A Nonlinear-Phase Model-Based Human Detector for Radar

A Nonlinear-Phase

Model-Based Human Detector

for Radar

SEVG_I Z. GURBUZ, Member, IEEE

TUBITAK Space Technologies Research Institute

and TOBB University of Economics and Technology

Turkey

WILLIAM L. MELVIN, Fellow, IEEE

Georgia Tech Research Institute

DOUGLAS B. WILLIAMS, Senior Member, IEEE

Georgia Institute of Technology

Radar offers unique advantages over other sensors for the

detection of humans, such as remote operation during virtually

all weather and lighting conditions, increased range, and better

coverage. Many current radar-based human detection systems

employ some type of Fourier analysis, such as Doppler processing.

However, in many environments, the signal-to-noise ratio (SNR)

of human returns is quite low. Furthermore, Fourier-based

techniques assume a linear variation in target phase over the

aperture, whereas human targets have a highly nonlinear phase

history. The resulting phase mismatch causes significant SNR

loss in the detector itself. In this paper, human target modeling

is used to derive a more accurate nonlinear approximation to the

true target phase history. The likelihood ratio is optimized over

unknown model parameters to enhance detection performance.

Cramer-Rao bounds on parameter estimates and receiver

operating characteristic curves are used to validate analytically

the performance of the proposed method and to evaluate

simulation results.

Manuscript received October 6, 2008; revised July 16, 2009,

November 12, 2009, and January 11, 2010; released for publication

March 11, 2010.

IEEE Log No. T-AES/47/4/942884.

Refereeing of this contribution was handled by R. Adve.

Authors’ addresses: S. Z. Gurbuz, TUBITAK Space Technologies

Research Institute, ODTU Campus, 06531 Ankara, Turkey, E-mail:

([email protected]); W. L. Melvin, Sensors and

Electromagnetic Applications Laboratory, Georgia Tech Research

Institute, Atlanta, GA 30332-0856; Douglas B. Williams, School

of Electrical and Computer Engineering, Georgia Institute of

Technology, Atlanta, GA 30332-0250.

0018-9251/11/$26.00 c° 2011 IEEE

I. INTRODUCTION

Human detection is valuable to humanitarian,military, and security applications. Visual, acoustic,vibration/seismic, infrared, and electromagnetic

sensors have all been used in human detectionsystems. However, radar offers unique advantagesover other sensors, such as remote operation in allweather and lighting conditions, greater range, andbetter coverage. The problem of human detectionwith radar may be broken down into two key tasks:

detecting the presence of a target and decidingwhether the target detected is human.Over the last decade, a variety of Fourier-based

techniques have been applied to detect human targetsin differing situations using radar, most of whichattempted to pick out probable human targets by

identifying low-frequency periodic motion [1—3].Such generalized techniques have the disadvantageof confusing human motion with the motion of otherslow-moving nonhuman moving targets, such as aslowly spinning ceiling fan.The first attempts at developing detection

algorithms that focused on features unique to humantargets began with work by Weir and Childress[4], [5] in 1997, who developed the concept of a “gaitvelocitygram”–the velocity profile of a human targetthat contained distinct features yielding informationabout the gait.

Frequency-based concepts were then refined withthe concept of the “radar gait signature”–a spectralanalysis of the gait signature, such as spectrograms,that has been shown to be characteristic of humans[6—12, 14]. Although Geisheimer, et al. [10] initially

used chirplet transforms to characterize the gaitsignature, spectrograms have been proposed as asimpler way of extracting biomechanical informationfrom the radar return. In 2002, Geisheimer, et al. [11]experimentally verified that the overall spectrogramfrom a human target matched the sum of spectrograms

constructed from the returns of individual body parts.The theoretical basis for this result was developed byVan Dorp and Groen [12], who divided the humanbody into 12 parts and computed the time-varyingrange for each part using a walking model developedby Boulic, et al. [13]. Van Dorp and Groen showed

that the human model-based simulated gait signaturesmatched the spectrograms derived from measureddata–an important result that will serve as the basisfor the human target model utilized in this paper.

This foundation has led to much work that exploits

the unique features of the human spectrogram. For

example, Otero [14] used the spectrogram to compute

features such as the stride and appendage/torso ratio,

and Greneker [15] proposed a spectrogram-based

suicide bomber detection system. The potential

of applying human motion models to the human

detection problem is further illustrated in recent work

by Van Dorp and Groen [16] and Bilik and Tabrikian

2502 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 4 OCTOBER 2011

[17], in which the Boulic walking model is used to

animate a person by estimating model parameters

from measured spectrograms, and to classify targets,

respectively.

Spectrogram and gait analysis techniques generally

aim at characterizing targets already detected

through Doppler or Fourier-based processing; they

are effective in high single-pulse (predetection)

signal-to-noise ratio (SNR) scenarios, where limits

on temporal aperture do not adversely affect signal

contrast. In other situations, those of particular

interest in this paper, human target detection presents

unique challenges because humans are slow-moving

targets with low radar cross section (RCS). Increased

temporal aperture is required to boost SNR, and

therein lays a key challenge: As the aperture increases,

the increasingly nonlinear target phase history leads

to significant integration loss when using a Fourier

(Vandermonde) basis. While detection in clutter is

a critical radar attribute–and a highly active area

of research for rigid body targets (viz., vehicles)

[18—20]–we first concern ourselves in this paper

with improving noise-limited human target detection

performance as a prerequisite to the more challenging

case of clutter-limited detection.

We can enhance coherent detection of human

targets by exploiting information that we know

a priori is specific to this target class: the uniqueness

of human gait. The complexity of human motion

manifests as nonlinear variation over the aperture

[21, 22].

The goal of this paper is to exploit the results of

gait analysis to derive a new, nonlinear-phase detector

such that the mismatch between the presumed target

characteristics used in the detector design and the

actual target response is minimized, and thus output

SNR and probability of detection are maximized.

II. SIGNAL MODELING

A. Received Signal

Consider a radar antenna transmitting a series of

chirped pulses at constant intervals in time and space

while moving along a straight path. In general, the

received radar signal is a time-delayed version of the

transmitted chirp signal. Then the return for a point

target may be expressed as

sr(n, t) = atrect

μt¡ td¿

¶¢ expfj[¡2¼fctd +¼°(t¡ td)2]g (1)

where the time t is defined as t= T(n¡ 1)+ t interms of the pulse repetition interval (PRI) T, the

pulse number n, and the time relative to the start of

each PRI, t; at is the amplitude as given by the radar

range equation; ¿ is the pulse width; c is the speed

of light; ° is the chirp slope; fc is the transmitted

center frequency; and td is the round-trip time delay

between antenna and target, defined in terms of the

target range R as td = 2R=c.

Exploiting the work of Geisheimer and Van Dorp

who showed that a human target could be divided

into parts and the total response obtained by simply

summing the responses of each part–i.e., that the

principle of superposition could be applied to human

modeling; the human body is subdivided into K body

parts, each of which is modeled as a point target

located at its centroid. Thus, the total return from a

human target may be expressed as

sh(n, t) =

KXi=1

at,irect

Ãt¡ td,i¿

!¢ expfj[¡2¼fctd,i+¼°(t¡ td,i)2]g (2)

where at,i and td,i are the amplitude and time delay of

the return of each body part.

The amplitude at,i, defined as

at,i =G¸pPt¾i¾n

(4¼)1:5R2ipLspLa

qTsys

(3)

and includes several factors that vary with target range

Ri and geometry. For instance, the antenna gain G

varies according to the angle of incidence, and the

atmospheric losses La vary with range. For simplicity,

we assume that these parameters are constant along

with the transmitted signal power Pt, the wavelength

¸, the system loss Ls, the system temperature Tsys,

and the noise standard deviation ¾n. The RCS ¾i is

modeled according to the approximate shape of the

body parts. Thus, the RCS of the head is computed

from the scattering amplitude and phase of a sphere

[12], while the RCS of the other body parts are

computed from the scattering amplitude and phase

of a cylinder [12].

B. Human Model

A human is a complicated target because of the

intricate motion of body parts moving along different

trajectories at different speeds. In this work, the

human body is divided into 12 basic body parts: the

head, upper arms, lower arms, torso, thighs, lower

legs, and feet. As indicated in Fig. 1, each point target

is taken to lie at the center of the corresponding body

part.

The time-varying position of each point target

may be computed using the kinematic model of a

walking human developed by Boulic, et al. [13]. In

the Boulic model, all positions are referenced to the

base of the spine, denoted by OS. Over the course

of one cycle (two steps), the vertical, lateral, and

translational position of OS varies sinusoidally. The

time-varying angles of the joints are also provided by

means of charts and equations in [13]. Together with

GURBUZ ET AL.: A NONLINEAR-PHASE MODEL-BASED HUMAN DETECTOR FOR RADAR 2503

Fig. 1. 12-point human model [21].

the dimensions of the human body, these time-varying

joint angles may be used to compute the time-varying

positions of each point target relative to OS and each

of the time delays (td,i) required in (2).

The received return from the human target, stored

as a slow-time, fast-time data matrix, is then pulse

compressed so that the peak occurs at the range bin

in which the target is present. Taking a slice across

slow-time at the range bin of the peak output,

xp[n] =

12Xi=1

at,i¿e¡j(4¼fc=c)Rd,i (4)

where Rd,i is the range from the antenna to the center

of each body part.

The new method proposed in this paper uses the

slow-time slice in (6) as the starting point for data

processing.

III. SNR LOSS CALCULATION

The fast Fourier transform (FFT) used in Doppler

processing provides the appropriate Vandermonde

structure leading to maximum output SNR when

the inputs likewise exhibit constant amplitude and

linear phase variation over the aperture. However, as

previously shown, the phase history of a human target

can be highly nonlinear, resulting in an inherent SNR

loss when processed by a linear-phase filter, such as

the FFT. For example, consider the phase history of

a typical human target walking at a 45± incidenceangle relative to the initial antenna-target vector as

shown in Fig. 2. In this case, we have removed the

linear phase component. The oscillatory components

of the phase history are evident and imply the target

response spreads over a range of velocities during the

dwell.

The SNR loss incurred may be quantitatively

analyzed as follows. Define from (4) the true target

Fig. 2. Target phase history after linear component is removed.

data as

s= [®0 ®1ejμ1 ¢ ¢ ¢®N¡1ej(N¡1)μN¡1 ]T (5)

where N is the total number of pulses transmitted, and

®i and μi are generalized amplitude and phase factors,

respectively. Note that in general both the amplitude

and phase factors vary with slow-time (n).

When filtered with the weight vector w, assuminguncorrelated noise with single pulse noise variance ¾2n ,

the output SNR is

SNR=jwHsj2¾2n jwHwj

: (6)

The maximum output SNR is then attained when

the received signal s is matched filtered against itself:

SNR0jw=s =1

¾2nsHs: (7)

However, because knowing the target return

exactly in advance is impossible, current systems

typically matched filter with a linear phase filter,

which may be expressed as

wLIN = [¯1 ¯2ejÁ ¢ ¢ ¢¯N¡1ej(N¡1)Á]T (8)

where ¯i and Á are the generalized amplitude and

phase parameters, respectively. Here, the amplitude

factor is left in a general form that varies with

slow-time, while only the phase has been restricted

to be linear. The output SNR for such a linear phase

filter is

SNRLINjw=wLIN =jwHLINsj2

¾2n

¯PN¡1n=0 ¯

2n

¯ : (9)

Thus, the SNR loss incurred from signal mismatch

is

SNR Loss =SNRLINSNR0

=jwHLINsj2¯PN¡1n=0 j¯nj2

¯sHs

· 1:

(10)


Fig. 3. Output SNR variation over dwell time normalized by

input SNR for the target phase history shown in Fig. 2 comparing

clairvoyant and FFT-based detectors.

There are three main factors that affect SNR loss:

phase mismatch, amplitude mismatch, and dwell

time. Fig. 3 illustrates the effect of these factors by

plotting the output SNR normalized by input SNR as

dwell time is increased for both the ideal, clairvoyant

detector and the FFT applied to the example shown

in Fig. 2. In this case, the data exhibit miniscule

amplitude variations of 2:64£ 10¡7§2:5£ 10¡7 overa 2-second dwell. The FFT computes a constant,

flat-line approximation to the true amplitude, i.e.,

¯n = ¯ for all n, but the amplitude mismatch–defined

as the SNR loss incurred due to amplitude differences,

assuming identical phase–is not nearly as significant

as the phase mismatch. Over a 2-second dwell, the

phase moves through 100 rad, or 5732 deg, of phase.

However, just a 1% error in the slope causes the

phase to shift by 2¼, an entire cycle. Thus, unlike the

amplitude, phase is extremely sensitive to mismatch;

the nonlinearities result in substantial SNR losses (the

difference in SNR between the clairvoyant and FFT

filter outputs) of up to 30 dB for a 2-second dwell

time.

As Fig. 3 indicates, collecting data over a longer

dwell in an attempt at increasing integration gain also

does not help to alleviate the inherent SNR loss in

Fourier-based, linear-phase detectors. Specifically,

while the output SNR continually increases with

dwell for the clairvoyant detector, the FFT exhibits

on average a slightly degraded trend so that the output

SNR does not increase with dwell time.

Thus, matched filtering with a more accurate

model of the signal phase history has the potential to

yield a significant reduction in output SNR losses and,

thereby, substantially improve detection performance.

For example, while the model-based optimized

nonlinear-phase (ONLP) detector proposed in this

paper exhibits an SNR loss of only 11 dB for a

0.5-second dwell time, the FFT suffers an SNR loss

of 21 dB.

The remaining sections of this paper focus on the

design and performance of the ONLP detector.

IV. DETECTOR DESIGN

A. Detector Formulation

For each range bin centered at rb, the detector must

make a decision between two hypotheses:

H0 : x= xI

H1 : x= s+ xI(11)

where xI is complex Gaussian interference with

covariance matrix RI and s is the true target signal,

and we’ve assumed that there is no range migration,

i.e., the entire target return is contained within

one range bin. However, because the interference

covariance matrix cannot be known a priori, an

estimate must be used instead. In this paper, we

assume clairvoyance and the noise-limited case:

RI = RI = ¾2nI. Achieving acceptable performance

in the noise-limited case is necessary before

considering the more general cases for RI and has

practical significance for a number of radar

configurations.

A likelihood ratio test [23—25] is used to determine

the detector decision rule. Under H0, the disturbance

is distributed as CN(s,RI) while under H1 the

disturbance distribution is CN(s,RI), where s= xp is

the target signal vector. Thus, the decision rule may be

expressed as

Decide H1 ifp(x;H1)

p(x;H0)> °! RefsHR¡1I xg> °0

(12)

where °0 =qRefsHR¡1I sg ¢Q¡1(PFA) and PFA is the

desired probability of false alarm.

Because the target return s is not known a priori,

a realizable detector must use an approximation. Our

goal in this paper is to develop a better approximation

to the target return than that of the linear-phase

FFT so as to minimize SNR loss and achieve better

detection performance.

The human-model based expression for the target

return in (4) is a very good approximation to the

true target signal s; however, this model is much too

complicated for use as an effective matched filter.

The model contains over 24 unknown parameters and

most of the kinematic expressions used to compute

the time-varying range of each body part are not

presented in closed form, but rather as graphs, which

must be combined with other charts or equations to

derive the time-varying position [13]. Thus, we next

derive a simpler, nonlinear approximation to (4) that

will serve as the basis for our proposed optimized

nonlinear phase (ONLP) detector.


Fig. 4. Antenna-target geometry.

B. Approximating Expected Target Return

The torso response is significantly stronger than

the response from the remaining 11 body parts

comprising the model [21, 22], so we will simplify

first by neglecting the motion of all body parts except

the torso, i.e., we will design the detector to match as

best as possible the response from the torso only.

Furthermore, because the overall SNR loss

is caused primarily by phase mismatch, we will

approximate the received signal amplitude at as being

a constant A, even though there is some variation

across slow-time due to slight variations in gain, RCS,

and other loss factors. We also approximate the range

term in the amplitude factor (3) with rb, the center of

the range bin at which the peak pulse compression

output occurred.

For the range term in the phase, however, we

cannot make as crude an approximation because

the phase is much more sensitive to errors than the

amplitude. A more accurate approximation to range is

obtained as follows.

Define r1 as the vector from the antenna to the

target’s initial position and rN as the vector from the

antenna to the target’s final position. For simplicity,

assume that the human motion is linear along a

constant angle μ relative to r1. Then the vector h

between the initial and final target locations represents

the human motion (Fig. 4).

Using the law of cosines, we may write

jrnj2 = jr1j2 + jhj2¡ 2jr1j jhjcosμ: (13)

Because in our application we assume jhj ¿ jr1j, andp1+ x¼ 1+ x=2 for small x,

jrnj ¼ jr1j ¡ jhjcosμ: (14)

Then, (4) may be written as

xp[n]¼A

r2bexp

£¡j(4¼fc=c)(r¡ hcosμ)¤ (15)

where r = jr1j and h= jhj.The human motion vector h may be more

explicitly expressed using the Boulic kinematic

equations, such as

h2 = OS2V+OS2L+(vTn+OSFB)

2 (16)

where

OSV = 0:015RV[¡1+ sin(4¼t%¡ 0:7¼)]OSL = AL sin(2¼t%¡ 0:2¼)

AL =

½¡0:032 for 0:5< RV< 2:3

¡0:128RV2 +0:128 for RV< 0:5

OSFB = Aa sin(4¼t%+2Áa)

Aa =

½¡0:021 for 0:5< RV< 2:3

¡0:084RV2 +0:084RV for RV< 0:5

Áa =¡0:127+0:731pRV:

Here, OSL, OSV, and OSFB represent the lateral,

vertical, and forward-backward motion of the center

of the torso relative to coordinate origin of OS (see

Fig. 1) located at the base of the spine. The Boulic

equations depend on only two variables: (1) RV, the

ratio of velocity (v) to thigh height (HT) and (2) t%,

a time index taken relative to the beginning of a step.

These variables may be expressed as

RV =v

HT(17)

and

t%=nT

1:346

pRV+ t0 (18)

where t0 is a constant indicating the point within the

stepping cycle that the first transmitted pulse reflects

from the target.

The expression for h may be simplified by

neglecting second order terms and approximating the

square root as

h¼q(vTn)2 +2vTnOSFB

= vTn

r1+

2OSFBvTn

¼ vTn+OSFB: (19)

This equation is consistent with the phase histories

plotted previously in Fig. 2. For small radial velocities

v the oscillatory term dominates, whereas for large v,

the linear term dominates.

Although generally speaking the phase will

contain multiple sinusoids as a result of the quadratic

components we previously neglected in (16), too

detailed a model will render the detector fragile under

noise, so just one sinusoid is used as a nonlinear

approximation to the true target phase.

Then, the ONLP approximation to the true target

phase is

xonlp[n] =A

r2bexp

£j(Mn+C1 +C2 cos(C3n+C4))

¤(20)

with M, the slope proportional to Doppler frequency;

C1, a factor dependent upon range; C2, the amplitude

of torso motion; C3, torso frequency; C4, torso phase;

and A, the amplitude as defined in the range equation.


These variables are unknown model parameters

over which the matched filter response will be

optimized.

C. Estimating Model Parameters

The maximum likelihood estimate (MLE) for

any parameter »i in a signal x with mean ¹ and

covariance matrix Cx in complex Gaussian noise may

be expressed as [26].

@ ln(p(x;»))

@»i= 2Re

½(x¡¹(»))HC¡1x

@¹(»)

@»i

¾:

(21)For »i = A,

@¹

@A=@xonlp

@A=1

r2bej6 xonlp : (22)

Thus, we set

@ ln(p(x;A))

@A=

2Re

½xHe¡j 6 xonlp ¡ AN

r2b

¾¾2r2b

= 0

(23)which in turn can be solved to find the MLE:

A=r2bNRefxHe¡j 6 xonlpg

=r2bNRe

(NXm=1

xme¡jxonlp,m

)(24)

Although (25) may also be used to estimate

the phase parameters, the resulting estimate is not

numerically robust. Thus, the MLEs for the phase

parameters are instead found by first explicitly

extracting the phase data. The phase of a complex

signal may be found by taking the ratio of the

imaginary and real parts. However, this operation also

transforms the disturbance distribution from complex

Gaussian to Cauchy, so the problem may be restated

as follows:

H0 : x= xI

H1 : x= tan( 6 xonlp)+ xI(25)

where xI is Cauchy distributed. Thus, the distribution

of x under H1 may be computed to be

p(x; tan( 6 xonlp)) =1=¼NQN¡1

m=0[1+ (xm¡ tan( 6 xonlp,m))2](26)

where xm ´ [x]m, the mth element of x.The MLE for an unknown parameter »i is given by

solving the following equation for each »i:

@

@»ilnp(x; tan( 6 xonlp),°) = 0 (27)

which can be reduced to

N¡1Xm=0

xm¡ tan( 6 xonlp,m)1+ (xm¡ tan( 6 xonlp,m))2

@

@»itan( 6 xonlp,m) = 0:

(28)

Computing the MLE estimates for all five phase

parameters thus requires solving a system of five

nonlinear equations, which is impossible to do in

closed form and is very computationally and memory

intensive even when solved numerically.

Therefore, we break the problem into two stages

by first estimating the linear component, and then

estimating the nonlinear term.

If in (16) we model human motion as being simply

that of a constant-velocity point target, then h= nvT

and

xp[n]¼A

r2bexp

£¡j(4¼fc=c)(r¡ nTvr)¤ (29)

where vr = vcosμ. The two unknown linear phase

parameters r and vr may then be found by solving the

least squares problem A» = b as

» = (ATA)¡1ATb (30)

where

A=4¼fcc

·T 2T ¢ ¢ ¢ NT

¡1 ¡1 ¢ ¢ ¢ ¡1

¸T» = [r vr]

T

b= [tan¡1(x1) tan¡1(x2) ¢ ¢ ¢ tan¡1(xN)]T:An initial estimate of the slope M may then be found

to be

Mi =4¼fccTvr =

4¼fccT»(2): (31)

This initial estimate is then numerically refined so as

to ensure that no residual linear component remains.

Having computed an estimate for the slope M, we

estimate the remaining parameters in the nonlinear

component of the phase from (28), which is now

reduced to a system of four nonlinear equations and

four unknowns, solved using numerical iteration,

searching over a discretized interval of parameter

values. Note that of these four parameters, the

frequency and phase shift coefficients C3 and C4 are

the most crucial for ensuring the best match between

model and data. Thus, to save computation time,

larger step sizes are used for C1 and C2, while finer

step sizes are utilized for C3 and C4.

The final form of our proposed ONLP detector is

RefxHonlpR¡1I xg> °0 (32)

where °0 =qRefxHonlpR¡1I xonlpg ¢Q¡1(PFA) and xonlp is

the ONLP response with MLE parameters; (32) is still

a linear detector because the elements of xonlp merelyform the weights with which we filter our signal.


Fig. 5. CRB and variance of parameter (C1) MLE with ONLP

approximation over 500 Monte Carlo trials, 0.5-s dwell time, and

varying single-pulse SNR.

D. Quality of Parameter Estimates

Obtaining adequate estimates of the unknownmodel parameters is critical to the performance ofthe matched filter detector. The estimation problem,however, is affected by several factors, including thedirection of motion (i.e., target geometry), dwell time,and SNR.As indicated by (19), the phase history of a human

target may be represented as the sum of linear andoscillatory components. Depending on the targetgeometry and walking speed, the shape of the phasehistory may vary. When the target moves roughlyperpendicular to the antenna-target vector, the radialvelocity will be nearly zero and the oscillatorycomponent is clearly apparent. Because the targetmoves along the antenna-target vector, the radialvelocity is almost identical to the true target velocity,so the linear component has more of an effect.The dwell time is also an important factor because

the amount of data collected limits the number ofcycles we can observe. For example, at extremelyshort dwells, the phase history may only have amild nonlinearity, similar to a bowed curve or aquarter-cycle of a sinusoid. Finally, SNR is alsoa critical factor because as the disturbance levelincreases, the true signal curvature is obscured,leading to degradation in our parameter estimates.For low SNRs, longer dwell times will be requiredfor good parameter estimates.The quality of the parameter estimates may be

assessed using the Cramer-Rao bound (CRB). It canbe shown that for a general signal x[n,μ] in complexGaussian noise with variance ¾2, where μ is a vectorof parameters to be estimated, the CRB is givenby [26]

var(μi)¸ [I¡1(μ)]ii (33)

with

I(μ)jij =2

¾2Re

(N¡1Xn=0

@xH

@μi

@x

@μj

)(34)

where I(μ) is the Fisher information matrix.Evaluating (32) for the expression of xonlp given

in (20), the desired CRB may be analytically

expressed as

var(μ)¸ diag½

1

4 ¢SNRB¡1(μ)

¾(35)

where the pulsewise SNR is defined as A2=2r2b¾2 and

B(μ) =

266666664

N S1 C2S2 C2S3 0

S1 S4 C2S5 C2S6 0

C2S2 C2S5 C22S7 C22 S8 0

C2S3 C2S6 C22S8 C22 S9 0

0 0 0 0N

A2

377777775S1 ´

N¡1Xn=0

cos(C3n+C4)

S2 Ń¡1Xn=0

nsin(C3n+C4)

S3 Ń¡1Xn=0

sin(C3n+C4)

S4 Ń¡1Xn=0

cos2(C3n+C4)

S5 Ń¡1Xn=0

nsin(C3n+C4)cos(C3n+C4)

S6 Ń¡1Xn=0

sin(C3n+C4)cos(C3n+C4)

S7 Ń¡1Xn=0

n2 sin2(C3n+C4)

S8 Ń¡1Xn=0

nsin2(C3n+C4)

S9 Ń¡1Xn=0

sin2(C3n+C4):

In Fig. 5, the CRB on C1 is plotted together

with the simulated variance of C1 under two cases:

1) the underlying data are exactly the same as the

model xonlp in (20) used to compute the CRB; and,

2) the underlying data are the synthetic human data

representative of true human motion as given in (4).

The MLE estimator achieves the performance of the

CRB for single-pulse SNRs above 5 dB. Notice that

for intermediate single-pulse SNR values, there is

a slight difference between the simulated variances

when the model data and synthetic human data are

used. For example, when the underlying data exactly


Fig. 6. Variation of linear phase parameter, M , MLE versus

number of pulses transmitted over 100 Monte Carlo trials.

match the model, the estimates follow the CRB for

SNRs above 0 dB. This 5-dB difference illustrates the

impact of modeling error on the estimates. However,

the fact that the variances match for most single-pulse

SNRs also validates the quality of the ONLP model in

terms of approximating the true data.

The dwell time–i.e., the number of pulses

transmitted during the entire data collection duration

times the PRI–also has a significant impact on the

quality of the parameter estimates. As illustrated

in Fig. 6, the longer the dwell time, the better the

estimate. Thus, when the SNR is very low, as is

typical of human targets, data must be collected for a

much longer time to achieve comparable performance

to targets with a higher SNR (or RCS).

In this case, the estimate for the linear phase

parameter M stabilizes after a 1.5-second dwell time

when the single-pulse SNR is 20 dB; but when the

single-pulse SNR is ¡20 dB, the estimate stabilizesafter 2.4 s. In other words, for this particular example,

to achieve the same quality of estimate, an additional

0.9 s of data must be collected.

The CRBs for other model parameters, viz., C3and C4 describing torso frequency and phase, indicate

the same dependence on single-pulse SNR and dwell

as shown for the CRB on C1. The effectiveness of

the estimators is validated subsequently when we

investigate the detection performance of the ONLP.

V. PERFORMANCE

Detector performance is evaluated by applying

the proposed ONLP detector to simulated radar

data as generated using (6). The receiver operating

characteristic (ROC) curves as well as the impact

of SNR, incidence angle, and dwell time on the

probability of detection (PD) is assessed. By incidence

angle, we mean the angle between the initial

antenna-target vector and the target motion vector.

Fig. 7. PD versus PFA for a human target with an incidence angle

of 135±, a dwell time of 0.5 s, and single-pulse SNR=¡30 dB.

Fig. 8. PD versus single-pulse SNR for a human target with an

incidence angle of 135±, a dwell time of 0.5 s, and PFA = 10¡6.

The results presented in Figs. 7—10 are generated for a

radar with the characteristics shown in Table I.

A. Receiver Operating Characteristics

ROC curves for the clairvoyant, FFT, and ONLP

detectors are shown for a human target walking

parallel to the x-axis and with an incidence angle of

135± in Fig. 7 for a single-pulse SNR of ¡30 dB anda 0.5-second dwell. The proposed ONLP detector

exhibits similar performance to the ideal clairvoyant

detector at a PFA of 0.5, whereas the FFT never

approaches ideal performance until the PFA is about

1. The ONLP performance exceeds that of the FFT

for all PFAs.

B. Probability of Detection Versus SNR

The performance improvement of the proposed

technique may also be seen in Fig. 8, which shows

the effect of singe-pulse SNR on the probability


Fig. 9. PD versus incidence angle for a human target with a

dwell time of 0.5 s, single-pulse SNR of ¡10 dB, and PFA = 0:2.

Fig. 10. PD versus dwell time for a human target with an SNR

of ¡10 dB and PFA = 10¡6.

TABLE I

Parameters of Simulated Radar Data

Parameters Value Parameters Value

No. of pulses 500 PRI 0.2 ms

Center frequency 1 GHz Pulse width 40 ¹s

Sampling frequency 20 MHz Transmitted power 1.8 kW

Bandwidth 10 MHz Nominal range 8,760 m

of detection for a fixed dwell of 0.5 s. The ONLP

detector yields about an 11-dB improvement in output

SNR relative to the FFT.

C. Impact of Target Motion on Detection

The ONLP method maintains this performance

gain regardless of the target direction of motion.

Fig. 9 shows the probability of detection variation

over the incidence angle for both the FFT and ONLP

methods. Note that performance of the FFT plummets

as the target’s motion increasingly aligns with the

radar-target vector. Even a small error in estimating

the phase history slope results in errors that accrue

with dwell time and severely degrade performance.

When the radial velocity is small, the phase history

is predominantly sinusoidal and the phase mismatch

errors are limited by the oscillation amplitude.

Because the ONLP method optimizes the matched

filter parameters, it maintains superior performance

over all incidence angles.

D. Probability of Detection Versus Dwell Time

The impact of dwell time on detection

performance is shown in Fig. 10. After a dwell of

about 1.2 s, the proposed ONLP detector achieves the

same performance as the ideal, clairvoyant detector.

However, the FFT-based detector is unable to detect

any targets even after twice the dwell time. This result

is consistent with expectations because the output

SNR versus dwell time plot of Fig. 3 also showed

that for human detection, FFT-based detectors do

not exhibit improved performance with dwell time.

By making a significantly better approximation to

the unknown matched filter, we are able to achieve

substantially better detection performance for a given

dwell time.

VI. CONCLUSION

A novel method for improving the performance

of matched filters for coherently detecting human

targets has been presented. A sinusoidal (ONLP)

approximation to the true target phase was derived

based on the Boulic human walking model, thereby

capturing the characteristic nature of human motion.

Results show a dramatic improvement in output SNR

and detection performance for the proposed method

relative to existing FFT-based techniques.

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Sevgi Z. Gurbuz (S’01–M’10) received the Ph.D. in electrical and computerengineering from the Georgia Institute of Technology in December 2009, the

M.Eng. in electrical engineering and computer science in 2000, and the B.S. in

electrical engineering with minor in mechanical engineering in 1998, both from

the Massachusetts Institute of Technology.

She is a senior research scientist and group leader of the Signal Processing

and Remote Sensing group of the Scientific and Technological Research Council

of Turkey (TUB_ITAK), Space Technologies Research Institute located in

Ankara, Turkey, as well as a faculty member in the Department of Electrical and

Electronics Engineering of the TOBB University of Economics and Technology,

Ankara, Turkey. Her research interests include radar signal processing, distributed

sensor networks, detection and estimation, image processing, and cognitive

remote sensing. She received an Air Force Reserve Officer Training Corps

(AFROTC) scholarship during her B.S. studies, a Charles Stark Draper

Laboratory Fellowship during her M.Eng. studies, and a National Defense

Science and Engineering (NDSEG) Fellowship during her doctoral studies. From

February 2000 to January 2004, she worked as a radar signal processing research

engineer at the Air Force Research Laboratory, Sensors Directorate, Rome, NY.

She is a member of Eta Kappa Nu and Sigma Xi.

William L. Melvin (S’90–M’94–SM’99–F’08) received the Ph.D. in electricalengineering from Lehigh University in 1994, as well as the M.S.E.E. and

B.S.E.E. degrees (with high honors) from this same institution in 1992 and 1989,

respectively.

He is Director of the Sensors and Electromagnetic Applications Laboratory at

the Georgia Tech Research Institute and an adjunct professor in Georgia Tech’s

Electrical and Computer Engineering Department. His research interests include

digital signal processing with application to RF sensors, including adaptive signal

processing for aerospace radar detection of airborne and ground moving targets,

radar applications of detection and estimation theory, and synthetic aperture radar.

He served as a guest editor for several recent special sections appearing in the

IEEE Transactions on Aerospace and Electronic Systems and acted as the Technical

Cochair of the 2001 IEEE Radar Conference and 2004 IEEE Southeastern

Symposium on System Theory. He received a “Best Paper” award at the 1997

IEEE Radar Conference. He has provided tutorials and invited talks at a number

of IEEE conferences and local IEEE section meetings on ground moving target

indication, STAP fundamentals, and space-based radar. He is a regular reviewer

for several IEEE and IET journal publications. Among his distinctions, he is the

recent recipient of the 2006 IEEE AESS Young Engineer of the Year Award, the

2003 U.S. Air Force Research Laboratory Reservist of the Year Award, and the

2002 U.S. Air Force Materiel Command Engineering and Technical Management

Reservist of the Year Award.

He has authored over 135 publications in his areas of research interest and

holds 3 patents on adaptive radar technology.


Douglas B. Williams (S’82–M’89–SM’03) received the B.S.E.E., M.S., and

Ph.D. degrees in electrical and computer engineering from Rice University,

Houston, TX.

In 1989, he joined the faculty of the School of Electrical and Computer

Engineering at the Georgia Institute of Technology, Atlanta, GA, where

he is currently professor and associate chair for undergraduate affairs.

There he is also affiliated with the Center for Signal and Image Processing

(csip.ece.gatech.edu) and the Arbutus Center for Distributed Engineering

Education (www.cdee.gatech.edu). He has served as an Associate Editor of the

IEEE Transactions on Signal Processing and the EURASIP Journal of Applied

Signal Processing, and he has been area editor–special issues for the IEEE Signal

Processing Magazine. He is currently on the IEEE Signal Processing Society’s

Education Technical Committee, and he has been a member of the Society’s

Board of Governors and Signal Processing Theory and Methods Technical

Committee.

Dr. Williams was coeditor of the Digital Signal Processing Handbook,

published by CRC Press and IEEE Press. He is a member of the Tau Beta Pi, Eta

Kappa Nu, and Phi Beta Kappa honor societies.


A Nonlinear-Phase Model-Based Human Detector for Radar

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