Genralized Correlation Method for Estimation of Time Delay

8/9/2019 Genralized Correlation Method for Estimation of Time Delay

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320 IEEERANSACTIONS ONCOUSTICS,PEECH,ND SIGNALROCESSING,

VOL.

ASSP-24, NO. 4, AUGUST

[7] J. L. Butler, “Comparative criteria for minicomputers,”

Instrum.

and omputer imulation esults for Gaussian andelevis

Technol., vol. 17, pp.67-82, Oct. 1970.

input signals,”

Bell

Syst. Tech. J., vol. 45,

pp.

117-142,

[ 8 ] B. Liu, Effect of finiteword ength

on

accuracy of digital

1966.

filters-Aeview,” IEEE Trans. Circuit Theory, vol. CT-18,

[ l o ] D. J. Goodmanand A . Gersho,“Theory of anadaptivequ

pp.70-677,OV.971.izer,” IEEE Trans. Commun., vol.OM-22, pp.037-1045,

[9]. B. O’Neal, Jr.,Deltamodulationuantizingoisenalytical Aug. 1974.

Theeneralizedorrelation Met r

of

Time

Abstruct-A maximum ikelihood (ML) estimator sdeveloped or

determining time delay between signals received at

two

spatially sepa-

rated sensors in the presence of uncorrelated noise. This ML estimator

canbe ealized as apairof eceiverprefilters ollowed by across

correlator. The ime rgument at whichhe orrelator chieves

maximum is the delay estimate. The ML estimator

is

compared with

several other proposed processors of similar form. Under certain con-

ditions the ML estimator is shown

to

be identical to one proposed by

Hannan and Thomson

[1 1

and MacDonald and Schultheiss 211.

Qualitatively, the role

of

theprefiiters s

to

accentuate he signal

passed to thecorrelatorat frequencies for which the signal-to-noise

(S/N) ratio is highest and, simultaneously, to suppress the noise power.

The same type of prefiitering is provided by he generalized Eckart

fiiter, whichmaximizes the

S/N

ratio of the conelator output. For

low

S/N

ratio, the ML estimator is shown

to

be equivalent

to

Eckart

prefiltering.

A

INTRODUCTION

SIGNAL emanating from a remote source and moni-

tored in the presence ofnoise at wo spatially sepa-

rated sensors can be mathematically m odeled as

Xl t>

= S l ( t ) +n 1 ( t )

(1

a)

x2( t )

=

as

1 (t +

0 )

+ nz

( t) , (1b)

where

sl t),

n l ( t ) , and nz( t ) are real, jointly stationary ran-

dom processes. Signal sl(t) s assumed to be uncorrelated with

noise n

( t )

nd

nz t ) .

There are many applications in which

it

is of interest to esti-

mate the delay D . This paper proposes a maximum likelihood

(ML) estimator and compares it with othe r similar techniques.

While the model of th e p hysical phenomena’ presumes sta-

tionar ity, he techniques to be developed herein are usually

employed in slowly varying environments where the character-

Manuscript received July 24, 1975; revised November 21 , 1975 and

C. H. Knapp is with the Department of Electrical Engineering and

G. C. Carter swith the Naval UnderwaterSystemsCenter, New

February 23, 1976.

Computer Science, Universityof Connecticut, Storrs,

CT

06268.

London Laboratory, New London, CT 06320.

istics of the signal and noise remain stationary only for f

observation time

T.

Furth er, he delay D and attenuatio

may also hange slowly. The estimator s, herefore,

strained to operate on observations of a finite duration .

Anothe r important consideration

in

estimator design s

available amount of

a

priori knowledge of the signal

noise statistics.

In

many problems, this information is ne

gible. For example, in passive detection, unlike the u

communications problems, the source spectrum is unkn

or only known approximately.

One common method of determining the time de lay D

hence, he arrivalangle elative to the sensor axis

[ l ]

i

comp ute the cross correlation function

where

E

denotesexpectation . The argument 7 that m

mizes (2) provides an estimate of delay. Because of the f

observation time, however,

R x I x , ( 7 )

an only be estima

For example, for ergodic processes (2, p .

3271 ,

an estim

of the cross correlation is given by

where T represents the observation inter@. In order to

prove the accuracy of the delay estimate

D,

t is desirab

prefdter

xl t )

and

xz t)

prior to the integration in (3).

shown in Fig.

1,

x i

may be fdtered through H i to y ieldy

i =

1 ,

2.

The resultant

y i

are multiplied, ntegrated,

squared for a range of time shifts, T, ntil the peak is obtai

The time shift causing the peak

is

an estimate of the

d,elay

D .

When the fdters

H I f ) = I f 2 (f)

=

1, Vf ,

the estim

D is simply the abscissavalue at which the cross-correla

function peaks. This paper provides for a generalized cor

tion through the introduction of the fdters H 1

f )

and H

which, when properly selected, facilitate theestimation

delay.

The cross correlation between

xl t )

and

xZ t)

s relate


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KNAPP A N D CARTER: G E N E R A L I Z E D CORRELATIONETHOD 32 1

Fig. 1 .

Received waveforms iltered,delayed,multiplied, nd nte-

grated for a variety

of

delays until peak output is obtained.

the cross power spectral density function by the well-known

Fourier transform relationship

J

R x l x 2 ( T ) = J

m

G x , x , ( f ) e j Z n f 7 @ - . 4)

When

x l ( t )

and

x 2 ( t )

have been filtered as depicted in Fig. 1,

then the cross power spectrum between the filter outputs is

given by [3,

p .

3991

G y , y , ( f >N l ( f ) H : ( f ) G X I . , ( f ) , ( 5 )

where denoteshe complex conjugate. Therefore,he

generalized correlation between x 1 ( t )nd x 2 ( t )s

( 6 4

where

,(f3 =H1 f )

K ? f )

(6b)

and denotes the general frequency weighting.

In practice, only an estimate

G,,

x z ( f ) of

G x I x z ( f )

an be

obtained from finite observations of x l ( t ) and x z ( t ) . Con-

sequently, the integral

*

is evaluated and used for estimating delay. Indeed, depending

on the particular form of

Qg(j’)

and the

priori

information,

it may also be necessary to estimate ,to n (6a)-(6b). For

example, when the role of the prefdters is to accentuate the

signal passed to the correlator at those frequencies at which

the signal-to-noise (S/N) r atio is highest, then

, ( f )

c8nbe

expected to be a function of signal and noise spectra which

must either be known

priori

or estimated.

The selection of

, ( f )

to optimize certain performance

criteria has been studied by several investigators. (See, for

example, [4] -[12]

.)

This paper w ill derive the ML estim ator

for delay D in the mathematical model (la) and (lb), given

signal and noise spectra. The results willbe shown to be

equivalent to (6a)-(6c) with an appropriate ( f ) . This

weighting turns ou t o be equivalent to ha t proposed in

[12] and under simplifying assumptions to that proposed

in [21] . The development presented here does not presume

initially that heestimatorhas he form (6c). Rat her, t is

shown that the

ML

estimator may be realized by choosing r

that max@izes (6c) with proper weighting, ,(f), and proper

estimate, G X l x 2 ( f ) .The weighting (ayielding the ML

estimate will be compared to other weightings that have been

proposed. Under certain conditionshe ML estimator is

shown to be equivalent to other processors.

PROCESSORNTERPRETATION

It is informative to examine the effect of plocessor weight-

ings on the shape of R y I y , ( r ) nder ideal conditions.For

models of the form of (l), the cross correlation of x l ( t ) and

x 2 ( t )

s

R x 1 x 2 ( ~ ) = ~ R , 1 , 1 ( ~ - ~ ) + R n , n z ( ~ ) .

( 7 )

GX]x 2 f ) = @ G I S

n 1

n, f

1.

(8)

If nl ( t ) and n2 t ) are uncorrelated (Gnl h, ( f ) =

0),

the cross

power spectrum between x l ( t ) and x z ( t ) is a scaledsignal

power spectrum times a complex exponential. Since multi-

plication in one domain is a convolution in the transformed

domain (see, for example, [13 ]), it follows for

G n I n , ( f )

0

that

The Fourier transform of (7) gives the cross power spectrum

(0 -iznfD

+

G

R x l x z ( ~ ) = ~ R , l , l ( ~ )

f i ( t - D > ,

( 9 )

where enotes convolution.

One interpretation of

9)

is that the delta function has been

spread or “smeared” by the Fourier transform of the signal

spectrum. If s l ( t ) is a white noise source, then its Fourier

transform is a delta function and no spreading takes place.

An importantproperty of autocorrelation unctions is tha t

Rss(7)


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322

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, ANDIGNAL PROCESSING,UGUST

[where the subscript R is to distinguish the choice of ,(f)],

yields'

Equation

(12)

estimates the impulse response of the optimum

linear (Wiener-Hopf) filter

w hc h best approximates the mapping of

x 1

t ) o x 2 t ) see,

for example, [141 , [151). If n 1( t ) 0, as is generally the case

for I ) , then

~ x l x l ~ f ~ = ~ , ~ s l ~ f ~ + ~ n l . l ~ f ~ ~14)

and

.

e

j271f7 df

15)

Therefore, except when G n l n l

f)

quals any constant (in-

cluding zero) times Gsls1(f) , the delta functio n will again be

spread ou t. The Roth processor has the desirable effect of

suppressing those frequency regions where G,,

n

(f) s large

and Gxl

x2 f)

is more likely to be in error.

The

Smoothed Coherence Tran sfom (SCO T)

Errors in GXLX2(j ) ay be due to frequency bands where

Gnzn,(f>

s large, as well as bands wh ere

Gnlnl(f)

s large.

One is,herefore, uncertain whether to form J / R ( f )

selects

l/Gxlxl f) or

J / R ( ~ ) =/Gx2x2(f);

ence, he SCOT [ I 1 1

J / s f ) =

1/d~X1Xl f)GX2X2 f). 16)

This weighting gives the SCOT

where the coherence estimate2

For H I (f> u and H2 f)

U G J T ) ,

the

SCOT can be interpreted through Fig. 1 as prewhitening F dters

followed by a cross correlation. When

G x l x lf)

Gxzx2(f),

'As

discussed earlier, I (f)may have to be estimated for this proces-

sor

and those which follow, because

of

a lack of priori information.

r, this ase,

(11)

must be modified by replacing Gxlxl(f)ith

2A

more tandard oherence stimate is ormedwhen

the

auto

Gx,x, f).

spectra must also

be

estimated, as s usually the case.

the SCOT s equivalent to

t h e

Roth processor. If n l ( t

and n 2 ( t ) 0, the SCOT exhibits the same spreading as

Roth processor. This broadening persists becausefn

apparent inability to adequately prew hiten the cross po

spectrum.

The Phase Transform (PH AT )

weighting [

161

To avoid the spreading evident above, the PHATuses

which yields

Forthemodel 1)withuncorrelatednoise i.e.,Gnlnz f)=0)

I G X , X * f ) l

=

f f G S l S I f ) .

Ideally when Gx x 2(f) ex1 2(f),

has unit magnitude and

R$, Jz(7) = 6 ( t - 0 ) .

The PHATwasdeveloped purely as an ad hoc techni

Notice that, for models of the form of 1) with uncorrel

noises, the PHAT

(20),

ideally, does not suffer the sprea

%atther processors do. In practice, however, w

Gxlx2(f)

GXlXZ(f),

f)

27rfD and the estim ate

R ' ,(T) will not be a delta function.Another appa

defect of the PHAT is that it weights ex,

(f)

s the inv

of Gsl 1(f). Thus, errors are accentuated where signal po

is smallest. In particular, if Gxl x (f) 0 in some freque

band, then the phase

e f )

is undefined in that band and

estimate of the phase is erratic, being uniformly distrib

in the interval

[-n,

] rad. For models of the form of

this behavior suggests that

J / p ( f )

be additionally weig

to compensate for he presence orabsenceofsignal po

The SCOT s one method ofassigningweight accordin

signal and noise characteristics. Two remaining proces

also assign weights or filtering according to

S/N

ratio:

Eckart filter

[5]

and the ML estimator or Hannan-Thom

(HT) processor [121 .

The Eckart Filter

The Eckart filter derives its name from work in this

done in

[5]

. Derivations in

[7] ,

[ 8 ]

, [17]

, and

[

181

outlined here briefly for completeness. The Eckart f

maximizes the deflection criterion, i.e., the ratio of the cha

in mean correlator o utput due to signal present t o the s

dard deviation of correlator ou tput due t o noise alone.


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KNAPP AND CORRELATION METHOD

323

long averaging time

T ,

the deflection has been shown [8] to

be

TABLE I

CANDI DATE

ROCESSORS

(24)

where L is a constant proportional to T, and

GsIsz ( f )

is the

cross power spectrum between s1 t ) nd s 2 ( t ) . For the model

( I ) ,

Gsl s2 (f)

aGsls,

(f) exp (j27rfD). App lication of

Schwartz's inequality t o (24) indicates that

H1

f)H? f) =

\LE f) e

+ j z n f D

(25)

maximizes d where

Weight

Processorame

rL

f) HI f) m f ) Texteference

Cross Correlation 1 [21-[41,~ 9 1

Roth Impulse

SCOT l / a l x , c f )Gx 2x , f) [I11 [I61

PHAT 1/IGXIX2 f)l [I61

Response

1/GX IX, f) 191

Notice that the weighting (26), referred to as theEckart

fdter, possesses some of the qualities of the SCO T. In par-

ticular, it acts to suppress frequency bands of high noise,

as does the SCOT. Also note hat heEck art filter unlike

the PHAT attaches zero weight to bands where G,, (f) 0.

In practice, theEckart filter requires knowledge or estima-

tion of the signal and noise spectra. For (l), when a

=

1 this

can be accomplished by letting

kE(f>=

~ x , x z f ) l C ~ x , , , f > -~ X i X Z f ) l 1

.

[ 6 x z x z f ) -~ x l x 2 ~ f ~ l l l ~ (2 7)

The first five processors in Table I can be justified on the

basis of reasonable performance criteria, whether heuristic or

mathematical.

In the next section, the ML estimator of the parameter D

is derived. It is shown to be identical to hatproposed.by

Hannan and Thomson [121

.

The HT Processor

To make the model (1) mathematically tractable, it is

necessary to assume that

s l ( t ) , n l ( t ) ,

and n 2 ( t ) are Gaussian.

Denote the Fourier coefficients of x i ( t ) as

in

[3, eq. (3.8)]

by

1 T / 2

x i ( k )

=

r

IT

i t )

-jktwA

d t ,

( 2 8 4

where

27r

T

0

=-.

Note that he linear transformation X i @ ) isGaussiansince

xi(t) is Gaussian. Furth er, from [3,eq. 3.13)], as T + m

and

K

such that

KWA=

o s constant

where s the Fourier transform

of x j ( t ) .

A more complete

discussion on Fourier transforms and their convergences

given in [3, p. 3811,

[4,

pp. 23-25], [19, ch. 11, and [20,

p. 111 . From [21 ], it follows for T large compared to

IDl

plus the correlation time of R,, s1 T), that

Now let the vector

=

[X1

x2 '> (3

0)

where ' denotes transpose. Define the power spectral density

matrix

Q

such that

E

[

X ( k )

X * (k)]

=

E

1

- Q x ( k W A ) .

: T

Properties of Qx ( k o a ) can be used to prove

OG y x , x 2 k ~ A ) 1 2 1, \ J k a A

[4, p. 4671 . The vectors X @ ) , k

=

-N, -N + 1,

. . . ,

N are, as

a consequence of (29), uncorrelated G aussian (hence, indepen-

dent) random variables. More explicitly, he probability for

X s X ( - N ) ,

X ( - N + l ) , * *

, X( N) ,

given the power spectral

density matrix Q (or thedelay , ttenuati on, and spectral

characteristics of the signal and noises necessary to determ ine

Q)

is


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324

IEEE TRANSACTIONS

ON

ACOUSTICS, SPEECH, ANDIGNA L PROCESSING, AUGU ST

1

where

J1 = x*'@)

; ' ( k O ~ ) x ( k ) T

N

(33)

k=-N

and

c

is afunction of IQx(kwA)l

[15,

p. 1851

.

Replacing

x(k)

y

- ?(koa)

from (28),

1

T

~1

=

X * ' ( k u A ) Q;' ~ w A )

~ O A ) .

(34)

N

1

T

k=-N

For ML estimation (see, for xamp le, 4] or [15 ]) , it is

desired to choose

D

o maximize

p ( X I Q ,0 .

In

general, the parameter

D

affects both

c

and J n (32).

Howev er, under certa in simplifying assumptions, c is constant

or is only weakly related to th e delay . Specifica lly, from

(1)

and (3

),

suppressing the f requency argument k w ~ ,

IQxI=(Gs ls l - tGn , n , ) ( a2Gs I s I +Gn, n , >

(4

In

order to relate these results to [I21 and interpret how

implement the ML estimation technique, note that for

x

and xz t ) eal,A*(f)

= A ( - f ) .

Then (42b) can be rewritten

Letting TGxl

x 2 ( f ) 6

zl(f)

,* ( f ) ,43) and (42c) can

rewritten as

For large T , 34) becomes

ca

J 1 X ' ( f )Q; (f) d f .

which will exist provided

lrl2(f)lZ

1; i.e., x l ( t ) and xz t)

cannot be obtainedperfectly rom one another by linear

filtering [14]

,

or equivalently for the mode l (1) tha t observa-

tion noise is present.

When Gn,

n 2(f) 0,

G x l x l ( f ) = G s l s l ( f ) + G n l n l ( f ) ~

(38)

G x z x 2 ( f ) = ~ 2 G s l s l ( f ) + G n 2 n , ( f ) ~

(39)

Gx,

x2(f) G I I f ) ,-iznfD,

(40)

and it follows that

Notice tha t he ML estimator or D willminimize J

J z

-t J 3 ,

but the selection of

D

has no effect on

J z .

Th

D

shouldmaximize

- J 3 .

Equivalently,when

2, f ) x z

isAviewedas T times the estimated cross power spectr

T GxIx , ( f ) , the

ML

estimator selects as the estimate

delay the value of r at which

achieves a peak.

The weighting in

(6),

where (as required for

Q;'

to exist) Iyl2 f)I2

1 ,

achie

the ML estimator. WhenG,,,,(f)lnd lrlz(f)I

known, this

is

exactly the proper weighting.

~

When the te

in (45b ) are unk nown , they can be estimated via techniq

of [22]

.

Substituting estimated weighting for true weigh

is entirely aheuristicprocedu re whereby the ML estim

can approxima tely be achieved in practice.

Note that, like the

HT

processor, the PHAT compute

type of transformation on

However, the HT processor, like th e SCOT, weights the ph

according t o the strength of the coherence.


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KNAPP AND CARTER:

GENERALIZED

CORRELATION METHOD

325

From [4, p.3791

,

1 -

IYI2

1

I Y I L1

var [e(f)]

T.- --

where L is a proportionality constant dependen t on how the

data are processed. Thus,

Comparison of (46b) and (20) ith (21) eveals that the ML

estimator is the PHA T nversely weighted according to the

variability of the phase estimates.

In interpreting he similarity of the HT processor to the

other processors listed in Table I, t canbe shown that if

Gn,nl(f)=Gn2n2(f)=Gnn(f)s equal t o a constant times

GS 1 (f),hen the last five processors

in

Table I are the

same except oraconstant,but he cross-correlation pro-

cessor ( (f) 1, V f ) is adeltafunction smeared ou t by

the Fourier transform of the signal (noise) power spectrum.

Interpretation

of

Low SI NR a t io ofML Estimator

Good delay estimation is most difficult to achieve in the

case of low

S/N

ratios. In order to compare estimates under

--

identical to the E ckart filter. Similarly, for low

S/N

ratio,

s(f)E 1/dGnInl(f)Gn2n2(f).

(49)

Therefore, if Q

=

1,

(50a)

Thus, under low

S/N

ratio approximations with

Q =

1 , both

the Eckart and HT prefdters can be interpreted ither as

SCOT prewhitening filters with additional S/N ratio weight-

ing or PHAT prewhitening fdters with additional S/N ratio

squared weighting.

VARIANCE

F

DELAYESTIMATORS

It can be shown , by extending a result from [21], hat the

variance of the time-delay estimate in the neighborhood of

the true delay for general weighting function (

f)

s given by

J

I (f)12(271f)2Gx1x1(f)GX2X2(f)[1

I Y W I ~ If

m

var [Dl

(51)

T 1

(2.1Tf)21Gx1x2(f)lJ/(f)~f12

low S/N ratio conditions, let

QI

= 1. Then,

Thevariance of the HT processor (substituting from (45b)

which agrees with

[21,

eq.

(28)]

f in (47b)

Gnln1(f)

Gn2 ,

f

>.

For low

S/N

ratio,

it follows that

That s,for

01 =

1 and low

S/N

ratio, heHT processor is

and using the definition of coherence) is

V ~ ~ H T [ E I( 2 ~

2nf)2IY(f)12/[1-r(f)121 df1-1.

(52)

In pa rticular, the HT processor achieves the Cram& -Rao lower

bound (see Appendix). It should be pointed ou t hat (51)

and (52) valuate the local variation of the time-delay esti-

mateand husdo no t account for ambiguous peaks which

may arisewhen the averaging time is not large enough for

the givenignal and noise characteristics . Indee d, when

T

is not sufficiently large, local variation may be

a

poor

indicator of system performance and the envelope of the


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326 IEEE TRANSACTIONS O N ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,UGUST 1

ambiguous eaks mu st be considered

[21,

p .401, 23] ,

[24, p. 411.Further, 51)and

(52)

predict system perfor-

mancewhenignalndoise spectral charac teristics are

known; or

T

sufficiently large, these spectracanbe esti-

matedaccurately. However, n general, (51) and (52) must

bemodified to accoun t for estimation ezrors; alternatively,

systemperformancecan be evaluatedby compu ter simula-

tion. Empirical verification ofexpressions for variancehas

not been undertaken by simulation, because to do

so

with-

out specialpurpose corre lator hardwarewould be computa-

tionally prohibitive.For xamp le, or given

G,l

s,

(f),

Gnlnl(f),nz,,(f),, and averaging time T , an estimated

generalized cross-correlation function can be compu ted, from

which only one numb er (the delay estimate) can be e xtract ed.

To empiricallyevaluate the statistics of the delay estimate

(whichwould bevalid only for these particular signal and

noise spectra) many such trials would need to be conducted.

We have co ndu cted one such trial (with

T

large) and verified

tha t useiul delay estimates can be obtained by inserting esti-

mates

lGxlx2 f)l

nd

l ?12 f ) 12

inplaceof the rue values.

(This might have been expected since the estimated optimum

weighting will converge to the ru e weighting as

T 00.

In

practice,

T

may be limited by the stationary properties of the

data and (52 ) may b e an overly optimistic pred iction of sys-

tem performance when signal and noise spectra are unknown.)

CONCLUSIONS

AND DISCUSSION

The HT processor has been shown to be an ML estimator for

timedelay under usual conditions. Under a low S/N ratio

restriction, the HT processor is equivalent to Eckart prefilter-

ingand cross correlation. Theseprocessorshavebeencom-

paredwith our other candidate processors to demonstrate

the interrelation of all six estimation techniques. The deriva-

tion of the ML delay estim ator, toge ther with its relation to

variousad hoc echn ique s of intuit ive appeal, uggests the

practical significanceofHTprocessing fordetermination of

delay and,hence, bearing. Finally,nterpretation of the

results leads one to believe that, if the coherence is slowly

changingas a unctio n of time, he ML estimationof he

sourcebearingwill still be a cross correlator precededby

prefdters hat must

also

vary according to time-varying esti-

mates of coherence.

APPENDIX

Cram.4-Rao Lower B ound o n Variance

of

Delay Estimators

The Cram&-Rao lower bound is given

[

15, p.

721

by

The only part of the logdensitywhich dependson

r ,

the

hypothe sized delay, is .J3 of (44). More explicitly,

If

GXtxz f)=G,l , z ( f ) le - i2n fD,

then since

E[~x,xz(f)]

Hence the minimum variance

is

(A

But this is the variancewhich the HT processor chieve

[see (52)]

.

R EFER ENC ES

A.

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Signal Ana-lysis by Homomorphic Prediction

Abstract-Two commonly used signal analysis echniques are inear

prediction and homomorphic filtering. Each has particular advantages

and limitations. This paper considers several ways of combining these

methods to capitalize on the advantages

of

both. The resulting ech-

niques, referred to collectively

as

homomorphic prediction, are poten-

tially useful for pole-zero modeling and inverse filtering of mixed phase

signals. Two of these techniques are illustrated by means of synthetic

examples.

INTRODUCTION

T

O classes of signal processing technique s wh ich have

been applied to a variety of problems are homomorphic

filtering or cepstral analysis [11,

[2]

and linear predic-

tionor predictive deconvolution

[ 3 ]

[S .

Separately, each

has particular advantages and limitations. It appears possible,

however, to combine them into new methods of analysis

which em body the advantages of both. In this paper we dis-

cuss several ways f doing this.

Linear prediction is directed primarily at modeling a signal as

the response of an all-pole system. Its chief advantage over

other identification method s is that for signals well matched to

the model it provides

an

accurate representation with a small

numberof easily calculated parameters. However, in situa-

tions where spectral zeros are important linear prediction is

less satisfactory. Furtherm ore, t assumes that he signal s

either minimum phase or maximum phase, butnot mixed

phase. Thus, for example, linear prediction has been highly

successful for speech coding [3], [ S I , [ 6 ] sinceanall-pole

Manuscript received April 3, 1975 ; revised September 30, 1975 and

February 3, 1976. This work was supported in part by the Advanced

ResearchProjects Agency monitoredby he O N R underContract

N00014-75-C-0951and in part by he NationalScience Foundation

under Grant ENG71-02319-A02. The work of G . Kopec was supported

by the Fannie and John Hertz Foundat ion. The work

of

J. Tribolet

was supported by the Instituto de lta Cultura, Portugal.

The authors are with the Department of Electrical Engineering and

Computer Science, Research Laboratory of Electronics, Massachusetts

Institute of Technology, Cambridge, MA 02139.

minimum phase representation is often adequate for t h s pur-

pose. It has also been applied in the analysis of seismic data,

although limited by he fact that such dataoften involve a

significant mixed phase com pone nt.

Homomorphic filtering was developed as a general method

of separating signals which have been nonadditively combined.

It has been used in speechanalysis to estimate vocal tract

transfer characteristics [7]-[9] and is currently being evalu-

ated in seismic data processing as a way of isolating the im pulse

response of the earth’s crust from the source function [IO]

-

[

121 . Unlike linear prediction, homo morphic analysis is no t a

parametric technique and does not presuppose a specific

model. Therefore, it is effective ona wideclassofsignals,

including those which are mixed phase and those characterized

by b oth poles and zeros. However, the absence of an under-

lying model also means that homo morphic analysis does no t

exploit as much structure in a signal as does linear prediction.

Thus, it may be far less efficient than an appropriate para-

metric technique when dealing with highly structured data.

The asic strategy for combining linear prediction w ith

cepstral analysis is to use homomorphic processing to trans-

form a generalsignal into one or more other signals whose

structures are consistent with the assumptions of linear predic-

tion. In this way the generality of homo morph ic analysis s

combined w ith the efficiency of linear prediction. In the next

section we briefly reviewsome

of

the properties ofhomo-

morp hic analysis tha t suggest this appro ach. We then discuss

several specific waysof combining the two techniques.

HOMOMORPHIC IGNAL ROCESSING

Hom omorphic signal processing is based on the trazsform a-

tion of a signal x ( n ) as d epicted in Fig. 1. Letting X ( z ) and

X z ) denote he

z

transforms of

x^(n)

and x ( n ) , the system

D* I

is defined by the relation

2 ( z ) = log

X z)

(1)

where the complex logarithm of X z ) s appropriately defined

Genralized Correlation Method for Estimation of Time Delay

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