I, .‘I. ....“.

. '11
0:. I”)!
I

5‘I '.
1V. ‘1':.
x‘ C
, 33431., 4.7,
. . .
‘.

x) ' . I, . ‘ .
{2}} (‘1‘: 6' '15 ‘
. .«fli'? ‘ ’7; $3)? "é, 1‘5 "9'
usék -
{335% fig . $3:
1'" t _.3-7;3 $9,»,
1‘” .

s’. 3.73:3).
- .

hi.
‘3'
5' - .
‘é \_‘

x I. ("1‘
“/P‘J. -(' "TI-:1"
‘ .' ‘ X ‘l

.. , . E- .‘ I :3ij ,
-' bl} ‘l . ,1 , s-(I 0K:- i‘. <\\‘>" "K:i7". ‘,‘:'l:‘
7‘ 1 > 2 if":
Q“ U:

‘l l I.
J_.’ '5" 138:". «(I
I .

I.

, ,3
li-S.
$5.1“ 3.43:7 '23.:
”£5 ‘ £3,119!
‘3' ~ ‘33:”-

a: ‘3'ma1 .
'urriiwqg 5a; 52' -
1-18 x, If); Am

f'mlfl
i" arr.:.-1:~ :2 ‘1, '. (.3?
. 1‘31?" ”3"”..«1'3‘; ./
I'I" L‘I1brru,-;f"{ ’

,3.” 1.. I’m

4’ ' Y" 2“ ,

6‘3" {'25: '

v"! .r
a

2:, 7N..-{,.,:.‘a.,l.‘£.'

l‘nt'fi‘l
«n.- -c-‘~'

n'
:2“)? p232}.
.x

~01 .
2 3“"
5‘

'rr. 3&9 ‘...‘&v "(3, M
4'1. " .:r* 7.3,: mug” 1,”;43fi.
'r-'{4‘5$°F '1' - 53,-"? "a“? 927%
‘L '.

o
-u

£5.14!” .. _ ..:;_.:..-
.1 fi'a..-...‘ «am-7--
$323” . 21 i"

n
" moJlmC

»I II; gar: “:‘;; 1;”?

{1:51, ‘3’”?! ‘2’. th:. ,g-u a“

A . .h- . r
_. . “'1',“ "3 . J ' .5':I,J“£:'
3- V. . .
'::.m'.m.
'21.“ 1'93“ "
.. “I u [9.64;
‘ ‘ :Y.'°":'3:41"q~
mm...“ l
C- 7;"

v-o I.-
.2):-

,IIJJ
\ 1\~.

 

\'A1}I:l

h
->t' \

\.
‘D

,2:‘
2%} ’4'.in
-.-
u. ”fig
.0} . "4'" 75‘... “.24. . "I." u. -" U
... 2,: ‘73...“a...7.‘_.:, -..,.--
—. Wilt!" km..- .r" .._
o - ,,

 

C' v. u—‘w v. _‘I- vv\ 4 .w';

 

, r
7

Wfié

“LITHIUM”! «Qmfiw

                                                                                                       

31293 00788 3667

 

This is to certify that the

thesis entitled

THE APPLICATION OF AN ADAPTIVE LEAST SQUARES LATTICE FILTER
IN THE DETECTION OF HEARTBEAT OCCURRENCES IN .
MEASUREMENTS FROM A REMOTE MICROWAVE VITAL LIFE SIGNS MONITOR

presented by

Patricia Ann Mahoney

 

has been accepted towards fulfillment
of the requirements for

Master of Science Electrical Engineering

degree in

 

 

Major professor

Date February 23 , 1989

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE

__J|
:|___
+7

 

 

 

 

 

 

 

 

 

 

 

==l|

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(—5-

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution

THE APPLICATION OF AN ADAPTIVE LEAST SQUARES LATTICE FILTER IN
THE DETECTION OF HEARTBEAT OCCURRENCES IN
MEASUREMENTS FROM A REMOTE MICROWAVE VITAL LIFE SIGNS MONITOR
By

Patricia Ann Mahoney

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering

1989

Q04 13950

ABSTRACT
THE APPLICATION OF AN ADAPTIVE LEAST SQUARES LATTICE FILTER IN
THE DETECTION OF HEARTBEAT OCCURRENCES IN
MEASUREMENTS FROM A REMOTE MICROWAVE VITAL LIFE SIGNS MONITOR
By

Patricia Ann Mahoney

A portable remote microwave vital life signs monitor has been built. Heart rate is estimated
by detecting Doppler shifts of a microwave signal that illuminates the chest wall of a human A
method of detecting heartbeats is based on modeling the microwave heartbeat signal as the output
of an all-pole filter that has been excited by a pseudo-periodic impulse train. An adaptive least

squares lattice filter is used.

The use of this detection method for a modified version of the monitor is verified in this
research. Estimates of the coefficients of the all-pole filter are found. The detection method
works well for only a limited set of Operating conditions of the modified monitor. The results of
the verification process suggest an alternative heartbeat detection method could be based on the

detection of changes in the statistics of the microwave heartbeat signal.

ACKNOWLEDGMENTS

The author wishes to thank Betsy Mates-Needham and Michael Francois for all of their
help. She is especially grateful to Betsy Mates-Needham for her help in the development of the

computer program in Appendix J.

iii

TABLE OF CONTENTS

List of Tables ....................................................................................................................... vi
List of Figures ...................................................................................................................... vii
Introduction .......................................................................................................................... 1
Research Objectives ............................................................................................................. 4
Verification Method and the Method Used to Estimate the All-Pole Filter Coefficients
.............................................................................................................................................. 7
Results .................................................................................................................................. 21
Heartbeat Detection with the Likelihood Variable ............................................................... 28
Conclusions .......................................................................................................................... 32
Recommendations ................................................................................................................ 35
Appendix A: Samples of Files form the Data Base ........................................................... 37 '
Appendix B: Prediction Error Variances Used to Determine All-Pole Filter Order .......... 42
Appendix C: All-Pole Filter Coefficients ........................................................................... 44
Appendix D: The Results of Thresholding the Likelihood Variable Sequence as a

Heartbeat Detection Method ................................................................................................ 46
Appendix E: The Unnormalized Pre-Windowed Least Squares Lattice Filter Parame-

ter Update Algorithm ........................................................................................................... 48
Appendix F: Computing the All-Pole Filter Coefficients from the Unnormalized Lat-

tice Parameters ..................................................................................................................... 51
Appendix G: A Fortran Implementation of the Unnorinalized Pre-Windowed Least

Squares Lattice Filter ........................................................................................................... 54
Appendix H: A Fortran Implementation of the All-Pole Filter Coefficient Update

Algorithm ............................................................................................................................. 59
Appendix I: The Normalized Pre-Windowcd Least Squares Lattice Filter ........................ 63
Appendix J: A Fortran Implementation of the Normalized Pre-Windowed Least

Squares Lattice Filter ........................................................................................................... 66
Appendix K: Review of the Development of the Microwave Vital Life Signs Moni-

tor ......................................................................................................................................... 72
Appendix L: Review of the Relationship Between Autoregressive Process Synthesis

and Adaptive Linear Prediction ........................................................................................... 87
Appendix M: Illustrating the Use of Adaptive Linear Prediction to Extract Irnpulses

from Processes Using Speech Processes as an Example ..................................................... 93

Appendix N: The Unnormalized Pre-Windowed Least Squares Lattice Filter ................... 98

References ............................................................................................................................ 106

\I

LIST OF TABLES

Table Bl. Prediction error variances used to determine all-pole filter order. ................... 43
Table Cl. All-pole filter coefficients. ............................................................... 45

Table D1. The results of thresholding the likelihood variable sequence as a method
of detecting heartbeats. ....................................................................................................... 47

LIST OF FIGURES

Figure 1. AR model all-pole filter coefficient Al for a chest wall microwave signal.

.............................................................................................................................................. 12
Figure 2 AR model all-pole filter coefficient A2 for a chest wall microwave signal
.............................................................................................................................................. 13
Figure 3 AR model all-pole filter coefficient A3 for a chest wall microwave signal
.............................................................................................................................................. 14
Figure 4 AR model all-pole filter coefficient A4 for a chest wall microwave signal
.............................................................................................................................................. 15
Figure 5. The effect of A on the likelihood variable. ........................................................ 16
Figure 6. The effect of it on the prediction error sequence, example 1. ........................... 18
Figure 7. The effect of A on the prediction error sequence. example 2. ........................... 19
Figure 8. Pole-zero plots of the AR model all-pole filters for the human subject files

and the inanimate object files. ............................................................................................ 22
Figure 9. Amplitude frequency response AR model all-pole filter for the human sub-

ject files and the inanimate object files. .................................................... 23
Figure 10. Recovered excitation process for a chest wall file recorded under ideal

conditions. ........................................................................................................................... 25
Figure 11. Recovered excitation process for a chest wall file recorded under more

realistic conditions. ............................................................................................................. 26
Figure 12. A chest wall file example of thresholding the likelihood variable. ................. 29
Figure 13. A leg file example of thresholding the likelihood variable. ............................. 30
Figure A1. Chest wall file with human subject at rest and holding breath. ...................... 38
Figure A2. Chest wall file with human subject at rest and breathing. .............................. 38
Figure A3. Chest wall file with human subject exercised and holding breath. ................. 39
Figure A4. Chest wall file with human subject exercised and breathing. ......................... 39
Figure A5. Leg file of a human subject. ........................................................................... 40
Figure A6. Inanimate object files. ..................................................................................... 41
Figure K1. The first Michigan State University microwave vital life signs monitor.
.............................................................................................................................................. 74
Figure K2 The modified Michigan State University microwave vital life signs moni-

tor ...................................................................................................................................... 85

vii

Figure M1. Block diagram of simplified model for speech production. ........................... 94

Figure M2. Deconvolution of the impulse response of an all-pole filter by linear
prediction. ........................................................................................................................... 96

Figure N1. Direct realization of an order M least squares linear predictor. 99

Figure N2. Lattice realization of an order M least squares linear predictor. .................... 100
Figure N3. Character of the least squares lattice filter exponential window for vari-
ous values of k .................................................................................................................. 103

viii

INTRODUCTION

There is an interest in developing a noninvasive vital life signs detector that will be used by
medical personnel working in hazardous environments. One requirement of the insu'ument is that
it is able to measure a human‘s heart rate at distances up to 6 inches from the body and through
protective clothing. A portable low energy microwave device has been developed to measure
heart rate by detecting pertubations of the chest wall due to the heart beating. The device
operates by illuminating the chest wall with a pulsed X-band microwave signal. Relative motion
betweenthedeviceandthechestwallisdetectedbyDoppIershiftsintherefiectedmicrowave
signal. Under realistic operating conditions, breathing, body, and background movements
obscure the heartbeat signal. The current challenge is to develop a real-time signal processing

technique that will extract heartbeat information from the microwave retums.

Past efforts to measure heart rate from the microwave signal have used various time and fre-
quency domain detection methods (Byme, Flymr, Zapp. & Sicgel, 1986; Byme & Siegel, 1985;
Byme, Zapp, Flynn. & Siegel, 1985; Hoshal, Ivkovich. Siegel, & Zapp, 1984; I-Ioshal & Siegel,
1985; Hoshal, Siege], & Zapp, 1984; Lin, Kiernicki. Kiemicki, & Wollschlaeger, 1979; Popovic,
Chan, & Lin, 1984). The heart rate estimation techniques can be classified into two groups, tech-
niques that rely the periodic nature of the heartbeat and techniques that attempt to identify indivi-
dual heartbeats in order to estimate heart rate. Detection techniques that rely on the periodic
nature of the heartbeat signal have had limited success because heartbeat occurrences are
pseudo-periodic. Time periods between heartbeats are not always constant. Autocorrelation is an

example of such a technique. Averaging techniques such as autoconelation also perform poorly

in tracking instantaneous changes in the heart rate.

Under ideal conditions. individual heartbeats can be easily identified in the microwave
heartbeat signal. A microwave heartbeat signal looks somewhat like an electrocardiogram
(EKG). The occurrence of a heartbeat in the microwave signal appears to be impulsive in nature.
Peak detection can be used to locate individual heartbeats in microwave heartbeat signals
recorded under ideal operating conditions. Peak detection fails under more realistic operating
conditions because the heartbeat signal is obscured by clutter due to breathing and background
movement. The search for a ”universal” heartbeat signature in the microwave signal has been
unsuccessful (M. Siegel, personal communication. 1988). The signature of the heartbeat changes
with orientation and distance of the microwave device with respect to the chest wall. The signa-

mrecanvaryfiompersonwpersonandfiomheanbeatmheanbeatmyme&8iegel,1985).

'I‘heextractionofanimpulsive signalfromaprocessesisalsoaproblem ingeophysicsand
speech analysis. Adaptive linear prediction is a useful technique in recovering impulse
occurrences from seismic traces and speech processes (Friedlander, 1982b; Lee & Morf, 1980;
Makhoul, 1975). Because of the similarity between the impulsive nature of heartbeat occurrences
in the microwave signal and pitch pulses in speech processes, Byme and Siegel (1985) adopted a
microwave heartbeat signal model and heartbeat dewction technique that are similar to the speech
process model and pitch pulse detector presented in Lee and Morf (1980). The detection methods
are based on adaptive linear prediction. The particular predictor used by Byme and Siegel was an
adaptive least squares lattice filter. a computationally efficient implementation of an adaptive
least squares linear predictor. Byme and Siegel showed that the adaptive least squares lattice
filter worked well in extracting heartbeats from microwave signals recorded under ideal and

adverse operating conditions.

Byme and Siegel (1985) worked with the Michigan State University Biomedical Signal

Processing Laboratory vital life signs monitor. Other work related to this monitor was done by

Byme, Flynn, Zapp, and Siegel (1986), Byme, Zapp, Flynn, and Siegel (1985), Hoshal, Ivkovich,
Siegel, and Zapp (1984), Hoshal and Siegel (1986) and Hoshal, Siegel, and Zapp (1984). A
comprehensive history of the development of a heart rate estimation technique for the Michigan
State University vital life signs monitor is given in Appendix K. A number of modification have
been made to the design of the vital life signs monitor since the Byme and Siegel (1985) study.
The modifications were made to improve the safety of the device. minimize power consumption.
increase the monitor’s dynamic range and sensitivity, and filter out signal components related to
breathing. The standard position of the monitor during the recording of test data has also
changed. The new monitor position provides data that will test the heart rate estimation tech-

niques under development more extensively.

'Ihe microwave signals recorded from the modified microwave unit appear to be very dif-
ferent from the microwave signals that Byme and Siegel (1985) used during the development of
theirheartbeatdctector. The purposeofthis researchis toverifythatthe Byme and Siegel model
and the use of the Byme and Siegel heartbeat detector are valid for the microwave heartbeat sig-
nals recorded from the modified monitor. The adaptive least squares lattice filter used in the
Byme and Siegel heartbeat detector has parameters that must be defined before the detection
technique can be used. Byme and Siegel did not give a systematic method of selecting values for
the lattice filter user defined parameters. Therefore, the bulk of this research effort is involved in
the selection of user defined lattice filter parameter values. A byproduct of this verification pro-
cess is an alternative heartbeat detection method based on the detection of changes in the statis-

tics of the microwave heartbeat signal.

RESEARCH OBJECTIVES

The primary objective of this thesis research is to verify that the Byme and Siegel (1985)
model and heartbeat detection method are valid for microwave heartbeat signals recorded from
the modified vital life signs monitor. Byme and Siegel models the microwave heartbeat signal as
the output of an all-pole filter that is excited by a process consisting of a pseudo-periodic impulse

train added to band-limited white noise. Equation 1 is the transfer function of the all pole filter.

Hdl-polcfilur(z)= l . . (1)
1+ A52”

where M is the order of the filter and A.- is a coefficient of the filter. The impulse train represents
the original heartbeat signal The all-pole filter represents the system the heartbeat signal passes
through. Adaptive linear prediction can be used to deconvolve the output of an all-pole filter that
has been excited by a white process or an impulse train (Friedlander, I981; Friedlander, 1982b;
Lee & Morf, I980; Makhoul. 1975). The excitation process is recoverable from the prediction
error sequence. Adaptive linear prediction can also be used to identify the coefficients of the all-
pole filter. Assuming that adaptive linear prediction will inverse the operation of an all-pole filter
on a process consisting of an impulse train added to white noise, Byme and Siegel applys adap-
tive linear prediction to the microwave heartbeat signal in order to recover the heartbeat
occurrences from the linear prediction residuals. The heartbeat signal should appear as a

sequence of large prediction enors.

The Byme and Siegel (1985) model is very similar to the autoregressive (AR) process
model. For the benefit of the unfamiliar reader, Appendix L gives a review of the AR model and
the use of linear prediction in the identification of AR model parameters. The adaption of the
Byme and Siegel model for the microwave heartbeat signal was motivated by the speech process
model used by Lee and Morf (1980) in their development of a pitch pulse detector. Appendix M
gives a simplified illustration of the use of linear prediction in the analysis of speech processes for
the benefit of the unfamiliar reader.

Byme and Siegel (1985) uses an adaptive least squares lattice filter as the linear predictor in
their heartbeat detection technique. The lattice filter is a normalized version of the the least
squares lattice filter used by Lee and Morf (1980). The lattice filter parameter update algorithm is
recursive.

A lattice filter parameter called the likelihood variable aids in the identification of large
prediction errors associated with heartbeat occurrences. The likelihood variable is related to the
log-likelihood function of the process that is input to the lattice filter. The likelihood variable is a
measure of the likelihood that successive data samples will come from the same Gaussian distri-
bution (Lee & Morf, 1980). The likelihood variable is a good detection statistic of the "unexpect-
edness" of the most recent input data points (Friedlander, 1982a). A function of the likelihood
variable is used in the lattice filter parameter update algorithm. The likelihood variable function
enables the lattice filter to quickly adapt to unexpected data (Lee & Morf, 1980). Byme and
Siegel found that the sequence formed by subtracting the value of the likelihood variable for the
previous time step from the current likelihood variable value could be used to isolate the large
prediction errors that are associated with heartbeat occurrences. The likelihood variable differ-
ence sequence is used to mask out prediction errors that are not associated with heartbeat

occurrences.

The first step of the Byme and Siegel (1985) heartbeat detection procedure is to pass the
microwave heartbeat signal through the lattice filter. The likelihood variable difference sequence
is formed. The prediction error sequence and the likelihood variable difference sequence are
multiplied together. The result is a sequence of isolated large prediction errors that represent pos-
sible heartbeats. The primary objective of this research is to verify that heartbeats in the
microwave heartbeat signals from the modified monitor can be extracted with the Byme and
Siegel procedure. The first step of the verification process will be to apply the lattice filter to
microwave heartbeat signals recorded from the modified monitor and observe the behavior of the

prediction error sequence and the likelihood variable sequence.

Byme and Siegel (1985) did not investigate the behavior of the configuration of the all-pole
filter in their microwave heartbeat signal model. The behavior of the all-pole filter configuration
might contain information that is useful in the development of a heart rate estimation technique.
Thesecondobjectiveoftlfisresearchistoestimatethecoefficientsoffireall-polefilterand

observe how they behave in time.

VERIFICATION METHOD AND THE METHOD USED TO ESTIMATE
THE ALL-POLE FILTER COEFFICIENTS

The Lattice Filter Parameter Update Algorithm

The Unnormalized Pre-Windowed Least Squares Lattice Filter from Friedlander (1982a, pp.
842-844) was used in the verification of the Byme and Siegel (1985) heartbeat detection method
for the microwave heartbeat signals recorded from the modified monitor. The prediction error
and the likelihood variable are directly available from the lattice filter parameter update algo-
rithm. The lattice filter parameter update algorithm and a Fortran implementation of the lattice
filter are given in Appendix E and Appendix G, respectively. For those who are unfamiliar with
least squares linear prediction and the lattice form, see Appendix L and Appendix N. The all-
pole filter coefficient estimates were obtained from the lattice filter parameters through an algo-
rithm given by Friedlander (1982a, pp. 845). The algorithm used to estimate the all-pole filter
coefficients is given in Appendix F. Appendix H gives a Fortran implementation of the algorithm

used to estimate the all-pole filter coefficients.

The Data Base of Files Recorded from the Modified Monitor

The verification process was facilitated by the formation of a data base of microwave sig-

nals recorded for various operating conditions of the modified microwave vital life signs monitor.

Samples of files from the data base are given in Appendix A. There are 1024 data samples in
each data file. The sampling rate was 64 samples per second. Microwave signals were recorded
for two kinds of reflective surfaces, human subjects and inanimate objects. The inanimate objects

consisted of a wool surface. a metal surface. and an open room.

In the experiments reported in this research. there were four human subjects. For each
human subject five microwave signals were recorded. Four of the five signals resulted from
reflecting a microwave signal off the chest wall of the subject. The fifth recorded Signal resulted
from reflecting the microwave signal off one of the subject’s legs. Each subject was rested and
holding his/her breath for the first chest wall file. The subjects were resting and breathing nor-
mally for the second chest wall file. In the third file, the subjects had been exercising and held

their breath. The subjects were exercised and breathing for the fourth chest wall file.

The monitor had the same position with respect to the subject for each chest wall file. The
subjects were seated. The monitor was pointed just left of center of the chest wall. Studies previ-
ous to this research showed that the strongest heartbeat signals can be found in microwave signals
recorded with the monitor positioned left of center of the chest wall (M. Siegel. personal com-
munication, 1988). The monitor was place about 6 inches from the subject. The subjects were

wearing street clothes.

Heartbeat reference signals were obtained from an in-house designed unit that measures
body surface potential between the hands of a human subject. The output of the device resembles
an EKG signal. A hand potential reference heartbeat signal was simultaneously recorded for each

chest wall file.

A microwave signal reflected from the bare calf of a leg of each subject was recorded. The
monitor was positioned 6 inches from the leg. A hand potential reference heartbeat signal was
simultaneously recorded. Movement detected in the returns of the microwave signal from the leg

due to the heart heating is expected to be insignificant.

The data base includes three files where the microwave signals were reflected from the inan-
imate objects. One microwave file was recorded for a microwave signal reflected from a wool
surface placed 6 inches from the monitor. The second file resulted from microwave reflections
from a flat metal surface placed 6 inches from the monitor. The third file is a record of returns

from a microwave signal sent into an open room.

The Selection of Lattice Order

The prOper order of the lattice filter must be chosen before the verification of the Byme and
Siegel (1985) model and heartbeat detection method for the modified monitor can proceed. The
order of the all-pole filter in the Byme and Siegel model will be the same as the proper order of
the lattice filter. The method used to determine the proper order of the lattice filter is based on
the optimization criterion of least squares linear prediction. In linear prediction the current value
Of a process is estimated by a linear combination of past values of the process. The coefficients
in the linear combination are chosen such that the sum of the squared prediction errors is minim-
ized. The prOper order of the linear predictor is the number of past data elements in the linear
combination. If the order of the least squares linear predictor is greater than the order of the pro.
cess, the linear predictor is using past data in its prediction of the current data point that the
current data point is not dependent on. The excess past process elements should not make a con-
tribution to the prediction of the current data item. The weight the linear predictor assigns to the
excess process elements should be close to zero. If the excess process elements make only a
small contribution to the prediction of the current process element, then the sum of the squared
error should not significantly change with an increase in linear prediction order beyond the order

of the process (Chandra & Lin, 1974). The order of the microwave heartbeat signals was

lO

determined by processing the signals with a high order lattice filter and then calculating the mean
of the sum of the squared prediction errors for each lower order section of the lattice filter. The
order of the process was chosen to be the order at which the mean of the sum of the squared pred-
iction errors. or the variance of the prediction enors. began to level off for increasing lattice filter

order.

A preliminary study was done in order to establish a range for the order of the microwave
signals in the data base. Data files recorded from the microwave signal being reflected from a
human chest wall. a human leg and the inanimate objects were processed through a lattice filter of
order 8. The exponential weighting factor, a parameter of the lattice to be explained in the next
section, was arbitrarily set to 0.95. The variance of the prediction error leveled off at order 4 for

each data file.

In all subsequent analyses of files from the data base. a lattice filter of order 6 was used.
The exponential weighting factor was set to 0.99 for reasons given in the next section. The vari-
ance of the prediction errors for order 1 through 6 were recorded for each file in the data base.
The results are tabulated in Appendix B. The results in Appendix B are the same as the results of
the preliminary study. The variance of the sum of the squared prediction errors for each file in

the data base appears to level off at order 4.

The Selection of a Value for the Exponential Weighting Factor, A

A second parameter of the lattice filter that needs to be defined before the lattice filter can
be used is the exponential weighting factor, A. The exponential weighting factor determines the
configuration of exponential window that weights the past data used in the estimation of the

statistics of the signal being processed by the lattice filter. As the value of A increased. the length

11

of the exponential window increases. A study of A’s effect on the lattice filter’s processing of

microwave heartbeat signals was made in order to chose a value for A.

A group of chest wall microwave signals from the data base were processed through the lat-
tice filter with A set at various values. A was set to values ranging from 0.99 to 0.85. A 4th order
lattice was used. The behavior of the output and parameters of the lattice filter was observed for
each file. Three consistent behavior pattems were observed. The behavior pattems occurred in
the estimated coefficients of the all-pole filter, the likelihood variable, and the prediction error

sequence.

The value of the all-pole filter parameters were updated for every new data element of the
file being processed. Figure 1 through Figure 4 illustrate how the four all-pole filter coefficients
vary in time for r. = 0.99, A: 0.95, and 2. = 0.90. For 2. = 0.99, the plots of the all-pole filter
coefficients are smoorh. As A decreases in value, the plots of the coefficients show large varia-
tions.

Figure 5 shows how the likelihood variable behaves for A = 0.99, A = 0.95, and A = 0.90. As
A decreases in value the variation of the likelihood variable increases. In most of the chest wall
files studied, a rise and fall of the likelihood variable can be correlated with a heartbeat
occurrence from the hand potential heartbeat reference signal. Although the behavior of the likel-
ihood variable reported in Byme and Siegel (1985) is similar, the rise and fall of the likelihood
variable associated with the occurrence of a heartbeat are faster in the Byme and Siegel study.

The rise and fall of the likelihood variable in this research appears to be more like a hump.

The behavior of the prediction error sequences for decreasing values of A was opposite the
behavior of the all-pole filter coefficients and the likelihood variable. As A decreases. the ampli-
tude of the largest prediction errors seem to decrease. The variation of the prediction error for
decreasing values of A is not as pronounced as the variations observed in the all-pole filter

coefficients and the likelihood variable. The variance in the prediction error for different values

(a)

(b)

(e)

(d)

12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-1.0 —
-l.5 —
-2.0 -
-2.5
I T I I
4 6 8 10
-1.0 -
-1.5 —
-2.0 a
-2.5 -
T I l l
4 6 8 10
-1.0 ~
-1.5 —
-2.0 —
-2.5 -l
j l I l
4 6 8 10
0.0 — ‘
I I I 1
4 6 8 10
“lime (seconds)

Figure 1. AR model all-pole filter coefficient A; for a chest wall microwave signal. (a) A, for
A=0.99. (b) A] for A=0.95. (c) A] for A=0.90. (d) chest wall microwave signal.

l3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.0 _
(a) 2.0 —
1.0 -
I I I I I I
0 2 4 6 8 10
3.0 a
(b) 2.0 -
1.0 a
I I I I I I
0 2 4 6 8 10
3.0 -
(c) 2.0 -
1.0 —
I I I I I T
0 2 4 6 8 10
(d) 0.0— '1 ' _ r I ~
I I I I I I
0 2 4 6 8 10
Time (seconds)

Figure 2. AR model all-pole filter coefficient A; for a chest wall microwave signal. (a) A; for
A=0.99. (b) A 2 for A=0.95. (c) A 2 for A=0.90. (c) chest wall microwave signal.

l4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0—
(a) -l.0-
-20—
I I I I I I
0 2 4 6 8 10
0.0- l
(b) 4.0—
4.01 W
I I I I I I
0 2 4 6 8 10
0.0— l
(c) -l.0-
-20—
I I I I I I
0 2 4 6 s 10
(d)0.0— °
, , .
I I I I I I
0 2 4 6 3 10
Time(seeonds)

Figure 3. AR model all-pole filter coefficient A 3 for a chest wall microwave signal. (a) A3 for
A=0.99. (b) A 3 for A=0.95. (c) A 3 for A=0.90. (d) chest wall microwave signal.

15

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0 .—
(a)
0.5 -
I I T I
4 6 8 10
1.0 J
(b)
0.5 —
I I I I
4 6 8 10
1.0 -
(e)
0.5 —
I I I I
4 6 8 10
(d) 0.0 —
I I I I
4 6 8 10
Time (seconds)

Figure 4. AR model all-pole filter coefficient A4 for a chest wall microwave signal. (a) A4 for

A=0.99. (b) A 4 for A=0.95. (c) A4 for A=0.90. (d) chest wall microwave signal.

 

1.0
0.8
0.6 _-
(a)
0.4 _
0.2 _
0.0 —

 

 

 

—.
—
—

 

1.0 -—
0.8 -
0.6 —
0.4 —
0.2 —
0.0 —

(b)

 

 

 

 

1.0 -
0.8 —
0.6 —
(C) 0.4 —
0.2 -

0.0 —

 

 

 

._.
0‘4

 

 

 

 

 

 

I I I I I ‘I
0 2 4 6 8 10
Time (seconds)

Figure 5. The effect of A on the likelihood variable. (a) likelihood variable with A=0.99. (b)
likelihood variable with A=0.95. (c) likelihood variable with A=0.90. ((1) hand potential heart-
beat signal.

17

of A is greater for some files than it is for others. Figure 6 and Figure 7 Show the error sequence
for two different files from the data base filtered with A = 0.99, A = 0.95, and A = 0.90. The vari-

ance is greater for the prediction error sequence in Figure 6 that it is in Figure 7.

The behaviors of the all-pole filter coefficients, the likelihood variable and the prediction
error sequence can be related to each other by considering the length of the exponential weighting
window of the lattice filter. As A increases, the length of the exponential window increases. For
larger values of A. the lattice filter’s memory of the process it is trying to predict is longer. More

past values are used by the lattice filter to estimate the statistics of the process.

The likelihood variable can be used as an indicator of the event that the most recent data
points received by the lattice filter are outliers from the Gaussian distribution of the data that the
filter has in its memory (Lee & Morf. 1980). As A decreases, the variance of the likelihood vari-
able increases for the microwave heartbeat signals. Generally. the periods during which the likel-
ihood variable rises and falls correlate with occurrences of heartbeats. The microwave heartbeat
signal appears to be different in character for periods that correlate with the occurrence of heart-
beats. If the length of the memory of the lattice filter is shorter than the time between heartbeats.
then the occurrence of a microwave heartbeat may appear to be a far outlier from the distribution
of the data points that just preceded it. Under these conditions, the value of the likelihood vari-
able should rise. If the memory of the lattice filter covers the occurrence of several heartbeats,
the occurrence of a new heartbeat may not appear to be such a distant outlier. The value of the
likelihood variable would vary less for a lattice filter with a long memory than for a lattice filter

with a short memory.

The behavior of the likelihood variable directly affects the estimation of the lattice filter
parameters used to calculate the all-pole filter coefficients. A function of the likelihood variable
is used as a gain in the update algorithm for the lattice filter parameters. The gain enables the lat-

tice filter to quickly adapt to changes in the process being predicted. If the likelihood variable is

l8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 A
o ._
(a) I
—20 —
-4o _
I I I I
4 6 8 10
20 - .
0 q z I
(b) , . ‘,
.20 _
—40 -
I I I I
4 6 8 10
20 _
l‘ '
0 — i 4
(C) i
-20 _
_40 _
I I I I
4 6 8 10
(d) 0- ,‘
I I I I
4 6 8 10
Time (seconds)

Figure 6. The effect of A on the prediction error sequence, example 1. (a) enor sequence with
A=0.99. (b) enor sequence with #095. (c) enor sequence with A=0.90. (d) hand potential

reference.

l9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

so _
(a) 0 — I
-50 _
F 7 I I I
2 4 6 8 10
50 —
(b) 0— ‘
.50 _
1 1 I I I
2 4 6 8 10
so ..
(c) O -
-5o ..
I I I I I
2 4 6 8 10
(d) 0.0 — .
I T I I I
2 4 6 8 10
Time (seconds)

Figure 7. The effect of A on the prediction enor sequence, example 2. (a) error sequence with
A=0.99. (b) error sequence with A=0.95. (c) enor sequence with A=0.90. ((1) hand potential

reference.

20

large, then the lattice filter responds by altering its parameters to minimize the sum of the squared
prediction errors. The information contained in the most recent data points received by the lattice
filter will be considered to be the target to which the filter adapts. If the parameters of the lattice

filter change, the estimates of the all-pole filter coefficients might show variations.

The variations in the prediction error sequences can be related to the length of the memory
of the lattice filter. When the memory of the lattice filter is short. the data the lattice filter is try-
ing to adapt to is localized. The likelihood variable is more sensitive to changes in the data the
lattice filter receives. If the likelihood variable is more sensitive to changes, then the lattice filter
is better able to track short term changes in the microwave signal such as microwave reflections
during the occunence of a heartbeat. Consequently, the lattice filter may be better able to minim-
ize the sum of the squared prediction errors. This would explain why the amplitudes of the larg-

est prediction errors decrease for decreasing lattice filter exponential window length

The value of A chosen for the verification of the Byme and Siegel (1985) model and bean-
beat detection method was A = 0.99. Long lattice filter memory was chosen because the large
prediction errors associated with heartbeats in the hand potential reference signal were more pro-

nounced. The all-pole coefficient sequences are smooth for A = 0.99.

RESULTS

The Byme and Siegel Model All-Pole Filter Coefficients

Estimates of the all-pole filter coefficients were found for each file in the data base. The
all-pole filter coefficient estimates were updated for each point in a file. A mean value of each
coefficient for each data base file was found from a sample of consecutive estimates of the
coefficients. Each sample consisted of 600 points starting at point 200 or 3 seconds into the
coefficient file. ‘I‘hepurposeofthedelayinthestartofthe sample wastocnsurethatthelattice
filter parameters had converged. Appendix C is a tabulation .of the all-pole filter coefficient sam-

ple averages for each file in the data base.

The all-pole filter coefficients of files for microwaves reflected from the human subjects are
close in value. The all-pole filter coefficients of files for microwaves reflected from inanimate
objects are close in value. The mean of the coefficients for the human subject files and the mean
of the inanimate object coefficients are given in Appendix C. The coefficients for the inanimate

object files are lower in value than the coefficients for the human subject files.

In order to determine if the difference between the inanimate object file all-pole filter
configuration and the human subject file all-pole filter configuration is significant, pole-zero plots
were made for the transfer functions of the all-pole filters for both file types. The pole-zero plots
are in Figure 8. The four poles of each all-pole filter appear as two conjugate pairs in the pole-
zero plot. The plot shows that the poles of the different file types are close. Figure 9a and Figure

9b are plots of the amplitude responses of the different file type all-pole filters. Although the

21

22

 

 

 

z‘

 

Harm Sawmill) = 24 _ 1934354» 248527— 1.6842 + .7273

H - - (2) " Z4
Inanimate Object - Z4 _ 1.35623 + 1.68022 _ 1.0132 + .5994

 

Figure 8. Pole-zero plots of the AR model all-pole filters for the human subject files and the
inanimate object files. H - Human subject AR model all-pole filter poles. I - Inanimate object
AR model all-pole filter poles.

 

154

10—

(a)

5-4

 

 

 

 

_4
_
—I
—
-—I
-_I
_l

 

15-‘

10—

(b)

 

 

 

 

I I I I I I
15 20 25 30

Frequency (Hz)

O—J
LII
H
C)

Figure 9. Amplitude frequency response AR model all-pole filter for the human subject files and
the inanimate object files. (a) frequency response for the human subject files. (b) frequency
response to the inanimate object files.

24

resonance peaks for the all-pole filter for the human subject file are greater in magnitude than the
resonance peaks for the all-pole filter of the inanimate object files, the resonance peaks are at

approximately the same frequencies

Recovery of the Heartbeat Signal from the Microwave Heartbeat Signal

The primary objective of this research was to verify that the Byrne and Siegel (1985) model
and heartbeat detection technique are valid for the microwave heartbeat signals recorded florn the
modified monitor. If the Byme and Siegel model is valid for the modified monitor microwave
heartbeat signals and linear prediction will inverse the operation of an all-pole filter on a process
consistingofanimpulsetrainaddedtowhitenoise,thenthepredictionerrorsofthelattice filter
shouldconsistofanimpulsen'ainaddedtowlfitenoise. Theimpulsetrainshouldcorrelatetothe

occunences of heartbeats in the hand potential reference signal.

Only a few chest wall files had prediction error sequences that resembled an impulse train
added to white misc. These files were recorded under ideal conditions. The human subject was
at rest and holding her/his breath. There was little movement of the chest wall due to breathing
or the body moving. Figure 10 shows the original microwave heartbeat signal and the prediction

error sequence for a chest wall file where the model is well fitted.

Heartbeat occurrences in chest wall files recorded under more realistic operating conditions
are not easily detectable from the prediction error sequences. For files where the subjects were
breathing, the prediction errors at locations of heartbeats were not distinguishable fiom other
prediction enors. Figure 11 shows an example of a prediction error sequence for a chest wall file

recorded while the subject was breathing and after the subject had exercised.

 

 

 

 

 

 

 

 

 

 

 

 

(a) 0" ' i ". l, ‘ ii i ii ‘
i ‘ I H J
I I I T I I
0 2 4 6 8 10
0.. i
(b)
I I I I I W
O 2 4 6 8 10
(d) O_‘ I | |I
I I I j I I
0 2 4 6 . 8 10
Time (seconds)

Figure 10. Recovered excitation process for a chest wall file recorded under ideal conditions. (a)
chest wall microwave signal. (b) prediction error sequence. (c) hand potential heartbeat signal.

26

 

(a) o _

 

 

 

 

(b) 0‘ " ‘ . I I A“

 

 

 

 

(d, 0_ II..II.II

 

 

 

 

 

 

0
~—
&
0‘
co
5

Figure 11. Recovered excitation process for a chest wall file recorded under more realistic condi-
tions. (a) chest wall microwave signal. (b) prediction error sequence. (c) hand potential heart-
beat signal.

27

The occurrence of large prediction errors that can be associated with heartbeats from the
hand potential reference file is not consistent for the chest wall files. The behavior observed that
was consistent fiom one chest wall file to the other was the behavior of the likelihood variable.
For most of the chest wall files, there is a hump in the likelihood variable sequence for most of

the heartbeats in the hand potential reference signal.

HEARTBEAT DETECTION WITH THE LIKELIHOOD VARIABLE

This section presents the results of an investigation into the feasibility of developing a
heartbeat detector using only the likelihood variable. Heartbeats were detected by applying an
arbitrary threshold of 0.5 to the likelihood variable sequences of all the chest wall files. If the
value of the likelihood variable remained at or above 0.5 for 4 consecutive points. a likelihood
variable cluster was formed. Note that the likelihood variable sequences were produced by a lat-
tice filter with A = 0.90. In order to verify that a file’s likelihood variable cluster sequence
represents the file’s heartbeat signal, each likelihood variable cluster was classifyed as a heartbeat
cluster or a non-heartbeat cluster depending on the location of the cluster in relation to the heart-
beats of the hand potential reference signal of the file. A cluster was classifyed as a heartbeat
cluster if the cluster occurred during the period between the reference heartbeat to the halfway
point to the next reference heartbeat. If a cluster occurred during the period between the halfway

point and the next heartbeat, the cluster was classifyed as a non-heartbeat cluster.

The number of heartbeat clusters, the number of non-heartbeat clusters, and the number of
heartbeats in the reference signal were tabulated for each chest wall and leg file from the data
base. Appendix D contains the tabulation of these results. The percentage of reference heartbeats
associated with a likelihood variable cluster was calculated for each file and was recorded as the
percentage of hits. The percentage of likelihood variable clusters not associated with reference
heartbeats was calculated for each file and was recorded as the percent of false alarms. Figure 12
shows an example of thresholding the likelihood variable sequence for a chest wall file. Figure

13 shows the result of thresholding the likelihood variable sequence for a leg file.

28

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) o -
I I I I I
8 10 12 14 16
0.5 -
(b)
I I I l I
8 10 l2 14 16
H n H F1 I! H I- F
(e)
_l ._ _J
I I I I I
8 10 12 14 16
J I - = ‘ . . i
(d) 0.0 g i . . ‘ ‘
I I I I I
8 10 12 l4 16
Time (seconds)

Figure 12. A chest wall file example of thresholding the likelihood variable. (a) chest wall
microwave signal. (b) original likelihood variable sequence. (c) likelihood variable cluster
sequence. ((1) hand potential heartbeat signal.

30

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) 0_ M
T I I I I
8 10 12 l4 16

0.5 —

(b)
I I I I I
8 10 12 14 16

I ’ I I ' i i I

(C)
I I I I I
8 10 12 l4 16

(d) 0.0 —
I I I I I
8 10 12 14 16

Time (seconds)

Figure 13. A leg file example of thresholding the likelihood variable. (a) leg microwave signal.
(b) original likelihood variable sequence. (c) likelihood variable cluster sequence. (d) hand

potential heartbeat signal.

31

For three of the classifications of the chest wall files, the percentage of heartbeats detected
is about 80% and the rate of false alarms is less than 20%. Thresholding the likelihood variable
at 0.5 is far from being an optimal detector. Figure 12 shows sections of the likelihood variable
sequence that were not identified as heartbeats yet by sight seem to be obvious heartbeats
occurrences. The 0.5 threshold may be too stringent of a requirement. Although some heartbeats
were not detected, the periodicity of the detected heartbeats could be used in the estimation of

heart rate through a technique such as autocorrelation.

The percentage of detected heartbeats for the leg data was low, 50%. The rate of false
alarms for the leg data, 30%, was higher then the false alarm rate for the chest wall data. The low
heartbeat detection rate and the higher false alarm rate for the leg data indicates that likelihood
variable cluster sequence resulting from thresholding a leg likelihood variable sequence at 0.5
does not correlate with the hand potential reference heartbeat signal Each leg file had 18 to 20
likelihood variable clusters while the number of heartbeats in the hand potential signals varied
from 35 to 20. The constant number of likelihood variable clusters for the leg files and the vary-
ing numbers of heartbeats in the reference files helps support the conclusion that the likelihood
variable clusters of the leg files do not correlate to the heartbeats in the hand potential reference

signal.

CONCLUSIONS

The primary objective of this research was to verify that the Byme and Siegel (1985) model
and heartbeat detection method are appropriate for microwave heartbeats signals recorded from
the modified monitor. The Byme and Siegel model seems to hold for only a few chest wall files
from the data base. The files were recorded under ideal conditions for heartbeat detection. The
subjects were still and holding their breath. Distinctive large prediction errors are coincident with
the occunence of heartbeats in the hand potential reference signals for these files. For files
recorded with more movement of the chest wall, the Byrne and Siegel model does not wear to
hold as well. In these files, the prediction errors that are near heartbeat locations are not distin-
guishable from prediction errors between heartbeat occurrences. The Byme and Siegel model
may not sufficiently characterize the microwave heartbeat signal when there is chest wall move-

ment due to sources other than the heart beating.

When linear prediction is used to deconvolve a process, it is assumed that the process is the
result of exciting an all-pole filter that has all of it poles and zeros inside the unit circle. This
implies that the systems linear prediction can inverse are minimum-phase. If the system the
heartbeat signal passes through is not minimum-phase, adaptive linear prediction will not exactly
deconvolve the microwave heartbeat signal. The prediction error sequence would probably not
resemble the original excitation process. If the microwave heartbeat signal is the output of a
non-minimum-phase system, the prediction enor sequences resulting from processing the

microwave heartbeat signal files recorded when the subjects were breathing may be expected.

32

33

Linear predictionmay not exactly deconvolve the microwave heartbeat signal even if the
minimum-phase all-pole filter model held for the microwave heartbeat signal. Ifthe all-pole filter
1 is excited by a white process or a single impulse, linear prediction can exactly deconvolve the
output of the all-pole filter. A white process or a single impulse are uncorrelated processes. Both
processes have flat spectra. An impulse train is correlated. The linear prediction may be biased
for a process that results from exciting an all-pole filter with a conelated input. The correlation
information of the excitation process may be absorbed in the linear prediction. The prediction
error sequence will be something other than the process that excited the all-pole filter. The pred-
iction enor will be more white that the original excitation process. If the microwave heartbeat
signal could be modeled as the output of an all-pole filter excited by an impulse train, the impulse
train may not be recoverable from the prediction errors because of imperfect deconvolution.

The all-pole coefficients were obtain in the ham of gaining new information that might help
in the develOpment of a heart rate estimation technique. It was found that the configuration of the
all-pole filter was uniform for the data base files. The data base files are the result of reflecting
microwave signals off very different surfaces. One thing that all the data base files have in com-
mon is the monitor. The all-pole filter in the Byme and Siegel (1985) model may be modeling

the monitor.

The likelihood variable was the most consistent source of information about the location of
heartbeats in the microwave signals of this data base. The fact that the isolation of prediction
errors associated with heartbeats in the Byme and Siegel (1985) heartbeat detection technique
relies on the likelihood variable sequence supports the conclusion that the likelihood variable is a
good source of information about the location of heartbeats in the microwave signals The inves-
tigation into the use of the likelihood variable alone in the detection of heartbeats showed that it
may be feasible to estimate heart rate from the likelihood variable sequence for the data base

chest wall files. The likelihood variable of the lattice filter is probably not the optimal way of

34

detecting changes in the statistics of a stochastic process. Other methods of detecting changes in
the statistics of stochastic processes should be considered before a method using the lattice filter

likelihood variable to detect heartbeats in the microwave heartbeat signals is developed.

In conclusion, the Byme and Siegel (1985) model and heartbeat detection technique should
not be accepted or rejected for the signals recorded from the modified monitor before the validity
of the assumptions of the model and detection technique are investigated. If the Byrne and Siegel
model is valid for signals recorded from the modified monitor. it appears that the configuration of
the all-pole filter is uniform for the files in the data base. The consistent behavior of the likeli-
hood variable for the files of the data base suggests that heartbeats in the microwave heartbeat
signal can be identifiable by the detection of changes in the statistics of the microwave heartbeat

signals.

RECOMMENDATIONS

The Byme and Siegel (1985) study and this research considered only one pitch pulse detec-
tor. Because of the similarity between the problems of finding pitch pulses in speech
processes and detecting heartbeats in the microwave heartbeat signals, other pitch pulse
detection techniques should be investigated. The study of other pitch pulse detection tech-
niques might reveal more information about the limitations of using linear prediction and
ways to overcome those limitations. Methods other than linear prediction that can be used

to detect heartbeats from the microwave signals may be found.

The consequences of using linear prediction to inverse the Operation of a non-minimum-
phase system should be investigated. This problem has been researched in the area of

speech analysis.

The consequences of using linear prediction to deconvolve a process that results from excit-
ing an all-pole filter with a conelated process should be investigated. This problem has

been research in the area of pitch pulse detection in speech processes.

If the development of a method of identifying heartbeats in the microwave heartbeat signal
by detecting changes in the statistics of the microwave signal is to be pursued, methods of

detecting changes in the statistics of stochastic process should be studied.

35

36

It is possible that the all-pole filter of the Byme and Siegel (1985) model is modeling the
monitor. It may be useful to see how changing the configuration of the monitor effects the
configuration of the all-pole filter. The configuration of the monitor may be affecting how

well linear prediction can perform in the detection of heartbeats in the microwave signals.

APPENDICES

APPENDIX A

SAMPLES OF FILES FROM THE DATA BASE

37

38

 

 

 

 

 

 

 

 

200
100—
(a) o- l
l
-100-
I I I I I
0 2 4 6 8
100-
(b) 0- ‘0 I
-100-
I I T j I
0 2 4 6 8
Time (seconds)

FigureAl. (Jrestwallfilewithhmnansubjectatrestandholdingheath. (a)microwavesignal. (b) hand

potential referencesignal

 

200.
100-

(a) 0-
-100A

 

 

 

12

14 16

 

100-

(b)
-100-

 

 

 

r
10

Time (seconds)

I
12

I I
14 16

Figure A2. Chest wall file with human subject at rest and breathing. (a) microwave signal. (b) hand
potential signal.

39

 

 

 

 

 

 

 

 

 

200-
(a) 0 .-
-200-
I I I I I
8 10 l2 l4 16
100 -I
0 ..
(b)
-100 _
I 1 I I I l
8 10 l2 l4 16
Time (seconds)

FigureA3. Chestwallfilewithhumanmbjectexereisedmdholdingbreath. (a)microwavesignal. (b)

hand potential reference signal

 

300
200 _
100-

(a) o .1
-100 _
-200 -

 

 

 

 

 

 

-1“)—

 

 

 

N-i

j
4

Time (seconds)

j
8

Figure A4. Chest wall file with human subject exercised and breathing. (a) microwave signal. (b) hand

 

 

 

 

 

 

 

 

100..
(a) o-
-100-

I l T I T

o 2 4 6 8
100-
50—
-sod
-1“)...

I 7 I I I

o 2 4 6 8

Time (seconds)
Figure A5. Leg file of a human subject. (a) mia'owave signal. (b) hand potential reference signal

41

 

50—
(a) 0—

-50 _

 

 

—100—

 

 

 

 

 

 

 

 

 

l l l ' l l
0 2 4 _ 6 8

Time (wconds)

Figure A6. Inanimate object files. (a) microwave signal for wool surface. (b) microwave signal
for metal surface. (c) microwave signal for open room.

APPENDIX B

PREDICTION ERROR VARIANCES USED TO DETERMINE

ALL-POLE FILTER ORDER

42

43

 

Table Bl. Prediction Error Variances Used to Determine All-Pole Filter Order

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Prediction Error Variance
File Type Order
1 1 2 l 3 I 4 1 5 6

Human Subjects at rest and holdingbreath.

Subject 1 1557 342 274 142 137 125

Subject 2 1129 454 349 167 152 144

Subject 3 4855 1217 886 474 421 376

Subject 4 2183 551 416 201 189 180
Human Subjects at rest and brean‘_g.

Subject 2 587 219 162 63 59 56

Subject 3 4195 ll 15 784 351 305 247

Subject 4 980 295 225 92 88 86
Human Subjects exercised and holding breath.

Subject 1 3564 1098 781 325 272 248

Subject 2 2531 859 549 302 274 235

Subject 3 5833 1285 936 476 409 336

Subject 4 1965 563 388 152 136 119
Human Subjects exercised and breathing.

Subject 1 2858 1006 687 359 319 252

Subject 2 1925 614 442 208 190 166

Subject 3 3287 1029 718 342 308 253

Subject 4 2237 ' 725 501 203 170 135
Human Subjects’ legs.

Subject 1 2026 558 407 174 153 130

Subject 2 2028 624 408 194 176 153

Subject 3 985 289 208 110 105 101

Subject 4 1 180 393 280 1 19 106 93
Inanimate Objects.
Wool Surface 659 316 276 201 198 188
Metal Surface 1415 566 484 292 275 228

Open Room 775 333 289 184 181 161

 

 

 

 

 

 

 

APPENDIX C

ALL-POLE FILTER COEFFICIENTS

45

 

Table C1. All-Pole Filter Coefficients

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fil T All-Pole Filter Coefficients
e ype A1 L A2 1 A3 1 A4
Human Subjects at rest and holding breath.
Subject 1 -l.864 2.397 -1.534 .7052
Subject 2 -1.898 2.298 -1.607 .7369
Subject 3 -2.055 2.562 -1.696 .7027
Subject 4 -1.984 2.512 -1.670 .7320
Human Subjects at rest and breathing.
Subject 2 -1.631 2.140 -1.467 .7745
Subject 3 -1.991 2.526 -1.716 .7363
Subject 4 -1.931 2.431 -1.651 .7550
Human Subjects exercised and holding breath.
Subject 1 -2.117 2.548 -1.739 .6786
Subject 2 -1.980 2.426 -1.647 .6597
Subject 3 -2.081 2.576 -1.684 .6735
Subject 4 -2.047 2.630 -1.806 .7732
Human Subjects exercised and breathin .
Subject 1 -l.775 2.360 -1.594 .7788
Subject 2 -2.116 2.589 -1.780 .7189
Subject 3 -2.1 1 1 2.604 -1.781 .7223
Subject 4 -1.899 2.438 -1.687 .7792
Human Subjects’ legs.
Subject 1 -2.025 2.551 -1.730 .7449
Subject 2 -2.049 2.561 -1.739 .7129
Subject 3 -2.143 2.575 -1.734 .6786
Subject 4 -1.991 2.482 -l.730 .7555
Human Subject Means -l.984 2.485 -1.684 .7273
Inanimate Objects.
Wool Surface -1.301 1.497 0.831 .5089
Metal Surface -l.480 1.861 -1.220 .6830
Open Room -1.286 1.683 -0.989 .6064
Inanimate Objects Means -1.356 1.680 -1.013 .5994

 

 

 

 

 

 

APPENDIX D

THE RESULTS OF THRESHOLDING THE LIKELIHOOD VARIABLE SEQUENCE

AS A HEARTBEAT DETECTION METHOD

46

47

 

Table Dl. The Results of Thresholding the Likelihood Variable Sequence
as a Method of Detecting Heartbeats

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Actual Number Number % Number %
File Heartbeat of of Hits of False Alarms
Count Clusters Hits False Alarms
Human Sub' cts at rest and holdifl breath.
Subject 1 20 19 17 85 2 10
Subject 2 17 17 19 100 0 0
Subject 3 23 24 19 83 5 21
Subject 4 24 23 19 79 4 18
Means 85 12
Human Sub'ects at rest and breathing.
Subject 2 14 20 14 100 6 30
Subject 3 23 20 18 78 2 10
Subject 4 25 21 17 68 4 19
Means 82 19
Human Sub' cts exercised and holding breath.
Subject 1 37 28 25 68 3 10
Subject 2 14 23 13 92 10 43
Subject 3 27 23 22 81 1 4
Subjgct 4 24 23 21 87 2 9
Means 81 17
Human Sub' exercised and breathing.
Subject 1 30 22 15 50 7 31
Subject 2 26 21 19 73 2 9
Subject 3 31 25 22 70 3 12
Subject 4 32 23 21 66 2 9
Means ‘ 64 15
Human Sub' ctslgs.
Subject 1 23 20 14 60 6 30
Subject 2 21 18 15 71 3 16
Subject 3 29 18 12 41 6 33
Subject 4 35 18 1 1 31 7 38
Means 50 30

 

 

 

 

 

 

 

 

APPENDIX B

THE UNNORMALIZED PRE-WINDOWED LEAST SQUARES LATTICE FILTER

PARAMETER UPDATE ALGORITHM

This appendix gives the parameter update algorithm used for the Unnormalized Pre-
Windowed Least Squares Lattice Filter. The algorithm came from Friedlander (1982a. 4p. 844).
The algorithm found in Friedlander is for the multi-channel case. This means that all of the vari-
ables in the algorithm presented in Friedlander are either vectors or matrices. The scalar case was
used in this thesis research. This means that all of the variables in the lattice filter update algo-

rithm were scalars. The scalar version of the algorithm is given in this appendix.

Input parameters:
M = maximum order of lattice
yr = data sequence at time T

it = exponential weighting factor

Variables:

R5; = sample covariance of forward errors

48

49

R5}, = sample covariance of backward errors
AP; = sample partial correlation coefficient
73; = 1 — 'ypfr = 1 - likelihood variable

5p; = forward prediction errors

53.] = backward prediction errors

Kg; = forward reflection coefficients

K53~ = backward reflection coefficients

The following computations will be performed once for every time step (T=0

Initialize:

803 = fort = YT

R6; = R6.r = 31261-1 + YTYT

'Y—CIJ‘ = 1

Do for p = 0 to min{M,T} -1

Ap+r.r = Mp+1.T-r + €p.Tfp.'r-r/Y.§-r.r-r "‘
76.1: = 76—13 - rp,Trp.T/R;§.T *
K6+1rr = Ap-H,T/R;;.T-l *
Ep+r,r = 89.1 - K5+r,rfp.T—1
R541; = R551 - KI§+1.TA;»1.T
It:

K§+1,r = Ap+r,T/R§,T

rp+1,T = rp,T—1 - K§+1.T£p:r

. . . . TMAX).

50

R5243 = Rp.T-l ' Ap+rch§+LT

Note: Only the variables A, R‘. R’, 7‘. r need to be stored from one time step to the other. They

are all set initially to zero. The quantities R’, 7°, r need to be stored twice to avoid "overwriting".

When the divisor x = ye, R’, R5 is very small, set l/x = 0 in the equations marked with *.

APPENDIX F

COMPUTING THE ALL-POLE FILTER COEFFICIENTS

FROM THE UNNORMALIZED LATTICE PARAMETERS

This appendix gives the algorithm that was used to compute the all-pole filter coefficients
from the Unnormalized Pre-Wrndowed Least Squares Lattice Filter parameters. The algorithm
came from Friedlander (1982a. p. 845). The transfer function of the all-pole filter has been

rewritten in [Bl] so that the notation corresponds to the algorithm presented in this appendix.

 

Hat...“ m,<z)= 1 , . [BI]
1 + .- AIM-z"

where p is the order of the all-pole filter. The following algorithm was used to update the estima-
tion of the all-pole filter coefficients from the lattice parameters for each time step. Recall that
for the latfice filter of order M all lower order lattice filters are available in its structure. This
means that the parameters of lattice filters of orders lower than M are available from the parame-
ter update algorithm of a lattice filter of order M. The following algorithm computes the
coefficients of the all-pole filter corresponding with the lanice filter of order M and the
coefficients of the all-pole filters of all lower orders. The all-pole coefficients are designated by

AI, ,- where p is the order of the all-pole filter.

51

52

Note that the algorithm in Friedlander (1982a, p. 845) is for the multi-channel case. This
means that the parameters of the algorithm are either vectors or matrices. The scalar case was
used in this research. Therefore. the following algorithm is for the scalar case. This means that

the parameters of the following algorithm are scalars.

Inputs:

M = maximum order of the lattice

rp = backwmd enor for the current time step

7', = likelihood variable for current time step

K; = forward reflection coefficient for the current time step.

K; = backward reflection coefficient for the current time step

R5: sample covariance forbackward errors forthe currenttime step
Variables:

Ani = all-pole filter coefficients

Bpjo Bp‘ir Cp,i

These computations will be performed each time it is desired to update the all-pole filter

coefficients.

Initialize:

B5..1=0 for p=0,...,M-1
(30.0 = 0

Fori=0,...,M

53
AOJ=BOJ=1 fori=0
AOJ=BOJ=O fori>0
Doforp=0,...,M-1
354 = BPJ ' prpJ'YS—r
Cw1.i=cp.i"poth§
Aw=Apr-K5+rB,;s-r

3pm = BPJ-l - KfilApr-ps 11

APPENDIX G

A FORTRAN IMPLEMENTATION OF THE UNNORMALIZED

PRE—WINDOWED LEAST SQUARES LA'I'I'ICE FILTER

The Unnormalized Pre-Wrndowed Least Squares Lattice Filter was implemented in two
parts. a data structure holding the lattice parameters and the parameter update algorithm. Micro-
soft Fortran was used for the implementation. The lattice data structure is contained in a labeled
common block. The lattice parameter update algorithm is contained in a subroutine. The param-
eter update algorithm is to be called for each new time step. The Fortran code for the lattice data

structure and parameter update algorithm is given below.

J+4444++A4A¢AAJJAIJJJ144444L+4A4+44+.L_L_A.-.A__A_L.A_A.-_n444.;44J444JJ.‘LJAJAJAMMJJJ

T1 T’V‘“ T—v *v-‘v TTTT‘TTTTTT“ TTV‘V‘T rTTw v. rrv-T-v-T-v—vv—T-v V v Tf’rT‘rTTTTT‘v—Tfj T'V'TTV‘TT‘I'V—TTTTTTTj

THIS FILE DEFINES THE UNNORMALIZED PRE-WINDOWED LEAST SQUARES LAT-
TICE ALGORITHM DATA STRUCI'URE. THE ALGORITHM WAS TAKEN FROM:

FRIEDMNDER. B.. LATTICE FILTERS FOR ADAPTIVE PROCESSING. PROCEEDINGS
OF THE IEEE. VOL. 70. No.8. AUGUST 1982. pp. 842-844.

DEFINITIONS:

COMMON/LATTICE/ - LABELED COMMON THAT CONTAINS THE LATTICE DATA
STRUCTURE

RE - ARRAY CONTAINING SAMPLE COVARIANCES OF FORWARD ERRORS. EACH
ELEMENT OF THE ARRAY CORRESPONDS TO THE COVARIANCE OF THE

54

55

FORWARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN
OFFSET BETWEEN THE ARRAY INDEX AND ORDER. ARRAY INDEX 1
CORRESPONDS TO ORDER 0.

RR - ARRAY CONTAINING SAMPLE COVARIANCES OF BACKWARD ERRORS. EACH
ELEMENT OF THE ARRAY CORRESPONDS TO TIE COVARIANCE OF THE FOR-
WARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN OFFSET
BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS
TO ORDER 0.

RRCOPY - ARRAY CONTAINING SAMPLE COVARIANCES OF BACKWARD ERRORS
FOR THE PREVIOUS TIME STEP. EACH ELEMENT OF THE ARRAY
CORRESPONDS TO TIE COVARIAN CE OF TIE FORWARD ERROR FOR A PAR-
TICULAR ORDER. NOTE THAT THERE IS AN OFFSET BETWEEN TIE ARRAY
INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS TO ORDER 0.

PARCORR -ARRAY CONTAINING SAMPLE PARTIAL CORRELATION COEFFICENTS.
EACH ELEMENT OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE
FORWARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN
OFFSET BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1
CORRESPONDS TO ORDER 0.

LHOOD - ARRAY CONTAINING TIE LIKELIHOOD VARIABLE. EACH ELEMENT OF
TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE FORWARD ERROR
FOR A PARTICULAR ORDER. NOTE THAT THERE IS AN OFFSET BETWEEN TIE
ARRAY INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS TO ORDER -1.

LHOODCOPY - ARRAY CONTAINING TIE LIKELIHOOD VARIABLE FOR THE PREVI-
OUS TIME STEP. EACH ELEMENT OF TIE ARRAY CORRESPONDS TO TIE
COVARIAN CE OF TIE FORWARD ERROR FOR A PARTICULAR ORDER. NOTE
THATTHEREISANOFFSETBETWEENTIEARRAYINDEXANDORDER.
ARRAY INDEX 1 CORRESPONDS TO ORDER -1.

EERR - ARRAY CONTAINING FORWARD PREDICTION ERRORS. EACH ELEMENT OF
TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE FORWARD ERROR
FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN OFFSET BETWEEN TIE
ARRAY INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS TO ORDER 0.

RERR - ARRAY CONTAINING BACKWARD PREDICTION ERRORS. EACH ELEMENT
OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE FORWARD
ERROR FOR A PARTICULAR ORDER. NOTE THAT THERE IS AN OFFSET
BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS
TO ORDER 0.

RERRCOPY - ARRAY CONTAINING BACKWARD PREDICTION ERRORS FOR TIE
PREVIOUS TIME STEP. EACH ELEMENT OF TIE ARRAY CORRESPONDS TO
TIE COVARIANCE OF TIE FORWARD ERROR FOR A PARTICULAR ORDER.
NOTE THAT TIERE IS AN OFFSET BETWEEN TIE ARRAY INDEX AND ORDER.
ARRAY INDEX 1 CORRESPONDS TO ORDER 0.

KB - ARRAY CONTAINING FORWARD REFLECTION COEFFICIENTS. EACH ELE-
MENT OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE FOR-
WARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN OFFSET
BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS
TO ORDER 0.

KR - ARRAY CONTAINING BACKWARD REFLECTION COEFFICENTS. EACH ELE-
MENT OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE

56

FORWARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN
OFFSET BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1
CORRESPONDS TO ORDER 0.

MAXORDER - MAXIMUM ORDER OF TIE LATTICE FILTER.

(NOTE: INDEX 1 CORRESPONDS TO TIE ZEROTH ORDER. EXCEPT FOR TIE LIKELI-
HOOD VARIABLE WHERE INDEX 1 CORRESPONDS TO ORDER -1. THIS INCON-
VENENT INDEXING METHOD IS DUE TO LIMITATION OF TIE FORTRAN COMPILER
BEING USED.)

JJJJ44444444J4L444+4444.1444...-444.‘.1...-JJAJJJJJJJJJJJJJAJJJAALJJJJJJJ 3.1.1...-

T'vv-v-va j V TTTTTTTTTTTT—rw—w—Tw—w T‘I—‘V‘T’V‘TTTTTTTTTTTTV‘TTTTTV—T" V-w j w w 1 -v -v TT‘vvw-w ~v TT‘V—TT

REAL RE(10). RR(10), RRCOPY(IO), PARCORR(10)
REAL LHOOD(IO). EERR(10). RERR(10), RERRCOPYUO)
REAL KE(10), KR(10), LHOODCOPYOO)

INTEGER MAXORDER

COMMON/LATTICE] RE, RR, RRCOPY, PARCORR. LHOOD. LHOODCOPY,
EERR. RERR. RERRCOPY. KE. KR. MAXORDER

42.1.... a4444444444444+¢4444¢4 .44...- IJJJJJJJJJJJJJJJJJJJJJJJJJ.‘JAJJJJAJJJAJA.‘

SUBROUTINE UPDATELATTICE( DATA, ORDER. WEIGHT. TIME, LIMLHOOD,
PARMLIM)

ALA44JJJAAJ.‘JLAAAAAJAAAALAJAAJJJJJJAJAAJAAAAA_A_A_A44++4A44444A+++444+4444

TTTTTTTTTTTTTT‘v—TTTT-v-Tv-v TT-v-‘r-v-w-vv-V - TTTw-w-TTT-v-w—w f“ T“ TT-v-TT-v—Tw-V—Tv-v-rv—‘r-v-‘rw wa ‘V'TT‘V‘

SUBROUTINE UPDATELATTICE

THIS SUBROUTINE UPDATES THE UNNORMALIZED LEAST SQUARES LATTICE
DATA STRUCTURE FOR A TIME STEP. THE UPDATE ALGORITHM WAS TAKEN
FROM:

FREDLANDER. B.. LATTICE FILTERS FOR ADAPTIVE PROCESSING. PROCEEDINGS
OF TIE IEEE, VOL. 70. NO.8 AUGUST 1982. pp 842-844.

INPUT:

COMMON/LATTICE] - LATTICE DATA STRUCTURE (SEE LATTICEDST)
ORDER - MAXIMUM ORDER OF THE LATTICE FILTER.

WEIGHT - EXPONENTIAL WEIGHTING FACTOR FOR PAST DATA.

57

LIMLHOOD - E TIE ABSOLUTE VALUE OF LHOOD OR LHOODCOPY IS LESS THAN
LIMLHOOD. LHOOD OR LHOODCOPY WILL BE CONSIDERED TO BE ZERO.

PARMLIM - IF ANY VARIABLES OF TIE LATTICE FILTER EXCEPT FOR LHOOD AND
LHOODCOPY HAS ABSOLUTE VALUE LESS THAN PARMLIM, TIE VARIABLES
WILL BE CONSIDERED TO BE ZERO.

DATA - A DATA POINT FROM A DATA SEQUENCE.
TIME - CURRENT TIME STEP

OUTPUT:
COMMON/LATTICE] - LATTICE DATA STRUCTURE

*ttttttttttttitit*tfitfitit.tIt*tfififiltfififittltttfififittfittfifittttttti********

SINCLUDE: 'LAT'IICEDS'T’

REAL DATA,WEIGIIT.LIMLHOOD.PARMLIM
INTEGER TIME,ORDER,P

** INITIALIZATION

EERR(I) = DATA

RERR(I) 8 DATA

REG) = WEIGHT‘REG) + DATA’DATA
“(D = REG)

LHOOD(1) = l

** LATTICE STAGE UPDATES
DO 100 P = 1.(MIN(ORDER,TIME))

E (LHOODCOPY?) .LT. LIMLHOOD) TIIEN
PARCORR(P+1) = WEIGHT‘PARCORR(P+1)
ELSE

PARCORR(P+1) = WEIGII'T‘PARCORR(P+1)+ EERR(P) "' RERRCOPY(P) /
LHOODCOPY?)

ENDE

E (RR(P) .LT. PARMIJM) THEN
LHOOD(P+1) = LHOOD(P)
ELSE
LHOOD(P+1) = LHOOD(P) - RERR(P)*RERR(P)/RR(P)

100

200

58

ENDE

rr= (RRCOPY(P) m. PARMLIM) THEN
KR(P+1) = PARCORR(P+1)* 0.0
ELSE ‘
KR(P+1) = PARCORR(P+1)/RRCOPY(P)
ENDIF

EERR(P+1)= EERR(P) - KR(P+1)*RERRCOPY(P)
RE(P+1) = RE(P) - KR(P+1)*PARCORR(P+1)

IF (RE(P) .LT. PARMLIM) THEN
KE(P+1) = PARCORR(P+1)* 0.0
ELSE
KE(P+1) = PARCORR(P+1)/RE(P)
ENDIF

RERR(P+1) = RERRCOPY(P) - KE(P+1)*EERR(P)
RR(P+1) = RRCOPY(P) - PARCORR(P+1)*KE(P+I)

CONTINUE
DO 200 P = IMAXORDER

RRCOPY(P) = RR(P)
RERRCOPY(P) = RERR(P)
LHOODCOPY?) = LHOOD(P)

CONTINUE

APPENDIX H

A FORTRAN IMPLEMENTATION OF THE ALL-POLE FILTER

COEFFICIENT UPDATE ALGORITHM

The all-pole filter coefficient update algorithm was implemented in two parts, a data struc-
ture holding the all-pole filter coefficients and the coefficient update algorithm. Microsoft For-
tranwasused fortlre implementation. Thecoefficients arecorrtainedin alabeled commonblock.
The coefficient update algorithm is contained in a subroutine. The Fortran code for the

coefficient data structure and the coefficient update algorithm is given below.

JJJJAJJJJJ.‘_A.‘_A__‘.A_l.l_l.‘_...._l__‘AJJJJJJ‘AMALJJJAJJJJJJ.‘_‘JJJJAJJJAJJJ A.._-_- A4.-4444444444

‘ w v T—v T“ -v Tw—TTT1 w w w T1 TTV f-r r—vva—v ‘v—va-w T-v—Tv-V TT—v-w-TVv-w—v-v—f-v—TT—v—w—w w f1 Ti v—T-v—‘v—fvw ‘v w ‘v w - -v 'v —v -v

LATCOEF.DST

THIS FILE DEFINE-ZS THE ALL-POLE FILTER COEFFICIENT DATA STRUCTURE. THE
ALGORITHM USED TO UPDATE TIIE ALL-POLE FILTER COEFFICIENTS WAS TAKEN
FROM:

FREDLANDER. B.. LATTICE FILTERS FOR ADAPTIVE PROCESSING. PROCEEDINGS
OF TIE E. VOL. 70. NO. 8. AUGUST 1982. pp. 844-845.

NOTE TIIAT TIE VARIABLES IN THIS DATA STRUCTURE SHOULD BE INITIALIZED
TO ZERO BEFORE TIEY ARE USED.

DEFINITIONS:

59

60

A(p,i) - MATRIX CONTAINING TIE ALL-POLE FILTER COEFFICIENTS. p
CORRESPONDS TO TIE ORDER OF TIE FILTER. i CORRESPONDS TO TIE
SPECIFIC COEFFICENT IN TIE p ORDER ALL-POLE FILTER. p = l
CORRESPONDS TO TIE ZEROTH ORDER.

B(p,i) - MATRIX CONTAINING PARAMETERS USED IN TIE ALL-POLE FILTER COEF-
FICIENT UPDATE ALGORITHM. p CORRESPONDS TO TIE ORDER OF TIE
FILTER. i CORRESPONDS TO TIE SPECIFIC COEFFICENT IN TIE p ORDER
ALL-POLE FILTER. p = l CORRESPONDS TO TIE ZEROTII ORDER.

BSTAR(P.i) - MATRIX CONTAINING PARAMETERS USED IN TIE ALL-POLE FILTER
COEFFICENT UPDATE ALGORITHM. p CORRESPONDS TO TIE ORDER OF TIE
FILTER. i CORRESPONDS TO TIE SPECEIC COEFFICENT IN TIE p ORDER
ALL-POLE FILTER. p = l CORRESPONDS TO TIE -1TH ORDER.

C(PJ) - MATRIX CONTAINING PARAMETERS USED IN TIE ALL-POLE FILTER COEF-
FICENT UPDATE ALGORITHM. p CORRESPONDS TO TIE ORDER OF TIE
FILTER. i CORRESPONDS TO TIE SPECIFIC COEFFICENT IN TIE p ORDER
ALL-POLE FILTER. p = l CORRESPONDS TO TIE ZEROTH ORDER.

44444444444.444444444444444444 A44444444444444444444444444A4.-444444444444

v 1 T“ 1'1 1T1 1—1—11 1‘1-1'1 TVV' 11111—1-1-11-1'1-1—1-1-1-1‘7-1-1-1 TTV 1 1 1 1—1 r1-1-1 1 1 1 1—1—1-1—1—1—1-1-1v-r-v—1-1 1 1 -r

REAL A(10,10), B(10.10), BSTAR(10.10). C(10.10)

COMMON/LATCOEF/ A. B. BSTAR. C

444444444444 144444444444444444.444444444444444444444444 A4.-.4444444444444

1-1-—v—‘r1~1—1—1-1-1-1-'v-1'1--v-1—1~1"r1-1-1-1—1 1—1—1—11 1-1—1-1-1—1-1-1-1'1—1-1—1-1-1-1-1—1 vv-v—1-1 1-1-1'1-1-11 1 1 1—1—1-1-TT1-1'1'1-

SUBROUTINE COEFCALCULA'I'ION(ORDER.PARMLM.LM.HOOD)

444444.44444444.4444444444444444444444444444444444444444444444444444444

1*1‘1 TT‘I‘T‘V‘T’I 1-1 vy— r—v-vrvr—v—T—v—T—rj . v. 1 1'1 f . 1 1 r1 -. TT-r-rT—v 1T1-1'1- r1'1'1'1'1-1—1-1-1- r1-1 ‘rTf1'1—1—1—1 1 1 1 1

THIS SUBROUTINE UPDATES TIE ALL-POLE FILTER COEFFICIENTS WITH TIE
UNNORMALIZED PRE-WINDOWED LEAST SQUARES LATTICE FILTER PARAME-
TERS. TIE COEFFICENT UPDATE AIflORITHM WAS TAKEN FROM:

FREDLANDER. B.. LATTICE FILTERS FOR ADAPTIVE PROCESSING. PROCEEDINGS
OF TIE EEE. VOL. 70. NO. 8. AUGUST 1982. pp. 844-845.

INPUTS:

COMMON/LATTICE] - LATTICE DATA STRUCTURE (SEE LATTICEDST TN APPENDIX
C).

61

COMMON/LATCOEF/ - ALL-POLE FILTER COEFFICENT DATA STRUCTURE (SEE
LATCOEF.DST).

ORDER - MAXIMUM ORDER OF TIE LATTICE FILTER.

PARMLIM - IF ANY VARIABLES OF TIE LATTICE FILTER EXCEPT FOR LHOOD AND
LHOODCOPY HAS AN ABSOLUTE VALUE LESS THAT PARMLIM. TIE VARI-
ABLES WILL BE CONSIDERED TO BE ZERO.

LIMLHOOD - E LHOOD OR LHOODCOPY FROM TIE LATTICE DATA STRUCTURE
HAS AN ABSOLUTE VALUE LESS THAN LIMLHOOD. LHOOD OR LHOODCOPY
WILL BE CONSIDERED TO BE ZERO.

OUTPUTS:

COMMON/LATCOEF/ - ALL-POLE FILTER COEFFICENT DATA STRUCTURE (SEE
LATCOEF.DST).

***fitfitflttfifitlfifltfi*tttttttttttiti!i.fittiltfitfitfitfitlltittttittttttfitttifitlt

$INCLUDE: ’LATTICEDST’
$INCLUDE: ’LATCOEF.DST’

REAL PARMLIM.]..IMIHOOD
INTEGER ORDERJP

DO 1000 I = l, ORDER-#1

IF(I.EQ.1)THEN
A(1,I)=1.0
80.1) = 1.0

' ELSE
A(l,I)= 0.0
30.1) =0.
ENDIF

D0100 P= LORDER

E (LHOODCP) .LT. LIMLHOOD) TIIEN
BSTAR(P,I+1) = B(P,I)
ELSE
BSTARCPJ-t-l) = B(P,I) - RERR(P)*CCPJ)/LHOOD(P)
ENDIF

62

E (ABS(RR(P)) .LT. PARMLIM) THEN
C(P+l,I) = C(P,I)
ELSE
C(P+1J) = C(PJ) - RERR(P)*B(P.D/RRCP)
ENDE

A(P+1.I) = A(PJ) - KR(P+1)*BS'TAR(P.I)
B(P+l,I) = BSTAR(P,I) - KE(P+1) "' A(P,I)

100 CONTINUE

1000 CONTINUE
RETURN
END

APPENDIX I

THE NORMALIZED PRE-WINDOWED LEAST SQUARES LATTICE FILTER

This appendix gives the parameter update algorithm used for the Normalized Pre-
Windowed Least Squares Lattice Filter. The algorithm came from Friedlander (1982a, p. 846).
In the normalized version of the lattice filter, the lattice parameters maintain an absolute value of
less than unity. The algorithm found in Friedlander is for the mum-channel case. This means
that all of the variables in the algorithm presented in Friedlander are either vectors or matrices.
'I'hescalarcasewasusedinthisthesisresearch. Thismeansthatallofthevariablesinthelattice
filter update algorithm were scalars. The scalar version of the algorithm is given in this appendix.
Please note that when Friedlander uses a T in a superscript on parameters in the normalized lat-

tice filter parameter update algorithm theT means transpose and not time.

Input parameters:

M = maximum order of lattice
y-r = data sequence at time T

A = exponential weighting factor

Variables:

63

ST = estimated covariance of y-r
Eprr = normalized forward prediction errors
I‘m-H = normalized backde prediction errors

K1,; = reflection coefficients

Initialization:

K,f,Saresettouro

The following computations will be performed once for every time step (T=0. . . . , TMAX).

Sr = 351-1 + YTYT

Eon" = 1'03 = SfV‘Y'r

Forp= 0,. . . ,min{M,T} - 1
Kpua' = F‘O‘prJ—r. fps-r. £133)
£9043 = F(Ep.Tr YpJ-r. K9143) "

Tp+r.'r = F (pm-1. Epm KprlJ) “

F(a,b,c) = [1 - CC]""[8 - Cb][1 - bbl’”

F"(a,b.c) = [l - cc]"‘a[1 - bb]” + cb

The function F(a.b,c) involves division. In the scalar case, when the divisor x is small, set U): = 1

in the equations marked by *.

65

A result of normalizing the lattice filter update algorithm is that the likelihood variable has
been folded into the algorithm. The likelihood variable does not appear in the normalized update
algoritlun. The likelihood variable and all other unnormalized lattice filter parameters are recov-
erable form the parameters of the normalized lattice algorithm. The relationship between the nor-
malized and unnormalized lattice parameters can be found in Friedlander (1982a, p. 863). The
relationships between the normalized lattice parameters and the likelihood variable and the for-

ward prediction error for the unnormalized lattice parameters follows.

Yfi-m-r = (1 - ‘Yp-r:r—r) = 130 - fix-113.14)

OT

(1 - ‘Yp.'r-r) = (1 - Yp-r.'r—r)( 1 - 1‘p.'r-r1‘p.'r-r) -
R5,? = S? :10 - Karon)”

6px = (1 - ’Yp—m-r)” R5,? Ear

APPENDIX J

A FORTRAN IMPLEMENTATION OF THE NORMALIZED

PRE-WINDOWED LEAST SQUARES LATTICE FILTER

The Normalized Pre-Wrndowed Least Squares Lattice Filter was implemented in two parts.
a data structure holding the lattice parameters and the parameter update algorithm. Microsoft
Fortran was used for the implementation. The lattice data structure is contained in a labeled com-
mon block The lattice parameter update algorithm is contained in a subroutine. The parameter
update algorithm is to be called for each new time step. The Fortran code for the lattice data
structure and parameter update algorithm is given below. (This is a Fortan version of a program

that was developed by Betsy Mates-Needham.)

*itfl!*1!******************************************************************1:

NLATI'ICEDST

THIS FILE DEFINES TIE NORMALIED PRE-WINDOWED LEAST SQUARES LATTICE
ALGORITHM DATA STRUCTURE. TIE ALGORITHM WAS TAKEN FROM:

FREDLANDER, B., LATTICE FILTERS FOR ADAPTIVE PROCESSING. PROCEEDINGS
OF THE EEE. VOL. 70, NO.8, AUGUST 1982. pp. 845-846.

66

67

DEFINITIONS:

COMMON/NLATITCE/ - LABELED COMMON THAT CONTAINS TIE LATTICE DATA
STRUCTURE

NEERR - ARRAY CONTAINING NORMALIZED FORWARD PREDICTION ERRORS.
EACH ELEMENT OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE
FORWARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN
OFFSET BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1
CORRESPONDS TO ORDER 0.

NRERR - ARRAY CONTAINING NORMALIZED BACKWARD PREDICTION ERRORS.
EACH ELEMENT OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE
FORWARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIERE IS AN
OFFSET BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1
CORRESPONDS TO ORDER 0.

NRERRCOPY - ARRAY CONTAINING N ORMALIZED BACKWARD PREDICTION
ERRORS FOR TIE PREVIOUS TIME STEP. EACH ELEMENT OF TIE ARRAY
CORRESPONDS TO TIE COVARIAN CE OF TIE FORWARD ERROR FOR A PAR-
TICULAR ORDER. NOTE THAT TIERE IS AN OFFSET BETWEEN TIE ARRAY
INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS TO ORDER 0.

K - ARRAY CONTAINING NORMALIZED REFLECTION COEFFICENTS. EACH ELE-
MENT OF TIE ARRAY CORRESPONDS TO TIE COVARIANCE OF TIE FOR-
WARD ERROR FOR A PARTICULAR ORDER. NOTE THAT TIIERE IS AN OFFSET
BETWEEN TIE ARRAY INDEX AND ORDER. ARRAY INDEX 1 CORRESPONDS
TO ORDER 0.

S - ESTIMATED COVARIAN CE OF TIE DATA SEQUENCE.

(NOTE: INDEX 1 CORRESPONDS TO THE ZEROTH ORDER. TIIIS INCONVENENT
INDEXING METHOD IS DUE TO LIMITATION OF TIE FORTRAN COMPILER BEING
USED.)

JJIIAAJ‘JJAJJAJJLALAEJIJ—‘JJJJJJJJJJJJIJJJIJLJJJJJJJJJAJJJJJJJIJJJJJJJAJ

REAL NEERR(10),NRERR(10), NRERRCOPYOO), K00), S

COMMON/NLATTICE/ NEERRNRERRNRERRCOPY,K.S

JJJJAJJJJJJJAJJJ.‘_‘JJJJJJJAJAJJJJAJJJJJJJJJJJJIJJl..__‘_-.l_l_lIIJJJJJJJJEJJJJJ

‘I‘U’U - 1 r—v-v-v'w VTTTT-v‘w v '1‘. T—rvf r-r-V'T-v T‘U’T‘V‘V v'v-v-I "' ’v v—T-v—‘v—T'v I r—V'T'v ‘v‘v‘w TTTI’TTTTTT“ T'U T‘v

68

SUBROUTINE NUPDATELATTICE(DATA,ORDER,WEIGHT,TIME)

itIll*fllit*****til******tll*****#****ttlflldfltt**********************************

SUBROUTINE NUPDATELATITCE

THIS SUBROUTINE UPDATES TIE NORMALIZED LEAST SQUARES LATTICE DATA
STRUCTURE FOR A NEW DATA POINT IN A INPUT DATA SEQUENCE. TIE UPDATE
ALGORITHM WAS TAKEN FROM:

FREDLANDER, B., LATTICE FILTERS FOR ADAPTIVE PROCESSING. PROCEEDINGS
OF TIE EEE, VOL. 70, NO.8 AUGUST 1982, pp 845-846.

INPUT:

COMMON/LATITCE/ - UNNORMALIZED LATTICE DATA STRUCTURE (SEE

LATTICEDST IN APPENDIX C)
COMMON/NLATTICE/ - NORMALIZED LATTICE DATA STRUCTURE (SEE
NLATTICEDST)

ORDER - MAXIMUM ORDER OF THE LATTICE FILTER

WEIGHT - EXPONEN'I'IAL WEIGHTING FACTOR FOR PAST DATA.
DATA - A DATA POINT FROM A DATA SEQUENCE.

TIME - CURRENT TIME STEP.

OUTPUT:

COMMON/LATTICE/ - UNNORMALIZED LATITCE DATA STRUCTURE (SEE
LATTICE.DST FROM APPENDIX C)

COMMON/NLAT'TICE/ - NORMALIZED LATTICE DATA STRUCTURE (SEE
NLAT'TICE.DST)

444+4+44+4JJEAAAAJJJAJAJLJJJJJ.‘_AJJAAAEJJEJJJJJJJJJJJ I AJJAJELLJALEEAJJ.‘
TT‘U‘j—V'T‘V‘V'TTTTTV‘IVTTTTTTTTTT‘V—‘TVTTV‘Uj—‘I uwwwwi’v'v’u‘v‘v‘uT‘v‘wT‘v‘T'v—VTVWWTT"V‘v'jv v v1

$INCLUDE: ’LATITCE.DST’
$INCLUDE: ’NLATTICE.DST’

REAL DATA,WEIGHT,SQRRE,PRODUCT,F,INVF
INTEGER TIME,ORDER.P

100

69

"' INIT'IALIZATION

E ((ABS(S) .LT. .00m01).AND.(ABS(DATA).LT. .000001)) TIEN
NEERR(1)= 0.0
NRERR(1) = 0.0
ELSE
S = WEIGHT‘S + DATA*DATA
NEERR(1)= DATA / SQRT(S)
NRERR(1) = NEERR(l)
ENDE

EERR(I) = DATA

PRODUCT = 1.0
LHOOD(1) = 1.0

DO 100 P = l,MIN(ORDER,TINE)

K(P+1) = INVF(K(P+1), NRERRCOPYW). NEERR(P))

NEERR(P+1) 8 F(NEERR(P).NRERRCOPY(P),K(P+1))
NRERR(P+1) = F(NRERRCOPY(P). NEERR(P). K(P+1))

* CALCULATING LIKELIHOOD VARIABLE AND UNNORMALIZED FOR-
WARD PREDICTION ERROR

" NOTE THAT TIE FOLLOWING CALCULATIONS ARE NOT PART OF TIE
NORMALIZED LATTICE FILTER UPDATE ALGORITHM.

LHOOD(P+1) = LHOOD(P)*(1 - NRERR(P)*NRERR(P))
E (LHOOD(P+1) .LT. 0.0) TIEN '
LHOOD(P+1) = 0.0
ENDE
PRODUCT = PRODUCT*SQRT(1 - K(P+1)*K(P+1))
SQRRE = SQRT(S)*PRODUCT
EERR(P+1) = SQRT(LHOODCOPY(P+1))*SQRRE*NEERR(P+1)

CONTINUE
DO 200 P = 1.MAXORDER

NRERRCOPY?) = NRERR(P)

70

LHOODCOPY(P) = LHOODCP)

200 CONTINUE
RETURN
END

**************Ill!!!I!!!itit*tilttt*fitfitttltlflltt*fiilttiltttttidl****************It

REAL FUNCTION F(A,B,C)

IF(C .GT.1.0)TIEN
C=1.0

ELSE E (C .LT. -1.0) TIEN
C= -l.0

ENDE

E (B .GT.1.0)TIEN
B = 1.0

ELSEE(B .LT. -l.0)TIEN
B = -1.0

ENDE

X1 = SQRT(1.0 - C‘C)
E (X1 .LT. .000001) TIIEN
INVX1= 1.0
ELSE
INVX1= 1.0/X1
ENDE

X2 = SQRT(1.0 - B*B)
E (X2 .LT. .ooooor) THEN
INVX2 = 1.0
ELSE
INVX2 = 1.0/X2
ENDE

71

F = INVX1*(A - C*B)*INVX2

RETURN
END

ItI!I!****it.I!*tttfi*fittttt*ttfititfittttfififldflfldflflltfit*ttttttt***************

REAL FUNCTION INVF(A,B,C)
REAL A,B,C

E(C .GT.1.0)TIEN
C=1.D

ELSE E (C .LT. -l.0) TIEN
C: -1.0

ENDE

E(B .GT.1.0)TIEN
B = 1.0

ELSE E (B .LT. -1.0) TIEN
B = -1.0

ENDE

TTTTTTTTTTT'f'V 'v TTTTT '— ff w TTTTTTT1 vTT‘v—TT" T-V'T'v-‘r—v TT—v—w—v‘ j ‘v ’1 TTTTT’YTTTva‘j 1 T‘V'T'v'

APPENDIX K

REVIEW OF THE DEVELOPMENT OF

THE MICROWAVE VITAL LIFE SIGNS MONITOR

The use of a microwave device in the measurement of human heart rate can be found in
Byme, Flyrur. Zapp. and Siegel (1986), Byme and Siegel (1985), Byme, Zapp, Flynn, and Siegel
(1985), Hoshal, Ivkovich, Siegel, and Zapp (1984), Hoshal and Siegel (1986), Hoshal, Siegel, and
Zapp (1984), Lin, Kiemicki, Kiemicki. and Wollschlaeger (1979), and Popovic, Chan, and Lin
(1984). The research presented in this thesis is an extension of the work done in developing a
heart rate estimation method for the Michigan State University Biomedical Signal Processing
Laboratory’s microwave vital life signs monitor (Byme et al., 1986; Byme & Siegel, 1985; Byme
et al., 1985: Hoshal, Ivkovich, et al., 1984; Hoshal & Siegel, 1986: Hoshal, Siegel, et al., 1984).
This appendix will review the history of the development of heart rate measurement techniques

for the Michigan State University unit.

The First Michigan State University Microwave Vital Life Signs Monitor

The first version of the Michigan State University microwave vital life signs monitor was
used by Byme, Flynn, Zapp, and Siegel (1986), Byme and Siegel (1985), Byme, Zapp. Flynn,

and Siegel (1985), Hoshal, Ivkovich, Siegel, and Zapp (1984), Hoshal and Siegel (1986) and

72

73

Hoshal. Siegel, and Zapp (1984). Figure K1 is a block diagram of the device." A description of
the device’s operation was found in Hoshal, Ivkovich, etal. (1984) and Hoshal, Siegel, etal.
(1984). In the monitor, a portable homodyne transceiver system is responsible for transmitting a
low-level microwave signal and detecting Doppler shifts in the returned signal. The microwave
transceiver emits a 10.5 GHz continuous wave at a level of a few milliwatts (more recent versions
of the instrument use pulsed transmitters). The returns of the microwave signal supply an analog
signal of only a few microvolts to the Doppler shift detector. After Doppler shift detection, the
low—level analog signal is sent through an amplifier that provides 60dB to 80dB of signal
amplification. After signal amplification, the analog signal is filtered with a bandpass filter. The
band of the filter was placed at 1-30 Hz. The selection of the 30 Hz cutoff was based on the spec-
tral analysis of signals resulting from reflections of microwave signals off the chest walls of
human subjects. There were no significant spectral components of the microwave heartbeat sig-
nals past 30 Hz (Hoshal, Siegel, & Zapp. 1984). The low end cutoff of 1 Hz is used to eliminate
any DC component in the signal and minimize breathing effects (M. Siegel, personal communica-
tion. 1988).

The amplified and filtered analog signal is sampled by an eight-bit analog to digital con-
verter. The sampling rate varied from 96 to 128 samples per second in the above studies. The

final destination of the digitized microwave signal is a microprocessor. The final heart rate esti-

mation algorithms will reside in the microprocessor unit.

In order to evaluate the success of any heart rate measurement method, a reference of the
actual heartbeat activity is needed. A heartbeat reference signal was successfully obtained from
an in—house designed unit that measures body surface potential between the hands of a human
subject. The output of the device closely resembles an EKG signal. Hand potential signals were

simultaneously recorded when microwave heartbeat measurements were taken.

74

..ozeoE new.» 8: .3? 02.3838 Emmi—5 38m 5%.an .8: 2:. .3— Ban—L

 

 

8.8280
.235
2 moi:

 

 

 

 

 

E

commooEqEEE

 

 

 

 

52395.
N= 8..
.EE as use

 

 

 

 

 

£5 a. 8.8
3535.

8.92

 

 

 

 

 

3:08:85.
.2950
26322.2

 

 

 

 

coatsm
958:3—

 

 

75

Review of the Development of a Heart Rate Estimation Method

for the Michigan State University Microwave Vital Life Signs Monitor

Peak detection.

Peak detection was the first method considered in the development of a heart rate estimation
technique. In an uncluttered signal, the microwave measurements of a heartbeat resembles a
heartbeat occurrence in an EKG. Large peaks in the microwave signal correlate with the

occurrences of heartbeats. The large peaks can be located with peak detection.

Peak detection is effective for only a restricted set of operating conditions. The subject
must be rested and breathing regularly during the recording of the microwave measurements. The
transceiver must be placed at only short distances away from the chest wall of the subject There
must be little interference from breathing or background movement in the Doppler shifted
encoded heartbeat signal. Under realistic operating conditions, the heartbeat signal may be
obscured by background noise and clutter. Peak detection techniques applied to the microwave
heartbeat signal are unreliable for realistic operating conditions (Hoshal. Ivkovich, Siegel, &

Zapp. 1984; Hoshal, Siegel, & Zapp, 1984; Byme & Siegel, 1985).

Correlation Techniques.

The next stage of the development of a heart rate estimation technique was based on utiliz-
ing the periodic nature of heartbeat occurrences in the microwave signal. Peak detection failed
because of the high level of clutter in the microwave signal. Hoshal. Ivkovich, Siegel, and Zapp
(1984) and Hoshal, Siegel and Zapp (1984) posed the following argument: If correlation tech-
niques are useful in detecting periodic signals that are completely obscured by random misc, then
correlation techniques may be useful in extracting the heartbeat signal from the microwave meas-

urements. Autocorrelation was performed on the Michigan State University monitor microwave

76

heartbeat signal by Hoshal, Ivkovich, etal . (1984) and Hoshal, Siegel, etal . (1984). Lin, Kier-
nicki. Kiernicki. and Wollschlaeger (1979) and P0povic, Chan, and Lin (1984) also used auto-
correlation to estimate heart rate from a microwave heartbeat signal. Hoshal, Ivkovich, eral.
(1984) and Hoshal. Siegel, era! . (1984) showed that autocorrelation could be used to accurately
measure heart rate, but the set of operating conditions used to test the autocorrelation techniques
was limited. A number of problems were encountered when heart rate estimation methods using

autocorrelation were more rigorously tested.

The ability of autoconelation methods to separate a signal from obscuring noise is depen-
dent on the signal being periodic. Human heartbeats are pseudo periodic events. The time period
between heartbeats may not be constant (Byme & Siegel, 1985: Hoshal & Siegel, 1986). The
less the heartbeats are periodic the more the autocorrelation peaks broaden. An accurate estima-

tion of the heart rate cannot be obtained from the broadened autoconelation peaks.

The length of the periods between heartbeats can change abruptly. The autoconelation time
window length must be restricted to the duration of a few heartbeats in order to track abrupt
changes in heart rate (Byme & Siegel, 1985; Hoshal & Siegel, 1986). When the periods between
heartbeats are regular, 3 longer autocorrelation time window is desirable since it would result in
more accurate estimates of autocorrelation peak locations (Byme & Siegel. 1985). A trade off
must be made in chasing a time window length if autocorrelation is to be used in estimating heart

rate from the microwave signal. This is a classical time-frequency resolution problem.

It was also found that the signature of a heartbeat occurrence in the microwave signal can
change not only from person to person but fiorn heartbeat to heartbeat (Byme & Siegel, 1985).
Autocorrelation techniques depend on repeated behavior. If the same pattern is not repeated in
the microwave signal for each occurrence of a heartbeat, the effectiveness of autocorrelation as a

heartbeat signal detector will be limited.

 

77

The ability of autocorrelation to extract the heartbeat signal from the microwave measure-
ments is further underrninded by the presence of periodic background components (Byme &
Siegel. 1985). The most dominant periodic component is breathing. It is difficult to filter out
breathing because there is no prior knowledge of a subject’s breathing rate during the microwave
measurement of the subject’s heart rate. Because the breathing rate may be close to the heart rate
and many times larger in amplitude, the problem of removing the breathing component is com-

pounded.

Variations in the heartbeat signal have limited the success of using autocorrelation to esti-
mate heart rate from the microwave signal. Hoshal and Siegel (1986) observed that variations in
the level of the microwave signal clutter also effected the performance of autoconelation. Auto-
correlation was unreliable in cases of low signal-to—noise ratios (SNR) or high clutter level
environments (Hoshal & Siegel, 1986). Experimental measurements showed that obtaining a
sufficiently high SNR cannot be done consistently. The SNR was strongly affected by the posi-
tioning of the microwave transceiver over the body. The unpredictable rate and strength of the

breathing motion also affected their ability to obtain a consistent SNR.

Hoshal and Siegel (1986) postulate that Signal and clutter variations will become an even
bigger problem when the microwave monitor is to be used in the field. The heart rate estimation
technique must be able to contend with uncontrollable background clutter and pathological heart
conditions. Hoshal and Siegel call for a robust signal processing methodology. A methodology
that can make a reasonable estimate of the heart rate from the microwave signal despite the possi-

ble variations in the heartbeat signal and background noise that might be encountered.

The Hoshal and Siegel Stochastic Model of the Microwave Heartbeat Signal.

The approach Hoshal and Siegel (1986) took to develop a robust heart rate estimation

method was to formulate a stochastic model of the microwave heartbeat signal in order to gain

78

knowledge about the Statistics of the microwave signal. The knowledge gained was then used to

aid in the development of an optimal processing technique.

Hoshal and Siegel model the microwave heartbeat signal as
y(t)=X(t)* h(t). (K1)

where * denotes convolution. y(t) is the microwave heartbeat signal. x(t) is a pseudo periodic

impulse train defined by
x(t) = 5(t-To) + tic—7'04“.) + + 6(t- $710+. . . , (142)

where T,- is a time series formed fiom successive beat-to-beat periods. 8(t) is the has a value of
on for t = 0. Otherwise, 50) is zero. I: (t) represents the time domain response characteristics of

a single heartbeat cycle.

The beat-to—beat period sequence, T,- , is modeled as a 4th order autoregressive process. The
parameters of the T,- model were based on time intervals between heartbeat occurrences in the
hand potential reference signals. A six pole, six zero pole-zero model is used for Mt). The
parameter estimation for h (t) was based on microwave signals taken from 10 subjects. Back-
ground noise was minimal during the recording of the signals. The subjects were lying on their
backs while the measurements were taken. Hoshal and Siegel (1986) do not state whether the
monitor was placed directly on the chest wall or at some distance from the chest wall of the sub-

jects.

The model was then used to estimate the power specu'al density of the microwave Signal.
Microwave signals were simulated for various means and variances of the model parameters.
The power spectral density of the simulated microwave signals were obtained. Distinct harmonic
peaks were seen in the power spectrum of each simulated microwave heartbeat Signal, even for

the signals with the worst cases of the parameters variances.

79

The time varying nature of a human’s heart rate and the harmonic peaks in the power spec-
tral density of the simulated microwave signals suggested to Hoshal and Siegel that adaptive
comb filtering may be an appropriate technique to estimate heart rate. As described in Hoshal
and Siegel (1986). the adaptive comb filter searches the components of the input signal spectrum
for a best fit to a specified number of harrnonically related signal components. Nonharmonic

components are attenuated. The best fit fundamental frequency is the estimate of the heart rate.

Results of preliminary tests on the adaptive comb filter’s ability to estimate heart rate are
given in Hoshal and Siegel, (1986). The adaptive comb filter was applied to simulated and actual
microwave data The simulated data consisted of a series of microwave heartbeats generated by
the Hoshal and Siegel model added to band-limited white noise. The real microwave data was
taken under the same conditions as the microwave data used to estimate the Hoshal and Siegel
model parameters. The adaptive comb filter gave accurate estimates of the heart rates for both

simulated and real data.

Although adaptive comb filtering shows promise as a method for estimating heart rate from
the microwave signal, further investigation is needed. The Hoshal and Siegel model is based on
data recorded under ideal conditions. The model needs to be expanded to include pathological
heartbeat behavior and the affects of breathing and background clutter (Hoshal & Siegel, 1986).
The ability of the comb filter to adapt to abrupt changes in heart rate needs to be investigated. It
must be determine that the adaptive comb filter technique can give accurate heart rate estimates
for microwave heartbeat Signals cluttered by breathing and background movements. The robust-

ness of the adaptive comb filter has yet to be proven.

The Byme and Siegel Stochastic Model of the Microwave Heartbeat Signal.

The approach that Byme and Siegel (1985) took to find a robust heart rate estimator is simi-

lar to approach taken by Hoshal and Siegel (1986). Byme and Siegel postulated a model for the

80

microwave heartbeat signal and then used the model to apply an adaptive filter in the estimation
of heart rate. The Byme and Siegel model is similar to the Hoshal and Siegel model but the

choice of adaptive filtering is different.

Byme and Siegel (1985) modeled the microwave heartbeat signal as the output of an all-
pole filter that is excited by a train of impulses added to white noise. The Byme and Siegel

model is shown in Equation K3.
y(t)=[1(t)+W(t)] "‘ Mt). (K3)

where * denotes convolution. y (t) is the microwave heartbeat signal. x(t) is the pseudo periodic
train of impulses. Equation K2 of the Hoshal and Siegel (1986) model can be used to define x(t)
of the Byme and Siegel model. w(t) is a band-limited white process. h(t) is the impulse
response of the all-pole filter. The response of the all-pole filter excited by one impulse should
resemble a microwave signal heartbeat. This model differed from the Hoshal and Siegel model in
two ways. The Hoshal and Siegel model used a pole-zero filter where the Byme and Siegel
model uses an all-pole filter. Secondly, the Byme and Siegel model includes white noise as a

component in the excitation process of the model.

The primary motivation behind using fire Byme and Siegel (1985) model for the microwave
heartbeat signal was the success of applying a similar model to the problem of detecting pitch
pulses in voiced speech. The particular pitch detection technique that prompted Byme and Siegel
to adopt the model in Equation K3 was developed by Lee and Morf (1980). Voiced sounds such
as vowel sounds can be modeled as the output of an all-pole filter excited by a pseudo periodic
train of impulses. The impulse train models the pitch pulses. The all-pole filter of the speech
model represents the vocal tract. In speech analysis, it is of interest to determine the period
between the pitch pulses. Linear prediction can be used to inverse the all-pole filter’s response
and recover the pseudo impulse train that excited the all-pole filter. The recovered impulses

appear as large prediction enors at the output of the linear predictor. (Appendix L reviews the

 

81

relationship between the Byme and Siegel model and linear prediction.) For different sounds, the
vocal tract will take on different configurations. In order to inverse the effects of the vocal tract
on the pitch pulses by linear prediction, the configuration of the vocal tract must be determined.
The proper configuration of the all~pole filter model might not be known or might change within
a speech process. Adaptive linear prediction is used to determine the unknown configuration of

the vocal tract model.

Byme and Siegel (1985) modeled the occurrence of heartbeats as a pseudo periodic impulse
train. The chest wall, microwave, microwave channel and monitoring unit are modeled with an
all-pole filter. If the Byme and Siegel model is valid for the microwave heartbeat signals, linear
prediction can be used to recover the heartbeat impulse train. Like the speech process, the system
the heartbeat impulse train excites might not be known or might change in time. Assuming their
model is valid. Byme and Siegel used adaptive linear prediction to determine the configuration of
the system model. Once the heartbeat impulses are recovered through adaptive linear prediction,
B yrne and Siegel hoped to estimate the instantaneous heart rate by measuring the period between

consecutive impulses.

In Lee and Morf (1980), the recovery of pitch impulses was enhanced by the use of a
parameter of the particular adaptive linear predictor they used. The parameter is related to the
log-likelihood function of the speech process input to the adaptive linear predictor. The parame-
ter is a measure of the unexpectedness of the most recent data points of the speech process (Fried-
lander, l982a). Sudden large changes in this variable were good indicators of the occurrence of a
pitch pulse. Byme and Siegel used this parameter in the same way to enhance the recovery of

heartbeat occurrences.

If the Byme and Siegel (1985) method of estimating heart rate is valid for microwave meas-
urements. the primary advantage of the method over autocorrelation is that the detection of the

heartbeats is not dependent on the periodicity of the heartbeat. With the Byme and Siegel

82

method, the occurrence of a heartbeat would be recognized by the occunence of a special event at
the output of an adaptive linear predictor: the presence of a large prediction error and an abrupt
large change in the likelihood parameter.

The Byme and Siegel (1985) heartbeat detection technique would have a computational
advantage over autocorrelation. Byme and Siegel (1985) experimented with a number of dif-
ferent adaptive linear predictors and found the best results to come from a normalized least-
squares lattice filter, a normalized version of the adaptive linear predictor used by Lee and Morf
(1980). The filter is a recursive algorithm. The parameters of the adaptive lattice filter are
updated for every new data sample input to the filter. Autocorrelation processing requires blocks
of data, therefore there is a time delay between estimates of the heart rate. The lattice filter struc-
ture allows for updates of heart rate estimation with every new input data sample.

The performance of the Byrne and Siegel (1985) heartbeat detection technique is impres-
sive. Microwave signals obtained from subjects lying flat with the microwave monitor mounted
directly on the their chests were processed. Byrne and Siegel obtained large prediction errors and
large changes in the likelihood parameter during the occurrences of heartbeats. The exact timing
of heartbeat occurrences was determined by hand unit reference signals. After the errors were
masked by the derivative of the likelihood parameter, peak detection was used to determine the
location of the heartbeats. The Ireartbeat signal was successfully recovered from. these
microwave signals. Note that autocorrelation could have been used to estimate the heart rate

because of the regularity of the data and the lack of clutter.

In order to more rigorously test the Byrne and Siegel heartbeat detection method, adaptive
linear prediction was applied to microwave heartbeat signals taken from subjects placed three feet
from the microwave monitor. The subjects were seated and had exercised. Having the subjects
sitting is considered to be a more hostile condition for heartbeat detection compared to lying

down. In the sitting position, it is felt that the heart impinges the chest wall to a lesser degree

 

83

than the if the subject is lying down (M. Siegel, personal communication, 1988). The heartbeat
signal was obscured in these measurements. Unlike the signals recorded for subjects lying down,
autocorrelation failed to give accurate heart rate estimates. The use of the normalized adaptive
lattice filter prediction error and likelihood parameter gave consistently accurate estimates of the

instantaneous heart rate of the subjects.

Current Interest in the Development of a Heart Rate Estimation Technique

The current interest in the development of a technique to estimate heart rate for the
microwave vital life signs monitor is to formulate a real-time implementations of the most
promising heart rate estimation techniques. The real-time implementations would be used on
board the heart rate monitor. The heart rate estimation tecluriques developed by Hoshal and
Siegel (1986) and Byme and Siegel (1985) were implemented and tested independent of the mon-
itor. Recorded data files were used. The initial purpose of this thesis research was to develop a
real-time implementation of Byme and Siegel adaptive least squares lattice filter estimation

method.

Although Byme and Siegel (1985) showed that the adaptive lattice filter technique worked
well under adverse conditions, the technique had not been fully tested. The technique was tested
with only a few data files and the results shown in Byrrre and Siegel (1985) were of cases in
which the technique worked very well (M. Siegel, personal communication, 1988). Before a
great deal of effort was expended in developing a real-time implementation of the Byme and
Siegel technique, more extensive off-line testing of the algorithm was performed.

A number of problems were encountered in the attempt to repeat the results of Byme and
Siegel (1985). The source of these problems was traced to differences in the character of current

microwave heartbeat signals and the data used in Byme and Siegel (1985). The differences might

be the result of two actions. First, a numwr of modifications have been made to the microwave

 

84

unit since the Byme and Siegel study. Second, the standard testing position of the microwave

monitor and subject has changed.

The Modified Michigan State University Microwave Vital Life Signs Monitor.

Figure K2 shows a block diagram of the current microwave vital life signs monitor. Four
major modifications were made. All the changes were made to the analog signal processing sec-
tion of the monitor. The transmitted microwave signal was changed from a continuous wave to a
pulsed microwave. This modification was made to improve the safety of subjects during expo-
sure of the microwave transmission and to minimize power consumption (M. Siegel, personal
communication, 1988). A logarithmic amplifier was added to increase the dynamic range and
sensitivity of the microwave transceiver. The logarithmic arnplifier increases the systems sensi-
tivity to small signals while large signals are not allowed to saturate the system. An automatic
gain control unit was added so that the input to the data processing section of the monitor would
have an even level. The band-pass filter was changed to a switched capacitor filter. This
modification allows the microprocessor unit of the monitor to control the character of the band of

the switched capacitor filter during operation of the monitor.

The cut-off frequencies of the switched capacitor filter remained constant during this thesis
research. The low frequency cut-off was 4 Hz and the high frequency cut-off was lSHz. The 4
Hz cut.off was chosen to filter out signal components related to breathing and still allow harmon-
ics of the heartbeat signal to pass. It is believed that most breathing components are found at or
below 4 Hz (M. Siegel, personal communication, 1988). The 15 Hz cut-off was used to reduce
noise. The latest spectrum analysis of the microwave heartbeat signal show that there were no

significant heartbeat signal components above 15 Hz (M. Siegel, personal communication, 1988).

The current standard positioning of the monitor and the subject are different than the posi-

tioning used in Byrrre and Siegel (1985). Current microwave data are recorded with the subject

85

 

.LQEoE 2&3 8: 22> 93322:. .3933: 35m came—82 35.5:— RF .NM Baum

 

 

 

85.50
5.5 83:65.4.

 

 

 

 

 

 

 

 

ssae<
25559:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5239.3 a: 2.... 35250
coau— ..o=oanau III! 335 camooEA—EEZ
85.5w 9 moses.
355 a Lozenges—h. 8a a
3.2.. .u._w...___.s - sea 1 saw”
2o: 93 nae—am F. < 2A— o>a3222 . c m

 

 

 

 

 

 

 

 

 

 

 

 

 

86

sitting. The monitor is placed six inches from the subjects chest wall. The monitor is placed

slightly to the left of the center of the chest wall. The subjects are sitting rather than lying down.

APPENDIX L

REVIEW OF THE RELATIONSHIP BETWEEN
AUTOREGRESSIVE PROCESS SYNTHESIS

AND ADAPTTVE LINEAR PREDICTION

An autoregressive (AR) process is modeled as
y(n)=gary(n-k)+W(n). (L1)

where y(n) is the AR process, the ag’s are the AR model parameters, w(n) is a white process,
and M is the order of the model. A sample of an AR time-series can be expressed as a linear
combination of past samples plus a sample from a white process. The number of past samples in
the linear combination is the order of the AR process. The time interval between samples is con-

stant.

Equation L2 shows that the AR process can be modeled as the output of an all-pole filter

excited by a band-limited white process.

 

Y(z) = 1 W(z). (L2)
I -gakz"‘

87

88

The model in (1.2) was formed by taking the z-transform of (L1) and treating W(z) as the input to
a system and Y(z) as the output of the same system. A fundamental goal of AR model analysis is
to determine the parameters of the all-pole filter. Linear prediction is can used to estimate the

coefficients of the AR model all-pole filter.

Linear prediction estimates the next data sample of an AR process by a linear combination

of past data samples.
r<n)= gram—k). (L3)

where for) estimates y(n) and b; is a linear prediction coefficient. The prediction error is

defined in (IA).
e<n)=y(n>— gbtym-k). (M)

where e(n) is the prediction error. By taking the z-transform of the error equation and treating
the prediction error as the output of a system and the AR process as the input ofthe same system,
(L5) shows that linear prediction can be viewed as the inverse operation of AR process synthesis

(Haykin, I984).

5(2),.[1-‘g‘bfl41 Y(z) (L5)

89

Linear prediction deconvolves the AR process. If the prediction error sequence is viewed as the
output of linear prediction. then applying linear prediction to an AR process is the same as pass-
ing the AR process through an all-zero filter. If the prediction coefficients are the same as the AR
model parameters, then the prediction error is the same as the white process input of the AR
model all-pole filter. The linear prediction coefficients can be used to estimate the AR model

parameters.

How are the linear prediction coefficients found? If the past values of an AR process and
the relationship between those past values are known, the only thing the one cannot predict about
the next value of the process is the sample from the white process. The prediction error will con-
tain the white process sample. Ifthe linearpredictorcan be used to extractthe partofthe process

sample that is conelated with the past process samples, the prediction error will be minimal.

IfthestatisticsoftheARprocessbeinganalyzedareknown,awaytooptimizethethe
selection of linear prediction coefficients is by minimizing the mean of the squared prediction
error, E {e 2(n )}. The parameters of the predictor are uniquely determined by the second-order
statistics of the process. The Yule-Walker equations give a linear relationship between the pred-
ictor parameters and the autocorrelation coefficients, {R;,r'=0,...M} where R,- =E {y (n )y (rt-4)}.
The optimal mean squared error predictor or the least squares predictor can be obtained by solv-

ing the Yule-Walker equations.

In most applications, the statistics of the process are not known and must be estimated from
data. Often, the process statistics are time-varying. In order to neck the changing statistics of a
non-stationary process, estimates of the second-order statistics and computations of the predictor
coefficients need to be continually updated. The problem of predicting a time-series without

prior knowledge of its statistics is called adaptive prediction (Friedlander, 1982a).

90

When the second-order statistics of the process being predicted are known, estimation of the

least squares predictor is a well defined problem. A well defined problem is also possible for the

estimation of least squares prediction parameters when all the information that is known about

the process is a finite number of data points, {y (n )} where n=l.2...JV . The problem as defined in

Friedlander (1981) is to fit the observed data to a linear model by choosing prediction

coefficients, 6,- , that minimize the sum of the squared prediction errors.

where

SSE = e201),

e(n)=y(n)+gézy(n-i).

(L6)

(L7)

The estimates of the coefficients are given by the homogeneous solution to the equations in (L8).

r -

e(l)

e(1V)

 

 

r .

y(1)

NV)

 

 

y(0) 0
y (0)

y(1~i-1) . . y(N.—M)

 

 

 

6:

6..

 

(L3)

 

(L8) can be more compactly written as

where

The solution of (L8) is given by

 

b

y (0)

 

e(1)

e(N)

 

y (0)

c=y+Yé,

0

yasi-r) . . y(N.-M)

é=fi‘lYTy,

91

 

 

 

where Y T is the transpose of Y and R is the sample covariance matrix.

§=YTY

 

6:

6'1.

 

(L9)

(L10)

(L11)

The equations that provide the solution to the problem of estimating the predictor coefficients of

a process with unknown statistics has the same form as the Yule-Walker equations, which were

derived for process with known statistics (Friedlander, l982a).

92

There are a number of efficient computational procedures for solving these equations. The
adaptive least squares lattice filter used by Byme and Siegel (1985) is an example. The useful-
ness of the relationship between AR modeling and adaptive linear prediction will be illustrated in

Appendix M by showing how this relationship has been used in the analysis of Speech processes.

 

APPENDIX M

ILLUSTRATTNG TIE USE OF ADAPTIVE LINEAR PREDICTION TO
EXTRACT IMPULSES FROM PROCESSES

USING SPEECH PROCESSES AS AN EXAMPLE

A simplified model for speech production is shown in Figure M1. The model consists of an
all-pole filter that is excited by either a quasi-periodic train of impulses or a white noise source
(Makhoul, 1975). Voiced sounds such as vowels are generated from nearly periodic impulse
sources. The impulse train represents a series of pitch pulses. The impulses are spaced at a fun-
damental period known as the pitch period. The white process produces the unvoiced sounds
such as the f in fish. The probability distribution of the white process does not appear to be criti-
cal (Haykin, 1984, chap. 1). The all-pole filter models the vocal tract. The identity of the sound

produced by both sources is determined by the parameters of the all-pole filter (Makhoul, 1975).

In speech recognition and synthesis, it is of interest to determine the character of the indivi-
dual sounds that make up a speech process. If the above model is used, the type of input source
that produces the sound must be known. For voiced sound, the pitch period of the impulse train
must be determined. The parameters of the all-pole filter are needed for both voiced and
unvoiced sounds. Therefore, the objectives in analyzing a Speech process is to estimate the

model parameters and recover the input process.

93

I i". .a'

 

demos—co... :89... .8 Bee... poem—9:1. .o 883:. soc—m .3). 2:3.—

 

 

 

 

 

 

 

 

 

 

 

 

 

9.58 889.5
.8 8.509
«.8058 8.8050
88... 38> 862 233
.25. 38>
$80k— :oooam A1 05 9.2322
BEE o_cA_-=<
Egon 88.5
.8 8.509
83.280
52.. 8.35.

 

 

 

moron— :BE a

95

Linear prediction is used in the analysis of speech processes. Parameter estimation per-
formed through least squares linear prediction works well provided the input to the all-pole sys-
tem is a zero mean white process. Linear prediction can be used to deconvolve a process that is
the output of an all-pole filter that has been excited by a white process. The white excitation pro-
cess is recoverable from the prediction error. For the analysis of sounds produced by a white
input, the constraints on the use of least squares linear prediction present no problem. But, the
periodic impulse source that generates voiced sounds is not white. Estimates of model parame-
ters may be biased and deconvolution may be imperfect for non-white input processes. For
example, if an input is coherent, the input may not be recoverable from the prediction error. The
model parameters would absorb the correlation information of the input process. The least

squares linear predictor would produce as nearly as possible a white prediction error sequence.

If the non-white input is known, then it is possible to unbias the estimate of the model
parameters (Friedlander, 1981). In speech analysis, the input to the all-pole model may not be
known apriori. In certain situations, it may be possible to estimate a non-white input from the
prediction error sequence. In particular, this may be true for the case where the input is an
impulse train as in speech processes (Friedlander, 1981).

Imagine that an all-pole filter of known parameters was excited by a single impulse. A
linear predictor with a configuration that inverses the operation of the all-pole filter would be able
to perfectly predict the impulse response of the all-pole filter except for the first non-zero value of
the response (see Figure M2). This initial non-zero value would show up in the prediction error
(Makhoul, 1975). Similarly, a large prediction enor is a good indicator of the occurrence of an
impulse exciting the all—pole speech model.

lrnagine that the configuration of the above all-pole filter was not known. The parameters
of the all-pole filter may be obtained from the impulse response of the filter. The parameters of

an unknown all-pole filter can be estimated when the autocorrelation coefficients are known for

 

96

 

.5533... 32... .3 .2... 0.2. .3 5 .o 8:83. 8.2.6. o... .0 5.5.9683 .NE 2:»...—

eeoeafi. 3...". o.o..-..<

 

 

 

 

5...". 5.8.3... 8.5%»: 8.2.5. .59.. 8.3....
A .9335 .35.. .o....... u.on.-..< I

 

 

 

 

 

 

 

 

 

cu.

 

 

 

 

97

the output of an all pole filter excited by a white process. Makhoul (1975) shows how the rela-
tionship linking the autocorrelation coefficients of the output of an all-pole filter are the same for
both the input of a single impulse or a white process. This result is expected because both a
deterministic impulse and a white noise process have identically fiat spectra. Makhoul observes
that the usefulness of this dualism between a deterministic impulse and statistical white noise is

shown in the modeling the speech processes.

It might appear that the pitch pulses of a speech process could be located by large prediction
errors, but there are some problems with this method. A signal impulse is a nonconelated pro-
cess were as an impulse train is correlated. The linear prediction estimates of the speech process
may be biased. The linear predictor may not perfectly deconvolve the speech process. This prob-
lem has been studied in the area of speech analysis. Friedlander (1981) shows how a particular
linear predictor, a least squares lattice filter, can be modified such that its predictions are not
biased when the linear predictor is used to deconvolve the output of an all-pole filter that has been

excited by an impulse train.

Another problem in relying on the presence of large prediction errors to indicate the loca-
tion of pitch pulse in a speech process is that the system that produces the speech process must be
minimum phase. The all-pole filter of the speech model has all of its poles and zeros inside the
unit circle. If linear prediction is used to inverse the Operation of a system that is not minimum

phase, the prediction error will probably not resemble the process that excited the system.

 

APPENDIX N

THE UNNORMALIZED PRE—WINDOWED LEAST SQUARES LATTICE FILTER

The Unnormalized Pie-Windowed Least Squares Lattice Filter is the same adaptive least
squares lattice filter used by Lee and Morf (1980) in their pitch pulse detector. The least squares
lattice algorithm is based on the Levinson—Durbin recursive method of computing the solution to
the Yule-Walker equations. The Yule-Walker equations provide a solution to the problem of
estimating linear prediction coefficients for processes with known statistics. Haykin (1984),
Friedlander (l982a). and Lee, Morf, and Friedlander (1981) show how the least squares lattice
filter is derived by applying the Levinson-Durbin method to the problem of estimating linear
prediction coefficients for processes with unknown statistics. Pre-windowed implies that it was
assumed that the input processes to the lattice filter is zero prior to t = 0. The procedure of
estimating linear prediction coefficients for processes with unknown statistics is usually done by

minimizing the sum of the squared prediction errors and is called least squares linear prediction.
Operating on a process with linear prediction can be viewed as being the same as passing
the process through an finite impulse response (FIR) filter or all-zero filter. The output of the
filter is the prediction error. It is usually assumed that a linear predictor is implemented in the
direct (or tapped—delay-line) form (Friedlander, l982a). Figure N1 shows a direct realization of a

least squares linear predictor. The direct realization coefficients, A.- ’s, are estimated by the 6’s of

98

 

2-;

 

E
o...»

. .Z

.0

.8.

£3.

no.3

.e

5

.38

o. S.

.3

a...

. no...

u...

a .a

p.

A...v

.8

 

 

 

 

 

 

 

2<

 

 

 

 

 

n<

 

 

 

 

fl

 

 

 

 

 

 

u<

 

 

 

.IN

 

 

 

 

 

 

.<

 

 

 

 

 

 

 

 

h.»

100

 

.823... 30.... 8.3....» 63. 2 .095 5 .o 523:3... 3.3.. .9. 2.6.".

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.4»

101

(L8). The é’s are the solutions to the least squares linear prediction problem for processes with
unknown statistics. The structure of the Unnormalized Pre-windowed Least Squares Lattice
Filter is shown in Figure N2. The K.- ‘s of the lattice implementation are called the reflection
coefficients. Appendix E gives the algorithm that is used to update the parameters of the lattice
filter. The direct realization of the least squares linear predictor and the least squares lattice filter
are mathematically equivalent. Values for the direct realization coefficients can be derived from
the lattice parameters. Appendix F gives an algorithm that can be used to calculate the direct

form coefficients from the lattice parameters.

Although the direct form and the lattice form of the least squares linear predictor are
mathematically equivalent, the differences in their structures may give one form a clear advan-
tage over the other in a specific use. A useful property of the lattice form is that the M th order
least squares lattice filter contains within its structure all lower order predictors. The pth order
lattice filter can be obtained from the first p sections of an Mth order lattice structure. When a
process is filtered with an M th order lattice filter, the prediction errors and lattice parameters of
all lower order lattice filters are available. This property is not shared by the direct realization of
the least squares linear predictor. Separate direct forms are needed if it is desired to simultane-
ously perform linear prediction on a process with different orders of linear predictors. Simultane-
ous access to different order linear predictor results may be useful in the real-time analysis of AR

processes with unknown order.

When compared with other adaptive linear predictors, the lattice form gives the Pre-
Windowed Least Squares Lattice Filter computational advantages. Friedlander (1982a) divides
adaptive processing methods into to categories: block processing and recursive techniques. The
Pre-Windowed Least Squares Lattice filter has a recursive parameter update algorithm. In block
processing, incoming data are divided into blocks which are then used to estimate the prediction

parameters. Block processing techniques update parameter estimation only once during a block

 

102

period. In recursive techniques, predictor parameter estimates are updated with every new data.
The recursive algorithm may be much more sensitive to changes in the input process that it needs

to adapt to.

A particular feature of the Pre-Wrndowed Least Squares Lattice Filter that enhances its abil-
ity to adapt to changes in an input process is the exponential weighting factor, A. In most appli-
cation, the statistics of a process are slowly time-varying (Friedlander, 1982a). Because the
linear predictor parameters depend on estimates of the statistics of the incoming process, it is
necessary for the linear predictor to be able to track time variations in the statistics of incoming
processes. 2. determines the character of an exponential window that defines the set of past input
data used to estimate the statistics of the input processes. The value of A. determines the shape
and length of the exponential window. Older data samples have less weight than newer data sam-
ples in the estimation of process statistics. The exponential weighting factor aids the lattice filter
in adapting to new trends in the input process by allowing it to forget past data values. Figure N3
illustrates the character of the exponential window for various values of 1.

Another important feature of the Pre-Wrndowed Least Squares Lattice Filter is the likeli-
hood variable, 7,. ,1 where n represents the order of the section the likelihood variable is being
calculated for and T represents time. The likelihood variable enables the lattice filter to adapt
quickly to sudden changes in the input process. The likelihood variable is closely related to the
log-likelihood function of the lattice filter input process. Lee and Morf (1980) defined this rela-
tionship in the following way.

Assume that {y, } is a zero mean Gaussian process. The joint distribution for

(yr, Jr.) is
DOT, . . . vyT-n)= IZfl'Rn I-” e{-%[y1'. '“ Walk-Al». ”(T-OJ? (N1)

where

 

103

 

.« .o 8...? 25...... .o. 32......» 3.5.53". .3... 8...... 33...... .93. o... .o 3.8.5.0 .nZ 2.5....

0.95m 8.5 .3. .e 2......

O Owl 8~l On—l 8N...

 

 

 

Ind

 

 

2...»?

.man—

104

Rn ED’T. ° ' ° vyT-n l’b’T. ' ° ' oyT—n1- (N2)

The log-likelihood function associated with (NZ) is proportional to
L =1n|Rn I + UT, ' ° ° ayT-rrJRn-ID’T. ' ' ' ryT-n]’- (N3)

The likelihood variable can be interpreted as the sample estimate of the second term in the log-
likelihood expression of (N3) (Lee & Morf, 1980). The likelihood variable. 7,. .1 is a measure of
the likelihood that successive data samples will come from the same Gaussian distribution (Lee
& Morf, 1980). The likelihood variable is a good detection statistic of the "unexpectedness" of
the most recent input data points (Friedlander, l982a). The value of 1,. ,1 ranges from O to 1. Lee
and Morf reported that whenever non-Gaussian type components are present in the data, 1...;-
tends to large values (close to l). The factor (1 - y. ,1) is in the denominator of a gain used in the
update recursions for certain lattice parameters (see Appendix B). As 7.1- approaches 1, the gain
goes to no. The gain enables the lattice filter to quickly adapt to unexpected data (Lee & Morf.
1980).

The likelihood variable aided Lee and Morf (1980) in the detection of pitch pulses in speech
processes. Their basic assumption was that the speech driving process consists of an approxi-
mately Gaussian part (for unvoiced mob) and a jtnnp component (for voiced speech). Large
linear prediction enors are good indications of the presence of a jump component or pitch pulse
location in the excitation processes. Lee and Morf found that the locating pitch pulses from pred-
iction errors could be done more accurately if the likelihood variable was used. Large changes in
the likelihood variable corresponded to the occurrences of pitch pulses in the speech process. Lee
and Morf showed how the derivative of the likelihood variable could be used as a mask in the
extraction of the pitch pulses from the prediction enor sequence. The likelihood variable was
used to separate the Gaussian components from the highly non-Gaussian jump components of the

speech driving processes. Byme and Siegel (1985) directly applied the Lee and Morf technique

105

to the problem of recovering heartbeat impulses from the microwave measurements.

A normalized version of the Pre-Windowed Least Squares Lattice Filter can be found in
Friedlander (1982a). The normalization of the least squares lattice algorithm assures that the
absolute values of the lattice filter’s output and parameters are less than or equal to unity. One
advantage of using the normalized least squares lattice filter is that it is easier to build a robust
implementation of the filter when the range of the filter’s parameters are known for every possible
operating condition. Another advantage is that the normalized version of the lattice filter is less
complex than the unnormalized version. After normalization, there are only three variables in the
lattice filter parameter update equations. The unnormalized lattice filter has six variables (Fried-

lander, 1982a).

A possible disadvantage in using the normalized lattice filter is that the prediction enors are
normalized. This may be a problem if one was trying to recover the exact excitation process that
was used to synthesis the process under analysis. Friedlander (l982a) shows how the unnormal-
ized lattice filter parameters and output can be calculated from the parameters of the normalized

lattice filter.

REFERENCES

 

 

REFERENCES

Byme, W., Flynn, P., Zapp, R., & Siegel, M. (1986). Adaptive filter processing in microwave
remote heart monitors. IEEE Transaction on Biomedical Engineering, BME-33 , 717-722.

Byme, W. J., & Siegel, M. (1985). Adaptive Filter Signal Processing in Microwave Heart Mon-
itors (Tech. Rep. MSU-ENGR-85-017). East Lansing, MI: Michigan State University,
Department of Electrical Engineering.

Byme, W., Zapp, R., Flynn, P., & Siegel, M. (1985). Adaptive filtering in microwave remote
heart monitors. IEEE Seventh Annual Conference of the Engineering in Medicine and Biol-
ogy Society, 1196-1199.

Chandra, S., and Lin, WC. (1974). Experimental comparison between stationary and non-
stationary formulations of linear prediction applied to speech. IEEE Transactions on Acous-
tics, Speech and Signal Processing, ASSP-22 , 403-415.

Friedlander, B. (1981). A modified lattice algorithm for deconvolving filtered impulsive
processes. Proceedings of the IEEE International Conference on Acoustics, Speech, and
Signal Processing , 865-868.

Friedlander, B. (l982a). Lattice filters for adaptive processing. Proceedings of the IEEE, 70,
829-867.

Friedlander, B. (1982b). System identification techniques for adaptive signal processing. IEEE
Transaction on Acoustics, Speech, and Signal Processing, ASSP-30 , 240-246.

Haykin, S. (1984). Introduction to Adaptive Filters. New York: Macmillan.

Hoshal, G., Ivkovich, S., Siegel, M., & Zapp, R. (1984). Remote noncontacting heart rate meas-
urement using microwave signals and digital processing. Computers in Cardiology, 409-
412.

Hoshal, G., & Siegel, M. (1986). Advances on Identification of Microwave Cardiograms ‘ for
Remote Heart Rate Estimation (Tech. Rept. MSU-ENGR-86-003). East Lansing, MI:
Michigan State University, Department of Electrical Engineering.

Hoshal. G., Siegel, M., & Zapp, R. (1984). A microwave heart monitor and life detection system.
IEEE Frontiers of Engineering and Computing in Health Care 1984, 331-333.

106

107

Lee, D. T. L., & Morf, M. (1980). A novel innovations based time-domain pitch detector.
Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Pro-
cessing,40-44.

Lee, D. T. L‘., Morf, M., & Friedlander, B. (1981). Recursive least squares ladder estimation

algorithms. IEEE Transaction on Acoustics, Speech, and Signal Processing, ASSP-29,
627-641.

Lin, J. C., Kiemicki, J ., Kiemicki, M., & Wollschlaeger, P. B. (1979). Microwave apexcardiog-
raphy. IEEE Transactions on Microwave Theory and Techniques, MIT -27, 618-620.

Makhoul, J. (1975). Linear prediction: a tutorial review. Proceedings of the IEEE, 63, 561-
580.

Popovic, M. A., Chan, K. H., & Lin, J. C. (1984). Microprocessor-based noncontact heart

rate/respiration monitor. IEEE Frontiers of Engineering and Computing in Health Care
1984, 754-755.

ill ‘rulll'li A‘Illr

ICHIan STRTE UNIV. LIBRARIES
"HI“WINIIIH‘IWWINWWI”INIWIWWHI
31293007883667