USING MATLAB TO CREATE AN IMAGE FROM RADAR Douglas Hulber Mahemaics Deparmen Norfol Sae Universiy 700 Par Avenue Uni 483 Norfol VA 3504-8060 dhulber@nsu.edu Inroducion. Digial imaging algorihms developed over he las hiry years have made remarable progress in our abiliy o mae accurae images of planes, ocean floors, and he ineriors of living organisms. Algorihms used in synheic aperure radar (SAR) have many similariies wih hose used in x-ray compuerized axial omography (CAT), magneic resonance imaging (MRI), ulrasound and acousic imaging, and radio asronomy. While hese disciplines have evolved in many complex ways, several ey aspecs of digial image processing can be undersood using conceps from a sandard undergraduae mahemaics curriculum. Ouline. The purpose of his paper is wofold: o describe he radar imaging process using conceps from undergraduae mahemaics courses o presen images generaed by he mahemaical sofware in MATLAB o give examples of a few of hese conceps. D equals r imes. The concep of echo-ranging saes ha, in a medium of nown propagaion speed, he round-rip fligh ime of a signal, muliplied by he signal propagaion speed, is equal o wice he range from he signal source o he signal reflecor. If c denoes he speed of propagaion, he range o he reflecor is c r = The simples problem: esimaing refleciviy of a discree se of reflecors of nown sizes and disinc ranges. Because he reflecors have disinc ranges, heir refleciviies can be esimaed by measuring reurned signal srengh a disinc arrival imes; hen refleciviies vary direcly wih received ampliudes and inversely wih cross-secional area. If wo reflecors were o have idenical ranges, heir echoes would be combined in some unnown way, and he respecive refleciviies could no be esimaed. The problem of closely spaced reflecors. Echo ranging wors well when reflecors are isolaed from one anoher, as when a radar is looing upward and he reflecors are isolaed aircraf. When an airborne radar is looing downward, echo ranging doesn' wor very well because ground reurns arrive in a coninuous sream. Furhermore, here is an inheren conflic (even wih a discree se of reflecors) beween he radar's abiliy o resolve range and is abiliy o deec wea echoes. The reason is: shorer pulses enable 10/01/01 11:53 AM 1
he discriminaion of more closely spaced reflecors, while longer pulses allow he radar o inegrae refleced energy over a longer period and hereby disinguish beween refleced signal energy and inerfering energy sources such as hermal noise. Tha is, longer pulses end o increase he signal-o-noise raio, or SNR. Figure 1 depics he problem of overlapping reurns when range differences beween successive reflecors become shorer han he duraion of he illuminaing ransmi pulse. Noice ha he duraion of each echo is equal o he duraion of he ransmied pulse. ransmied pulse ime ime delay echo echo 1 longer pulse lenghs increase he oal refleced energy longer pulse lenghs increase he amoun of overlapping reurn Figure 1 Range resoluion vs. oal illuminaion energy The erm pulse compression denoes various mehods of encoding he phase of he ransmied signal o effecively shoren he lengh of he ransmi pulse. In a chirp waveform, for example, he ransmied energy sweeps hrough a wide range of frequencies, allowing he radar receiver o lisen while he ransmier is acive (ransmiing and receiving a he same frequency simulaneously and in he same proximiy can desroy sensiive circuiry). The duraion of a frequency-modulaed (FM) chirp can be on he order of 500 imes he effecive duraion of simple pulse processing scheme having he same range resoluion. The chirp ransmied waveform can be wrien as p( ) = a( ) exp( iβ + iα ) (1) where a() represens a real-valued, posiive funcion called he signal envelope or windowing funcion. The echo from reflecors wih refleced energies x a ranges r is represened by a sum of delayed signals s ( ) = x a ( r / c ) exp( iβ ( r / c ) + iα ( r Processing he echo. In engineering erms, he reurn signal is mixed wih delayed inphase and quadraure versions of he ransmied FM chirp and low-pass filered. In mahemaical erms, he received signal s() is muliplied by he complex conugae of a replica of he ransmied signal p() o obain / c ) s ()p *() = x a( ) exp( iβ + iα )exp( iα ), (3) ) () 10/01/01 11:53 AM
where = r /c. Nex he signal (3) is passed hrough an analog-o-digial converer o collec samples for digial processing. Using as a sampling index for discree -values, (3) becomes s ( ) p *( ) = x a( )exp( iβ + iα ) exp( iα ) (4) If here are m samples aen of he received waveform, and n unnown refleced energies, hen (4) consiues a se of m linear equaions in n unnowns, i.e., n where = 1 A, x = b, = 1,.., m or Ax = b (5a) A, = a( )exp( iβ + iα )exp( iα ) for = 1,..,m and = 1,...,n (5b) and b = s ( ) p*( ) for = 1,...,n (5c) When he and are uniformly spaced and he coefficien marix is approximaed by he complex exponenials exp(-iα ), he mapping represened by A in (5a) is called he discree Fourier ransform (DFT). When A represens a DFT, i is uniary so ha is inverse is he ranspose of is complex conugae. The model embodied in (5) is only an approximaion, so ha he soluions {x } have limiaions even hough (5) is numerically well-posed. The reason is ha, in he limiing case of a coninous ground refleciviy funcion and an infiniesimal sampling inerval, he finie duraion of he chirp waveform (he finie suppor of a(*)) causes he rue values of he refleciviy funcion o be disribued among he soluions {x } by means of a convoluion called he poin spread funcion (psf). For example, if he envelope a(*) is a simple sep funcion wih range {0,1}, hen he psf is of he form sin(u)/u. As he duraion of he chirp waveform increases, he suppor of he psf narrows and he accuracy of he soluions {x } increases. Cross-range resoluion: he wo-dimensional Fourier ransform. The nex sep in our developmen is o leap anoher dimension furher ino a mulipliciy of daa collecion and record he reurns from a number of chirped waveforms. Our model for his larger inerval of collecion includes he following assumpions. Each chirp collecion inerval is modeled as a ime during which he radar plaform is saionary (he plaform moves, bu displacemen is negligible) 10/01/01 11:53 AM 3
Beween chirp collecion inervals he radar plaform moves bu is anenna urns wih respec o he line of plaform moion (slews) in a way ha eeps is line-of-sigh coinciden wih a fixed poin on he ground called he scene cener The reference poin for he range measure se { } or {r /c} of Equaions ()-(5) is changed from one chirp collecion inerval o he nex so ha he ime delays / spaial offses are always referred o he scene cener. The range from he radar plaform o he scene cener is large enough so ha spheres of consan range are reaed as sraigh lines where hey inersec he ground. Figure below depics hese assumpions. Wih he noaion inroduced in Figure 3 we will rewrie (3), bringing i ino he form of a double inegral. We firs mae he following subsiuions in (3): = r /c x = f(x,y ), where f(x,y) is unnown refleciviy and is reflecor area a( ) = 1 inside he scene of ineres exp(-iβ + iα ) = 1 as i is compensaed by hardware pre-processing r = x cos θ + y sin θ, where θ is he angle ha he radar line of sigh maes wih he scene reference axis (Figure 3). scene cener: =0 here for each chirp waveform daa raecory of plaform moion Lines of sigh for differen chirps lines of consan ime delay for one chirp waveform Figure : Plaform posiions, lines of sigh, and scene cener Combining hese we obain α s ( ) p *( ) = f ( x, y ) exp i ( x cosθ + y sinθ ) c (6) Nex, we assume ha he reflecor surfaces form a pariion of he scene of ineres, so ha (6) can be undersood as a double inegral. In paricular, he new form is of a Fourier ransform in wo dimensions: 10/01/01 11:53 AM 4
[ i ( xx yy) ] dx dy s( ) p * ( ) = f ( x, y) exp +, (7a) scene where α α ( X, Y ) = ( cosθ, sinθ ) (7b) c c is a measure of spaial frequency wih unis in each dimension of cycles per uni lengh. In (7b), as he reurns of one chirp waveform are processed, he variable runs hrough a represenaive discree se of values as indexed by in (5a,b,c). As he radar plaform moves along is raecory (Figure ), he variable θ (Figure 3) runs hrough a discree se of values called he collecion aperure. In he (X,Y) spaial frequency domain, his discree se of poins aes on he appearance of seas in an amphiheaer (Figure 4). (0,0): scene cener (x,y ) : posiion of reflecor r : proecion of reflecor posiion on o he chirp waveform ime-delay axis θ Figure 3: Posiions in scene coordinaes and proecions ono one ime-delay axis 8 6 4 0 - -4-6 -8 8 10 1 14 16 18 0 Figure 4 Locaion of daa collecion poins in spaial frequency coordinaes 10/01/01 11:53 AM 5
The deerminaion of he refleciviy f(x,y) is hen based is correspondence wih a Fourier dual F(X,Y). The wo funcions are paired via he wo-dimensional Fourier ransform as i( xx + yy ) i( xx + yy ) F( X, Y) = f ( x, y) e dxdy and f ( x, y) = F( X, Y ) e dxdy (8a,b) From (7) and (8) we have α α F ( X, Y) = F( cosθ, sinθ ) (9) c c The polar reformaing algorihm is a commonly used mehod of recapuring f(x,y) from F(X,Y), in which values of F, wih collecion poins depiced in Figure 4, are inerpolaed o a recangular grid in (X,Y) space, and hence o a recangular grid in (x,y) space. The recangular grid in he spaial frequency domain is chosen as a way saion for he informaion because of he exisence of a very fas ransform from he recangular (X,Y) grid o he spaial (x,y) domain nown as he Fas Fourier Transform, or FFT. MATLAB images for a single poin reflecor. We conclude wih a pair of MATLAB images which depic an idealized se of radar reurns from a single reflecor a scene cener. Figure 5 depics daa on inpu from he receiver, Figure 6 a oupu o a screen.. For boh figures, he marix of ampliudes is 51-by-51. 50 100 150 00 50 300 350 400 450 500 100 00 300 400 500 Figure 5: Ampliude of colleced daa pairs (I,Q) for a poin reflecor 10/01/01 11:53 AM 6
50 100 150 00 50 300 350 400 450 500 100 00 300 400 500 Figure 6: Image of he poin reflecor Figure 5 porrays ampliudes of colleced complex-valued reurns from he radar receiver: ligher color indicaes greaer ampliude. The horizonal axis is indexed by in (4), i.e., he ime index wihin a chirp collecion inerval. Wihin each row of pixels, his ime is referred o he ime ha he midpoin of a chirp waveform reurns from he scene cener; i is called "fas ime." The verical axis in Figure 5 couns successive chirp waveforms, i.e., successive values of hea in (6) and (7). This coun of loo angle (or waveform processing inerval or aperure index) is called "slow ime." Figure 6 gives he final image of he poin reflecor, i.e., he refleciviy funcion f(x,y) in geographic coordinaes, range along he horizonal axis, cross-range (disance along he plaform raecory) along he verical. The smearing along he horizonal is caused by a poin spread funcion as discussed above. The smearing along he verical is caused by a similar poin spread effec, influenced by he number of cross-range daa collecion poins. 10/01/01 11:53 AM 7
References Jacowaz, C.V., D.E. Wahl, P.H. Eichel, D.C. Ghiglia, P.T. Thompson, Spoligh Mode Synheic Aperure Radar: a Signal-Processing Approach, Kluwer Academic, Boson, 1996 Soumeh, Mehrdad, Synheic Aperure Radar Signal Processing wih MATLAB Algorihms, John Wiley and Sons, New Yor, 1999 10/01/01 11:53 AM 8