Theoretical limits and a practical estimator for joint estimation of respiration and heartbeat rates using UWB impulse radio

(1)

Teoretical Limits and aPractical Estim-ator for

oint

Estim-ation

Res

iration

an

ear

tbea Rates

sing uWB Impu

se

Rad

o

Sinan Gezici, Member

IEEE,

and Orhan

Arikan,

Member,

IEEE

T..

Abstract- In this paper, Cramer-Rao lower bounds are derived 4 forjointestimation ofrespirationand heartbeat ratesviaimpulse

radio ultra-wideband signals. Generic models are employed for ____I displacementfunctions due torespirationandheartbeat, and the - T

bounds areobtained for the cases of known and unknown channel Fig. 1 Transimitted signalstructureforjoint estimation of respiration and coefficients. In addition, atwo-step suboptimal estimator is pro- heartbeat parameters.

posed,which is based onjointtime-delayestimation followedbya

least-squares approach. Itis shown that the proposed estimator is

respiratory

detection of hidden humans

using

UWB

signals

asymptotically optimalunder mild conditions. Simulationstudies

areperformed to evaluate the lower bounds andperformanceof is implemented. A mathematical framework for estimation of the proposed estimator for realistic system parameters. vital signal parameters is established in [4], which employs Index Terms- Ultra-wideband (UWB), impulse radio (IR), the Fourier transform and motions filters for estimation of Cramer-Rao lower bound(CRLB), least-squares (LS)estimation. respiration andheartbeat rates.

In [7], theoretical limits for estimation of vital signal

I. INTRODUCTION parameters are derived under the assumption that there is

AtrtesCommission's one

periodically

moving object

in the environment.

Although

After theUSFederal Communications

this

mnodel

can be

employedl

for

estimatioln

of

respilratioln

rate (FCC's) approvalonthe limiteduseofultra-wideband (UWB)

paramoet

jont

estimation

ofh

respiration

rates

technologyinFebruary2002[1],communications andimaging parameters'

joint

estmation ofheartbeat and

respiration

rates systems that employ UWB signals have drawn considerable

requires

an extended

analysis,

which takes both

respiration

attention. Largebandwidths of UWB signalsnotonlyfa ilitate and heartbeat relatedparametersinto

account.

Inthis

paper,

we

high-speed data transmission, but also result in high time

derive Cramer-Rao

lower

bounds

(CRLBs)

for

joint

estimation

which important for accurate ranging and location ofrespirationand heartbeat parameters, andproposean

asymp-resolution, which

important

for accurate

ranging

and location

totically

optimal

estimator (under certain conditions), which estimation [2].

Commonly, impulse

radio (IR) systems, which is composed of

time-delay

and least-squares (LS) estimators. transmit very short duration

pulses

witha low

duty cycle,

are We

numerically

evaluate the CRLBs for

practical

systems,and

employedtoimplementUWB systems [3]. In an IRcommuni- . . '

ations system, atrain of

pulses

issentand information isusu- re perforn e of the proposed estator wth the

ally conveyed by

the

positions

or_{1- - **}the_{1 *}

amplitudes

_* _{T T rsXT}ofthe

pulses.

d

_nTe

₁

herivedrCRLBs

_{remainder of the} _paper _{iS organized} _as _follows. _In Inaddition to communications

applications,

IR-UWB systems

Section

II, the

signal

model used for parameter estimation

have also been considered for medical applications [4]-[8]. . p

Specifically, high spatial resolution of UWB signals faclitates is prsne. In Seto I,temxiu ieiod(L Specifically, high spatialresolution of UWB

signalssolution

is obtained and CRLBs are derived for generic time

detection of

objects

and estimation of vital signal parameters, displacement functions. In Section IV, a two-step estimator such as respiration and heartbeat rates ofhumans, in a given based on time-delay and LS estimation is proposed and its environment. Estimation of vital signalparameters canbevery o

impotan inmanscnaros,incldin serchng eope uder

optimafilty

properties

are

investigated. Then,

numnericalL

exam-important

in many

scenarios,

including.searchingpeonit

ples and simulation results are presented in Section V, and

debris afteranearthquake, through-the-wallhealthmontorng

concluding

remarks are made in Section VI.

ofhostages, and non-invasive patient monitoring [4].

Compared to the Doppler based techniques for estimation II. SIGNAL MODEL

of vital signal parameters [9] [10] IR-UWB signaling has Consider a

sequence

of

pulse

bursts as shownin Figure 1, advantages such as high penetration capability which fa- which is

mathematically expressed

as

cilitates through-the-wall applications, and low transmission N

power. In [5], possible medical applications ofUWB signals 1 N-i

are addressed, and their

penetration

and reflection

properties

t

T--.

- - ), (1)

are investigated. The channel characteristics for the signals k=O

refilectilng

fromn

a

humanar

chest are studied in [6], which also where N is the number of bursts, Ts is the burst period, and preselnts an algorithmL for respiration rate

estimnatioln. Inr

[8], w(t) is a burst of pulses, which consists of Alt pulses and is

t)The authors are with the Department of Electrical and Electronics 1Since reflections from a heart are much weaker than those from a human

Engineering, BilLkent University, BilLkent, Ankara TR-06800, Turkey,emlail: chest,the respiration rate estimation problem can omlitthe signalLsreflecting {gezici oarikan}@eehbilkent.edu.tr from the heart; hence,itcanhestudied in the framework proposed in [7].

(2)

given by since morenuisance parameters exist for the latter.

M-1

III. MAXIMUM

LIKELIHOOD

SOLUTION

AND

CRLB

w(t)= p(t-jTp), (2) CALCULATIONS

j=° Let the

unknown

parameters

of the received

signal

in (3)

with p(t) denoting the transmitted pulse and

Tp

being the be represented as

interval between consecutive pulses. It is assumed that T0 >

A

[Ob

Oh

ab

ahl,

(5)

T. where

Tw

denotes the width of

p(t).-The burst of pulses in (I) is aimed at an object being whichhasK (=S

K+K2+2)

components.Let

fxA(t)

represent

monitored and reflections are collected by areceiver. Pulses thereceivedsignal in the absence ofnoise; i.e.,r(t) =r;X(t)+

ineach burstare employed to obtain a reliable channel profile oTn(t).Then, thelog-likelihoodfunction of A can beexpressed

(i.e., to improve signal-to-noise ratio (SNR)), and comparison as [7]

of channel profiles obtainedfromconsecutive bursts is used to 1 T

estimate certain parameters of the object in the environment. A(A) = 2c- 2 ] r(t)c- f(t) 2dt, (6)

In thispaper, it is assumedthat the received signal consists 0

oftwo signal components (reflections), one due to respiration where c denotes a constant independent of A, and T - NT and the other due to heartbeat, and additive white Gaussian is the observaion

intterval.

noise; i.e., Considering real signals, the ML solution can be obtained

from

(6)

as

N-1

()z [i (. - k, - ) AL=i argmaxj

[2ijt)TA(t)

-

(t))]

dt, (7)

..V

(t

-

kTs

-

hk

(Oh))

+

(n(t),

(3)

which

requires

an exhaustive search over the parameter space

for A.

where

Tn(t)

denotes zero-mean white Gaussian noise with From

(6),

the components of the Fisher

information

matrix unit

spectral density,

ab and ah are channel coefficients for

(FIM)

I an be obtained as follows [12],

[7]:

the respiration (breathing) and heartbeat signal components, 2 1 IT 2

respectively, and 9k(Ob) and

hk(01)

represent, respectively,

B]

E (&A(A) l _I AT

(t)

dt_

(8

the time displacement functions induced by respiration and j

OAi

J j )2J t

(Ai8

heartbeat of the person being monitored. Note that the

un-known parameters of the displacement function

gkk(0b)

are for i 1. K, and

denoted by

0O),

which consists of K1 scalar parameters; i.e.,

rToA()

A(A) 1 TUrA(t)

of,(t)

Ob

=

[0b,l

..

Ob,K,].-

Similarly, the unknown parameters [Iij = dt,

of th dis 1 c m ntf nction Z '61 ' b 0 - t

0~~~~~Ai

OAj

J

r2

JOAi

OAj

of the

displacement

function Il

(Oh)

are given by Oh -

(9)

[Oh,i

I *.

Oh,K2I.

For example, if the displacement function for

therespiration is a sinusoidal function [6], gk(Ob) becomes for it j, where Ai is the ith component of A, and

[B]ij

denotes the element of matrix B in the ithrowand jth column.

Yk(Ob)

=Tfb+

Absin(2rFfbkTS +c), (4) From (3), (8) and

(9),

the FIM can be obtained for various where

Ob

is given by

Ob

=

[fb

Tb Ab

01]

displacement functions. In the following, two different

scenar-Since thesignal componentreflecting from the heart travels ios are considered depending on the available informationon

a

longer

distance than the one

reflecting

from the chest, we the channel coefficients.

assume that hk(Oh) > gk(Ob) Vk. We also assume that the A. Case-I. Known Channel

Coefficients

range of the displacement functions and the pulsep(t) satisfy First, we assume that the channel

coefficients

abi

and

Cah

are

(MI-

1)Tp

+T,

+

max{hk

_k,1

(Oh)

-

91(Ob)

} <

TS

sothat there is known at thereceiver. This scenario approximates the situation

no overlap between consecutive pulsebursts, which is usually in which a number of initial bursts are used for channel

thecase in practice. estimation, and the estimates for Cab and ah obtained using

Note that the signal model in (3) is not very realistic for those initial bursts are employed for the remaining estimation wideband pulse-based systems, since it ignores the multipath

period.

components from the other objects in the environment. How- For known ab and ah, the unknown parameter vector A ever, it is still an

important

first step towards

understanding

redu es to A

=[O

Oh].

Then,

from

(3),

(8)

and

(9),

the

FIM

of a real system as the main ideas in the analysis can be can be obtained, after some manipulations, as

extended to

multipath

s enarios. Also, this model gets more

lbbh

Ih

-.

(l

accurate when directional antennas are used for transmission

TbTh

Ihhj (10)

and reception, and/or an

effiLcient

clutter removal algoriLthm

[11l]

is applied before the parameter estimation process. Fi- whelre

nally, the;:theoret...icalliiS:on-::th acuay ofprmter N- &gk...(0)b&ek (Oh)e(

estimation obtained using the signal model in (3) provides

[Ihhhj=

Nea2 E ab 0j(

(3)

for i,j

=1,...,K,

parameter vector A is

given by (5),

which consists of

K,1

+

2 N-1 K2+2 components.

[Ihhl

a1l

£

hk

(Oh)

Ohk

(Oh)

(12) From

(3), (8)

and

(9),

theFIMfor this case canbe obtained,

NTLo=

Z_k=_o &

0h,i

&0h.j

o I after_some

manipulation,

_as

for , 1, ...,K2, and [blb

Ibh

lba

N-1 12 bh lhli Iha . )

[Ibhpij

N

NibT2

2

JY(hk(Oh)

gk(0b))

k0,i

0h,j b l I

k= (h ) b 1 where

lbb, lhh

and

lbh

are as in

(11)-(13),

lb,e

is

a K1 x 2

(1L3)

matrix given by

fori =1, K1 and j 1,. ..,K2, with R(x) representing o- N1g(b)

the autocorrelation of the first derivative of the

pulse

burst

[-boi

- 2

w(t)

in

(2);

i.e., N

=-oo

_{ab_}

N-1 _-

_&gk(Ob)

R -_

VSJ

"t+x'"t)dt (14) lb _ji _b R( (Oh) -gk(Ob)) (19)

-W~ §V}CWV}, k)

[bozN2N-T2

k0b,i

00 ~k=O b

and

£= R(0).

for i = 1, ...,

IKI,

J111

is a K2x 2matrix given by

Then, the CRLB for the covariance ofan unbiased estimate ^ N

ofA can be

expressed

as [12]

[ia1il

aohE

v

&hk(Oi)

^ -1 L hoeltl -

~~~~~~

~~N(T2

L

00h,i

Cov{A} >

I1-'

(15) k=o

N-1

where B > C means that B C is

positive

semi-definite.

['h1i2

=

N2

X R

(k(Ob)

-

hk(Oh))

(20)

Let the first elements of

Ob

and

Oh

denote the ratet

Nr

0h,i

(frequency)

parameters to be estimated; i.e.,

respiration

_{and for} _i

= 1,....,

K2,

and

I,

iLs

expressed

as

heartbeat rates, respectively. Then, the CRLBs for estimating f

fb

=

01)1,

and fil =

01

1 can be obtained, by

utilizing

the

[JaIi=

[-I]22_

E

inversion formula for blockmatrices, as cr

F -1iT \-i1l N-1

Var{fb} > [(Ibb- bhIhh 'bh) j11

['c12

= [Icc21= NT2 1R(hk(Oh) -%k(Ob)), (21)

Var{fh}> [(Ihh

-

hI)

)

wi

(1t6)

ko

>

_ hh bh

bl)

bh

)

116

with

R(x)

representing the autocorrelation function of the

where

fb

and fh are unbiased estimates for

fb

and fh, pu b f< w w x)dt,

respectively. and E=

R(0).

Ifthe time-delay between the signal components reflecting In general, the CRLBs forestimatingtherateparameters

fl)

from the heart and the chest satisfies the condition that and

fh

can be obtainedas

hk

(Oh)

-

9k(Ob)

> A,

Vk

such that

R(x)

= 0 for x >

Al,

Var{fb}

>

[1-J]

Varf

. ]K K

(22)

the FIM in (10) reduces to a block diagonal structure, since 2

it can be shown from (113) that

Ibh

= 0. Then, (116) can be

However, under

certain

conditions,

simpler expressions

than

expressed as

(22)

can

be

obtained.

Note from

(19)

and

(20)

that if the

delay

Var{

f >

[i-j]

Var{

fh

>

[i-,

(17)

difference between the

signal

components

reflecting

from the

Vaef

,-, - bbL1)131l ) teJ -L I1I1 1hl ' ' heart and the chest cavity satisfies the condition that

hk

(Oh)

-In

[5],

the round

trip delay

difference between the

signal

gk(Ob)

> A2 Vk such that R(x) = 0 for x > A2, then

componentsreflecting from the skin and the heart ispredicted

lb,

=

Ih,

= 0. In that case, (22) reduces to (16) of

Case-i.

tobe around 1.7nsbasedonnarrowband models. If the result In other words, under the stated condition, the knowledge of is close to this value for UWB signals, the bound in (17) is the channel coefficients does not affect the theoretical lower valid for sub-nanosecond UWB pulses, which means that the bounds.

same accuracyforrespiration rate estimation can be achieved

as inthecaseofperfectly known heartbeat related parameters,

IV.

SUBOPTIMAL SOLUTION and vice versa. In other words, unknown parameters of the

respiration and heartbeat related signals do not deteriorate After

obtaining

the theoretical limits for

respiration

and the estimation accuracy of each other if UWB pulses with heartbeat rate estimation, this section considers a suboptimal

sufficiently short durations are employed. estimator that asymptotically achieves the CRLBs under cer-tain conditions. Note that theML solution in (7)is anoptimal B. Cafse-2. Unknown Chaznnel Coefficients solution; however, it requires correlation of the received signal, Now we consider the case in which the channel coefficients over the observation interval [0,TI, with a template signal for the siglnal

com[ponenlts

reflectilng

from[

the chest cavity alnd Ai(t) for various values of the

param[eter

A, which requires, the heart are considered

unlknownl.

Iln this case, the unknown in general, optimizatioln over a (K, + K2 + 2)-dimelnsional

(4)

LSEstimator f A.

Optimality

Properties

LSEstimator 'fb

Time Delay b

|Although

the

proposed

estimator is

suboptimal

in

general,

Estimationa:y - it can be shown that it is asymptotically optimal undercertain

6htimatlon

<

conditions.

Towards this end, the following result is

helpful:

LS Estimator fh Lemma 1

[14]:

Consider the signal model

L

Fig. 2. A two-step suboptimal solution forjointestimation ofrespiration rk(t)aiW(t -

5i)

oTIk(t),(24)

and he beatrates. 1

for t E [0,

T§],

wherenk(t) iszero mean white Gaussian noise space, which has prohibitive complexity for practical imple- with unit spectral density, and

6i's

are the time-delays to be

mentations. estimated.

Due to the complexity of the ML estimnator inL (7), a two- The ML estimation of the multipath delays canbe obtained

step suboptimal

estimator, as shown in

Figure

2, is

proposed.

by finding

di's

that maximize j

rk(t)

Ei=1

oaiw(t-

6i)dt.

This

algorithm

first

performs

delay

estimation of the

signal

Then, the MLdelay estimates canbe modeled

for

sufficiently

componentsdueto

respiration

and heartbeat in eachburst, and large SNRs, as then estimates the desired signal parameters from the delay

estimates in the first step via an LS

algorithm.

t = 2

Assuming

that the channel

parameters

are known at the for i = 1,... ,L, where 77 =

['1

.l

/L]

is a zero mean

receiver23 as in Case- of Section III-A let b _ (0b) multivariate Gaussian random variable with the inverse ofits and 1,k

hk

(Ol)

represent

the

time-delays

in the kth burst covariance matrix given by

of the

signal

components due to

respiration

and heartbeat, 4

2[

- j

R_

-

)

(26)

respectively (k =0,

1,..

17.,N-

1).

Notethat the estimation of /SR 2 (

6b,k

and

6h,k

in_{X .: . .}a

given

burst is_.a

special

_. case of

multipath

_,

~~~~~~~~~~fo

_r _i ₇ _{y Note that}

_in1

_(26),

_R(x;)

_is _as

_in?

_(14),

_SNRi=

delay

estimation, studied

extensively

_{[4.Itefrttpterpsdeiaoprom.Ea/}in the literature

[13],

fo2/

_withoethti (6 (x

sasi214

N

£denoting theenergy ofwQt) and 3iseffective

est1mation

of

te-l

in each

bsti[14].

bandwidth

defined

as

132 =ff2W(f)

df/f

bW()2

2df

estimation oftime-dlaysachin burs

4with

W(f)

denoting the Fourier

transform

of

w(t).

Let the delay estimates related to respiration and heart-

From Lemma

1,

it

can

be

deduced

that the ML

estima-beat components obtained in the first step be 6b tion of the

time-delays

in a

given

burst k for the

signal

[6t,O

6b,

I*...

*b,N-11

and

6h

=

[6h,O

1i, **...

*1i,NI-

I In the components due to respiration and heartbeat can be obtained

second step, these delay estimates are employed in the fol- by correlating the received signal in that burst; i.e., r(t) for lowing LS estimators to estimate the parameters related to t

[kTs,

(k

+

1)TsI,

with

a

template signal for

various

delay

respiration and

heartbeat:

values. In addition, the estimation errors can be modeled by

N-1 2 a multivariate Gaussian random variable thatdepends on the

Ob

arg min (k

g-9k(0b))

signal characteristics and the noise level. Therefore, the delay

=O estimates

in

the first step of the proposed algorithm

can

be

N-1 2 modeled as

O argh

minJ

(hk

Jk (23)_I(Oh)) 6bk

[k(Ob)+

1

fb,k

Ohk=O

_L8h,kj

_hk

_(Oh)]

+ _N _[f/h,kj^

(-27)/

Note that this two-step estimator is usually considerably

simpler and more practical than theML solution in (7) since

it does not have to perform correlations over the whole r g b k g ^ r b~~~~'qb,[ (7 ]

[47F2

2

SNRb

Ct,ah

1k

1

observation interval for all different values of A. Instead, [k 0

232 SNRhj

it first

performs

a search over a two-dimensional space for (28)

delay

estimates in each burst. Then, it uses these

time-delay

estimates from different bursts for the LS estimation for k =

0,1,...

N - 1, with

SNRb

EEa./.

2b2,

SNR11

of the desired parameters. For comparison purposes, it can

Ea2/u2

and k

=R(Ik

(Oh)

-gk(0b)).

be observed that for small number of bursts and K1 > K2, From (27) and (28), the following result can be obtained. the complexityof the proposed algorithmis dominatedby the Note that the noise components for different bursts are

in-search over a K1 dimensional space, whereas thecomplexity dependent since they are affected from independent noise

of the ML algorithm in (7) is determined by the search over realizations.

a

(K1

+

K2)

dimensional space. Proposition 1: Considera set of time-delay measurements

cz

n(27)

and

(28) for k

=0,

1,.

..,N

N-1. Assume that

21n fact, the knowledge of the ratio between the channel (0) kA1R(.The

3WewilLlLsee how thisassumuption canLbeapproximratelLyrealized in practical the CRLB for the covariaznce maztrix o.f azn unbialsed estimazte

(5)

FIM is givenby

(10)-(13).

0.035 _Case-i,_T

=0.5ns

Proof: See

Appendix

A.

t * I:: -w tl 0-03

~~~~~~~~~~~~~~~~~~~~~~~~~Case-1,

=1sns

T

Note that under the

assumption

in

Proposition

1, the ML

Case-i,

T=2ns

solution for [Ob

Ohl

can be obtained from (27) and (28) as Case-2,

TW=0.5

ns

0.02 ^ Case-2,T =1ns

N- w; r (::E2 .i h Case-2,I0.02.0w--W . 0,TW=2ns

[Ob Ohl

=arg ml

KL

Z a b

Yk(Ob))02w

[09-,

Oh] _ L E

. +ah (h,k-

hk(Oh)

(29)

0.01 ..

which is equivalenttothe secondstepof the proposed solution

in

(23).

Inotherwords, theLSsolution in the secondstepisthe 0.005

ML solution for the

signal

model in

(27)

and

(28).

Since the 0

ML estimator

asymptotically

achieves the CRLB [12], which

00

5 10 15 20

is

equal

to the CRLB in Section III

according

to

Proposition

SNR (dB)

1, the LS solution in

(23)

provides

an

asymptotically optimal

Fig. 3. CRLBs for respiration rateestimationfor

Case-I

andCase-2. solution forestimating Ob and Oh under the conditions stated

I' 1'}:1. ' . -D. ' -1:) (^E w wi s s D . . Case-i, T00.5ns

inLemma Iand

Proposition

0.-Cae1,T=05n

Although the proposed estimator in the previous section is 0.8 r .Cs- w05nCase-2,T =1ns

asymptotically optimal under ertain onditions, it assumes the 0.6 Case-2,T =2ns

knlowledge

of the

chanlnel

coefficients alt) an:d ath, whichmay 0.5.

not be true

inl manay

situations.Therefore,in: apractical system, Ea7_0.4 _.;... the channel coefficients should be estimated from theiinitial.

burst(s), which can then be used in the implementation of 0.3-- \ ---\7''-''''-'-''-'-'-the proposed approach, since they are assumned to be constanlt o.c. <--- - C T5

during the observation ilnterval t [0, T]. Inl other words, al) --- 7''''.7_

Andhouhcanhe estimated frtmaom theireceved usiga (et)ior 0.1s

t

a

y0,

NT1i where N > 1 is the cnumber of bursts used ford _{L : c s g c _} 4₄₅ ₄₀ 50T55₅₀ ₅₅ _{~~~~~~~~~~~~~~~~~~~~~~~~~~~~60}

estimatiol of the channel coefficients SNR (dB)

Since theestimates for thechannel coefficients will include Fig. 4. CRLBs for he beat rate estimation forCase-I andCase-2. error is practice, the proposed algorith cannlot achieve the

CRLB in reality. HEowever, for small estimation errors, the ns, Ab =6.67x 10-11 s, Ah =1.667x 10-11 s, (b3=J

performaoce will still be quite close to the optimal under the 0, b

21,

°h an.d0.0015 N = 500.From. the figures, conditions stated in Lemra land Proposition 1. it is observed that the CRLBs for Case-I and Case-2 match very closely for all SNR and TN values, since the time-delay V. SIMULATION RESULTS difference between the respiration and heartbeat

components

Ince

this

sestion, numerical

studies

aned

simulationlls

are per-

is

larger than 1 ns. In

addition,

the

accuracy

inacreases

as the

formed to evaluate the CRLB expressions derived in Section pulse width is decreased, which is expected sihce higher time III, and to investigate the performance of the proposed esti- resolution results in better localization of the object. Also, it is

mator in Section IV. observed thathigher SNRs5 are needed for reliable heartbeat

Consider a system thatusesthe Gaussian mono-cycle p(t) rate estimation than thoseaneeded for reliable respiration rate

)n

th

section

estimation, which is

due

to

the

addition,al

loss (about 28 dB)

(irmd4to evaluat /h E wexressios usrivedto a stitcaused by the

propagaodn

of the siegnalfrom the

skhin

to the energy o

thne

pulsein the simulatons, aon t determiones throeheart aonrdback to the skin [5].

pulse

widethioT 2V.

[15erd]. In Figu 5,theSame pareterseain reviouse senarteatio

In the simulatiosns the displacementfusnctions k

(b)

and were

employedo

and the lower bounds for

respiration

rate

hk(Oh)

are

modeled as estimation are plotted versus

pulLse

width for various SNRs.

gk(2b)

wbrAbpSi

iu(2tfbkT

oab)

u (30) It is observed again

that

as the

pulse

width

decreases,

the

energyofhe(ulseh)

n

theA

simulatins,andT

h (31)rs t accuracy of the estimation increases. Moreover, the slope

decreases as the SNR

ipnacreases,

whinch

means that for high

In Figure 3 and 4, the CRLB expressions in (1L6) ad (22), SNRs, the loss in accuracy when using wider pulses (pulses correspording to Case-i and Case-2 respectively, are plotted with

sealler

bandwidths) is

slaller

than the loss for low

for

various SNRs and pulse widths4. The system parameters SNRs. Similar

observatiohnas

are

mlade

for the CRLB versus are Ts = 0.1 s, fb =0.5 Hz, fh =1.1 Hz,Tb = 0, Th 1.7 pulse width curves for the heartbeat rate estimation (not

4Square-roots of the expressions are plotted and the lower bounds are In order to obtain higher SNR values longerobservation intervals and obtained in units oflH:z. more pulse combining can be employed.

(6)

0.035 the assumption of lk(Oh) -gk(Ob) > AI (whichmakes sure

O Case-1,SNR=0dB o

Case-1,SNR=10 dB that the

off-diagonal

terms in (28) are zero), as

0.03 Case-1,SNR=20 dB

Case-2,SNR=0 dB N1 2 2 2 2

Case-2,SNR=10dB NpjA5<- e [SNRbb,k-gk(0b)) +SNRh(tk-hk (Oh))]

0.025 Case-2, SNR=20 dB -J 1

k=O

,,0.020

0.02 ... -

~~~~~~~~~~~~~~~where

is aconstant independent ofA. l32f: :(2

E < Fromv« (32), the elements of theFIM I can be obtained from

E 0.015

> ,/ 1 -

(A0~~~~~~~~~~~~~~~~~0logP,\(0i)0

0.005 E

T (ns).and ^ &

0 0l a 15X

=ElOp(6)

I

0

logp()

iX

j(34)

0. 1 Tw(ns) 15 2I]i ASi

OA)

,J34

Fig.5. CRLB versus pulse width for respiration rate estimation.

From the definition of the effective bandwidth, it can be

0.014. LSsolution| | ! shown that

4r2/s2SNRb

=

Ea4//N

band 4 2ji2SNRh

\ : : x ~~~~~~~~~LSsolution b

0.012>\ O CRLB

Ec(2/or2.

Then, from

(32), (33)

and

(34)

can be calculated,

after some manipulation6, and it can be shown that I

0.01 ... ... .... ... .... .... ... .... -b lb

LT\\ j In other words, I is equal toJ1 in Section

III-A,

I 0.008 - hich completes the proof.

...R

rr 0.006

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T,

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estimation

scenLario.

All the parameters are as

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Conoference

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24-27,

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b and

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unlifor:mly

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Froma

the

maodel given:

by (27)

an:d

(28) for the

timne-delay

Mllitary Commun7. Coif.

(MILCOM) vol.2,pp. 529-533, Oct. 1L998.

estimnates,

thedistributioln of6 [6o 6i

N-i1],

with