sea lion echo ranging

16.6, 16.7 Received 12 December 1966

Sea Lion Echo Ranging*

H. N. SmtVER AND T. C. •POULTER

Stanford Research Institute, Menlo Park, California 94025

Recordings of a short series of sea lion pulses are analyzed. The approach used in the analysis is to infer echo-ranging capabilities based upon the full utilization of the signal properties. The analysis has led to the hypothesis that, in addition to echo-ranging pulses and possible communication signals, the sea lions--at least in some circumstances--generate and transmit signals that are used in the detection of target returns of echolocation pulses. In this framework, the actual echolocation pulse is the precursor pulse, and the main pulse is generated as part of the detection procedure. The resulting cross-correlation detection approximates the optimum detection technique. Finally, it is concluded that, for the signals considered, range may be measured to within approximately =!=4 cm and that, for range rates of interest--say, up to 5 m/sec--the signals provide essentially no range-rate discrimination.

INTRODUCTION

HE study of the echo ranging used by many biological species has been approached in several ways, most frequently by biological studies and sophis- ticated experimentation. The approach used here is an application of communication theory to the study of sea lion echo ranging. It has been particularly influenced by the work of Cahlander on bat signals. 1 Our approach is based on the premise that the sea lion has evolved a good echo-ranging system. The biological aspects of the system are de-emphasized, and emphasis is placed upon the signal design and an inferred receiver fully utilizing the signal properties.

In Secs. I and II, the properties of echo-ranging signals are examined in terms of the effect of a target upon the signal and of the capability of the signal to measure range and range rate. In Sec. III, the properties of sea lion signals are analyzed, and a signal detection procedure is hypothesized. Finally, the capability of the sea lion to measure range and range rate is discussed.

I. ECHOLOCATION TARGET RETURN

In this Section, we examine some of the effects of a moving target upon a signal that has been reflected from the target. The channel through which the signal

* This research was supported under the Independent Research Program of Stanford Research Institute.

x D. A. Cablander, "Echolocation with Wide-Band Waveforms: Bat Sonar Signals," MIT Lincoln Lab. Tech. Rept. 271 (6 May 1964), AD-605 322.

428 Volume 42 Number 2 1967

is transmitted is assumed to be deterministic and non-

dispersive, and the target is assumed to be a point target. An effective point target (with linear dimensions small, relative to signal wavelengths) is assumed, so that we do not need to consider the signal-time dis- persion introduced by the target. Since it is conceivable that the echolocation signal may be used for the deter- mination of target velocity as well as range, we concentrate upon the effects of these phenomena upon the target return. Signal attenuation is neglected, since it is not of direct concern.

If a signal source, a receiver, or both, are in motion relative to the medium in which the signal is propagating, the received signal is modified by what is generally known as the Doppler effect. Suppose that a source and its collocated receiver are moving with a velocity vs and that a target is moving with a velocity vt, as indicated in Fig. 1. The velocity of propagation of the signal in

SOURCE AND

RECEIVER TARO ET

• v s • v t

I- Fzo. 1. Echolocation geometry.

the medium is designated c. The frequency as seen by the target is given by

(1)

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00:41:19

SEA LION ECHO RANGING

As the signal is reflected, the target acts as a source, and the received frequency is

fr = [ (cq-vs)/ (cq-vO'] ft

= (cq-vs/cq-vt)(c-vt/c-v•)f•. (2)

Next, we define a paramater a as

c2+(vt--vs)c--v•v• (3) o

If we further define the range rate v as the difference between the target velocity and the source velocity and assume that both these velocities are small as com-

pared to c, then

-' l q- (2v/c). (4)

For a sinusoidal transmitted signal of frequency f,, the frequency of the received signal is

f,= fda. (5)

For nonsinusoidal signals, each frequency is modified as if it were a sinusoidal signal. For a real signal s(t), the Doppler effect may be examined by considering its Fourier transform,

S (f) = f • (t) e -,2,•tdt. (6)

The Doppler effect modifies each frequency in the Fourier transform by the factor 1/a and, as a result, yields the modified time waveform

s'(/) = f_• =as(at). (7)

Thus, the significant difference between the echo and the transmitted waveform is a time-scale change.

The major effect upon signals used for echolocation is that of time delay. If a signal is transmitted from a source, reflected from a target, and detected at a receiver collocated with the source, the resultant signal delay is a direct measure of the distance or range•between the source and the target. If the velocities and range are as specified in Fig. 1, then the time delay is given by

r = 2Rc/(d-- cv-- vsvO

-' (8)

where, as before, c is the velocity of signal propagation, and the range rate is

v=v•-vs. (9)

In addition, for the approximation, we have made the assumption that both v, and vt are very small as compared to c.

For a transmitted signal s(t), the received signal has the form s(t--r). In terms of S(f), the Fourier transform of s(t), the delayed signal is

S(f)ei•'•'ct-•)d f. (•0)

Thus, in the frequency domain, the delay is introduced by multiplying S(f) by the factor exp(--i2,rfr).

Although we have considered the Doppler effect and the time-delay effect separately so far, the target return is affected by both these phenomena stimultaneously. It is given by

y(t) = f_ S e-i•I•ei2•Itd f =asEa(t--r)-]. (11)

We have neglected any signal attenuation and have assumed that there is a point target and that the propagation channel does not introduce any distortion.

II. SIGNAL DETECTION AND PARAMETER ESTIMATION

A. Basic Receiver Operation

A principal objective of this program was to determine how well the sea lion can measure range and range rate, on the basis of a set of measured sea lion pulse transmissions. To approach this problem, we must postulate a form for the receiver, which, in the absence of detailed information, may or may not model the actual detection procedure used by the sea lion.

As the basic receiver element, we postulate the matched filter, the linear filter that maximizes the output signal-to-noise ratio for any white additive noise. (For detailed discussion of matched filters, one may refer to Ref. 2. An introductory discussion has been given by Turin. a ) Although the matched filter may also be developed for additive noise with an arbitrary spectrum, a we limit our attention to the case where the additive noise has a uniform power spectrum over frequency limits large compared to signal spectrum. If the transmitted signal is designated s(t), then the impulse response of the matched filter is

h(t)=s(--t). (12)

If we designate the received signal y(t), the response of this filter at a time t is

z(t)= f_ y(x)s(x--t)dx. (13) •' "Matched Filter Issue," IRE Trans. Inform. Theory 6, 310-

417 (1960). a G. L. Turin, "An Introduction to Matched Filters," IRE

Trans. Inform. Theory 6, 311-329 (1960).

The Journal of the Acoustical Society of America 429


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SHAVER AND POULTER

The matched-filter response may be recognized as a cross-correlation function. If the received signal is the return from a point target with possible range rate, we have

y(t) = ks[a (t- r)•q- n (t), (14)

where k represents the signal attenuation, a is due to the Doppler effect (discussed in Sec. I), and n(t) represents the additive noise. Suppose that the transmitted signal energy is

fl E= s•'(t)dt; (15)

then the energy in the signal portion of the received signal is given by

k 2 k%2[a (t- r)•dt=--E. (16)

Therefore, the signal a«s[a(t--r)• has an energy E. The matched-filter response may be written as

J --oo \Or TM

and

where

q- f_ n (x)s (x- t)dx (17)

z(O)=--7X(r,v)+ n(x)s(x)dx, (18)

X(r,v)=-- s[a(x--r)']s(x)dx. (19) E 0o

The quantity X(r,v) is the normalized cross correlation between two signals of equal energy--one representing the transmitted signal, and the other the return from a target moving relative to the signal source.

The value of the parameter c for sound propagating in sea water is 1505 m/sec, and , will generally lie between +5 and -5 m/sec. The parameter a will thus be near 1.0 for all cases of interest. The quantity X(r,v) may be thought of as a filter response with the filter matched to the transmitted signal, but not to the return signal. By examining this function for various degrees of mismatch in both the r and v parameters, we may gain an understanding of how well our hypothesized receiver may discriminate in range and range rate.

We have chosen our receiver to have the form of the

matched filter, stating that a receiver of this form maximizes the output SNR, and we have assumed the noise to have a uniform power spectral density. If, in addition, we assume that the noise is Gaussian, the matched-filter receiver maximizes the probability of detection for a fixed probability of false alarm (Neyman-

Pearson criterion). 4 With the Gaussian noise assumption and application of the method of inverse probability, 5.ø we also arrive at a receiver specification that is the cross-correlation receiver or matched filter. (The matched filter is theoretically identical to the cross- correlation receiver, although the physical implementa- tions may differ.)

B. Signal Function

We have seen, in the preceding section, that the ability of the receiver to discriminate in range and in range rate depends upon the ratio of signal energy to noise-power density and upon the quantity we have designated X(r,v), and which we term the signal function. (This terminology has been used by Cahlander. • Apparently, 7 it was first used by Woodward a for the narrow-band approximation of the quantity we wish to describe. In our case, however, such an approximation may not be valid.) Recall that the signal function is given by

(20)

where, for simplicity, we have normalized our two signals so that each has unity energy.

Let us now define the Fourier transform pair,

So(f)= f_ so(t)e-V"•S'dt, (21)

so(t) = f_ So(f)e+i2'•S'd f. (22)

Then, by using the fact that the signal So(t) is purely real, an alternate form for the signal function is

(23)

4 W. W. Peterson, T. G. Birdsall, and W. C. Fox, "The Theory of Signal Detectability," IRE Trans. Inform. Theory 4, 171-212 (1954).

a P.M. Woodward and I. L. Davies, "Information Theory and Inverse Probability in Telecommunications," Proc. IEE 99, Pt. III, 3•-$$ (1952).

• M. 1. Skolnik, Introduction to Radar Systems (McGraw-Hill Book Co., New York, 1962), pp. 427-430, 471-474.

• Ref. p. 31. s p.M. Woodward, Probability and Information Theory, with

Applications to Radar (Pergamon Press, Inc., New York, 1953).



00:41:19


By separating the last integral into two ranges, and by using the fact that so(t) is real, we obtain

= f0 s0(-V)s0*

+f0 S0(a•f)S0* e••df (2•)

=2Re ••So(alf)So*(•)ei•df}

From the definition of a and for range rates very small relative to the velocity of propagation,

a«'-lq-v/c, and 1/ai-'l-v/c. (25)

Thus, with these restrictions on the range rate,

X(r,v) -' 2 Re So f_fv

So* -Jr--- e-i9"•Y•(i+•/*)d ß X (f c) f} (26) In those cases where So(f) is nonzero only in small

regions of f near =t: f0 (narrow-band cases), the signal function becomes

X(r,v) '- 2 Re /0 © X So*(f-+-f•)e-e•s•(•+•/*)df}. (27)

If we now define the Doppler shift

•= 2for/C, (28)

which, as Turin points out, a is meaningful only for narrow-band bandpass signals, the signal function may be written as

x(r,•)-' 2Re{foøøSo(f--i)So*(f-+-i) X e --i2a'f r (1-t-v/2/o)df]. (29)

We now consider the relationship between the signal function and the ambiguity function that has evolved from Woodward's work s and has been used extensively for signal design and analysis/.6 especially in applications to radar signals (see, for example, Klauder9). In

the literature, minor variations are found in the definition of the ambiguity function. We choose a form used by Klauder, 9 among others, which, in our terminology, becomes

A(r,v)= z* t-- z t+ ei•edt, • 2/

(30)

where z(t) is the complex envelope of the real signal So(t) (see Appendix A). Several alternate forms for the ambiguity function are developed in Appendix A. One form of particular interest is

A (r,r)= e-e•so•[foøø So(f--i)So*(f-+-i)ei2•d f --f0 I•l/u So(f--i)So*(f-+-i)eiU•-r•dfl. (31)

When fo>>lz, l/2, and so(t) is a narrow-band signal, then

So f-- So* f-+- e e•i•d f --' A (r,•,)e i2'•o•, (32)

and it follows that

X(r, vc/2fo)-' 2 Re{A (--r,v)e-'U•o•}. (33)

Therefore, with the restrictions that have been indicated, there is a simple relationship for the signal function in terms of the ambiguity function.

III. SEA LION ECHOLOCATION

Sea lions emit a series of pulses, and it is fairly well established that they use these signals for echolocation? In this study, we have chosen to select a relatively short recording of a series of these pulses and to examine in detail the properties of the signals. By performing this detailed study, we hope to gain a greater understanding of how the sea lion actually processes the signals and how well the sea lion can measure range and range rate under ideal conditions. On the first point, we de- emphasize the physiological aspects of the sea lion and concentrate upon the signal design and an inferred receiver based upon the full utilization of the signal properties. Finally, by restricting ourselves to this short (but typical) signal sample, we enhance our capability for detailed signal analysis. However, our conclusions do not imply that at some other time the signal structure and, consequently, the inferred receiver processing, may not be quite different.

9 j. R. Klauder, "The Design of Radar Signals Having Both High Range Resolution and High Velocity Resolution," Bell System Tech J. 39, 809-820 (1960).

x0 T. C. Poulter, "The Use of Active Sonar by the California Sea Lion," J. Auditory Res. 6, 165-173 (1966).



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SHAVER AND POULTER

GO0 "'l I' I I • ' • 1 i ' I ' Precursor Main Pulse I

400 - I I -

c• '\\ I _

< - i"vVV-V II j • • I• 'v'vj•' Vl '•JV' •-200 - / - • /

-400 -

.6000 , I , I , I , I , I , f , I , I 2 3 4 5 6 7 8

TIME • msec

Fro. 2. Sea lion Pulse 1.

A. Sea Lion Echolocation Signals

Analog recordings of sea lion signals made by Poulter have been used in this program. These recordings of the California sea lion have been analyzed from a different point of view by Poulter. n A short section of the analog recordings was converted into digital form for computer analysis. The signals were effectively low-pass filtered to the range 20-72 000 cps and sampled at a rate of 144 000 samples/sec. The digital record representing 768 msec contained a total of 21 sea lion pulses.

The recordings were made with the hydrophone slightly off the direct path and between an approaching sea lion and a fish that had been placed in the water. The recordings thus represent the sea lion sound emissions as received at the hydrophonesthe sum of the original emissions and the multipath reflections. Eighteen of the sea lion pulses are reproduced as Appendix B. Each record has two significant regions, designated the precursor pulse and the main pulse. These two regions, which are not precisely defined, are shown in Fig. 2. As indicated, the end of the main pulse is determined approximately by the envelope, unless it is clearly terminated (see for example, Pulses 10 and 11, in Appendix B). The signal preceding the precursor pulse is noise; the signal following the main pulse is due to multipath transmissions of the source signal to the hydrophone.

B. Receiver Structure

In Sec. III-A, we have indicated the conditions under which the matched filter is optimum. In view of the signal structure, we initially hypothesize a receiver utilizing two matched filters as indicated in Fig. 3. We are not now concerned with the problem of how a sea lion might realize such a receiver configuration but only with what the two matched-filter responses are and ultimately with how well range and range rate can be measured.

A computer program that provides the matched- filter response to the recorded signal has been developed

n T. C. Poulter, "The Sonar of the Sea Lion," IEEE Trans. Ultrasonics Eng. 10, 109-111 (1963).

RECEIVER

INPUT

PRECURSOR PULSE

MATCHED FILTER

MAIN PULSE MATCHED FILTER

OUTPUT .,

OUTPUT ,. ,

B

FIC,. 3. Hypothesized receiver configuration.

(this is discussed in Appendix C). The filter is matched to either the precursor pulse or the main pulse. The responses for Pulses 1 and 3 are given in Figs. 4 and 5, respectively. The responses have been normalized so that the value at zero shift (r=0) is unity. The responses for the main-pulse matched filters indicate what might be expected in the way of responses to multipath transmissions to the hydrophone. On the other hand, the responses of the precursor matched filter show the strong cross correlation between the precursor pulse and the main pulse and, similarly, between the precursor pulse and the received multipath transmissions of the main pulse. Although this leads us to reject the hypothesized receiver structure, it provides an im- portant clue to an alternative structure. We have examined the precursor pulse matched-filter response for two more pulses (Fig. 6). For Pulses 5 and 8, the

6j_ J PRECURSOR PULSE MATCHED- - -- m FILTER RESPONSE

4

-

-2

-4

-2 0 2 4 6 8 I0 12 14

0.5 [I Iljt - [I,,,,,hllllll,,., ......... ,.,,,,.,...,, ..... ,,,,,, ...... ' t.,,,.i ,,,,,,,- ..... ....,.,.,......,,,,,,,,....

ø"t "1111" - 1.0

-2 0 2 4 6 8 I0 12 14 TAU• msec

Fro. 4. Matched-filter responses of sea lion Pulse 1.



00:41:19


strong cross correlation between the precursor pulse and the main pulse is not so clearly evident from the pulses themselves (Appendix B); however, from the responses shown in Fig. 6 it is apparent.

c. •ew Receiver Structure

We hypothesize a receiver that utilizes the transmitted main pulse as a locally generated cross-correlation signal, rather than for direct echolocation. At the time that the main pulse is generated, the sum of this main pulse and any target echoes that might have been received as a result of the precursor pulse transmission is present at the receiver. Suppose that this sum signal is applied to a nonlinear device--perhaps in the sea lion ear--and the resultant signal is averaged. As a consequence of the nonlinear operation, the product of the two signals is generated. Thus, the nonlinear operation in conjunction with the averaging provides a form of cross-correlation detection that closely approximates the optimum-detection procedure.

We base the hypothesized new receiver structure primarily upon the strong cross correlation between the precursor pulse and the main pulse. Many questions may be raised concerning the plausibility of the hypo-

thesized receiver. By continuing with our speculation, we attempt to find answers to some of the questions.

A question immediately arises: Why is the energy in the ranging pulse (the precursor pulse) small as compared to the locally generated cross-correlation signal (the main pulse)? One possible answer is that the spatial directivity of the sea lion deteriorates as the signal energy is increased. That is, perhaps the sea lion can achieve some angular resolution by emitting the precursor pulse predominantly in a predetermined direction. Further, the directivity of the main pulse makes little difference, since it is used as part of the receiver, and it may be made as large as necessary to provide the desired dynamic range of levels for further processing.

Another interesting question is: Since the main pulse is emitted for a limited time, does this mean that the sea lion is looking for targets specifically within two range limits? The answer may be yes; however, by speculating that the sea lion detects signals in the manner indicated, we do not mean to exclude further use of the signals. For example, in addition to the angle-sensitive ranging for close targets, the main pulse might be used for less directire, coarse ranging to distant targets.

The use of the active cross-correlation receiver re-

quires proper phasing between a target return and the

- ,I PRECURSOR PULSE MATCHED- - 2 PULSE 5 - [[[ FILTER RESPONSE - '

I llrl ,, 2.0 -

1.0- [ MAIN PULSE MATCHED- - I FILTER RESPONSE 1.5 UL

0.5- -

0.

.... ,,,,, ...... ..... ...... , .......... I t,ll

-0.5 -I.0

1.5

i0 I •.0 , , I , I , I • ! , I • I • ... • , • , • , 1 , I , • ,

-2 0 2 4 6 8 I0 12 4 -• 0 Z 4 6 8 I0 IZ TAU --- msec TAU • msec

Fzo. 5. Matched-filter responses of sea lion Pulse 3. Fro. 6. Matched-filter response of precursor pulses.



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SHAVER AND POULTER

400 PULSE 2__ .... Ill [• , I1•! , , 200 I' • • • • !1• ,.

II I I III II I

o •V-••. T .... , ,, ,,, ' , ,,',.•,, ,• ii I I I II • II •

-200 • • • • I •

-600, , I , • , L .... Z__•_ • , J• ,

,

600 ' • ' • ' • ' i ' i ' [ '

PULSE • • • • •

_ PULSE 5 5 , " 200

o ' • I -400

-600 , ' •0 ' •5 o ,. ,. TIME

-vvvv'v V/VV" _

' i '

PULSE 4-

PULSE 5

msec

FIG. 7. Phase comparison of adjacent pulses.

main pulse. For this reason, adjacent sea lion pulses have been plotted on the same graph (Fig. 7). By ad- justing the precursor pulse phase to be the same for adjacent pulses, the pulse-to-pulse phase relationship for the main pulse region is shown. There is an approxi- mate 90 ø change from Pulse 1 to Pulse 2, a 180 ø change from Pulse 2 to Pulse 3, and no change from Pulse 3 to Pulse 4. For Pulses 4 and 5, there seems to be no change for the first part of the main pulse but then a change to approximately 180 ø for the second part of the main pulse. The sea lion has probably moved relative to a target during the interpulse interval (43, 48, 37, and 53 msec for the pulses shown). However, there does seem to be an attempt to make an adaptive ad- justment of the phase of the main pulse relative to that of the precursor, either to maintain the proper phase or to search for targets at the different phases.

D. Target-Resolution Capability

Since there is evidence that the sea lion attempts to use phase information--i.e., that an attempt seems to be made to adjust the phase of the cross-correlation signal adaptively in the detection of targets--the central spike of the signal function is indicative of the resolution capability of the sea lion. The signal function for precursor Pulse 3 has been computed for v=0 (Fig. 8). For this pulse, the range resolution is on the order of 4-4 cm. If phase information were not used, the resolution would depend upon the envelope of the signal function and would probably be on the order of 10 times worse.

The signal function has also been examined for range rates 4-5 m/sec. These velocities were chosen as typical

of the maximum velocity that might be associated with targets that would be of interest to the sea lion. For the precursor pulse, it may be concluded that the signal function is essentially unchanged for these velocities. Consequently, it may be further concluded that, for the signals considered in this program, the sea lion cannot distinguish between range rates appropriate for typical targets.

V. SUMMARY AND CONCLUSIONS

For a point target and a distortionless propagation channel, a target return from an echo-ranging signal may be characterized as an attenuated and delayed version of the transmitted signal with a possible time- scale change. The delay is a direct measure of the target range, and the time-scale change is due to the range rate.

The ability of the sea lion to measure range and range rate may be examined in terms of the signal function (mismatched matched-filter response). Under not very restrictive conditions, the signal function is directly related to the radar ambiguity function. By using the signal function we may determine how well the sea lion could measure range rate using an optimum detector. This then provides a bound on how well the sea lion could actually perform these operations.

A short series of sea lion pulses has been recorded and converted into digital form. The signal function has been computed for several pulses by selecting various portions of the signal to represent the impulse response of the filter (a matched filter). This analysis has led to the hypothesis that, in addition to echo-ranging pulses and possible communication signals, the sea lions--at



00:41:19


RANGE RESOLUTION • cm -60-40-20 0 20 40 60

J.o

^^h

-I.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 TAU • msec

TARGET RESOLUTION- PRECURSOR PULSE 3

FIO. 8. Target resolution for Precursor Pulse 3.

least in some circumstances--generate and transmit signals that are used to aid in the detection of target returns of echolocation pulses. In this framework the actual echolocation pulse is the precursor pulse, and the main pulse is generated as part of the detection procedure. The resulting cross-correlation detection approximates the optimum-detection technique. Such a detection procedure requires the proper phase between the target-return and the cross-correlation signal. Addi-

tional analysis of the pulse-to-pulse changes in the phase relationship between the precursor and main pulse further supports the hypothesis of an active cross- correlation receiver. The question of the relative level of the ranging pulse and the cross-correlation signal has been discussed in terms of a possible nonlinear (level-sensitive) directivity associated with the signal emissions.

The approach taken in this program has been to infer a receiver based upon the full utilization of the signal properties. This approach leads us to the possi- bility that, in addition to angle-sensitive ranging for close targets (the main pulse being used as part of the receiver), the main pulse may also be used for less directire coarse ranging of distant targets.

On the basis of a small set of pulses, the capability of the sea lion to measure range and range rate has been examined in detail. In view of the apparent use of phase information by the sea lion, it is concluded that, for the signals considered here, range may be measured at best to within approximately 4-4 cm. It is also concluded that, for range rates of interest, say, up to 5 m/sec, the sea lion has no range rate discrimination. Other sea lion signals may differ, of course, and thus, imply different capabilities.

Appendix A: Alternate Forms for the Ambiguity Function

The ambiguity function may be defined as

T

A (r,•):/_• z*(t--i)z(t+2)ei2•tdt , (A1)

where z(t) is the complex envelope of the real signal so(t), and is given by

z(t)= (e-'2•/øF2)[so(t)+i•o(t)]. (A2)

The frequency f0 is the chosen center frequency of so(t), and

1 f; so(x) •o(t) = - •dx (A3) •r

is the Hilbert transform of the real signal. In the frequency domain, the ambiguity function

may be written as

A(r,v)= Z* 3- Z - e'•/•df, (A4)

where

and

Z(/)=/_

So(fq--fo), f>--fo 0 , f_<--fo

(AS)

So(f) = f so(t)e-V'•Y'dt. (A6) By simplifying these equations, the ambiguity function is given by

1•1/2 v ß v -/0 ß (A7)



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SHAVER AND POULTER

Appendix B' Sea Lion lulses

6O0

400

200

0

-200

-400

-6001

.,,I PULSE I

-200 -400

,

i i [ i ' • ' • ' i '

F PULSE 7

S -,,oo[- =• -6001 , '•. 600 • ' • ' • ' • ' • ß • ' • '

ß

• 400 PULSE I0 _

• 200

-200

-400

-600

6OO

400

200

0

-200

-400

-6OO

600

i i

PULSE 8

' i ' i '

PULSE II

' 1

PULSE

, I

' ] ' i ' i ' i ' i ' i ' i '

PULSE 9

400

2OO

0

-200

-400

, I , i ,

i '

PULSE 18 '

I

PULSE 17

-600 ' i ' I , • • , • , • , z , • , • , 0 2 3 4 5 6 7 0 i 2 3 4 5 6 7 0 I 2 $ 4 5 6 7 8

T I M E • msec



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Appendix C' Computation of the Signal Function

A computer program has been developed that searches the digital recording for a particular sea lion pulse and then transfers a chosen set of L signal samples into temporary storage. Given the set of L samples of a signal equally spaced in time, s(t•), s(t2), ..., s(tr), the signal may be either plotted or processed further. Since we are particularly interested in the matched- filter response, we choose a positive number N_<L (N corresponds to the filter memory in terms of sample

number) and compute N N

X(r,v)=a« • s(&)s[-a(&--r)-]/• s•'(&), (C1) i =1 i =1

where

a=(c+v)/(c--v). (C2) It is assumed that s(t) is zero if t<tl or t> try. For values of t such that tl_<t_<tr•, the value of s(t) is determined by linear interpolation between samples. The computed function X(r,v) may be either plotted or tabulated as a function of r and for a specific value of v.



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sea lion echo ranging

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