application of frequency correlation function to radar target detection

Application of FrequencyCorrelation Function to RadarTarget Detection

ALAA E. EL-ROUBY

ADIB Y. NASHASHIBI, Senior Member, IEEE

FAWWAZ T. ULABY, Fellow, IEEEUniversity of Michigan

Analysis of high-resolution 35 GHz synthetic aperture radar

(SAR) imagery of terrain reveals that when point targets, such as

vehicles, are viewed at angles close to grazing incidence, they are

often difficult to distinguish from tree trunks because the radar

cross section (RCS) intensities of the vehicles are comparable to

the upper end of the RCS exhibited by tree trunks. To resolve

the point target/tree trunk ambiguity problem, a detailed study

was conducted to evaluate the use of new detection features

based on the complex frequency correlation function (FCF). This

paper presents an analytical examination of FCF and its physical

meaning, the results of a numerical simulation study conducted to

evaluate the performance of a detection algorithm that uses FCF,

and the corroboration of theory with experimental observations

conducted at 35 and 95 GHz. The FCF-based detection algorithm

was found to correctly identify tree trunks as such in over 90% of

the cases, while exhibiting a false alarm rate of only 3%.

Manuscript received February 5, 2001; revised March 7 and July26, 2002; released for publication September 4, 2002.

IEEE Log No. T-AES/39/1/808640.

Refereeing of this contribution was handled by L. M. Kaplan.

This work was prepared through collaborative participation inthe Advanced Sensors Consortium sponsored by the U.S. ArmyResearch Laboratory under the Federated Laboratory Program,Cooperative Agreement DAAL01-96-2-001.

Authors’ current addresses: A. E. El-Rouby, Intel Corporation; A.Y. Nashashibi, Radiation Laboratory, University of Michigan, Dept.of Electrical Engineering and Computer Science, 3228 EECS, 1301Beal Ave., Ann Arbor, MI 48109-2122, E-mail: ([email protected]);F. T. Ulaby, Dept. of Electrical Engineering and Computer Science,University of Michigan, 1301 Beal Ave., Ann Arbor, Michigan48109-2122.

0018-9251/03/$17.00 c° 2003 IEEE

I. INTRODUCTION

For a number of remote sensing applications,it is important to be able to detect man-madetargets against the terrain background thatsurrounds them. Examples of these applicationsare: automotive collision-avoidance systems andmilitary target-detection systems. An automatictarget recognition (ATR) algorithm is consideredsuccessful if it can detect targets with a high detectionprobability while simultaneously exhibiting a lowfalse-alarm-rate (FAR).Analysis of high-resolution 35 GHz synthetic

aperture radar (SAR) imagery of terrain showsthat at near grazing incidence, man-made targetsare often difficult to distinguish from tree trunks.Fig. 1 shows a polarimetric 33.6 GHz SAR image

Fig. 1. 33.6 GHz SAR image of wooded area observed atgrazing incidence angle of 3:4±.

at 3:4± from grazing incidence for a wooded areacontaining a number of ground vehicles within it. TheSAR image exhibits a number of bright pixels thathave radar cross section (RCS) values on the orderof magnitude of those associated with man-madetargets, but many of these pixels are associatedwith tree trunks. Similar observations, based onbackscatter measurements recorded by a ground-basedscatterometer, have been noted at 95 GHz. The resultsof the experimental work described in Section IVreveal that the backscattering coefficient associatedwith ground vehicles is comparable to the upper endof the backscattering coefficient associated with treetrunks in Fig. 2. It is clear from these observationsthat the RCS alone is not sufficient for unambiguouslydistinguishing vehicles from tree trunks, and that adifferent feature is needed.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 39, NO. 1 JANUARY 2003 125

Fig. 2. Illustration of overlap between range of backscattering coefficient of ground vehicles and those of tree trunks. (a) Sections oftrunk ground scenes. (b) Sections of ground vehicles.

Recently, researchers have explored the useof the correlation function of the scattered fieldto detect an object buried under a random roughsurface [1—6]. Correlation functions are computedby taking averages over the product of two scatteredfields. The two fields can be measured over differentspatial realizations, frequency points, angularorientations, or polarizations resulting in spatial,frequency, angular, and polarization correlationfunctions, respectively. A key step in calculating thecorrelation function involves estimating averages.Spatial, frequency, and angular averaging havebeen proposed [2, 3]. It has been shown throughnumerical and experimental investigations that theangular correlation function (ACF) with frequencyaveraging is superior for detecting a buried objectunder a random rough surface when compared withan RCS-based technique or to other correlationfunctions and/or averaging techniques [3, 4]. TheACF considers the correlation between the scatteredfields observed at two or more combinations ofdifferent incident and scattered angles. It has beenshown that the values of the ACF are small exceptalong a line called the memory line of incident andscattered directions [3]. Through the proper choiceof at least two different combinations of incident andscattered angles (away from the memory line), theACF of the buried object can be made significantlyhigher than that of the random surface [2] and thepresence of the buried object can be determinedwith high success rate once a threshold level on thevalue of ACF is properly selected. It has been shownthrough numerical simulations that ACF improves thecross-range resolution while the frequency correlationfunction (FCF) improves the range resolution inSAR correlation imaging [5]. The ACF has noadvantage over other correlation functions if a singleincident and scattered combination of angles is used.In fact, it is straightforward to show analyticallythat the ACF value reduces to the first data point(zero frequency shift) of the frequency correlationfunction when a monostatic radar illuminating the

target/clutter scene at a single incidence angle isused. This paper is concerned with the detectionof range-confined targets, such as tree trunks,embedded in clutter, using a wideband monostaticradar operating at grazing incidence. As a result,the FCF is selected over the ACF in pursuit of newdetection features.We examine new detection features for overcoming

the confusion between tree trunks and man-madetargets. The new detection features are based on thecomplex FCF. The FCF was first introduced in [1]and its advantage in improving the range resolutionin SAR correlation imaging was demonstrated in[5]. More recently, Sarabandi and Nashashibi [7]have demonstrated that the geophysical parametersof homogeneous clutter can be inferred from the FCFof wideband backscatter data. In their formulation,the FCF was defined as the ensemble average of theautocorrelation function of the backscattered fieldcollected over many spatially independent samplesof the homogeneous clutter (multilook data). Fortarget detection applications, decisions are oftenmade based on single-look data. To eliminate anyconfusion, we introduce in Section II a slightlydifferent definition of the FCF aimed at single-lookanalysis, we interpret its physical meaning, andthen derive general expressions that are applicablefor the FCF of a point target, distributed clutter,and the combination of a point target containedin distributed clutter. The last case, namely that ofa single major scatterer embedded in distributedclutter, represents the main focus of this study.Next, in Section III, we identify four FCF-baseddetection features that are sensitive to the structureof the target. This analysis is based on numericalsimulations of the radar return from one or morepoint targets embedded in background clutter. Inaddition, we develop and evaluate the performanceof a detection algorithm based on these features.Then, in Section IV we report on a data-measurementcampaign conducted to examine the use of the FCF asa detection feature.

126 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 39, NO. 1 JANUARY 2003

II. DEFINITION OF THE FCF AND PROBLEMFORMULATION

In this section, we define the FCF and derivegeneral expressions for the FCF of the backscatterfrom various types of targets (single scatterer,distributed clutter, and the combinations of individualscatterers in distributed clutter). Also, physicalinterpretations of the behavior of the FCF and itsrelationship to the time-domain signal is introducedand used to gain more physical insight into thedetection problem.Mathematically, the FCF is the function generated

by convolving the frequency response of a targetX(f) with a conjugated, shifted version of the sameresponse, X¤(f+Â),

FCFX(Â) =Z 1

¡1X(f)X¤(f+Â)df: (1)

It can also be expressed in discrete form as

FCFX(m¢f) =Nf¡mXi=1

X(i¢f)X¤((i+m)¢f)¢f (2)

where ¢f is the frequency sampling resolution, m¢fis the frequency shift, and B =Nf¢f is the totalsystem bandwidth. The discrete form representationis particularly useful since for most practical systemsthe bandwidth is finite and the frequency responseof a target is measured usually at discrete frequencypoints. Fig. 3 demonstrates the typical behaviorof the magnitude of the FCF function (normalizedto its peak value). The FCF is a maximum at zerofrequency shift and it decreases in magnitude as thefrequency shift increases. An important measureof the rate of frequency decorrelation is the FCFbandwidth (FCFBW). The FCFBW is defined hereas the bandwidth at which the FCF drops to e¡1

of its peak value. It is used later as one of thedetection features. Another important property ofthe FCF is

FCFX(¡Â) = FCF¤X(Â) (3)

which implies that the FCF is fully characterized byits positive half-domain, including the zero frequencyshift. This property follows from the definition of FCFgiven by (1).In many signal processing applications, the

autocorrelation function is used to provide informationabout the structure of the signal. In case of the radarbackscatter, the FCF can be used to infer informationabout the dependence of the target’s scattering centerson frequency and on their relative positions withinthe target. To illustrate this point further, considerthe simple case of two identical scatterers separatedby a distance x0 as shown in Fig. 4(a). Assume thata plane wave is incident on the two scatterers at an

Fig. 3. Magnitude of typical FCF function (normalized here toits peak value).

Fig. 4. Example demonstrating sensitivity of FCF to effectivedistance between two identical scatterers. (a) Geometry and (b)magnitude of the normalized FCF for different values of x0 andµi. Expression in (4) used to generate plots of f0 and B set to34.5 GHz and 0.5 GHz respectively. Scatterers were separated bydistance x0 along x-axis and illuminated from direction defined by

angle µi.

angle µi, in which case the total backscattered fieldcan be expressed as

Es(f) = (ei°x0f sinµi + e¡i°x0f sinµi)

where f is the radar frequency, ° = 2¼=c, and c is thephase velocity in free space. Upon substituting theexpression for the total scattered field into (1) andintegrating over the frequency range f0 · f · f0 +B(where f0 is the start frequency and B is the systembandwidth), the following expression for the FCF ofthe two scatterers can be easily obtained:

FCF(Â) = 2cos(°DÂ)(B¡Â) + 2°D

sin(°D(B¡Â))

£ cos(°D(2f0 +2Â+B)): (4)

EL-ROUBY ET AL.: APPLICATION OF FREQUENCY CORRELATION FUNCTION TO RADAR TARGET DETECTION 127

Fig. 5. (a) Radar illumination arrangement in which radarparameters R, ®, and µi distort measured FCF. (b) Application of

fixed range gate Rg to limit system effect.

In (4), the parameter D (D = x0 sinµi) correspondsto the effective distance between the two scatterersmeasured along the direction of incidence. As Dincreases in value (by either increasing µi or x0)the rate of decorrelation of the FCF increases asdemonstrated in Fig. 4(b). Of particular interest is thecase when the direction of incidence is perpendicularto the plane where the two scatterers are located (µi =0±). From the radar’s point of view, the two scatterersare located at the same rangebin with zero relativedistance (D = 0). It should be noted that when µi = 0(hence, D = 0), the expression for the FCF simplifiesto FCF(Â) = 4(B¡Â), resulting in the lowest rate offrequency decorrelation (see Fig. 4(b)). The absenceof x0 from the simplified expression indicates that theFCF is sensitive, in general, to scatterers located atdifferent rangebins.

A. The Effect of Radar System Parameters on the FCF

In general, the measured frequency responseof any given target X(f) is governed by two typesof factors: 1) a system factor that accounts for theillumination parameters (incidence angle, radarbeamwidth, range to target, etc.), and 2) a targetfactor that is only a function of the target’s physicalparameters (such as scatterers sizes, number density,orientation, relative positions, etc.). The effect of thesystem parameters on the total radar return becomesmore significant when the observed target is randomlydistributed, as in most terrain clutter. It is clear fromthe radar/target setup depicted in Fig. 5(a) that thetotal radar return depends not only on the targetparameters and incidence angle µi, but also on theradar beamwidth ® and on the range to the target R.As a result, the measured FCF is also a function of ®,µi, and R.Sarabandi and Nashashibi [7], have demonstrated

that the FCF of the total radar return (betweenpoints A and C in Fig. 5(a)) can be expressed as amultiplication of two independent terms, a systemdependent term FCFsys and a target dependent termFCFtrg (FCFtot = FCFsys ¢FCFtrg). The FCFtrg isa function of the relative depth positions of thescatterers and the various scattering mechanismsgoverning the total backscatter response. The FCFsysis a function of the radar parameters ®, µi, and R.

Aside from the general expressions derived in [7],the FCFsys behaves essentially like the FCF of twoidentical scatterers positioned at points A and C andits decorrelation rate is a strong function of D (seeFig. 5(a)), which is given by

D = R cosµi

24 1

cos³µi+

®

2

´ ¡ 1

cos³µi¡

®

2

´35 :

For a given radar beamwidth and a particularincidence angle, an increase in range translates into anincrease in D and into a higher decorrelation rate inFCFsys (effectively, in FCFtot too). In order to retrieveFCFtrg, the FCFsys contribution to the measured FCFtotmust be removed. This can become a cumbersometask, especially when R is an unknown variable. Analternative approach is to assign a fixed range gateRg (with Rg ¿ AC) in which the radar returns fromseveral rangebins dx are collected (Fig. 5(b)). Withthis approach, the FCFsys is governed by Rg (D = Rg)which is independent of R. This is particularly usefulwhen FCF-based detection features are used to detectconfined or “point” targets (i.e., confined within oneor two rangebins like the case of tree trunks) at neargrazing incidence. With a constant Rg, the FCFsyscontribution to FCFtot is constant and can be ignoredwhen comparing the FCFs of different parts of theradar scene.

B. Relation Between FCF and Radar Return in theTime Domain

The time-domain expression for the FCF can becomputed by applying the Fourier transform to theexpression in (1):

fcfX(t) =Z 1

¡1ei2¼Ât

·Z 1

¡1X(f)X¤(f+Â)df

¸dÂ

=Z 1

¡1X(f)

·Z 1

¡1X¤(f+Â)ei2¼ÂtdÂ

¸df:

(5)If we let f+Â= u, Â= u¡f, dÂ= du, and thensubstitute these quantities in the preceding equation,the fcfX(t) can be written as

fcfX(t) =Z 1

¡1X(f)

·Z 1

¡1X¤(u)ei2¼utdu

¸e¡i2¼ftdf

=Z 1

¡1X(f)e¡i2¼ftdf

Z 1

¡1X¤(u)ei2¼utdu

=¯Z 1

¡1X(f)e¡i2¼ftdf

¯2= jx(¡t)j2: (6)

Thus, the FCFX(Â) and jx(¡t)j2 constitute a Fourierpair. Similar to other Fourier pairs, the FCF


Fig. 6. Impact of character of time domain radar return on FCFof point target and distributed target.

rearranges the information embedded in the time-domain response by transforming it to the frequencydomain where certain distinctions can be drawnbetween the different targets.Consider a wideband radar (of bandwidth B and

an effective range resolution dx= c=2B) operatingat near grazing incidence and illuminating the areadepicted in Fig. 5(b). In addition, assume that rangegating has been applied to the radar return froma segment of the illuminated area Rg, (RgÀ dx).This segment is chosen such that radar returns fromseveral rangebins (at dx increments) are included.The frequency response of the range-gated area canbe computed by applying the Fourier transform to thegated time domain response. Let us consider the FCFof three types of targets: 1) a single point target, 2)a statistically homogeneous distributed target, and3) a hybrid target consisting of a single dominantscatterer embedded in a distributed background. Incase of a “point” target occupying a single rangebinwithin Rg (or in general a collection of scattererslocated within the same rangebin), the magnitudeof the frequency response will be constant over thebandwidth B as depicted in Fig. 6. This is certainlytrue at millimeter-wave frequencies (MMW) where thesignal bandwidth often constitutes a small fraction ofthe radar operating frequency. The resulting FCF willhave a maximum FCFBW that depends only on B.Furthermore, changing the width of the range gate Rgwill not affect the behavior of the FCF.Unlike point targets, the frequency response of

a target composed of many scatterers distributedover the same segment Rg will vary over B (seeFig. 6). This variation can be attributed to the factthat the frequency response of the distributed targetis the coherent sum of the frequency responses ofseveral scattering centers distributed over Rg. Thiscoherent sum results in an interference effect inthe total frequency response that is a function ofboth the strengths of the scattering centers and theirrelative positions in range. As a result, the FCF of adistributed target is expected to decorrelate at a higher

Fig. 7. Comparison between mean FCFs of three tree-relatedsections measured at near grazing incidence. They are trunk,

ground-trunk, and ground in front of trunk. Mean FCFs, measuredat 35 GHz (with 750 MHz bandwidth) over 50 independent

realizations of tree trunk-ground scene, were normalized to theirrespective peak values at zero frequency shift ¢F.

rate than the FCF of a point target as depicted inFig. 6. In addition, increasing the width of the rangegate Rg leads to the inclusion of additional scatterersand results in a faster decorrelation rate.For the hybrid target, the total frequency response

is the coherent sum of the responses from both thepoint target and the distributed target. The resultingFCF function not only depends on the propertiesof each individual target but also on their relativestrengths. The effect of the relative strengths of thetargets can be best examined through numericalsimulations which are examined in the followingsection. The mean FCFs of three tree-related sectionsmeasured at near grazing incidence are shown inFig. 7. The three sections are the trunk, the groundin front of the trunk, and the ground-trunk corner,which represent the three cases discussed earlier,namely those of point, distributed, and hybrid targets,respectively. The mean FCFs, measured at 35 GHz(with 750 MHz bandwidth) over 50 independentrealizations of the tree trunk-ground scene, werenormalized to their respective peak values at zerofrequency shift ¢F.

C. The FCF of N Scatterers

Next, we derive a general formula for the FCF ofthe radar backscatter from a radar-illuminated cellthat contains, in general, a set of N scatterers thatare arbitrarily positioned within the cell. Consider anarea illuminated by a plane wave Ei of polarization q(q= v or h). In addition, let the direction of incidencebe along the x axis and at a grazing angle µg as shownin Fig. 8. The p-polarized component (with p= v or h)of the scattered field from a scatterer ¢r away fromthe center of the illuminated area can then beexpressed as

Esp =ei(2¼f=c)(ki+ks)¢r

jrj SpqEiq (7)


Fig. 8. Illustration of geometry for general case of backscatteringby terrain.

where ki and ks are the incident and scattereddirections, respectively (in the backscatter direction,ki = ks and ki = xcosµg ¡ z sinµg), r = r0ki+¢r is thedistance between the radar and the scatterer (with r0being the distance between the radar and the centerof the illuminated cell), Spq is the pq element ofthe scatterer’s scattering matrix, and c is the phasevelocity in free space.Taking the zero-phase reference plane to be at the

center of the illuminated area (r0 meters away fromthe radar), and approximating jrj ¼ r0, the scatteredfield can be expressed as

Esp(f) = Sopqe

2i(2¼f=c)¢r¢ki (8)

where Sopq = Spq=r0, and Eiq was set to unity for

simplicity. For N scatterers randomly positionedwithin the illuminated area, the total backscatteredfield is the coherent sum of the scattered fields fromall N scatterers

Es(f) =NXk=1

Esk(f) =NXk=1

Sok e2i(2¼f=c)(¢rk ¢ki) (9)

where ¢rk = xkx+ zkz specifies the position of the kthscatterer. The resultant FCF(Â) can be expressed as

FCF(Â) =Z 1

¡1

ÃNXk=1

Esk(f)

!¢Ã

NX`=1

Es¤` (f+Â)

!df

=NXk=1

NX`=1

FCFk` (10)

where

FCFk`(Â) =Z 1

¡1Esk(f)E

s¤` (f+Â)df: (11)

As mentioned before, the radar backscatter responseis measured over a finite bandwidth B, and that atMMW frequencies the bandwidth is usually a smallpercentage of the operating frequency. Accordingly,explicit expressions for FCFkk(Â) and FCFk`,k 6=`(Â), for

positive Â can be derived:

FCFkk(Â) =½ jSok j2e¡i(2(2¼Â=c)(xk cosµg¡zk sinµg ))(B¡Â) 0· Â· B0 Â > B

(12)FCFk`,k 6=`(Â) =8>>>><>>>>:Sok (S

ol )¤e¡i(2(2¼Â=c)(xl cosµg¡zl sinµg ))¢

¢ei(2(2¼f0=c)[(xk¡xl)cosµg¡(zk¡zl )sinµg ])¢¢ ei2(2¼=c)(B¡Â)[(xk¡xl)cosµg¡(zk¡zl )sinµg ]¡ 1i2(2¼=c)[(xk ¡ xl)cosµg ¡ (zk ¡ zl)sinµg]

0· Â· B

0 Â > B

(13)where, as mentioned before, f0 is the lower frequencylimit of the radar system bandwidth (i.e., the radarsystem frequency range extends from f0 to f0 +B).The expression for FCFkl(Â) given by (13) is a

complicated nonlinear function of xk, zk, Sok , xl, zl,

and Sol . In the case of clutter, these parameters arerandom variables that change from one spot to thenext. The total number of scatterers N containedin the illuminated cell is also a random variable.The complexity of (10) makes it very difficult tofurther investigate the properties of the single-lookFCF analytically. Nevertheless, a closer look at theexpressions in (10)—(13) reveals: 1) the peak valueof FCFkk(Â) occurs when Â= 0 and its magnitudedecreases linearly with frequency shift increase,and 2) the zero frequency shift is not necessarilythe point at which FCFkl(Â) is a maximum. Thisis because of the lack of phase conjugation atzero frequency shift which, in turn, is a result ofthe dependence of the relative phase angle on thepositions of both scatterers k and l with respect tothe zero reference plane. Hence, it is expected thatthe contribution of the cross-correlated components,FCFkl, to the total FCF to be much smaller thanthe contribution of the self-correlated components,FCFkk. By ignoring the cross-correlated components,a simpler expression for the total FCF can be used togain further understanding of the behavior of the FCF,while the exact expression can be used in numericalsimulations to guarantee accurate results.

III. NUMERICAL SIMULATIONS AND DETECTIONALGORITHM

The sensitivity of the FCF to the structure of thetarget (i.e., point target vis-a-vis uniformly distributedclutter) was demonstrated in the previous sectionusing physics-based arguments. This sensitivitypoints towards the potential application of the FCFin target detection problems. The developmentof an FCF-based point target detection algorithminvolves the following steps: 1) identifying features


Fig. 9. Magnitude of normalized FCF(Â) as function of frequency shift Â for: (a) clutter only, (b) one point target, and (c) two pointtargets present in an illuminated area. In computing (b) and (c) no background clutter was assumed.

in the FCF that are sensitive to the differences instructure between the clutter and the point targets, 2)developing decision rules based on these features, and3) combining these rules into an optimal detectionalgorithm. For the simple case of detecting anddiscriminating between statistically homogeneousclutter (where no point targets are present) and oneor two point targets (where no background clutteris present), the expressions derived in Section IIcan be used to develop analytically an FCF-baseddetection algorithm. In fact, the clear differencesbetween the FCFs for the three cases shown inFig. 9 renders the analysis rather simple. However, amathematical development of the detection algorithmbecomes rather difficult when a combination of oneor more point targets, embedded in a homogeneousbackground clutter, is considered. To alleviate thisdifficulty and demonstrate the applicability of the FCFfor target detection applications, numerical simulationsare used in this section to generate the FCF of oneor more point targets embedded in a uniformlydistributed homogeneous clutter. The simulated data is

used to identify detection features and decision rules.It is also used for testing the detection algorithm.

A. Numerical Simulation Procedure

Consider the setup shown in Fig. 8 in which aradar system of bandwidth B illuminates an area ofthe terrain Lx£Ly at a grazing angle µg. Assume thatone or more point targets are randomly located in theilluminated area and are embedded in a uniformlydistributed and statistically homogeneous backgroundclutter. In addition, let the illuminated area bedivided into M small pixels of dimensions ¢x£¢y.Moreover, let the point targets be positioned randomlyat the centers of these pixels. The total scattered field(and its FCF) of the simulated setup can be computedusing the expressions in (9)—(13). Note that the totalnumber of scatterers in this case is M plus the numberof embedded point targets.Due to the random nature of clutter, the FCF

must be computed for many independent realizationsof the radar scene. In each realization, both the


Fig. 10. Normalized magnitude of the FCF(Â) of point target embedded in clutter when target-to-clutter ratio is: (a) 0.35 and (b) 19.

locations of the point target and the clutter responseare randomly assigned. In recent studies [8, 9], itwas demonstrated that at near grazing incidence theMMW radar backscatter response of statisticallyhomogeneous clutter complies with the Rayleighstatistical model. In other words, the phase of thescattered field is uniformly distributed and its powerobeys an exponential probability density function(completely defined by its mean value). To incorporatethis fact into the numerical simulations, a randomnumber generator is used in each realization of theradar scene to generate the magnitude and phaseof the scattered field of each clutter pixel subjectto the constraints that both the resulting phase isuniformly distributed and the scattered field poweris exponentially distributed with a given meanpower Ps.While the radar return from individual clutter-filled

pixels can be small compared with the return from apoint target, the effect of total clutter on the FCF canstill be substantial. This is due to the fact that the totalbackscattered field is the coherent sum of the fieldsscattered from all M clutter-filled pixels as well as thefields scattered by the point targets. As demonstratedlater in this section, the target-to-clutter ratio (TCR)plays an important role in the detectability of thetarget using FCF-based algorithms. In this paper, theTCR is defined as

TCR=jSop:t:j2M:Ps

where jSop:t:j is the magnitude of the scattered field of agiven point target and Ps is the clutter’s mean scatteredpower per pixel.An illuminated area of dimensions Lx = 11:5

m and Ly = 2 m was chosen for the numericalsimulations reported here. The radar was assumed tooperate over a 2 GHz bandwidth (hence, 7.5 cm inrange resolution) and at a grazing angle µg = 10

±. In

effect, the illuminated area consisted of 153 rangebins.Only few of these rangebins contained both pointtargets and clutter in them while the rest had clutteronly. The FCF of the illuminated area was computedfor each realization, using the expressions given in(9)—(13), at 801 equally spaced frequency pointsspanning the entire bandwidth.The simulated FCFs of a point target embedded

in homogeneous clutter are shown in Fig. 10 for twodifferent values of TCR. When TCR is small (= 0:35),the total backscattered field is dominated by clutterand the corresponding FCF, shown in Fig. 10(a),closely resembles that of pure clutter (see Fig. 9(a)).On the other hand, when TCR is large (= 19) the totalbackscattered field is dominated by the point targetand the corresponding FCF, shown in Fig. 10(b),closely resembles that of a single point target with nobackground clutter (shown in Fig. 9(b)). It should benoted here that the numerical simulations, performedover different combinations of point targets and TCRs,have shown that the tail of the FCF is more sensitiveto the presence of the point targets in the radar scenethan the mainlobe of the FCF. This can be attributedto the fact that the tail of the jFCFclutterj is small inmagnitude and its mean slope is small too. Hence, theeffects of the point targets on the FCF tail will be felt,even for a small TCR.

B. FCF-Based Detection Features

Based on an extensive set of simulated data fordifferent combinations of point targets and clutterover a wide range of TCR values, four features wereidentified in the FCF that are sensitive to the presenceof point targets. Hence, they are good candidatesfor target detection applications. These features arethe FCF bandwidth (FCFBW), the average slopeof the tail (AS), the squared error (SE) at the tail,and the ratio of AS to SE. As mentioned earlier,


Fig. 11. Spread in the values of four features computed using FCF of 1000 statistically independent realizations of clutter (no pointtargets).

Fig. 12. Effect of TCR on the four features when one point target was embedded in clutter. At any given TCR, 50 independentrealizations of the radar scene were used.

the FCFBW is defined as the frequency shift ¯ atwhich the magnitude of the FCF drops to e¡1 of itsvalue at zero frequency shift (FCFBW= ¯ and forFCF(¯) = (FCF(0)=e)). The AS is defined here asthe average slope of the middle section of the jFCFj(beyond the mainlobe) while the SE is defined asthe average of the difference-squared between thejFCFj and the line of AS at the middle section. Term“mainlobe” refers here to the portion of the FCFextending between Â= 0 and Â= ¯ while the “tail”

refers to the portion of the FCF extending betweenÂ= ¯ and Â= B (B is the bandwidth). We have optedto exclude the two ends of the “tail” while computingthe AS. For example, the section used to computeAS in Fig. 10a extends between Â= 0:18 GHz andÂ= 1:8 GHz.Figs. 11 and 12 demonstrate the sensitivity of

the four features to the presence of point targets ina given radar scene. In Fig. 11, 1000 realizationsof the clutter (no point targets) were used, while in


Fig. 12, 50 realizations of the case where one pointtarget is embedded in clutter were used at any givenTCR. Additional plots for two, three, four, and fivepoint targets embedded in clutter were generated forthe four detection features that are reported here.These plots were found to be consistent with thetrends observed in Fig. 12. Based on all the simulateddata, it was observed that the presence of pointtargets in a radar scene (with TCR> 0:5) results ina FCFBW that is greater than that of clutter alone.In addition, as long as the TCR is greater or equal to0.12, the value of AS when point targets are presentis smaller than the AS when only clutter is present(AS =¡1£ 10¡4). While no clear offset was observedbetween the values of the SE of the one-point targetcase and the no-target case (mean value at 1:6£10¡3),a significant offset was observed between the SE ofthe one point target case and the two, three, four, andfive point targets cases. Hence, the SE can be usedto determine the presence of two and higher pointtargets. Furthermore, the fourth feature, namely, theratio, was observed to be most of the time smallerthan ¡0:12 in case of one-point target and greaterthan ¡0:12 in cases where more than one-point targetwere embedded in the background clutter. Hence,the ratio can be used along with the SE to determinewhether one or more point targets are present in anilluminated area.

C. Detection Algorithm

In the previous section, four FCF-based detectionfeatures were introduced and their sensitivities to thepresence of point targets were demonstrated. Usingthese four detection features, a detection algorithm,whose flow chart is shown in Fig. 13, was developed.The algorithm was designed to determine the presenceof 0, 1, and 2—5 point targets in the illuminated areabased on single-look data. In case of more than 5point targets, the algorithm determines that no pointtargets are present.The algorithm uses a set of threshold values in

making the decisions listed in the flow chart (seeFig. 13). These threshold values, which were derivedfrom the numerically simulated data, are: BW0 =0:015 GHz, Slope0 = 0:95£10¡4, Slope1 = 6:0£10¡4, Err0 = 0:0015, Err1 = 0:01, and Ratio0 =¡0:12.The Slope0 and Slope1 are threshold values for theAS of the FCF tail, while Err0 and Err1 are thresholdvalues for the SE at the FCF tail. Both Slope1 andErr1 were introduced specifically to determinewhether more than 5 point targets were present inan illuminated area. It should be noted that thesethreshold values were derived from the numericalsimulations performed for the radar arrangementspecified earlier (i.e., for a given µg, BW, Lx, Ly, etc.).Another set of threshold values will be needed if adifferent arrangement is considered.

Fig. 13. Flow chart of the FCF-based detection algorithm.

The performance of the detection algorithm wasevaluated by testing it against the data generated vianumerical simulations. All the independent realizationsof the illuminated area, that were generated for agiven configuration (i.e., for a given number of pointtargets and TCR value) were passed through thealgorithm and its decisions were compiled into the0, 1, or 2—5 groups. Then a “percentage of indicating”number was derived for each decision and plotted asshown in Fig. 14. The following were

1) When no-point targets were embedded in theclutter, the detection algorithm had 0% FAR. Inother words, the detection algorithm made a correctdecision in all the cases where no-point targets werepresent.2) When one-point target was embedded in clutter

(see Fig. 14(a)), the detection algorithm was able toarrive at the correct decision 100% of the time as longas the TCR¸ 0:4. The ability to properly determinethe presence of one-point target decreases with thedecrease in the TCR.3) In cases where two, three, four, and five

point targets were embedded in clutter (the firstand third cases are shown in Figs. 14(b), and 14(c),respectively), the detection algorithm was able toarrive at the correct decision for large values of theTCR.4) When more than five point targets were

embedded in clutter (e.g. Fig. 14(d)), the outcomeof the algorithm indicated that no-point targets werepresent (as desired) at least 95% of the time.


Fig. 14. Performance of detection algorithm for cases where 1, 2, 4, and 10 point targets were embedded in clutter. Data used werecomputed assuming a 2 GHz system bandwidth. (a) 1 point target. (b) 2 point targets. (c) 4 point targets. (d) 10 point targets.

Fig. 15. Performance of detection algorithm for cases where 1, 2, 4, and 10 point targets were embedded in clutter. Data used werecomputed assuming a 0.5 GHz system bandwidth. (a) 1 point target. (b) 2 point targets. (c) 4 point targets. (d) 10 point targets.


D. System Bandwidth Effect

A reduction in the radar system bandwidth couldhave a significant impact on the performance ofan FCF-based detection algorithm. A closer lookat (12) reveals that the total FCF of an illuminatedarea is influenced by a multiplicative factor (B¡Â)common to all of its self-cell components. This factoressentially dampens the magnitude of the FCF asfrequency shift Â is increased. A reduction in theradar system bandwidth leads to an increase in thedampening effect. Hence, both the FCFBW andthe AS of the FCF tail will be different for the newbandwidth. It is expected that a smaller bandwidthwill result in a smaller AS and a smaller FCFBW.Unless the threshold values used in the detectionalgorithm were tweaked for the reduced bandwidthcase, the algorithm will suffer from a higher misseddetection rate especially at lower TCR values.Numerical simulations were performed over 0.5

GHz bandwidth for the same setup described earlier.The resulting data was used to test the detectionalgorithm (see the flow chart in Fig. 13). The resultsof this test are summarized in Fig. 15, where ahigher false detection rate was observed at low TCRvalues.

IV. EXPERIMENTAL VERIFICATION

The ability of the FCF-based target detectionalgorithm in detecting the presence of a point target,such as a tree trunk, was verified experimentallythrough a series of outdoor measurements that wereconducted on a set of trees using two scatterometersystems operating at 35 and 95 GHz. The two fullypolarimetric radar systems were used to measure thebackscattered return at 301 equally spaced discretefrequency points spanning a bandwidth of 0.375 GHz.The effective antenna beamwidth of the 35 and 95GHz radars were 2± and 1:4±, respectively. A detaileddescription of the two systems can be found in [10,pp. 142—143] and [11].Using a dual-axis Gimbal, the two radars

performed vertical scans on 50 randomly selectedground-trunk combinations, as depicted in Fig. 16.Three different sections were identified in eachscan: the trunk, the corner between the trunk andthe ground, and the ground in front of the trunk. Inthese vertical scans, backscatter measurements wereconducted at 14 different scan angles (1± increments).At the beginning of each scan, the two radars werepositioned at the same height and distance awayfrom the trunk to be scanned, and they were pointedat the corner between the trunk and the ground.Since the corner was chosen as a reference point, itscorresponding scan angle was set to 0± (while in factthe radars were tilted to 4± grazing angle). Scan anglesbetween 1± and 8± correspond to the trunk section,

Fig. 16. Experimental setup where MMW polarimetric radarswere used to scan vertically different sections of tree-ground

scene.

Fig. 17. Dynamic range of single-look FCFBW of differentsections of tree measured at 95 GHz. Marks are used for

single-look FCFBW while lines are used to denote the meanFCFBW at each scan angle. FCFBW for the VV- and

HH-polarized returns are plotted.

while scan angles between ¡1± and ¡5± correspond tothe ground section.Trunks of different bark roughnesses were chosen

and their circumferences and tilt with respect to theground were measured. The measured tilt angle variedbetween ¡9± to +2:6± while the trunk circumferencesvaried between 1.2 and 3.1 m.Another data set of the tree canopies, which was

acquired during a separate measurement campaign(using the same radars and from the same range), wasused to complement the vertical scans. In effect, thebackscatter responses from all sections of the tree(including the canopy) and from the ground in frontof the tree were compiled.Both 35 and 95 GHz data were processed and

analyzed. It was observed that the dynamic range ofthe copolarized reflection coefficient, ¾o(vv) and ¾

o(hh),

of the tree trunk, at any given frequency point withinthe bandwidth, cannot be used to separate the trunkfrom those of the trunk-ground corner, those of theground, nor those of a man-made target, as shown inFig. 2. On the other hand, the single-look FCFBWvalues demonstrated a substantial separation betweenthe majority of the trunk measurements and that of allthe other nearby targets, as shown in Fig. 17. In this


Fig. 18. Performance of proposed detection algorithm whenapplied to 35 and 95 GHz data.

figure, the marks refer to single-look FCFBW valueswhile the lines refer to the mean FCFBW at each scanangle.The proposed FCF-based detection algorithm was

applied to all measurements and the percentage ofmeasurements which the algorithm indicated as trunks(whether false or true) at each angle were calculated.The resultant percentages of indicating a trunk areplotted in Fig. 18 as a function of the scan angle.The percent value represents the FAR when the radarwas pointing at the ground, or at the tree canopy.In addition it represents the detectability of trunkswhen the radar was actually pointed at the trunk. Thealgorithm was able to detect successfully the presenceof a tree trunk in over 90% of the time at 35 GHzand slightly worse at 95 GHz. In addition, its FARwas less than 3% at 35 GHz and is a bit higher at95 GHz.The backscattered responses of two different

ground vehicles positioned at different locationsin a bare soil field (i.e., man-made target withterrain in the background) were extracted from theSWOE data set collected by the Army ResearchLaboratory (ARL) at 95 GHz [12]. This data wasused to evaluate the FCF of these vehicles andthe corresponding single-look FCFBW, shown inFig. 19. The presence of multiple scattering centerson the vehicles and their spread in range are majorcontributors to the low FCFBW values that wereobserved. The same detection algorithm tested earlieron tree data was applied to the ground vehicle data.The algorithm had a zero false alarm rate, i.e. thealgorithm determined correctly that both vehicleswere not tree trunks in all tested cases. It should bepointed though that data used for developing thedetection algorithm was acquired using a differentradar system (although at 95 GHz) from the radarsystem used to measure the return from the groundvehicles.

Fig. 19. FCF bandwidth of ground vehicles measured at 95 GHz.

V. CONCLUSIONS

In this paper the problem of possible confusion,from the power-based detection algorithmsperspective, between man-made targets and tree trunkswas examined and a new approach was introduced toresolve this possible confusion. The new approachis based on using FCF-based detection features todiscriminate between point targets and distributedones. The application of the FCF in detection wasexamined both through numerical simulations andexperimentally. An analytical study of the FCF and itsrelationship to the time-domain return was presentedand the effects of the parameters the radar system andgeometry on the behavior of the FCF were discussed.Numerical simulations of point targets embedded inbackground clutter were used to identify four differentFCF-based detection features which were used todevelop an FCF-based detection algorithm. Whenapplied to data acquired of different sections of treesat 35 and 95 GHz, the algorithm was able to detectthe presence of trunks 90% of the time at 35 GHz,with a slightly worse performance at 95 GHz. Inaddition, its FAR is less than 3% at 35 GHz and isa bit higher at 95 GHz. Also, the algorithm’s abilityto discriminate tree trunks from other nearby targets,including man-made targets, was demonstrated.

REFERENCES

[1] Young, C. Y., Ishimaru, A., and Andrews, L. C. (1996)Two-frequency mutual coherence function of a Gaussianbeam pulse in weak optical turbulence: An analyticsolution.Applied Optics, 35 (1996), 6522—6526.

[2] Tsang, L., Zhang, G., and Pak, K. (1996)Detection of a buried object under a single random roughsurface with angular correlation function in EM wavescattering.Microwave and Optical Technology Letters, 11, 6 (1996),300—304.


[3] Zhang, G., Tsang, L., and Kuga, Y. (1997)Studies of the angular correlation function of scatteringby random rough surfaces with and without a buriedobject.IEEE Transactions on Geoscience and Remote Sensing, 35,2 (Mar. 1997), 444—453.

[4] Chan, T., Kuga, Y., and Ishimaru, A. (1997)Subsurface detection of a buried object using angularcorrelation function measurement.Waves in Random Media, 7 (1997), 457—465.

[5] Zhang, G., and Tsang, L. (1998)Application of angular correlation function of clutterscattering and correlation imaging in target detection.IEEE Transactions on Geoscience and Remote Sensing, 36,5 (Sept. 1998), 1485—1493.

[6] Zhang, G., Tsang, L., and Pak, K. (1998)ACF and RCS of electromagnetic wave scattered by aburied object under a two-dimensional rough surface.Journal of the Optical Society of America (A), 15, 12(1998), 2995—3002.

[7] Sarabandi, K., and Nashashibi, A. (1999)Analysis and applications of backscattering frequencycorrelation function.IEEE Transactions on Geoscience and Remote Sensing, 37(July 1999), 1895—1906.

[8] Ulaby, F. T., Nashashibi, A. Y., El-Rouby, A. E., Li, E. S.,De Roo, R., Sarabandi, K., Wellman, R. J., and Wallace,H. B. (1998)95 GHz scattering by terrain at near grazing incidence.IEEE Transactions on Antenna and Propagation, (Jan.1998), 3—13.

Alaa El-Rouby received the B.Sc. and M.Sc. degrees in electrical engineeringfrom Cairo University, Giza, Egypt, in 1993 and 1996, respectively. He receivedhis Ph.D. degree in electrical engineering from the University of Michigan, AnnArbor, in 2000.He is currently a test development engineer at Intel Corporation. His research

interest includes millimeter-wave polarimetric scattering and remote sensing. Herecently started to focus his research on signal integrity and power delivery inhigh-speed digital circuit boards and packages.

[9] De Roo, R. D., Ulaby, F. T., El-Rouby, A. E., andNashashibi, A. (1999)MMW radar scattering statistics of terrain at near grazingincidence.IEEE Transactions on Aerospace and Electronic Systems,35, 3 (July 1999).

[10] El-Rouby, A. E. (2000)MMW Scattering by tree trunks and surroundingenvironment: Modeling & analysis.Ph.D. dissertation, University of Michigan, Ann Arbor,2000.

[11] Nashashibi, A., Sarabandi, K., Frantzis, P., De Roo, R. D.,and Ulaby, F. T. (1999)A novel design of an ultra-fast wideband polarimetricradar.Presented at the 1999 IEEE AP-S InternationalSymposium and USNC/URSI National Radio ScienceMeeting, Orlando, FL, July 11—16, 1999.

[12] Welsh, J. P. (1994)Smart weapons operability enhancement (SWOE) jointtest and evaluation (JT&E) program final report.SWOE Report 94-10, U.S. Department of Defense, Aug.1994.


Adib Y. Nashashibi (S’82–M’95–SM’01) received the B.Sc. and M.Sc. degreesin electrical engineering from Kuwait University, Kuwait, in 1985 and 1988,respectively. He received the Ph.D. degree in electrical engineering from theUniversity of Michigan, Ann Arbor, in 1995.He is presently an Assistant Research Scientist at the Radiation Laboratory

at the University of Michigan. His research interests include microwave remotesensing, polarimetric millimeter-wave radars, calibration and measurementtechniques, electromagnetic wave propagation and scattering in random media.

Fawwaz T. Ulaby (M’68–SM’74–F’80) received the B.S. degree in physicsfrom the American University of Beirut, Lebanon, in 1964 and the M.S.E.E. andPh.D. degrees in electrical engineering from the University of Texas, Austin, in1966 and 1968, respectively.He is the Vice President for Research and Williams Distinguished Professor

of Electrical Engineering and Computer Science at the University of Michigan,Ann Arbor. His current interests include microwave and millimeter-wave remotesensing, radar systems, and radio wave propagation. .Dr. Ulaby has authored ten books and published more than 500 papers and

reports on these subjects. He is the recipient of numerous awards, including theEta Kappa Nu Association C. Holmes MacDonald Award as “An OutstandingElectrical Engineering Professor in the United States of America for 1975,”the IEEE Geoscience and Remote Sensing Distinguished Achievement Awardin 1983, the IEEE Centennial Medal in 1984, The American Society ofPhotogrammetry’s Presidential Citation for Meritorious Service in 1984, theNASA Group Achievement Award in 1990, and the 2000 IEEE ElectromagneticsAward. He served as President of the IEEE Geoscience and Remote SensingSociety from 1980 to 1982, Executive Editor of IEEE Transactions on Geoscienceand Remote Sensing from 1983 to 1985, and as General Chairman of severalinternational symposia. In 1995, he was elected to membership in the NationalAcademy of Engineering, and currently serves as Editor-In-Chief of the IEEEProceedings.


application of frequency correlation function to radar target detection

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