"Event Compression Using Recursive Least Squares Signal

"Event Compression Using Recursive Least Squares Signal Processing" Webster Pope Dove TECHNICAL REPORT 492 July 1980

Massachusetts Institute of Technology Research Laboratory of Electronics Cambridge, Massachusetts 02139

This work was supported in part by the Advanced Research Projects Agency monitored by ONR under Contract N00014-75-C-0951-NR 049-328 and in part by the National Science Foundation under Grants ENG76-24117 and ECS79-15226.

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BEFORE COMPLETING FORM

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12. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

1. REPORT NUMBER

4.

PAGE

. TYPE OF REPORT & PERIOD COVERED

Subtitle)

Technical Report

EVENT COMPRESSION USING RECURSIVE LEAST SQUARES SIGNAL PROCESSING 7.

6. PERFORMING ORG. REPORT NUMBER

Webster P'. Dove

N00014-75-C-0951 10.

PERFORMING ORGANIZATION NAME AND ADDRESS

S.

Research Laboratoiy of Electronics Massachusetts Institute of Technology 02139 Cambridge, MA

12. REPqRT DATE

July 1980

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bstrct ont-d in Block 20, if

id

differant

rot Rport)

if neceswy end dentify by block nmmbe)

recursive least squares event compression linear prediction 20.

ABSTRACT (Continu- on r".,se

aid.

if nceearmy and identt

b

block nu.r)

see other side

DD ,,R.

1473

report)

Unclassified

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20. Abstract

This work presents a technique for time compressing the events in a multiple event signal using a recursive least squares adaptive linear prediction algorithm. Two event compressed signals are extracted from the update equations for the predictor; one based on the prediction error and the other on the changes in the prediction coefficients as the data is processed. Using synthetic data containing three all-pole events, experiments are performed to illustrate the performance of the two signals derived from the prediction algorithm. These experiments examine the effects of initialization, white gaussian noise, interevent interference, filtering and decimation on the compressed events contained in the two signals.

___

UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(Whof Dote

Entered)

- 2 EVENT COMPRESSION USING RECURSIVE LEAST SQUARES SIGNAL PROCESSING by Webster

Pope Dove

Submitted to the Department of Electrical Engineering and Computer Science, on 15 July 1980, in partial fulfillment of the requirements for the Degree of Master of Science in Electrical Engineering. ABSTRACT This work presents a technique for time compressing the events in a multiple event signal using a recursive least squares adaptive linear prediction algorithm. Two event compressed signals are extracted from the update equations for the predictor; one based on the prediction error and the other on the changes in the prediction coefficients as the data is processed. Using synthetic data containing three all-pole events, experiments are performed to illustrate the performance of the two signals derived from the prediction algorithm. These experiments examine the effects of initialization, white gaussian noise, interevent interference, filtering and decimation cn the compressed events contained in the two signals.

THESIS SUPERVISOR: TITLE:

Professor

Alan V. Oppenheim f

Electrical Engineering

- 3 -

I/

To Monica

f

- 4 -

Acknowledgements I give my thanks to all the people in the Digital Signal Processing Group at MIT. Many fruitful discussions with them at odd hours contributed to this work. Of particular help were Dr. Lloyd Griffiths with his insight into least squares algorithms and Steve Lang for infinite patience in listening to my ramblings. Last and most I thank Prof. Alar. Oppenheim for the original problem suggestion, much needed guidance during the investigation and the pressure necessary to produce this document. If I can acquire a tenth of his clarity of thought while I am here, I will consider myself lucky.

- 5-

CONTENTS

Abstract ............ Dedication

2

...........................

3

..........

Acknowledgements

1.

...........................

INTRODUCTION

.................... ...........

...................................

1.1 Previous Work .....

2.

3.

LINEAR PREDICTION

6 ,......

9

.............................

13

2.1 The Covariance Method .... ................. 2.2 Recursive Least Squares .. 2.3 Discussion ...............

14 17 38

EVENT COMPRESSION WITH RLS

39

3.1 3.2 3.3 3.4 3.5 3.6

4.

..................

4

....................

The Data Model .......... .................. Event Compression Signals ......... Experimental Results .... .................. Linear Filtering ........ ................. Decimation .............. ................. Summary ................. .................

CONCLUSIONS

I.......

..................................

40 48 54 112 130 144

146

- 6 1.

INTRODUCTION

various

In which

contain

detected.

signal

pulses

or

Problems

analysis.

All

to

improved of

Such

processing

a signal

of events in an separations,

the

and

event make

an

or

and

SONAR

seismic

data

need

for

a

signal in

the

This type

successive events

by

locating

increasing

the

events

the

easier.

procedure, which reduces the duration

input sequence without

is

event

changing

compression

deconvolution,

their

relative

algorithm.

matched

Examples linear

filtering,

deconvol ution.

Let

us

compression. targets

time.

may cause events

illustrate

Consider

requirements

the

each

homomorphic

predictive

in

share

detectability

impulsiveness

multiple

and

arise

located

RADAR

can reduce the overlap between

of processing

include

include

detection

fields

these

be

shorter events in the processed data.

into

lead ing

pitch

must

that

type

signals

problems,

technique which can compress events occuring

processing raw data

of

events this

of

speech

rangefinding,

processing

with

in

application

the

the RADAR

problem or

of

SONAR.

transmission, the

source

finding To

of the

reduce

pulses

are

event

range peak

of

power

dispersed

In addition, the reflection functions of the targets the in

returns the

to

be spread

received

signal

out must

targets are to be located accurately.

even more. be

Therefore

compressed

if

the

-

Surface disturbance

at

mechanical

7 -

seismograms

are

the

surface

of

or

an

vibrator

with

created the

by

earth,

explosion,

generating either

and

a

with

recording

a the

subsequent seismic vibrations with an array of geophones also located

at

they are layers used

the

As waves

surface.

partially reflected

to

locate

the As

reflectivities. the source

pulse and

individual

reflection

at boundaries

into

the

between

earth,

differing

recordings made at the surface can be

ideally the

and

propagate

depths in

the

the

these

of

boundaries

RADAR situation,

extension

functions

of

leads

that to

the

and

duration of

duration

the

their

need

by

for

the

event

compression of the data. One approach the

time

these before

between

to vocal pitch

successive

estimation pulses.

glottal

to measure

However,

pulses are filtered by the vocal tract impulse emanating

from the lips, the

period

compress

to

the

the next cycle. pulses of

the

It

speech

since

response

speech waveform does not

offer well defined points at each cycle the

is

is

from which to measure therefore

waveform

necessary to

without

changing

their positions for this scheme to be effective. Another

example

is the

which motivated this thesis. sonic

well

figure 1.1,

log

is

to

lower

problem

of sonic

The procedure for a

tool,

shown

well logging generating

schematically

a in

down an oil well and then raise it at a fixed rate

Well Log Diagram

I

I

i

HOLE 'I

I

ROC K

I

I

I

I %

I

R

61 1

CO 0

SH E AR

WAT E R--1-->/

l%1

\

T I

I

I

5

I

I

I

Figure 1.

II__

-

9 -

back to the

top of the well.

During its ascent the ultrasonic

transmitter

located

bottom

(roughly once

the

every foot

for

waveform at the

at

in

receiver

of

the

tool

is

pulsed

of well depth) and the pressure the

top of

the

tool

is

recorded

for later analysis.

In

a

problem, there take

in getting

model

simplified

very are

three

paths

for

from the transmitter

which has a characteristic velocity.

of the

the

physics

ultrasonic

of

this to

energy

to tne receiver, each of The slowest path is that

of a pressure wave traveling up the mud with which the well is packed

(to prevent collapse); the medium velocity path is that

of a shear wave coupled to the rock wall, and the fastest path is for the compression wave traveling up the

rock wall.

The

quantities of interest to geologists are the velocities of the salear wave and compression wave, which could be determined by knowing times

their of

times

of

waves

these

arrival. it

is

To

ascertain

necessary

to

the

arrival

compress

the

in the source data since the overlap between

individual events

them is considerable. 1.1

Previous Work Approaching the problem of event compression typically

entails

modeling

the

physical

fashion and then designing

an

in

an

appropriate

algorithm which

is

expected

situation

solve the problem for data which fits that model.

____II

__

to

Subsequent

-

investigation realistic

of

the

10

-

performance

environment

may

then

of

lead

the to

algorithm

alterations

in in

a the

model, the algorithm or both. For detecting

example,

events

that

events

they

are

known

separated

some

(the

set

In addition, he

the

(this

Unfortunately,

in

either

the

is

event

situations

of

of

source

the data's

he

derives

event at

a

each point

compressed

are not

that

set

the

(in particular

statistics

a

knowledge of

beginning of an

noise

data model

by

assumptions

the

of

distance and

determined

these

being

some

is

problem

combination

requires

From

ratio for

data

logging)

linear

the

His

by some minimum

unknown

statistics.

his

analyzes

the context of RADAR.

exponentials

likelihood in

are

each

waveform). noise

in

[Young,1965]

signal). sonic

well

known, detailed

information about the events is not available or the number of available

exponentials

from

which

they

could

be

composed

of

compressing

is

large making this formulation inappropriate. A seismic

common

data

convolving reflector

is

made

a fixed series.

source wavelet the

approach

source By

the

assuming wavelet

acquiring

series and

(or

problem the

data

with

an

was

estimate

to deconvolve and

something

[Tribolet,1977]

generated by

impulsive

an accurate

it would be possible

reflector

[Ulrych,1971]

by

to

have

close used

to

seismic of

the

recover it).

Homomorphic

-

techniques

to

estimate

[Peacock,19693

the

-

11

and

wavelet

used linear prediction to

series

reflector

and

estimate the wavelet.

Similar work has also been done on the speech pitch estimation problem the

of

convolution

impulse

excitation

a

vocal

to

recover

an

of

The nature

series.

impulsive

or

signal

speech

response

tract

and

a

glottal

methods

these

is

is

is applied to the data in

that a single deconvolution operator order

the

that

by assuming

(Markel,19721

nearly

impulsive

As such these techniques will only be effective

sequence.

if the events

in the data have similar spectra. In this thesis we apply a recursive least square (LS) adaptive

linear

prediction

to

algorithm

compression,

event

using synthetic data from a data model based on an abstraction of the sonic well logging problem. independent

data

have

time

invariant

event

spectra,

filtering

Because the events in this

would not

compression effective

be

by

(the inverse By using

filter for each arrival would have to be different). an

algorithm

adaptive

we

perform

time

linear

event

varying

compression on the input data sequence. To from

the

implement event compression we derive

RLS

algorithm,

one

is

the

two

signals

post-adaption prediction

error at each point and the other is a measure of the changes in the prediction coefficients as the data two

signals are compared

experimentally

is processed.

using

synthetic

The data

-

12 -

corrupted with white

gaussian noise.

of other

on

include

processing prefiltering

event

the

To illustrate the impact

compression,

data,

some

decimation

of

experiments

the

data

and

postfiltering the error signal. The next chapter equations and with

a

issues

discussion

covariance

method

is an analytical

involved of

of

in linear prediction.

linear linear

presentation of the

prediction prediction

and

It starts

describes

the

[Makhoul,1975].

The

following section presents a derivation of the Recursive Least Squares

algorithm

(from the

covariance

method

equations)

and

the last section of the chapter examines the

problem of proper

initialization

chapter

experimental

of

the

comparison

mentioned above, and the

recursion.

important

of

The

the two

third

event

offers

compression

an

signals

the thesis concludes with a discussion of

results

of

these

exper iments

and

some

suggestions for future work on this problem.

-

-

2.

1

-

LINEAR PREDICTION

Linear question:

prediction

attempts

Given a sequence

coefficients

for

formulation

answer

of data, what

predicting

sequence with a linear

to

the

value

is

of

the each

combination of previous

is presented

in equation

the

following

best set of p sample

of

samples?

the This

(2.1)

p sk=

where the for

{Sk}

cn are

L CnSk-n n=l

is

the

the

dat3 sequence,

coefficients of

determining

squared

(2.1)

which

prediction

{gk }

the

over

the

predictions and

predictor.

coefficients

error

are

are

the

The

best

chosen

is error

criterion the

total

region.

Specifically

E=

(s

k

-k

)

k

Q

(2.2)

k

and

the coefficients

are

chosen to

minimize

the

total squared

error E.

1. In the general case one could use an arbitrarily chosen set of previous samples, but this discussion assumes that they are the p contiguously previous samples to the one to be predicted.

__

-

The

choice

of

differentiates between

14 -

the

the

error

two most

prediction.

If Q extends fom -

padded

zeroes

with

or

region

Q

is

common methods of

to +

extrapolated

what linear

(the data sequence is

in

other

some

way)

the

resulting set of equations describe the Autocorrelation Method of

prediction.

linear

involves

In

this

inverting a Toeplitz

case

solving

for

the

matrix of autocorrelations and

is usually done by some variation of Levinson's Inversion for example

of

(see

[Makhoul,1975]).

The method

ck

other linear

common

method

prediction.

in As

use

is

described

the in

covariance the

next

section, it requires that only the available data sequence be used to generate the predictor. set

by requiring

Therefore the limits of Q are

that all the sk mentioned

in eqs.

(2.1) and

(2.2) lie in that finite set of available signal points.

2.1

The Covariance Method The equations defining the Covariance method are:

p

Sk=

cnsk-n

(2.3)

n=l

____

- 15 -

k

e

E=

(s k=k

-k)

2 = minimum

(2.4)

S

Here, p is the number

of predictions coefficients and k

ke are the endpoints cf the prediction region Q.

and

By defining

the following structures

S

S

kS 1

ks- P

S -
--.

.

II

12.5

i

-

i

--

--

4

I

PTS/DIV

Figure 3.33a It

'ER

. 298

/Div

'

__

Afx

fA

_

I

I

i[

t

,l

-

U

I

- -

-

12.5

PTS/OiV

1.40

/DIV

-

I'

I

--- --

--- -

._ HOlI

L

RLS ERROR

' V

iRo VER

12.5

I

I

I

I .

]

i\ r \-,\Q'

%

q

.

-I 1 J .

-

PTS/DIV

0.525E-01 /DlV

__

DECIM 4/1

10 1,N /

n

VV\I V VI

12.5

VER

BURSTI SIGMAR.O1, 5 POLE LPF.

LL

--

A-- P, V

-

',J,V

(12 COEF'S)

/ , JV '-W/ f

A

Ax_ , - -

-

,

V Wv\,

V

A

-

-

V

11 ---\kl\

I#

-#

_

-4!f* ..= f,

\

RLS COEFFICIENT CHANGE

- -,

·

-·

PTS/Di V

Figure 3.33b

----

-----

-------- -

I 1

i' K

0.303

VER ,

_~ _

_

/OIV

,

~

A f .

BURSTI SIGMA=.i, $ POLE LPF, DECIM 4/1

It

VV' { v I_

I

VI

,\ __

V'J'

/

,li\

VI K'

\J

_

y'

12.5

PTS/DIV

0.818E-01 /DIV

12.5

t

rER

0.245

RLS ERROR

(12 COEF'S)

PTS/DIV

/DIV

RLS COEFFICIENT CHANGE

J

i

HOR

12.5

PTS/D IV

Figure 3.33c

_

__

-

3.6

Summary

preceding

The

change signal) events

when

data.

If

pulses,

the

addition, order

exceeds

ratio despite

the

duration

the

predictor

the length

that and

both the

of

these

coefficient

locating the positions of apparent

not

in

the

input events

40db, the compressed overlap

severe

events of

error

for

are

positions

compressed

the

of

S/N

and

visible

(the RLS

provide a means

those

the

indicate

examples

signals

compressed

event

are

144 -

are

well

compressed

input

the

In

separated. events

appears

and

of

to

is be

on

the

fairly

independent of the existance or nature of previous events. The

following

are

some of

the

important

results

these experiments. The starting location of the RLS iteration the of sizes appears to only affect the a large there is compressed events. However, initial pulse in the coefficient change signal due to the sensitivity of the algorithm to the This means that the first few data points. lengths of predictor data must have several the if first event the preceding noise coefficient change signal is to be used for event compression. Additive noise in the data does not hve a substantial effect on the pulse shape of the However, it does cause compressed events. noise in the RLS error and coefficient change signals, which seems to be in proportion to Unfortunately, the noise level in the data. there is a large variation in the size of compressed events with this technique; making it difficult to establish a S/N criterion for event compression.

--

___

of

-

145 -

Compressed events do not influence each other substantially if the predictor has time to settle between them, even though the bursts which cause them may be severely overlapped. Figures 3.23a-h showed this by changing the relative position of the second and third bursts in an input sequence. The resulting compressed events had constant shape so long as they did not overlap. Therefore, the faster the predictor settles, the closer events can be to one another and still be resolved. Th i s al so means that events containing large numbers of zeroes (as in the FIR filtering examples) will not compress well ih this .thech-iQue unless they are widely separated. Note that a burst containing a large number of zeroes does generate a compressed event, but the event has longer duration than it would if the zeroes were not present. While filtering the data may be necessary to reduce aliasing, we found that it did not improve the quality of the event compressed signals. If filtering must be performed, all-pole filtering should be used since the zeroes introduced by FIR filtering tend to lengthen the compressed events. We found that decimation does increase the responsiveness of the predictor to each event, but the relative event to noise ratios in the location signals do not improve. In addition, the predictor settling time becomes a larger fraction of the event spacing, thereby increasing the likelihood of interevent interference. Consequently, decimation should be avoided; and in fact over-sampled data is likely to provide higher quality compressed events.

_

______I__IIIILCCC______

-

4.

146 -

CONC LUSIO!S

this thesis we have develope4 an event compression

In technique compress model

based

on

the

events

of

differing

by the

assumed

examined

the

RLS

RLS

algorithm, which

provided

spectra;

RLS method

prediction

(i.e.

error

can

effectively they fit

the We

all-pole events).

as

an

event

compressed

signal and

in addition developed the coefficient change signal

for

compression.

event

,

positions minor

that

but or

effects

on

should

technique

variations

the

in

that

S/N ratio

is

the

ordering,

of

event

the

those

high

this >

(e.g.

event

have

only

Therefore,

this

algorithm

events.

compressed useful

be

in

position

starting

indicate

experiments

is effective only if the

technique 40db)

Our

situations

where

the

events are close to all-pole and the noise level is not high. This

thesis

was

initial

an

investigation

into

the

feasability of using the RLS algoritm for locating events with differing discover

When

spectra.

we

to

synthesized

be

work

to

use

the

in

the data.

if

we

wanted

to

it did, where

Given that need, the best course

algorithm

examples, where we

of the events RLS

this

if the technique had any value and

should further work be done. seemed

began

on

a

large

knew the number

These examples

number

of

and positions indicate

that

is indeed a viable means for performing event compression.

However, there are many questions

_

touched

upon

in

the

course

_______I_____IIL___C__L__

-

147 -

of this thesis which need more detailed investigation. One

of

most

the

that the events in the data cannot be predictor

settling time, if

other. remains with

issue

The

of

unsolved.

noise

level,

and

point),

dependent settling

what

than the

spaced closer

determines

with

decreases 3.23a-h

settling

time

indicate that it increases predictor that

indicate of

this

previous

length

a

(up to

it is not strongly

events.

the

Also,

on

the

position

time

for

a given event is strongly dependent on the

"character"

of

that

dependence

of

the

characteristics means

thesis is

they are to be resolved from each

Our experiments

figures

this

important results of

for

event.

More

predictor

would

be

settling insofar

useful;

the

decreasing

investigation

settling

time

on as

it

of

the

of

the event

the

a

provides

predictor,

thereby reducing the necessary separation between events. Another

possible

avenue

of

investigation

is

that

of

alternatives to the RLS algorithm as presented in this thesis. Some variations of this moving

region

Q

over

adaptive algorithm exist which which

the

total

minimized, rather than an expanding Still

other

data.

methods

These

involve

modifications

squared

region

energy

allow

E

is

as was done here.

exponentially weighting would

us.>

the

the past

algorithm

to

be

used over large amounts of data without the predictor becoming totally

insensitive

to

the

new points

(as

_______________IC______

the

error

region

-

148 -

grows the predictor

tends to become less sensitive).

applications

that

capability

importantly,

reducing

the

predicted might allow the new

events,

thereby

might

amount

of

algorithm

reducing

be

essential;

previous

to more

the

minimization

expanding or moving would permit the

might the

be

settling

region

more

data

time.

to

be

Our

to own

the region of

effective

at each

more

readily adapt

feeling is that a means for dynamically changing error

In some

than

iteration,

simply

since

that

region of error minimization to be reduced at

a new event to shorten the settling

time, and

then lengthened

between events to reduce noise. We

found

the

coefficient

for

event compression, but our

all

the coefficients

and

a

change

signal

signal

used

to

equal

be useful

weights

for

fixed decay time for the filters.

Is there an optimum choice for the weights of the coefficients in

the change signal?

Could Kalman filtering be used to track

the coefficients more effectively? alternative

than

example

adaptive

signal

an

simply high

might make

decay

the

Perhaps

passing

time

compressed

for

there is a better

the

coefficients.

the

coefficient

events

For

change

for the second burst

in the Burstl signal more visible. Finally, real

data.

a means

there

is

no

substitute

for

experiments

on

The original motivation for this work was to find

for compressing

--111---

-

sonic well

log

-- -----

data.

__

Unfortunately,

-

subsequent

149 -

tudy of the well logging problem revealed that the

structure of the signals

recorded

in

that situation does not

fit the model that was assumed for this work.

Experimentally

we found that this

those

but

given

surprising. event

technique

their

complicated

Therefore,

compression

the assumed model

did not compress

to

the

data

application

which

is needed.

structure,

We

of

that

was

this

method

more closely corresponds hope

to

perform

using acoustic cardiac data in the near future.

II

signals,

__

not of to

experiments

-

150 -

References

Deczky, A.G. (1972). Synthesis of Recursive Digital Filters Using the Minimum-p Error Criterion", IEEE Trans. on Audio and Electroacoustics, vol. AU-20, no. 4, pp. 257-263 Eykhoff,P. (1974). SYSTEM IDENTIFICATION: Parameter and State Estimation, John Wiley and Sons, New York, p. 235 Makhoul, J. (1975). Linear Prediction: A Tutorial Review', Proc. IEEE, vol. 63, no. 4, pp. 561-580 Markel, J. (1972). The SIFT Algorithm for Fundamental Frequency Estimation",IEEE Trans. on Audio and Electroacoustics, vol. AU-20, no. 5, pp. 367-377 McClellan, J., Parks, T., Rabiner, L. (1973). A Computer Program for Designing Optimim FIR Linear Phase Digital Filters", IEEE Trans. on Audio and Electroacoustics, vol. AU-21, no. 6, pp. 506-526 Satorious, E.H. (1979). On the Application of Recursive Least Squares Methods to Adaptive Processing", Intl Workshop on Appl of Adaptive Control, Yale University Tribolet, J. (1977). Seismic Applications of Homomorphic Signal Processing", PhD Thesis M.I.T. Ulrych, T. (1971). Applications of Homomorphic Deconvolution to Seismology", Geophysics, vol. 36, no. 4, pp. 650-660 Young, T.Y. (1965). "Epoch Detection-A Method for Resolving Overlapping Signals",Bell Sys. Tech. Journal, vol. 44, pp. 401-426