. 7. Dependence of range standard deviation on sensor separation distance: nonideal medium. (a)u = 0.01. (b) u = 0.01.

enables us to obtain a compromise value of Ax. Thus, for the examples presented here, it may be seen that a reasonable choice of Ax = 10 m for u = 0.01 and AX = 100 m for u = 0.001 can be made, without undue degradations in the BSD, RSD, and the corresponding biases.

V. CONCLUSIONS

It can be concluded that the concept of an optimum separa- tion distance is a very important parameter in the design of an array for range and bearing estimation based on time-delay

PRASAD et ai.: TIME-DELAY ESTIMATION IN A IN MEDIUM 8 Or

SN R : O d B

True ronge : 5000 meters 641- 6~ = 0 01

48

< 32

CO

CD

16

J _ _L

12 15

SEPARATION DISTANCE ( rnelers 1 (a)

8 - 0

I/)

<£

CD

4.8

3-2

1-6 -

0 0

SNR : O d B

True ronge :5000rneters

<r = 0-001

40 80 120 160 200 240

SEPARATION DISTANCE ( meter s I (b)

Fig. 8. Dependence of range bias on separation distance. (a) u = 0 01.

(b) u = 0.001.

measurements. The optimum separation, however, depends not only on the scattering loss coefficient but also whether we intend to minimize the bearing error or range error standard deviation, or the corresponding biases. Hence the design of the array calls for an appropriate compromise. Fortunately,how- ever, the performance curves are quite flat near the minima, so that a reasonable compromise is easy to obtain.

REFERENCES

[I1 Speck1 Issue on Time Delay Estimation, If?El? Trans. Acoust., Speech, Signal Process, vol. ASSP-29, June 1981.

[21 c. H. Knapp and G. c. Carter, "Thegeneralized correlation method for estimation of time delay," IEEE Trans. Acoust., Speech, Sig- nal Processing, vol. ASSP-24, pp. 320-327, Aug. 1976.

[3j A. H. Quazi, "An vol.ASSP-24,overview of the time delay estimate in active and passive systems for target localization," IEEE Trans. Acoust..

Speech, Signal Processing, vol. ASSP-29, pp. 527-533, June 1981.

60 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 1, FEBRUARY 1985 [4] P. M. Schultheiss and E. Weinstein, "Lower bounds on the local-

ization errors of a moving source observed by a passive array,"

IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 600-607, June 1981.

G. C. Carter, "Time delay estimation," F’h.D. dissertation, Univ.

[5]

[6]

Connecticut, Storrs, 1976.

A. H. Quazi, "Lower bounds on target localization errors for various signal and noise characteristics," presented at Neuvieme Colloque sur le Trait du Signal et Ses Applications, Nice, France, May 16-20,1983.

[7] A. N. Venetsanopoulos, "Modelling of the sea surface scattering channel and underwater communications," in Communication Systems and Random Process Theory, J. K. Skwirzynski, Ed. The

Netherlands: Sijthoff and Noordhoff, 1978.

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is in charge of the Signal Processing Group at the Centre for Applied Research in Electronics (CARE). He was a visiting Research Fellow at the Loughborough University of Technology, Loughborough, England, from August 1976 to August 1977, where he was involved in developing algorithms for adaptive processing for high-frequency arrays. His teach- ing and research interests are radar/sonar signal processing, communica- tions, and computer-aided design of digital systems. Currently, he is engaged in research in the areas of sonar and seismic signal processing, underwater communications, and array signal processing.

M. S. Narayanan, photograph and biography not available at the time of publication.

Surendra Prasad (S72-M75) received the B.Tech. degree in electronics and electrical com- munication engineering from the Indian Institute of Technology, Kharagpur, India, in 1969, and the M.Tech. and Ph.D. degrees in electrical en- gineering from the Indian Institute of Tech- nology, New Delhi, India, in 1971 and 1974, respectively.

He has been teaching at the Indian Institute of Technology, New Delhi, since August 1971, where he is presently an Assistant Professor and

Sampath R. Desai was born in Dharwar, India, in 1954. He graduated from the National De- fence Academy in 1974, did his basic electrical engineering courses at the Naval College of En- gineering, Lonavala, and received the M.Tech degree from the Indian Institute of Technology, Delhi, in radar and communications.

Presently, he is serving in the Indian Navy with the rank of Lieutenant. His current re- search interests are sonar signal processing and sonar systems.