IRS 2012, 19th International Radar Symposium, May 23-25, Warsaw, Poland

Short Time Pulse Bistatic Forward Scattering Inverse Synthetic Aperture Radar Imaging

Andon Lazarov D epartment of Informatics and Technical Sciences

Bourgas Free University Bourgas, Bulgaria: City, Country

lazarov@bfu.bg

Abstract - In this work Bistatic Inverse Forward Scattering Inverse Synthetic Aperture Radar (BFISAR) concept is addressed. Geometric description of BFISAR topology with stationary transmitter and receiver, and moving target is given. Kinematic vector equations are derived. BFISAR signal

mathematical model with short pulse modulation is described. It is proven that BFISAR signal formation and image reconstruction can be interpreted as direct and inverse projective operation respectively. Through Taylor expansion of the projective operator basic image reconstruction procedures, including phase correction, Fourier transform range compression

and Fourier transform azimuth compression are defined. Numerical and natural experiments to verity mathematical models derived are carried out.

Keywords-component; formatting; style; styling; insert (key

words)

I. INTRODUCTION)

In recent years BFISAR principle enjoys considerable attention [1-3]. One of the most prospective BSAR applications is in early warning and prevention of maritime border intrusion. The concept of a forward scattering micro-sensors radar network for situational awareness and netted forward scattering micro radars for ground targets detection and identification by quasi-optimal signal processing is considered in [4]. Forward scattering radar power budget analysis for ground targets and forward scatter RCS estimation for ground targets are presented in [5]. The solution of problems of forward scattering radars for vehicles classification and automatic ground target classification using principle component analysis and neural network is suggested in [6-7].

The problem posed in this work is to analytically describe the discrete geometry of BFISAR topology and based on it to derive a mathematical model of short time pulse BFISAR signal and implement an image reconstruction procedure.

II. BFISAR GEOMETRY

A. Selecting a Template (Heading 2) The geometry of BFISAR topology is presented in Fig. I.

Consider UWB stationary transmitter and receiver both located on the land or sea surface and as a mariner target a ship all

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Christo Kabakchiev, Todor Kostadinov Institute of Information and Communication Technologies

Bulgarian Academy of Sciences Sofia, Bulgaria

kostadinov. todor@yahoo.com

situated in a Cartesian coordinate system Oxyz. The target presented as an assembly of point scatterers is depicted in its own coordinate system oxyz. Vectors R sand R r are the position vectors of the transmitter and receiver respectively in a coordinate system Oxyz.

R'

I'

Figure 1. Example of a figure caption. (figure caption)

The vector Rijk is the position vector of ijkth point

scatterer in the coordinate system OXYZ and Roo (p) is the current position vector of the mass center of the target at the time instant p. Based on the geometry in Fig. 1 the following kinematical vector equations hold. Range distance vector from the transmitter to the mass center of the target

s s s (N

) R (p)=R - Roo(p)=R - Roo(O)- V "2-P Tp,(I)

where V is the target's velocity vector, N is the full number of emitted pulses, T p is the pulse repetition period.

Range distance vector from the mass center of the target to the receiver

r r r (N

) R (p)=R - Roo(p)=R - Roo(O) - V "2-P Tp (2)

Range distance vector from the transmitter to the ijkth point scatterer of the target

978-1-4577-1837-3/12/$26.00 ©2012 IEEE

R S ijdp) = RS(p)+ARUk =Rs - Roo(O) ­

- V( � -p )Tp +ARijk (3)

Range distance vector from the ijkth point scatterer of the target to the receiver

Rr Uk (p) = Rr (p) - ARUk = Rr - Roo (0) ­

-v(� - Pjrp - ARijk (4)

The round trip range distance transmitter- ijkth point scatterer of target-receiver is defined by

(5)

Expressions (1-5) are applied in modeling of BFISAR signal.

III. PULSE WAVEFORM AND BFISAR SIGNAL FORMATION

Consider 3-D ISAR scenario (Fig. 1) and a generic point g from the target illuminated by sequence of short monochromatic pulses, each of which is described by [8]

s(t) = A. rec{f j exp(j.rot) ,

t 1,0::0;-<1, rect-= T I t

T 0, otherwise.

(6)

where A is the amplitude of the emitted signal, ro = 2n":" is A

the angular frequency; c = 3.108 m/s is the speed of the light in vacuum; A is the wavelength of the signal; T is the timewith of the emitted pulse.

The signal reflected by the generic point scatter can be written as

t - t (p) Sg(p,t) =ag rect

g exp{jro[t - t g(p) ] }, (7)

T

t -tg (p) jl 0::0; t -t g (p) < 1 rect = ' T ' T

h . 0, ot erwlse. 2Rg(p)

where tg(p) = --- is the time delay of the signal, g c

stands for the discrete vector coordinate that locates the generic point scatterer in the target area G, ag stands for the

magnitude of the 3-D discrete image function, T = t mod Tp is the slow time, p denotes the number of the

emitted pulse , Tp is the pulse repetition period,

t = T - pTp is the fast time, presented as t = k.T, where k

is the number of range bin, where the ISAR signal is placed. The demodulated ISAR signal from the target area is

k.T -t (p) s(p,k) = I ag rect

g .exp{-jrotg(p) ] }. (8)

gEG T

The expression (8) is a weighted complex series of finite complex exponential base functions. It can be regarded as an asymmetric complex transform of the 3-D image function ag, g E G , defined for a whole discrete target area G into 2-D signal plane s(p, k) .

IV. IMAGE RECONSTRUCTION FROM A SHORT PULSE

BFISAR SIGNAL

Eq. (8) can be rewritten as

Formally for each kth range cell the image function can be extracted by the inverse transform

N { 4n ] Gg = I s(p,k) ex j-Rg(p) p=l A

(10)

where p is the number of emitted pulse, N is the full number of emitted pulses during CPI. Because s(p, k) is a 2-D signal, only a 2-D image function Gg can be extracted. Eq. (10) is a symmetric complex inverse spatial transform or inverse projective operation of the 2-D signal plane s(p, k) into 2-D image function Gg, and can be

regarded as a spatial correlation between s(p, k) and

exp[j � Rg(P)] . Moreover, Eq. 10 can be interpreted as a

total compensation of phases, induced by radial displacement Rg (p) of the target. Taylor expansion of the distance to the generic point, Rg(p) at the moment of imaging is

(11 )

where rg , Vg , ag and hg is the distance, radial velocity, acceleration and jerk of the generic point, respectively at the moment of imaging. Due to range uncertainty of generic points placed in the kth range resolution cell, r g can be assumed constant, and (10) can be written as

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Eq. (12) stands for a procedure of total motion compensation of every generic point from kth range resolution cell. The range distance rg does not influence on the image reconstruction and can be removed from the equation (12) , i.e.

2 For each kth range cell the term -vg stands for the Doppler A

2 2 frequency whereas terms as iag' ihg .... , denote the higher

order derivations of the time dependent Doppler frequency, defined at the moment of imaging. If the D oppler frequency of generic points in the kth range cell is equal or tends to constant during CPI the equation (13) reduces to the following equation of radial motion compensation

clg =

P�

1 S(p ,k ) ex{J2rc{ivg }PTp)] . (14)

D enote � v g = p.I1F D , where I1F D = _I - is the D oppler A N�

frequency step; p is the unknown D oppler index at the moment of imaging; then the complex image function clg = clg(p,k) in discrete space coordinates can be written as

clg(p,k) = I s (p ,k)exP(J2rc pp). p�1 N

(15)

The equation (15) stands for an 1FT of s (p,k) for each kth range resolution cell and can be considered as phase and/or motion compensation of first order.

2 2rc 2rc D enote a1 = iVg, a2 = Tag, a3 = r;:hg, then (13) can

be rewritten as

D enote CIJ(P)=a2(pTp)2+ ... +am(pTp)m

as a phase correction and/or motion compensation function of higher order, then

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clg (p, k) = I [s(p, k) exp(J CIJ(p) H exp(J2rc pp) (17) p�1 N

where clg(p,k) denotes the complex azimuth image of the

target, p denotes the index of the azimuth space coordinate of the generic point scatterer from the target at the moment of imaging. The polynomial coefficients am , m = 2, 3, are calculated iteratively via applying image quality criterion. Eq. (17) can be interpreted as a BFISAR image reconstruction procedure implemented through inverse Fourier transform (1FT) of a phase corrected BFISAR signal into a complex azimuth image clg (p, k) for each kth range cell. In this sense the ISAR signal s(p, k) can be referred to as a spatial frequency spectrum whereas clg(p,k) can be referred to as a spatial image function defined at the moment of imaging. Based on Eq. (17) two steps of image reconstruction algorithm can be outlined.

Step 1 Compensate the phases, induced by higher order radial movement, by multiplication of s (p, k) with the exponential term exp[jCIJ(p)] , i.e.

s(p, k) = s(p, k) exp[JCIJ(p) ] (18)

Step 2 Compensate the phases induced by first order radial displacement of generic points in the kth range cell by applying 1FT (extract complex image) , i.e.

clg(p,k) = I S(p ,k ) .exP(J2rc pp) p�1 N

(19)

Complex image extraction can be extracted by inverse fast Fourier transform (lFFT). The algorithm can be implemented if the phase correction function CIJ(p) is preliminary known. Otherwise only 1FT can be applied. Then non compensated radial acceleration and jerk of the target still remain and the image becomes blurred (unfocused) . In order to obtain a focused image a motion compensation of second, third and/or higher order has to be applied, that means coefficients of higher order terms in CIJ(p) have to be determined. The definition and application of these terms in image reconstruction is named an autofocus procedure which can be accomplished by optimization step search algorithm (SSA).

V. EXPERIMENTAL RESULTS

The experiment has been conducted by the team of Birmingham University on 23 of March 2010 in which a medium yacht is crossing the transmitter-receiver line. The meteorological conditions included rain and a south � east wind with a speed of 4.8 m/s.

A. Parameters a/the measurement equipment Radar's carrier frequency: 7.5 GHz, pulse's bandwidth: 0.1

GHz, pulse width: 10-8 s, pulse repetition period: 1 MHz,

sampling frequency: 400 Hz, distance between the transmitter and the receiver: 100 m.

The amplitude signal on the exit of the D oppler radar channel is presented in Fig. 2.

Doppler Channel

t :� .5DL-----�1D�--���----���----�4D�----ffi�----�ffi·

Time/s

Figure 2. Experimental measurement data - Output signal in Doppler channel

One dimensional image reconstruction based on the measurement pulses with bandwidth equal to 0.1 GHz using standard Matlab fast Fourier transform is implemented. Results are demonstrated in Fig. 3. In Fig. 3, a, an image with inverse spectrum after FFT in presented. In Fig 3, b a final image with a frequency shift is depicted.

Normalized Frequency (x'J fad/sample)

(a) Weith Power Spectral Density Estimate

� r----r----�--�r-�-r----�---.

o Frequency (kHz)

(b)

Figure 3. BFISAR image obtined by FFT before frequency shifting (a) and after frewuency sfifting (b)

Because of radar range resolution equal to 100 m. and short synthetic aperture length the yacht is depicted as a point target.

VI. CONCLUSION

In this work BFISAR geometry, signal models and image reconstruction algorithm have been discussed. BFISAR topology with stationary transmitter and receiver, and a moving target has been analytical described. Kinematical vector equations and a mathematical model of BFISAR signal with short time pulse are derived. It has been illustrated that BFISAR signal formation and image reconstruction can be interpreted as direct and inverse space projective operation, respectively. Phase correction, range compression Fourier transform and azimuth compression Fourier transform are defined through Taylor expansion of the projective operator. Experimental results using BFISAR scenario with a sea target and short time pulse signal demonstrate correctness of BFISAR geometry description, signal modeling and target image extraction.

ACKNOWLEDGMENT

Authors express their gratitude Prof. M. Chemiakov for support and experimental results obtained by his team from Birmingham University.

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