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ARMY RESEARCH LABORATORY

Focusing of Dispersive Targets UsingSynthetic Aperture Radar

by John McCorkle andLam Nguyen

ARL-TR-305 (Reprinted in March 2010)

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August 1994

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Army Research LaboratoryAdelphi, MD 20783-1197

ARL-TR-30S (Reprinted in March 2010)

Focusing of Dispersive Targets UsingSynthetic Aperture Radar

by John McCorkle and Lam Nguyen

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August 1994

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Focusing of Dispersive Targets Using Synthetic Aperture Radar5b. GRANT NUMBER

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John McCorkle and Lam Nguyen 35E55B

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14. ABSTRACT

The focusing of synthetic aperture radar (SAR) data presents problems to the designer. This report addresses the SARfocusing problem with special attention to focusing an area in the near field of the synthetic aperture over a decade or more ofbandwidth in a manner that preserves target resonance characteristics. A method for solving this image formation problem isdescribed, along with a computationally efficient algorithm that is applicable to real-time processing with motioncompensation, Simplified program examples are given, as well as a complete program listing that executes on several single w

chip array processors simultaneously. An error analysis shows quantitatively when the depth of focus is adequate to preservelong-duration target resonance ringing effects for a given target Q, geometry, and bandwidth.

15. SUBJECT TImMS

SAR, UWB, focusing radar17. LIMITATION 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON

16. StCURITY CLASSIFICATION OF: OF ABSTRACT Of' PAGESLam Nguyen

a. REPORT I b. ABSTRACT I'TillS PAGE UU 9419b. TELEPHONE NUMBf:R (/lidude <lr~(l ~·"Je)

Unelassified Unclassified Unclassified (30 I) 394-0847

Standard Form 2911 (Rev. 11/98)

Pre~eribed by ,\NSI Std, Z39.18

2

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3

chart 24vie·w , , " , 32

b"'''''';' end 32cIE;mllll~. above , 33

reflecteor, Hamrnirrg above axis .

'0''''''0' end view " .reHectclf, Hammirlg end .

nE;lHHll~. above axis .ICllCl;ror. fiarnnling above axis , 35

~.,;mw,s. end .

~.b.w."b. end

page3. for error 174. as a aperture , .5. as a function of f) = 2 .6. as a of f) with =6 177. Error at end as a of m, \vhere r:= 0 " , 178. Error at end as a fnnction of m. where T = 4a 179. and 24

17.

13.14.15.

11.

2.nomenclature 8

. 29fJUlilllUlldl load versus for :;;:: 3 and ;;;: 2304 , .

3. SUbl'outinelVlJ'JUle_l~UUP .

0".......... 0' , , , , ••• £.,/

4

ra-

ob-

sowide.

are ro,'atlng

of

bet\veen the sensor and a

bins

from the

',n,fc'n,"c observed

IH,'iUiS, subsur­nnWITW ve"ULae:v, and Deonie scanned

treats the

the ree:en/euSnTIU:Hllell: an an,ertm'e that is

""'ULJIl, as the as!pel~t

thetic anorllm'o

a rotatirlgto other sv;;te:m

excellent rerr:e'0!

a sonar ~ or an

face

In the

,,,n[\)I'::\ reso-

pr,ocesl,ir:g have been limited

microwavethe context

in twoare iSDtroPlC

natIOn io workthe bJ~in"lUl:e must be

sensors.

fre:quler:cy in the sensol's Thethe lowest

the wa'Vehll1gthor less)

widewnioh'm,' curvature errors to the conventional

based must be relitrictc:d to small palClJeS

area.

rowwidthrvrl!'< of aD rferal ofho,wever. have

5

6

two are not is bandlwidth that 4Hv"r,

discriminaticn based on si~lnaltnJle 41141 ",n. and it is bandwidth that mllkclsformation more is to examine

signatunls and to

a ThisR;'''H'~ 1 shows a sketch cf

the

Image areawith polar grid

to form theis to the center of the ap-

parameter i. The k markslines. The points are marked off

dis,tarlce between the and thearea is denoted Both azimuthal

lines are to the center of the *' Tablethe nomenclature used in this report

Propagation

path d . . =dI,k,) -1,2,-2

1-1-1L1/-+-1-+-1-+-1-11---1j",-3 j=-2 j",-t j"O j",1 j=2 1'=3

1 sumnl1a;

SA.R systems use a 'v'ACHEY'W'

of a are that for aequally spaced samples the and these are optimallyfooused and on a single bearing line. These features make it forpostprooessors to look for ringing signatures.)

paradigm is used in discussions of SAR processing, orfocusing, Referring to figure 1, assume that the radar platform is mov­

at velocity v as it collects data along the aperture. Therefore, the dataare spaced equally in occurring at a rate governed

v and the PRF. (This sampling across the aperture is sometimes referred to

i ",3

jot ::;

j", i

i", 0

1=-1

i "'-2

i"" --3

Figure 1. llasic I1igl1land image area

Flight path

* This grid is convenient for postpmcessing. An x-y grid can also be computed with the techniques described.

7

outputs

func1ion of time (secOflds from theporr!t!,," in the aperture. The an'rlog-to-d

of the

Denotes a receivedof the transmitted

reprerent this

nomenclature.Tuole L

Denotes that the function an estimate. The exrlmiple denotes an estimate for a received

porilt"," in the

like thepascitlem in the acca_

pmc!;!,m of the radar and themewcS! betweenDenotes the distancearea to be cmrli1Cn

Particularizes the function be al oarticlliar QCilmetriesaFk:fture, Of the linc. Of the

Denotes thesyllth;ellc aperture, to the

the radio energy to travel fromin the area, and back.

c The of

Time shift between range bin i and range bin i + 1 the center of the apenure

";"nCl;VH freque,m;y of the radar in hertz,PRF_u

Pulse

Relative bandwidth J1::::: and -+

UWB Ultra-wide bacnd'N"lt". where fl.- 1.

Wrrvclenolh in meters,

of

If the radar is at a frE;qtren,cy thcn the echoH";U"_0 area will have a Dl>PIJ!er as the

Tne shift wili be UDW,lrd while theto zero as the moves to a

be downward as the !,um'_"

at the

cornplJtincg a SAR ad-so that the new data set

\vith the circle atas if the plactfc'rmthe center the is acenter the unlqC!e L'oppl"r nmlilp of zero over the entire anfCr[rme

pOlm3 have other

A iSlypclCaJly

llI':0l'ie areaand the distance the radar to the center suffi-

This addresses the case where none of these ret;tnictiorls

8

array:

identi­of

theone can see thatacross

t", 0 .

patch. Nfllie:?

formatted data iswhich is a ""nn,"

is identical to

+ t)

)each tormattm2 makes a different

sum:mai!lt!l1 is used to calculate thebcams are

and the

h,y,wver be looked at as a SliHl()n'lrYstored and cornbined in a com-

sj

foeus,;s the

can

f

ncmn,)c.r has to a convenient ,,"ren,m_

its convenience breaks down at wide relative band\vidths.

of N antennas whose outp!ut,

the D()D1Jie:t and snllg)!lill

9

cal at

of the data pomts.

once

entire

beam width compare with that obtained from "conventionai"antenna half-power

hpanwJirllh of a array is approximately where L is the of theThe heams formed the processing described above follow this role

and have a width proportional to frequencies have wide andhave narrow beams. has conse-

the time domain as a source moves through a beam. Althoughresponse at the center of the beam is a narrow pulse, as one

moves from the the response time-do-main broadening occurs because more and more high-frequency islost as the source moves out of the narruwing high-frequency beam.

from Doppler to the stationary array approach may lead to"phased array" terms. Such an approach is a mistake when

bailJelwidth and/or geometry are not restricted. Whenever the isnot restricted to the far field of an aperture, plane-wave simplifications

Since aFourier tech­

the near field. In adelfiniing element

one element

J then a

pUSHeu,"" or the network. Towith rC(iptrct to another element

down. This breakdownFourier forms beamsmrllle" becomes less and less useful as '"rap"UWB becomes J!J("iljJlJE;IIC"

needed to a wide-bandwidthseveral the best to use in the cquane,ns dc"rin'n

tenna are the time and distance: the time shift the lines thatthe and distance between antenna elementso Thus itis to think in terms. This ti:me-based framework results

clel';vrll;r1r\S that are freqwemiy and gersmietl'y

Cl2SSIccase is

that beam uU5'"O,

to form an endfireIn the S1nlnlest

el(oment, that waveform would bethat

in the opposite nm'c!!on

\vould say that the most critical aSlpe(ct of

nulls. Aantenna

desirable antennaspacing twe elements atbeam in one direction and acase the element WiiH'htl,na is +1 andeould time-steer thethe elements hFf",·p

lo\-vs more treedons.than the null weuld in the re'''TierIf we bandwidth effectsa function of the + and ~waveform is the

Ilene", to that waveform forms a main lobe in that

GratinlZ lobes are as lobes whose is to themain beam, this leads to in the UWB case.Are there lcbes? That If the response to the UWB IS

then the answer is no. Since the response on the"'IS"'" than the at other the definition

the antenna is a Unear It behaves anat a if the respOl1se

0'5,"U', the answer can be yes. If the elements are morethan A/2 at the in the UWB wavefnrnnthen there wiil be a lobe at that Allthe cw antenna analysis holds and remains useful. One

however, be careful when applying terms like lobes that pre-SUIDPI)Se a narrow-band signal. For SAR, the element spacing (velocity!

needed a function of what sidelobes arc permissible at thevarious frequency components the UWB

10

to

nu'mb,er. a.11;;' huw·

is afrequen<:y·domain c!haracteristi(;s can also be in the

lllj,m'g or resonant response. In case-time dOmtlinthe to as

hi,;torlcal SAR systems have not to respond tonn'j!;ll]j!;, no was to

describ"s and analyzes a procedure to focus a over ultra·wide bandwidths, such a way as to preserve the resonant response of tar-

even when are in near field.

102

2rrfr/c

/I\rRe~nance ·1·

region I 'Optical

/ \ \ 1/\ I' r\~ion

Ra'l!·ig~7 \ I \ V fJ V vregoo i

/ V! ,

/,

0.050.3

0.2

2

5

!2b 0.5

2. Radar crosssection of a ,pi,.....

11

is

SUililm'ilLiH must be done so

eqiwoclCiS, Freqiue'nC)I-lr,depollcient add-

assume that anadvai1tage of all the echo ~"~~I,J

all

at all

one must sum

is ac,~ornpilisfled as

+I) t 2: 0 '

the localed al I""V'lCi'm

term the 5; so that thereiipC)WIS slarls al t " 0 at all pUB"" in the anen'uno, The

"lilll~lp t,lnr!p.racross the "",prlllrF

The abeve discussion assumes Ihat a

ered true,

HHIJUI'C re,ipomae 1S

not change

the

iSCilf(spic f(cqtllT,em,ent, and SUiPpi1se that aH(V1FiC area, A horizontal UllAHC,

We could take ,rlclnb

scattererUll'P"", is an aniisotronlcbehavior to eXitend the detectlO,n andradar

Supp(sse we

Ineach one matched tc the

anrri'llfp In order to

tOi:o:gnize that teal large",

seattSf(;rS; this fact should not be

12

ne',de,d. The imr'actconsiders that the mass

reduces the ~N;hip~ case and ~n·cpntc

to traverse

Mcan

words. sup-

rl?cnihe and

to measure

the

and j = 0

it

def'oculses behind the

we have aif there was

rcasel:j=O, 1,. s (T ,) for ~ ~ .

) i.m,,) lcase 2: m = 0, Vj, Vi, 'Ilk J

a=

+

a to beon the

has been defined to be to the centerthe i can be of as a

time t t is units of seconds, i is in ofand they are related by a--one range bin equals a seconds.

we can

Dr()olem is

that instead of ca"ct"aLlllg

one value is calclllate,],the ringing can be retained in this

rc,juce,j. The aim thisUal:l!lJ'Y the error bounds when a 2-D image h.k(r) is

But

the

one

We make an approximation to eUl,j"']()l!

over a limitedilia inclucie

+ -s,(T k') for 'rIj,m = -1.L (5)J - l+m"J

The be said of the in

L It is at center of the U= m.

~ It is at m = 0 regardless of j.L..

3. It gets worse as m deviates further zero.

4, It is worst at the end

A solution is de,;ired to equationstant. In a system,

the form where Y is a con-be mapped to ",,:nncre range bins, so let

y=na . (6)

Sul)stitutJingequations (5) and (6) into (1), we define a 2-D "image" f(i,k) as

(0) = 2:j

= +na)E(7)

If one of a ringing target, then equation (7) can be described as al­lowing the point of perfect focus to be adjusted to any depth n the ring.For example, if n = 0, then the focus point would be at the leadingedge (the sample) of the response. n '" 0, say n " 3, then thethird sample the ringing response would be focused.

Next consider the The indexing is performed such that f(i,k) al-represents the edge the response from a target located at

(i,k) regardless whether it is or not. is found,we now wish to find the discrete of the target ringing. The sampleswill be counted as m " 0 the sample, m " 1 for the second, and soon. We these samples i The mth

value is just + which follows from equations and (7) as

$ +

+

or

14

perfect) when m = n.name "short impulse re-

sponse apiprc)ximallicm is used vel'"'l''' apprc}ximl,tic)fi needs to remainaccurate over the duration of a T"noef',

",.t'~'C'~

Examples ofApplying Approximation

To illustrate the use of equation (8), we follow these steps:

1. Fix n.

2. Place a target (for the purposes of the illustration) at say i " 237 in range onthe kth bearing.

3. Use equation (7) to calculate fti,k) for all (i,k).

The cases of interest are where n " 0 and where n '" O. Let us consider eachseparately.

where n:: 0

Where n " 0, use equation (8) to find the ringing response of the target. Theresponse for the first three points of the focused impulse response is

h37,(O)-fm_Hm,(O) = fm,,(O) = [(237,k) (case for m=O),

f231'(a)- ,m.,cO) = /,36,,(0)= [(238,k) (casefor m = 1) ,

[237. ,(2a) = /'37_"_,(0) =[m.,cO) =[(239, k) (casefor m =2) .

Since n " 0, the amplitude of the leading edge (m " 0) of the impulse reosponse of that target is perfectly focused. As m increases, the approximationgets worse. So the approximation is useful as long as the target resonancedies before the approximation gets too bad.

4.2.2 Case where n '" 0

Where n '" 0, perfect focus is na seconds past the leading edge of the targetimpulse response. Suppose, for this example, n " 2. The first four datapoints for the impulse response are

/'37,,(0) - [m_ff+m,,(na) = [",,,(2a) = [(237, k) (case for m = 0) ,

h37,,(a)- fz37_"~.,(na)= h36.l2a) = [(238, k) (case for m = 1) ,

h37, ,(2a) = /'37_n_jna) =h37,,(2a) =[(239, k) (case for m =2) ,

h37, ,(3a) - fz37_"~.,(na) = h38,,(2a) = [(240, k) (case for m = 3) .

Note that the leading edge is not perfectly focused, as it was when n " O.The important point here is that one can choose n " 0 to allow the leadingedge to defocus slightly for the sake of keeping later points in better focus.

15

eSIJOIlSe Approximation

and low frequen(:ies

the transient echo of acycles.

uencies dampbe collected.

3 illustrates the of the The as-sumes at = (237,0), then R isdistance the center is the distance from theend of the to the focused data

for the impulse = 1) in themit>,U'!" reSDonse IS - aT + or +

If £ is the difference (error) between the approximation and the actual,

£ =

+ ) (9)+ma. eose .a i

As an example of finding the approximation error, suppose R = 2 km, e=90°, = 1 and m = 10. How degrees off (round trip) would theend be at 1 these numbers equation (9), we find

£ = 0.0447 m. So the round phase error at 1 GHz is 107°. Figure 4 is aof error as a of the position in the aperture for e=

50°, and 30e. peculiar is that the error at 76°. This

peaking is a result of at a range of only the aperture length.Figures 5 and 6 show the error as a function of the beam angle Ai m = 5,

and 20 at two ranges.

Figure 7 is a plot of the error at the right end point the aperture as a func­tion m, with T::; O. Since the error at the end is greater than the errorat the left end for A below this is the worst case. 8 is thesame plot but with r: = na and n = 4. In case, the the

m = is out focus aboutoccurs when m = n = 4,

UUUiHJ> over which aby the approximation

also show that making r: " 0 can doublelarl'U'S The optimum T, depends

resonance.

16

, I

_~ I~.!:-9~90, ,

J l L L _I ; I I

, , I___ l L L _

, , II I ,

_J. l. L_, , ,, , I____ 1. _

I,oL:'=-2~~+_"~~~~-.J

-400 -200 a 200 400Position in aperture (m)

120

NJ:(!J- 801il'5

I20

"BI

~L,/2------+i~...;j j .. -'i!: j~-l j"O j~l j~2 j~3

I{ L, ~ 1

Figure 3. Geometry for calculation ofapproxi~mation error.

Figure 4~ Error as a function of position inaperture.

Figure 5~ Error as a function of Bwith RILs =2.

9

Figure 6. Error as a function of {} wilh R/L, =6.

25

~: :I ,

20 , 20I

I ~ _~_~L~:~~ ____ , ,-!- - - - - +-- -- 20 - - - - -1--, I

N liJ : :15 J: ,15, (!J ,

1510! I ,

1il Z -;-----:----- , - - - -1--

]0 '5 I ,10

" I , ,----i---

~ 'T----r--- r- - - - - -!--, , ,5 I , ,

Q 5,

---r-- 5 ----l ;-----;---1,II

060 60 0 20 40 60 8040

9

20

250 ,---c,r---;-,----,-----,--,I ,

ffiI RJLs ""2 j

N 200 20 4-----1---

I --- 1 !o 15 r I

J-' ,-W 150 ;22..-r1-----15 171! r:fi ~ l±J I !i 100 I - - - - ~ - -

5Or---- ;

j:!'" 80

Parameter m Parameter m

7. Error at right """rl:lJreend as.function where 1:

Figure 8. Error at aperture end as afunction of m where r:::: 4u.

7

Fast IOc:usmg rawwhere is ~a¥f0a~'0rl a set of polynomial coeffi-

that will be used and performing the fo-of eqlJation the coefficients found in stage 2 to

the index corresponding to the rm'n,-, time

1

A method to efficientiy time is needed, The dataare collected an converter that a vector of 1'1 numbers

at each aperture We ~all this vector Sj . A time shifttherefore, simply a shift in the index i,

, the time obtained indexing on the original N-pointvector is not fine enough, Finer resolution is gained a two-stage interpo-lator. a is used to produce a new K-point

It is implemented by insertion Z - 1 zeros betweenthe original after which the new sequence is

passed a low-pass FIR (finite impulse response) filter. The processresults in a new vector with the time-shift resolution, Thelength, in this case, is K =ZN. We can extra fine resolution by using afloating-point index with linear interpolation to find a value betweenany two data points in the interpolated data vector. For example, if thedex i were 6.3, then the interpolated value would just be (),7s/6) + 0.38/7).

The focusing algorithm that follows uses these techniques in the followingsequence, which is typically referred to as back projection.

L Read in one N-point vector s;(i) of raw data,;

2, Do an Z-point interpolation, to form a new K-point vector Sj(i),

3, Iterate for all pixels:

a. Find the floating point index needed for a pixeL

b. Find the data value for that pixel either by rounding the index and grab­bing the value, or by linearly interpolating between adjacent values.

c. Sum into that pixel the data value obtained.

Fast Index Calculation

Typically, the greatest computational load in back-projection focusing is infinding the index. Let P(I,k,j) be the exact index needed. Then equation (7),the 2-D focusing equation, becomes

fi.' = ~ Sj (P(I,k, j))j

(10)

Calculating P(i,k,j) exactly involves 3-D trigonometric solutions for everypixel in the image at every position in the aperture. Such a calculationwould be prohibitively large. For practical implementation, a secondary ap­proximation is used to speed the computations. Extremely efficient compu­tational methods exist for finding evenly spaced solutions to polynomials.Therefore, OUf approach is to define a polynomial in three variables G(I,k,j),where G(I,k,j) - P(I,k,j), and use G to compute the index.

The first issue that arises is choosing the order of the polynomial needed foreach variable. To understand the method, suppose that a second-degreepolynomial is adequate for each of the three variables I,k,j. Now the prob­lem can be restated as follows: for a given image pixel (i,k), and a givenaperture position (J), find the coefficients am for m =0.. .26 such that

G(i,k,j) - P(i,k,j)

= q", .+ q, .1 + q, k l'., ,) ,K,j ... , ,J

= [c(j , +c, ,k+c,k']+[c3 · +c, k+cs k']I+[c6 . +c. k+cs k 2 ]1 2," '1 -,j ,j ,j ,j ,j I,J ,j

= [(a" + aj + a2j') + (a3 + a,j + asj')k + (a, + + agj')k'J

+ [(a') + + a j + (a 12 + a13 j + aJ4j')k + (a'5

+ a16 j + a17 j')k2 Ji

+ [(a,S + + + + + a2)')k + + +

the equation written in this 1t IS

index as nested threeas the middle and i as the inner

to see that one oan codewith j as the outer k

the exact

have

a solu~

the

G,

index neededthe width of tile

k, + + + + + + + +

which is

1 a a a ° a a a a p(a,a,a)

1 a a' a a a ° a ° P(O,a,O)

1 2a 4a' ° ° 0 ° 0 0 p(a,2a,0)

1 3a 9a' 0 a 0 ° 0 ° P(O,3a,0)

1 4a 16a' 0 0 ° 0 ° ° P(O,4a,0)

1 ° ° b ° ° b2

° ° P(b,O,O)

1 a a' b ab a2b b2 ab' a2b' P(b,a, a)

1 2a 4a' b 2ab 4a'b b' 2ab' 4a2b2 P(b,2a,0)

1 3a 9a2 b 3ab 9a2b b2 3ab2 9a2b2CO,D P(b,3a,0)

1 4a 16a2 b 4ab 16a2b b2 4ab2 16a'b2cI,o P(b,4a,0)

1 ° ° 2b ° 0 4b2 0 0 c2,o P(2b,0,0)

1 a a2 2b 2ab 2a2b 4b2 4abz 4a2b' c3,o P(2b,a,O)

1 2a 4a' 2b 4ab 8a'b 4b' 8ab' 16a2b2c4 ,o = P(2b,2a,0)

1 3a 9a 2 2b 6ab 18a2b 4b' 12ab2 36a 2b2c5,o P(2b,3a,0)

1 4a 16a2 2b 8ab 32a 2b 4b2 16ab2 64a 2b2c6,o P(2b,4a,0)

1 0 ° 3b 0 0 9b2

° ° c7 ,o P(3b, 0,0)

1 a a2 3b 3ab 3a2b 9b2 9ab2 9a2 b2Ca,D P(3b,a, 0)

1 2a 4a2 3b 6ab 12a2b 9b2 18ab2 36a 2b2 P(3b,2a,0)

1 3a 9a 2 3b 9ab 27a 2b 9b2 27ab2 81a 2b2 P(3b,3a,0)

1 4a 16a2 3b 12ab 48a 2b 9b2 36ab' 144a2b2 P(3b,4a,0)

1 0 ° 4b 0 ° 16b2

° 0 p(4b,a,0)

1 a a2 4b 4ab 4a2b 16b2 16ab2 16a2b2 P(4b,a,0)

1 2a 4a 2 4b 8ab 16a2b 16b' 32ab2 64a 2b2 P(4b,2a,0)

1 3a 9a2 4b 12ab 36a2b 16b2 48ab2 144a2b2 P(4b,3a,O)

1 4a 16a2 4b 16ab 64a'b 16b' 64abz 256a'b2 P(4b,4a,0)

21

22

ture

Now Cis as C= and is adex into the data This index is either rounded,so that the nearest IS ctJe)sen, or is used a linear be-tween the two closest so that a data

Two for the Themethod is to the same orocess as shown but for the problem"That proce~s ' Cto hold the 27 co(:lllclenisCll"":~ill" B to hold the idea! values more than one position in the aper-

five positions); and enlarging A to match"

The second method is to solve vector C at a number ofthe Then construct a polynomial in j for each coefficient

solution does not involve inverting the large A matrix but will result ina less than optimum solution in the least-squares sense"

One advantage of this second method is it lends itself to correcting forairplane motion" Since A is independent of the airplane position, it is in­verIen once" The multiplication to find the index polynomial coeffi­cients can be applied at every aperture position or short subaperture if de­sired" The algorithm thus allows real-time UWB motion compensation"

Po,lynomial Calculation

Now that we have a polynomial to find the index, we need a fast way tocompute the polynomial. Directly calculating an NIh degree polynomialusually requires N adds and N multiplies" If, however, a sequence of solu-

is with the changed in increments (the case wehave here), more efficient means are available" An algorithm by Nuttall [8]

a method that can be applied to calculate the index recur-without the multiplications" The procedure is as follows:

L XCi) = qo +

Xl (i) = XCi) -

+ be the polynomial to be solved, where i = 0, " 2,

3" heretore a recursion can be set up where

X: =Xi (i-l)+ and

XCi) = X, (i -1) + Xi (i)

starting values for the recursion are

X(O) = q" and

Xl (0) = q, - q, + 2q,i = q, - q,"

23

The samea

can also be ex:plmljedall routines

of aOllendix Buses lhis leclhni.aue dir.ecIJ]y

ynOlniallO

lhe is broken UW'Wll. names usedin the SJJhrnllti

chart is shO'Nn

'5 are calculatedthe

notation,

Lboxes = 2

t

Hi. Coefficientgel,er'at,n flow c",rt.

24

Output header! fl, Degree[O].,Degree(n - i)

I", 0 (subapenure count)

m '" 0 (boxes across)

least squares fitDegree[n]

~ L z.:n.m,h,'Hp",o

n "" n+1no until an n done

h "" h+1no until all h done

file

Pseudo-inversematrix6x25

on the errorsIlUm'llJ rlearH' for the

same.a lower rI"orr'p.

Dr!!OnlT" that OPlnr.1'M"" the fnicwdnacaused

fast focusing simply app]]es the polynomial solutiontechnique to the indexing and does signal summation lnto all thepixels in the image. As outlined above, index is done

nested we displays ofinner_Loop, Middle_Loop, and Outer_Loop as pseudo-ooded ex­

amples of how the focusing routine is written using the recursive techniquewith nesting, A second-degree polynomial is used throughout for illustra­tion purposes, These subroutines assume that the image is broken into anumber of boxes where each box has its own set of coefficients,inner_Loop (display 1) is the simplest subroutine and follows the derivationof the recursive formuia directly, It essentially increments i to focus a singleline in a box,

Code CommentsInner loop subroutine

float *f;

float *s;POlmerto first pixel in a line; f(i_start,k)

Pointer to vector from jth aperture position

float Pointer to coefficients vector for index calculation

of actualwhen index < 0

of the box along i axis is lyixes.

Performdata),

Set up initial conditions.~f+

T = q[2]+q[2];Rl = q[1H[2];RO = q[O];

If RO < --{l.S then repeat

{f++;

Rl==RO = RO+Rl};

2. Subroutine

are

Set up initial conditIons for

Perform summatIon on filst line,

for rest of lines>

tty next line in bOlL

mcrernerlted re­_Loop

to

recursive approach meausiucreulented re~

zero.

all boxes.

Read data for first aperture position.

Sel up to loop

for j thread

f::;;; fo - \U'O"._"'~_box :::::-2;

for ~ 010 k_'Do:(es-l;box:= box + 1;

f = f + UVA_""U._~'

for =0 to U,oxes-lbox ~ box + 1,

Initial condition for Cj fix current box.

o to kaOll,s-l t

box::: box + 1;

f:::: f + uU"_'"~_'''

forbox=box+ 1;f:::;;f+

IS re-results:

nc,lvnr,mi"! order on the'>"Prtem" (j "worst-case DCtSlitlon

error in the f]ctatin£-p,}!ngeerr'Otl'yptrotdwees the fnllmu:

In\vith a record

was conducted to e1F.lp"minF the

and

12:3

228964

2191

27105315

4 657

the nested reeursive !etetJ11lClue as il1 the example subrou-we the load as shoWI1 il1 table 2,

28

Grouping terms, we get

L = (inner loop adds) + (middle loop adds) + (outer loop adds)

= IKJB(i' + l)+KJB(i'+ l)k'+JB(i'+ l)(k'+ 1)j',

From this equation, it is clear that a low-order polynomial is crucial in theinner loop. Conversely, the order of the polynomial in the outer loop can behigh iittie impact on the overall speed, Table 3 shows computa­tional load for various configurations with j' =3 and J =2304.

Table 2. Coml'ut2,tlonal Symbol Definitionload, ::!-:::;:-::.::-.-;:.:;.:.==,--;--,:-:-:---------------

Polynomial order for i indexk' Polynomial order for k indexr Polynomial order for j indexJ Number of points in the apertureJ Number of range bins (pixels) In box (iylxes)K Number of azimuth bins (pixels) in box (kyixes)B Number of boxes

J(i' + 1); adds per Inner_Loop callL1 KILo + (i' + l)k1; adds per Middle_loop callL J[L1 + (i' + l)(k' + 1)}1; adds per Outer_Loop (adds per image)

512512515513

k'i

Boxes acro~ ITotal pixels

across

Table 3. Matrix s!lowlng <omputatlon.llo.d versus partitioning forj' = 3 and J = 2304.

1 2 3 4

27 103 256 512

19 5 2 1

TolalBoxes pixels

i ~ 1 down down

] 227 18 4086

2 817 5 4085

3 2043 2 4086

L = 9.711

I,S = 21,888

B = 342

, L = 1451Is = 9120I B = 95,IL=19.33

I,S= 4864

B = 38

L = 9.786 L = 9.769 L = 9.810S = 8640 S = 4608 S = 28808=90 B=36 8= 18

L = 1458 L = 14.51 L = 14.53S = 3600 S = ]920 S = 1200B = 25 B= 10 8=5

L= ]9.4J L= ]9.3J L = ]9.32S = ]920 S = 1024 S= 6408 = 10 B=4 B:;:;.2

L ~ total number in giga·adds.B::: lotal number ~n,~~."e in the image.S = the coe,lj'icl'enttahles = " + ljli' +

29

30

6.5 Implementation Notes

The algorithm developed here requires no multiplication except in the pre­processing (interpolation) stage. There already exist DSP (digital signalprocessor) chips whose architecture is ideal for the FIR interpolation task.The bulk of the computations, however, occur in the summing and indexcalculation stage. Two facts make the summing algorithm attractive. First,adders take considerably less area to implement in a VLSI (very-large-scaleintegration) chip than multipliers. Second, parallel operation is extremelysimple; the image can simply be broken into boxes with a separate proces­sor working independently on each box. In this case, only the coefficienttable would be different between processors, and a broadcast mode wouldbe needed to pass the signal data to each processor. Therefore, design of acustom LSI chip appears practical and could result in real-time speeds.

If an off-the-shelf DSP chip with a parallel adder and multiplier is used toperform this algorithm, then other options become available. For example,the multiplier can be used in the summing algorithm to do interpolation onthe indexing rather than rounding. The multiplication required can be doneduring other add cycles so that it takes zero time. Another possibility is tocalculate the index polynomial directly instead of recursively. Finally, sinceall the operations can be fixed point, the address-generator ALU (arithmeticlogic unit) can sometimes be used in parallel with the floating-point units.

•

7. Beam Patterns and Sidelobe StructureTo develop an intuitive understanding for the beam shape, we assembled 16figures to display, in the time domain, the main-lobe and sidelobe beam pat­terns. We simulated a resonant target by generating a data record for everyposition in the aperture. The simulated scene was a single-point target. Thetarget bearing (see fig. 3) was squinted 15° off broadside (8 = 105°). Rangeto the target R'was 750 ft, the aperture length Ls was 385 ft, and the apertureheight was 60 ft. The point target had an impulse response of

set) -

sin(2.nft)[0.5 + 0.5cos(4.nft)]

Sin(2.nft)exP[(4/1_ t)O.23/]

1for O<t<-

4/1

fort;,:-4/

(15)

The data simulated a 2-GHz sample rate. A record length of 2048 sampleswas made for each aperture position. Each record was preprocessed with atimes-8 interpolator, producing 16K records. Interpolation was done by thestandard FIR filter method. A 255-tap Parks-McClellan low-pass filter witha 950-MHz cutoff was used to do the interpolation. Equation (7) (with n =0) was used to produce the focused beams. A Hilbert transform was used toobtain the magnitude that is plotted in the figures. Plots were made on adecibel scale. Four 3-D plots were made, showing two viewing angles (onaxis and above axis) and two aperture weighting functions (Hamming andrectangular) for each of four targets with different resonant frequencies (50,200, 400, and 900 MHz). These are shown in figures 11 to 26.

The on-axis plots were made so that amplitude values could be easily readoff the plots. The above-axis plots were made to reveal the sidelobe struc­ture and the ring-down time of the target. The axis labels were shifted sothat 0 range was at the point target and 0° was the bearing centered on thetarget. Notice that all plots are 3-0, even the on-axis plots. The varioushorizontal lines in the on-axis plots are the beam pattern on the lih rangebin. These horizontal lines are identical to those shown in the above-axisplots; they are just being viewed "end on."

These figures are unique in that they show how the sidelobes spread in timeas one moves off the main beam. Because of the sharp rise time of the tar­get echo, the contribution from the near and far ends of the aperture can beseen as the early and late peaks in the sidelobe structure. Hamming weight­ing was applied to the array so that one can see the effect of weighting onthe sidelobe structure. It is interesting to note that although the peaksidelobe levels drop only marginally, the average or integrated sidelobe lev­els are significantly lower when tapered aperture weighting is used. Theweighting also reduces the near and far peaks in the sidelobes since theweighting reduces the contribution of the array ends.

31

Since the antenna is a linear system, it is no surprise that the beam patternbehaves basically as classic antenna theory predicts. The aperture beam pat­tern is a function of (1) how long the aperture is relative to wavelength, and(2) what kind of weighting is used to sum the points in the aperture. If uni­form aperture weighting is used (a straight summation), then the typicalsin(f31/J)/(fJlp) pattern occurs, where 1/J is the angle off the main beam, and (3ex: Ls/A. (3 establishes the width of the main beam. As the wavelength getssmall, or as the aperture gets long, the beam gets narrow.

Figure 11. SO·MHzpoint reflector, equalweighting, end view.

o

5 4 3 2 I 0 -I -2 -3 -4Azimuth (degree)

1­- 5

Figure 12. SO·MHzpoint reflector,Hammingweighting, end view.

32

o

,....,l:Q~o""'­Q'~::l.....~-0Q.N

e'-<

-3 -4 -5

Figure 13. 50-MHzpoint reOector, equalweighting, above axis.

o

~ .....-......-

F1gure 14. 50-MHzpoint reOector,Hamming weighting,above axis. ~ .

~"0-g~-~<~~'..e-

. .~--..-

33

Figure 15. 200-MHzpoint reflector, equalweighting, end view.

o

---.l:Q o"0­'-' '

~,5 4 3 2 I 0 -I -2 -3 -4 -5

Azimuth (degree)

Figure 16. 200-MHzpoint reflector,Hamming weighting,end view.

34

o

4

Figure 17. 200·MHzpoint reOector, equalweighting, above axis.

Figure 18. 200·MHzpoint reDector,Hamming weighting,above axis.

'"

~"'""g~-:ae<~

fo'..e-

35

Figure 19. 400-MHzpoint reflector,Hamming weighting,end view.

Figure 20. 400-MHzpoint reflector,Hamming weighting,end view.

36

o

Figure 21. 400-MHzpoint reOectort equalweighting, above axis.

"

Figure 22. 400-MHzpoint reflector,Hamming weighting,above axis. =?

~~

37

Figure 23. 900-MHzpoint reflector, equalweighting, end view.

Figure 24. 900-MHzpoint reflector,Hamming weighting,end view.

o

o"'0;:l....-_0c..'"8 '<0 000,..,OJ ,

8-i= )~'. __L'.+2 I . 5

o

I .0 .5 0Azimuth

i:

If·

. 5 1 . 0(degree)

- 1 . 5 - 2 . 0

38

o"'0;:l....-_0c..'"8 '<0 000,..,OJ ,

8-o

'", 2

It

1 .5 1.0 .5 0Azimuth

I""f:!

.5 1. 0(degree)

ill,f "". I

-1.5 -2.0

Figure 25. 900·MHzpoint reflector, equalweighting, above axis.

Figure 26. 900-MHzpoint renector,Hamming weighting,above axis.

o

o..

"~.e.'" ..., ......t.,.

o

39

8. Summary

40

This report identifies a frequency-independent processing algorithm to get aperfectly focused impulse response from any object at any position from anultra-wide-bandwidth synthetic aperture radar. The report also presents anapproximation that reduces the computational complexity of the algorithm.An error analysis of this approximation demonstrates its applicability to fo­cus even high-Q objects in the near field of an aperture. An implementationthat is both computationally efficient and applicable to real-time motioncompensation is also presented. Finally, plots are presented demonstratingthe capability of the algorithm to focus ringing targets over an 18-to-lbandwidth.

References1. William M. Brown and Ronald J. Fredricks, Range-Doppler Imaging with

Motion through Resolution Cells, IEEE Trans. Aerosp. Electron. Syst.AES-S, No.1 (January 1969), 98-102.

2. J. L. Bauck and W. K. Jenkins, Tomographic Processing ofSpotlight-ModeSynthetic Aperture Radar Signals with Compensation for Wavefront Curva­ture, IEEE In!. Conf. Acoust. Speech Signal Process. (11-14 April 1988).

3. Edwin D. Banta, Limitations on SAR Image Area Due to Motion ThroughResolution Cells, Correspondence, IEEE Trans. Aerosp. Electron. Syst.AES-22, No.6 (November 1986), 799-803.

4. D. Mensa and G. Heidbreder, Bistatic Synthetic-Aperture Radar Imaging ofRotating Objects, IEEE Trans. Aerosp. Electron. Syst. AES-18, No.4(July 1982),423-431.

5. Dale A. Ausherman, Adam Kozma, Jack L. Walker, Harrison M. Jones, andEnrico C. Poggio, Developments in Radar Imaging, IEEE Trans. Aerosp.Electron. Syst. AES-20, No.5 (July 1984),363-400.

6. J. L. Walker, Range-Doppler Imaging ofRotating Objects, IEEE Trans.Aerosp. Electron. Syst. AES-16 (January 1980),23-52.

7. J. A. Landt, E. K. Miller, and M. Van Blaricum, WT-MBAILLLIB: A Com­puter Program for the Time-Domain Electromagnetic Response of Thin­Wire Structures, Lawrence Livermore Laboratory (6 May 1974).

8. Albert H. Nuttall, Efficient Evaluation ofPolynomials and Exponentials ofPolynomials for Equispaced Arguments, IEEE Trans. Acoust. Speech Sig­nal Process. ASSP-3S, No. 10 (October 1987), 1486-1487.

41

Appendix A. Coefficient Generator Program

43

Appendix A

Contentspage

A-I. Main Program to Calculate Coefficients (main.c) 45A-2. Main Coefficient Generator Routine (coefgen.c) 46A-3. Subroutine Find Ideal Index Vector (index.c) 48A-4. Geometry File - Declare All Global Constants (coefgen.h) 49A-5. Declare All Global Variables (coefgen.var) 50A-6. Matrix Pseudo Inverse Coefficients (initmat.c) 51A-7. Initialize All Variables (varinit.c) 55A-S. Find Least Squares Polynomial Fit (poly_fit.c) 57A-9. Matrix Inverter (inverse.c) 59

44

Appendix A

A-I. Main Program to Calculate Coefficients (main.c)

This is a listing of the code that calculates the coefficients needed by thefast focusing algorithm. It takes as inputs the geometry of the synthetic ap­erture radar (SAR) and image area, and it outputs the coefficient vectorsrequired.

/***************************************************************************

Program: coefgen

Description: This is the main program to generate coefficientsfor fast focusing algorithm.

*****************************************************************************/

#include <malloc.h>#include <stdio.h>#include <math.h>#include "coefgen.const"#include "coefgen.var"

mainO{long cacheset;FILE "fp,"tstartjp;char fn[80J;long i;ia=(BOX_SIZE_AZIMUTH-4)f4;ib=(BOX_SIZE_RADIAL-3)f4;

f" initialize variables "fcoefgen_varinitO;

j" initialize pseudo inver matrix"j

initmatO;

printf("Enter coefficient a output file name ===> ");scanf("%s",fn);fp=fopen(fn,"w");if(fp==NULL){

printf("Error open output file \n");exit(!);

}/*

45

I*geometry file*1I*list of variables*1

Appendix A

printf("Enter data acquisition timing file name ===> ");scanf("%s",fn);

*11*tstart_fp=fopen("delaytime.dat","r");if(tstartJp==NULL){printf("Error open input file \n");exit(O);

}read_delaytime(tstart_fp);*1coefgen(fp);}

A-2. Main Coefficient Generator Routine (coefgen.c)#include "coefgen.h"#include "coefgen.var"#include <stdio.h>#include <math.h>extern indexO;extern mxmul_O;extern poly-fitO;1* Note that Reference point is now taken at coordinate kpixel= n_azimuth/2-1 *1/* ipixel= n_radial/2 -I *11* At every position data were taken so that the reference point is at center *1/* data buffer *1

void coefgen(fp)FILE *fp;{float input[NPOINT_APER/4];long i,j;long section,kbox,ibox,position,position_start,posi tion_stop;long ipoint,kpoint;

for(i=O;i<NPOINT_APER/n_section;i++){input[i1=(f10a t)i;

}for(section=O;section<n section;section++){position_start=section*n_aper/n_section;position_stop=position_start+n_aper/n_section;

46

/

Appendix A

for(kbox=O;kbox<NBOX_AZIMUTH;kbox++){for(ibox=O;ibox<NBOX_RADIAL;ibox++)

{printf("processing section %d ibox %d kbox %d\n",section,ibox,kbox);for(position=position_start;position<position_stop;posi tion++)

{i******Calculating Actual index for all samples points in a box**********/

i=O;for(ipoint=O; ipoint<NPOINT_ BOXSAMPLE_ RADIAL;ipoint++){for(kpoint=O;kpoinl<NPOINT_BOXSAMPLE_AZIMUTH;kpoint++)

{ipixel=ibox*BOX_SIZE_RADIAL+ipoint*ib;jpixel=kbox*BOX_SIZE_AZIMUTH+kpoint*ia;

d=«float)position-«float)n_aper/2.-1.»*(aper_Iength/«float)n_aper-l.»; /* distance from radarto center of aperture */

x=sqrt(rcenter_ref*rcenter_ref-radar_height2);r_ref=sqrt(radar_height2+x*x*c_theta_center_ref*c_theta_centerJef+(d­

x*s_theta_centerJef)*(d-x*s_theta_center_ref);rmin=r_ref·d_rcenter*«float)n_radiaI/2.-1.);

index(&ipixel,&jpixel,&findex[i]);i++;

}}

/* generate coef for this box at this position */mxmuls(mat,&mat_c_stride,&stride,findex,&findex_c_stride,&stride,coef,&coef_c_stride,

&stride,&n_coef,&i_one,&n_boxsample);

/* save in coefc[i)[position) */for(i=O;i<COEF_SIZE;i++)

{coefc[i) [position-position_start)=coef[ i);}

/* for debug only *//* printf("pos= %d tstart= %.3e\n",position,2.0*rmin/c); *//* end debug */

} /* position */for(i=O;i<COEF_SIZE;i++)

{printf("Doing polyfit for coef %d\n",i);

poly_fit(input,&coefc[ i) [O),n_aper/n_section,deg[ i),&coefa[i) [0]);

47

Appendix A

for(j=O;j« deg[i]+ 1);j++){fprintf(fp,"%e\n",coefa[i][j]); 1* save ali] to file *1

}}

} 1* ibox *1} 1* kbox *1

} 1* section *1

} 1* subroutine *1

A-3. Subroutine Find Ideal Index Vector (index.c)/*************************************************************************/

Subroutine:Input:

ipixeljpixela_spana ofsetdd_rcenterdr8rmincenterhgthgt2length

Output:findex

index.c

pixel radial coordinatepixel angle coordinate

angle spanned by the patchoffset angle of the center line

distance from middle positionsampling range

(sampling range)/8min range rom center position

height of buildingsquare the heightlength of the building

floating point index to ivalue of that pixel

************************************************************************/

#include "coefgen.const"#include "coefgen.var"#include <math.h>

index(ipixel,kpixel,findex)int *ipixel,*kpixel;float *findex;{

theta=a_ofset-a_span/2.+(float)*kpixel *d_theta;c_theta=cos(theta);s_theta=sin(theta);rcenter=rmincenter+(float)* ipixel *d_rcenter;

48

•

Appendix A

x=sqrt(rcenter* rcenter-radar_height2);r=sqrt(radar_height2+x*x*c_theta*c_theta+(d-x*s_theta)*(d-x*s_theta));*findex=(r-rmin)/dr8;

}

A-4. Geometry File - Declare All Global Constants(coefgen.h)

#define NPOINT_AZIMUTH 600 j* number of bearing lines *j#define NPOINT RADIAL 4095 j* number of radial lines *j#define NPOINT APER 2304 j* # of positions in aperture *j#define NPOINT_DATA 2048 j* number of original data points *j#define NPOINT_DATA_INTER NPOINT_DATA*8 j* number of data points after

interpolated *j

#define PI 3.141592654#define RADAR HEIGHT 60. j* radar height in feet *j#define A OFSET -15.0 j* offset angle from center line *j#define A_SPAN 54.0 j* spanning angle of the patch *j#define RMINCENTER 550. j* range from center position to

nearest point on the patch *j

#define RCENTER_REF 802.0 j* range (ft) of ref. point from center pos. *j#define THETA_CENTER_REF -15.0 j* angle (deg)of ref. point from center pos. *j#define SAMPLING_RATE 2.0e09 j* sampling frequency of signal *j#define SAMPLING]ERIOD 5.0e-1O j* ts=ljfs *j

#define BOX SIZE RADIAL 195- -#define BOX SIZE AZIMUTH 100- -#define NBOX_RADIAL NPOINT_RADIALlBOX_SIZE_RADIAL#define NBOX_AZIMUTH NPOINT_AZIMUTH/BOX_SIZE_AZIMUTH

#define SECTION SIZE 4 j* # of sections for one aperture *j#define COEF SIZE 6 j* # of coeffs for curve fit *j#define MAXDEG 3 j* maximum degree for poly. fit *j#define NPOINT_BOXSAMPLE_RADIAL 5 j* # of samples in a box in radial

direction *j

#define NPOINT_BOXSAMPLE_AZIMUTH 5 j* # of samples in azimuth directionfor every box *j

#define NPOINT BOXSAMPLENPOINT_BOXSAMPLE_RADIAL*NPOINT_BOXSAMPLE_AZIMUTH

1* # of samples in a box for curve fit *j

49

Appendix A

A-5. Declare All Global Variables (coefgen.var)

struct complex { float r,i;}; 1* define a complex type *1

*1

float nyquist_filter;float nyquist_data;

1* Data Variables *1float pixel [NPOINT_AZIMUTH][NPOINT_RADIAL]; 1* array of imagefloat data[NPOINT_DATA]; 1* original data bufer *1float data_inter[NPOINT_DATA_INTER]; 1* interpolated data buffer *1struct complex data_inter_fft[NPOINT_DATA_INTER/2]; 1* Real->Complex Forward FIT

of dataJnter *1float filter_coef[NPOINT_DATA_INTER]; 1* filter coefficients *1struct complex filterjft[NPOINT_DATA_INTER/2]; 1* Real->Complex Forward FIT

offilter_coef *11* value of FIT of filter at Nyquist point *11* value of FIT of data at Nyquist Point *1

1* Geometry Variables *1float a_ofset; 1* offset angle from the center line *1float a_span; 1* spanning angle of the patch *1float d_theta; 1* delta angle *1float x,theta; /* coordinate of a pixel *1float c_theta,s_theta; 1* cosine and sine of theha *1float d; 1* distance from radar to center position

positive to the right, neg. to the left *1float rcenter,rmincenter,rmaxcenter; 1* range from center position *1float rcenterJef; 1* range from center pos. to ref point *1float theta_center_ref; 1* angle ofref. point from center position *1float r_ref; 1* range from any pos. to ref. point *1float d_rcenter; 1* sampling distance from center position *1float dr8; 1* (sampling distance)/8 *1float r,rmin,rrnax; 1* range from any position *1float aper_Iength; 1* length of aperture *1float radar_height; 1* height of radar *1float radar_height2; 1* square the height of radar *1

1* Radar Variables *1float c; 1* speed of wave *1float ts; 1* sampling period of signal *1float fs; 1* sampling frequency of signal *1

long n_azimuth;long n_radial;long n_aper;long pix_size;

1* # of points in azimuth direction *11* # of points in radial direction *11* # of positions in aperture *11* # of points in pixel array *1

50

long n_data;long n_data_inter;long n_data_inter_half;long n_boxsample;long n_coef;long n_section;

1* number of original data points *11* number of interpolated data points

1* 1/2 # of interpolated data points1* # of samples in a box for curve fit

1* # of coefficients curve fit *11* # of sections for one aperture *1

*1*1*1

Appendix A

1* SSL VAriables *1float f_zero;float Cone;long i_one;long stride;

1* floating point zero1* floating point 11* integer 1

1* stride for supercard ssl

*1*1

*1*1

1* working variables *1long n-position;float e_index,a_index,error,maxerror,minerror;long ierr_max,jerr_max,ierr_min,jerr_min;long column_max,row_max,column_min,row_min;long ia,ib;long ipixel,jpixel;float c_theta_centerJef,s_theta_center_ref;float

mat[COEF_SIZE][NPOINT_BOXSAMPLE],findex[NPOINT_BOXSAMPLE],coeqCOEF_SIZE];float coefc[COEF_SIZE][NPOINT_APER/4]; 1* c coef. for each section *1float coefa[COEF_SIZE][MAXDEG+l]; 1* a coef. for each c *1long deg[COEF_SIZE]; 1* degree for each coefc *1long mat_c_stride,findex_c_stride,coef_c_stride;float tstart[NPOINT_APER]; 1* time from trigger to start data acq. at *1

1* each position *1

A-6. Matrix Pseudo Inverse Coefficients (initmat.c)#incJude <stdio.h>#incJude <math.h>#incJude "coefgen.const"#incJude "coefgen.var"

initmatO{int i,j;float a,b;

a=((float)BOX_SIZE_AZIMUTH-4.)/4.;b=((float)BOX_SIZE_RADIAL-3. )/4.;

51

Appendix A

mat[0)[0]=93./175.;mat[O][1 ]=27./175.;mat[0)[2]=(-9.)/175.;mat[0][3]=(-3. )/35.;mat[0][4]=9./175.;mat[0][5]=62./175.;mat[0][6]=18./175.;mat[0][7]=(-6.)/175.;mat[0][8]=(-2.)135.;mat[0)[9]=6./175.;mat[0)[1O]=31./175.;mat[0)[1l]=9./175.;mat[0)[12]=(-3.)l175.;mat[0)[13]=(-I.)/35.;mat[0)[14]=3./175.;mat[0)[15]=0.;mat[0)[16]=0.;mat[0][17]=0.;mat[0)[18]=0.;mat[0)[19]=0.;mat[0)[20]=(-31.)/175.;mat[0)[21]=(-9.)1175.;mat[0)[22]=3./175.;mat[0)[23]=1./35.;mat[0)[24]=(-3.)/175.;

mat[1 )[0]=(-81.)/(175. *a);mat[1)[1]=39./(350.*a);mat[1 )[2]=12./(35. *a);mat[1 )[3]=81./(350.*a);mat[1 ][4]=(-39.)/(175.*a);mat[1 )[5]=(-54.)1(175. *a);mat[1 ][6]=13./(175.*a);mat[1 )[7]=8./(35. *a);mat[1 )[8]=27./(175. *a);mat[1 )[9]=(-26.)/(175 .*a);mat[1 )[10]=(-27.)/(175.*a);mat[1 )[11 ]=13./(350.*a);mat[1 )[12]=4./(35. *a);mat[1 )[13]=27./(350.*a);mat[1 )[14]=(-13.)/(175. *a);mat[1 )[15]=0.;mat[1)[16]=0.;mat[1 )[17]=0.;mat[1][18]=0.;

52

mat[1](19]=0.;mat[ 1](20]=27./(175.*a);mat[l ](21]=(-13.)1(350. *a);mat[l ][22]=(-4.)/(35.*a);mat[1 ][23]=(-27.)/(350.*a);mat[l ](24]=13./(175.*a);

mat[2](0]=3./(35.*pow(a,2.»;mat[2](1]=(-3.)/(70.*pow(a,2.»;mat[2](2]=(-3.)1(35. *pow(a,2.»;mat[2][3]=(-3.)/(70. *pow(a,2.»;mat[2](4]=3./(35. *pow(a,2.»;mat[2](5]=2./(35.*pow(a,2.»;mat[2][6]=(-1.)/(35. *pow(a,2.»;mat[2](7]=(-2.)I(35.*pow(a,2.»;mat[2](8]=(-1.)/(35. *pow(a,2.»;mat[2][9]=2./(35.*pow(a,2.»;mat[2][10]=1./(35.*pow(a,2.»;mat[2](11 ]=(-1.)/(70.*pow(a,2.»;mat[2][l2]=(-1.)/(35. *pow(a,2.»;mat[2][13]=(-1.)/(70.*pow(a,2.»;mat[2][14]=1./(35.*pow(a,2.»;mat[2](15]=0.;mat[2][16]=0.;mat[2](17]=0.;mat[2](18]=0.;mat[2][19]=0.;mat[2](20]=(-1.)/(35. *pow(a,2.»;mat[2](21 ]=1./(70.*pow(a,2.»;mat[2](22]=1./(35.*pow(a,2.»;mat[2][23]=1./(70. *pow(a,2.»;mat[2](24]=(-1.)/(35. *pow(a,2.»;

mat[3](0]=(-31.)/(175.*b);mat[3](1]=(-9.)/(175. *b);mat[3][2]=3./(175. *b);mat[3](3]=1./(35.*b);mat[3](4]=(-3.)1(175.*b);mat[3][5]=(-31.)/(350. *b);mat[3][6]=(-9.)/(350. *b);mat[3](7]=3./(350.*b);mat[3](8]=1./(70.*b);mat[3](9]=(-3.)I(350.*b);mat[3](1O]=0.;mat[3][ 11 ]=0.;

Appendix A

53

Appendix A

mal[3][12]=0.;mal[3][13]=0.;mal[3][14]=0.;mal[3][15]=31./(350. *b);mal[3][16]=9./(350.*b);mal[3][17]=(-3.)/(350. *b);mal[3][18]=(-1.)/(70.*b);mal[3][19]=3./(350. *b);mal[3][20]=31./(175. *b);mal[3][21]=9./(175.*b);mal[3][22]=(-3.)/(175.*b);mal[3][23]=(-1.)/(35.*b);mal[3][24]=3./(175.*b);

mall4][0]=27./(175.*a*b);mal[4][l]=(-13.)/(350.*a*b);mal[4][2]=(-4.)/(35.*a*b);mal[4][3]=(-27.)/(350. *a*b);malt4][4]=13./(175.*a*b);malt4][5]=27./(350. *a*b);malt4][6]=(-13.)/(700.*a*b);mal[4][7]=(-2.)/(35.*a*b);mal[4][8]=(-27.)/(700.*a*b);malt4][9]=13.(350."a*b);malt4][10]=0.;mal[4][ll ]=0.;mal[4][12]=0.;mal[4][13]=0.;mal[4][14]=0.;malt4][15]=(-27.)1(350.*a*b);mal[4][16]=13./(700. *a*b);mal[4][17]=2./(35. *a*b);mal[4][18]=27./(700. *a*b);malt4][19]=(-13.)/(350. *a*b);mal[4][20]=(-27.)I(175.*a*b);mal[4][21 ]=13./(350.*a*b);mal[4][22]=4./(35. *a*b);malt4][23]=27./(350.*a*b);mall4][24]=(-13.)/(175.*a*b);

malt5][0]=(-1.)1(35.*pow(a,2.)*b);mal[5][1]=1./(70.*pow(a,2.)*b);mal[5][2]=1./(35.*pow(a,2.)*b);mal[5][3]=1./(70.*pow(a,2.)*b);malt5][4]=(-1.)1(35.*pow(a,2.)*b);

54

Appendix A

mat(5)[5]=(-1.)I(70. *pow(a,2.)*b);mati5](6]=1./(140.*pow(a,2.)* b);mat(5)[7]=1./(70. *pow(a,2.)*b);mati5)[8]=1./(140.*pow(a,2.)*b);mati5](9]=(-1.)/(70.*pow(a,2.)* b);mat(5)[1O]=0.;mat(5)[1l]=0.;mat(5)[12]=0.;mat[5][13]=0.;mat[5][14]=0.;mat[5)[ 15]=1./(70. *pow(a,2.)*b);mat[5][16]=(-1.)I(140.*pow(a,2.)*b);mat[5][17]=(-1.)1(70. *pow(a,2.)*b);mat[5)[18]=(-1.)/(140.*pow(a,2.)*b);mat(5][19]=1./(70.*pow(a,2.)*b);mat[5)[20]=1./(35.*pow(a,2.)*b);mat[5][21 ]=(-1.)1(70.*pow(a,2.)*b);mat[5][22]=(-1.)/(35.*pow(a,2.)*b);mat[5][23]=(-1.)/(70.*pow(a,2.)*b);mat(5][24]=1./(35.*pow(a,2.)*b);

}

A-7. Initialize All Variables (varinit.c)/***************************************************************************

Subroutine: coefgen_varinit

Description: initialize all neccessary variables

****************************************************************************/

#include <stdio.h>#include <math.h>#include "coefgen.const"#include "coefgen.var"coefgen_varinitO{n_azimuth=NPOINT_AZIMUTH; 1* # of points in azi. direction *1n_radial=NPOINT_RADIAL; 1* # of points in rad. direction *1pix_size=n_azimuth*nJadial; 1* Pixel array size *1n_data=NPOINT_DATA; 1* # of orig. data points *1n_data_inter=NPOINT_DATA_INTER; 1* # of interpolated. data points*1n_data_inter_half=NPOINT_DATA_INTER/2; 1* lI2 # of inter. data points *1

55

Appendix A

n_aper=NPOINT_APER; /* # of points in the aperture *fn_boxsample=NPOINT_BOXSAMPLE; f* # of samples in box for curve fit *fn_coef=COEF_SIZE; f* # of coeffs for curve fit *fn_section=SECTION_SIZE; /* # of sections for one aperture*f

f* Radar Variables *ffs=SAMPLING_RATE; 1* Sampling Frequency *1ts=SAMPLING]ERIOD; 1* Sampling period *1c=3.e8; 1* Speed of wave *1

1* Geometry variables *fa_ofset=A_OFSET*PI/180.; 1* Offset angle from center line *1a_span=A_SPAN*PI/180.; /* Spanning angle of the patch *fd_theta=a_span/«float)n_azimuth-I.); 1* delta theta *faper_Iength=(n_aper-l)*2.0*.0254; 1* length of aperture *1radar_height=RADAR_HEIGHT*12. *0.0254; /* height of radar *1radar_height2=radar_height*radar_height; 1* square the height *1rmincenter=RMINCENTER*12.*0.0254; 1* min range from center position

to reference point *1d_rcenter=ts*c/(2.0*2.0); f* sampling range *1

1* project back to 2*n_data samples *1dr8=ts*cf(2.0*8.0); /* interpolated 16K buffer *f

f*theta_center=a_ofset-a_spanf2.+((float)n_azimuth/2.-1.)*d_theta;c_theta_center=cos(theta_center);s_theta_center=sin(theta_center);rcenter_ref=rmincenter+dJcenter*«float)n_radial/2.-1.);*frcenterJef=RCENTER_REF*12. *.0254; 1* range of ref. point from center pos. *1theta_center_ref=THETA_CENTER_REF*PIf180.; f* angle of ref. point from center

position *fc_theta_center_ref=cos(theta_centerJef);s_theta_center_ref=sin(theta_centerJef);

f* SSL Variables *fCzero=O.; 1* floating point zero *1stride=l; f* stride for ssl *1f_one=1.0; f* floating point 1 *1i_one=l; 1* integer 1 *1mat_c_stride=n_boxsample; /* column stride for mat *1findex_c_stride=1; /* column stride for findex *fcoef_c_stride=l; 1* column stride for coeff *1deg[O]=3; 1* degree for first coefc *1deg[l ]=3;

56

Appendix A

deg[2]=3;deg[3]=3;deg[4]=2;deg[5]=2;

/* xgints_(pixel,&f_zero,&pix_size,&stride); OliO clear pixel[][] */

}

A-S. Find Least Squares Polynomial Fit (poly_fit.c)1***************************************************** ********************

Subroutine: poly_fit.cDescription:

Perform data fit to polynomialInput: x[n] input array

y[n] input arrayn number of elementsdeg degree of poly.

output coeff[deg+ l] coefficients of poly**************************************************************************/

#include <stdio.h>#include <malloc.h>#include <math.h>void poILfit(x,y,n,deg,coeff)float x[],y[],coeff[];long n,deg;{double *dx,*dy,*dcoeff;double *dX,*dXtrans,*dA,*dB, ode;long m,i,j,one;one=l;m=deg+l;dx=malloc(n*sizeof(double»;if(dx==NULL){printf("Memory allocation Error!! !\n");exil(l );}

dy=malloc(n*sizeof(double»;if(dy==NULL){printf("Memory allocation Error !!!\n");exit(1);}

57

Appendix A

dcoeff=malloc(m 'sizeof(double»;if(dcoeff==NULL){printf("Memory allocation Error!! !\n");exit(l );}

dX=malloc(n'm'sizeof(double»;if(dX==NULL){printf("Memory allocation Error !!!\n");exit(l );}

dXtrans=malloc(n'm'sizeof(double»;if(dX==NULL){printf("Memory allocation Error!! !\n");exit(l );}

dA=malloc(m'm'sizeof(double»;if(dA==NULL){printf("Memory allocation Error !!!\n");

exil(l);}

dB=malloc(m'm'sizeof(double»;if(dB==NULL){printf("Memory allocation Error!! !\n");exil(l);}

dC=malloc(m'sizeof(double»;if(dx==NULL){printf("Memory allocation Error!! !\n");exit(l);}

for(i=O;i<n;i++){dx[i]=(double)x[i];dy[i]=(double)y[i];}

for(i=O;i<n;i++){dX[i·m]=l.O;

58

}for(j=l;j<m;j++){for(i=O;i<n;i++){dX[j+i *m]=pow(dx[i],(double)j);}

}mxlrans(dX,n,m,dXlrans);mxmuld(dXtrans,&n,&one,dX,&m,&one,dA,&m,&one,&m,&m,&n);mat_inverse(dA,dB,m);mxmuld(dXlrans,&n,&one,dy,&one,&one,dC,&one,&one,&m,&one,&n);mxmuld(dB,&m,&one,dC,&one,&one,dcoeff,&one,&one,&m,&one,&m);for(i=O;i<m;i++){coeff[i]=(floal)dcoeff[i];}

free(dx);free(dy);free(dA);free(dB);free(dC);free(dX);free(dXlrans);free(dcoeff);}

A-9. Matrix Inverter (inverse.c)#include <sldio.h>

mat_inverse(source_mal,desl_mal,size_mat)double source_mal[J;double dest_mat[J;long size_mal;{long n,i,j,k,ki,counl;double b,b1;double err=O.OOOl;double a[lOOJ(lOO];n=size_mal;counl=O;

Appendix A

59

Appendix A

for(i=O;i<size_mat;i++)

{for(j=O;j<size_mat;j++)

{a[i+ l][j+1]=source_mat[count++];

}}

for(i=l ;i<=n;i++){

for(j=n+ l;j<=2*n;j++){

if«j-n-i)==O){ a[i][j]=l.O; }

else{a[i][j]=O.; }

}}

for(k=l;k<=n-l;k++)

{b=a[k][k];ki=k;for(i=k+ l;i<=n;i++){if«abs(b)-abs(a[ i][k]))<0)

{b=a[i][k];ki=i;}

}

if«abs(b)-err)<O){printf("Error matrix inverse, matrix is singular\n");exit(l);

}

if«ki-k)!=O){for(j=k;j<=2*n;j++)

{

60

•

bl=a[kJUJ;arkJ[jJ=a[kiJ[jJ;a[kiJUJ=bl;

}}

for(j=k+ l;j<=Z*n;j++){arkJ[jJ=a[kJUJIb;

}

for(i=k+l;i<=n;i++){for(j=k+ l;j<=Z*n;j++){a[iJUJ=a[iJUJ-a[iJ[kJ*arkJUJ;

}}

}for(j=n+ l;j<=Z*n;j++)

{a[nJUJ=a[nJUJ/a[nJ[nJ;}

for(k=n-l;k>=l;k=k-l){for(j=n+ l;j<=Z*n;j++){for(i=k+ l;i<=n;i++){arkJUJ=a[kJUJ-a[kJ[iJ*a[iJ[jJ;

}}

}

count=O;for(i=O;i<size_mat;i++){for(j=O;j <size_mat;j++){dest_mat[count++J=a[i+1][size_mat+j+1J;}

}}

Appendix A

61

Appendix B. UWB SAR Focusing Program

63

Appendix B

Contentspage

B-1. Main Program for SUN Computer (focus.c) 65B-2. Main Program for Multiple Array Processors (sc.c) 70B-3. Routines Used to Interpolate (inter.c) 738-4. Initialize Interpolation Filter (filtinitc) 77B-5. Main Back Projection Focusing Routing (bp.c) 78:8-6. Declare All Global Constants (Focus.h) 82B-7. Declare All Global Variables (Focus.var) 83B-8. Initialize All Global Variables on Both Processors 85

64

-Appendix B

B-1. Main Program for SUN Computer (focus.c)This is a listing of the code that focuses the ultra-wideband (UWB) syn­thetic aperture radar (SAR) data. It takes as inputs the coefficient vectorsgenerated by the code listed in appendix A and the SAR data, and it outputsa focused image.

1***************************************************** **************

program: focus.cdescription:

this programs reads radar data from file, interpolatesdata using convolution in freq. domain, and projectsdata back to the polar grid on the ground. The indecesused in back projection are computed from the filewhich contains coefficients for a particular geometry

*******************************************************************/

#include <stdio.h>#include "focus.h"#include "focus.var"

float data[NPOINT_DATA); /* data buffer for host */struct cstype *cs[NO_SUPERCARD);

main 0{float tstart_err[NPOINT_APER); 1* tstart-tstart_trunc for each position */FILE *inputfile; 1* input file contains radar data */FILE *outputfile; 1* output file contains focused image */FILE *fiIterfile; /* input file contains filter coefficients */FILE *coeffile; /* input file contains back projection */

/* coefficients */FILE *tstart_err_file; /* input file contains tstart-tstart_trunc */FILE *junkfile;char inputname(80);char outputname(80);char filtername(80);char coefname[80);char tstart_err_name(80);char temp_string(20);long section,position,box,coefc_order,coefa_order;long i,j; 1* just working variable */

65

Appendix B

float fJ'osition;long data_ready=1; /* maibox to indicate data ready host---> SC */long data_consumed=2; 1* mailbox to indicate data consumed host <--- SC */long msg; 1* message read from mailbox */long one=1; /* nonzero message to put in mailbox */long cacheset=1; 1* cacheset=O turn off cache */long numpar=1;float pixel_zero[BOX_SIZE_AZIMUTH][NPOINT..:.RADIAL];

/** '* open supercards ***********.** ********* ** ** *** ** ** ** ***Ift** Ift**/for (i=O;i<NO_SUPERCARD;i++){cs[i]=(struct cstype *)xlubgn_(O,&cacheset,"sc.lo");}

/* * '* initialize supercard variables '" '" * '* * '* '* ** '* '* '* '* * *'* '* * ******* * '* '* * '*Ifocus_varinitO;

/*** initialize supercards' id ***********************************/for(i=O;i<NO_SUPERCARD;i++){cs[i]->supercard_id=i;}

/* * * Enter Input ***** *** ** "'* *** ** ** ** ***** ** *** ** ** ****** *** *** **/printf("Enter input file name ( .raw )===> ");scanf("%s",inputname);printf("Enter output file name ( .focus)===> ");scanf("%s",outputname);

1*printf("Enter back projection coefficient file name (coefa.dat)===> ");scanf("%s",coefname);printf("Enter start time difference file name (delaytime_diff.dat)===> ");scanf("%s",tstart_err_name);*/strcpy(coefname,"coefa.dat");strcpy(tstart_err_name,"delaytime_diff.dat");

printf("Do you want hamming weight accross the aperture (y or n) ===> ");scanf("%s",temp_string);

for(j=O;j<NO_SUPERCARD;j++){cs[j]->ham_flag=O;

66

if(tern p_ string[O] =='y'){es[j]->ham_flag= 1;}

}

strepy (fiI tername,"fil ter.dat");if((inputfile=fopen(inputname,"rb"))==NULL)

{printf("Error open input file\n");eXil(l );

}if((outputfile=fopen(outputname,"wb"))==NULL)

{printf("Error open output file\n");exit(1);

}if((fil terfile=fopen(filtername, "r"))==NULL)

{printf("Error open filter file\n");exit(l);

}if((coeffile=fopen(coefname,"r"))==NULL)

{printf("Error open back projection coefficient file\n");exit(l);

}if((tstart_errjile=fopen(tstart_err_name,"r"))==NULL)

{printf("Error open tstart difference file\n");exit(l );

}

/*** Read filter coefficients **********************************//* •• into 1st supercard and copy to others supercards *.* •• *.*.*/printf("»> Reading filter coefficients\n");for(i=O;i<FILTER_LENGTH;i++){

fscanf(fil terfile," %f",&cs[O]->filter_coef[i]);}

forG=l;j<NO_SUPERCARD;j++){for(i=O;i<FILTER_LENGTH;i++)

{

Appendix B

67

Appendix B

csUl->filter_coef[i]=cs[O]->filter_coef[i];}

}

1*** Read in tstart_err for all positions to first supercard *****11*** and then copy to other supercards ***************************1printf("»> Reading tstart_err data \n");for(i=O;i<NPOINT_APER;i++){fscanf(tstart_err_file,"%d %f",&j,&(cs[O]->tstart_err[i]));}

for(j=l;j<NO_ SUPERCARD;j++){for(i=O;i<NPOINT_APER;i++){csUl->tstart_err[i]=cs[O]->tstart_err[i];}

}

1*** Read in coefficients a for back projection ******************11*** to the first supercard and then copy to other supercards ****fprintf("»> Reading backprojection coefficients \n");for(section=O;section<SECTION_SIZE;section++){for(box=O;box<NBOX_RADIAL*NBOX_AZIMUTH;box++){for(coefc_order=O;coefc_order<COEF_SIZE;coefc_order++){for(coefa_order=O;coefa_order« cs[O]->deg[coefc_order] +1);coefa_order++){

fscanf(coeffile,"%f",&(cs[O]->coefa[section][box][coefc_order][coefa_order]));for(j= 1;j<NO_ SUPERCARD;j++)

{csUl->coefa[section][box][coefc_order] [coefa_order]=

cs[O]->coefa[section] [box][coefc_order][coefa_order];}

} /* coefa_order *1} 1* coefc_order *1

} 1* box *1} 1* section *1

1* ** Start supercards ******* ** *** ** ** ***** ** ***** ***** ** ** ****** */

68

j1'$j,

\1

f* position *1f* section *f

printf("»> Starting supercard programs \n");for(j=O;j<NO_SUPERCARD;j++){xrcall_("sc" ,&numpar,&cs[j]->dummy);

}

1*** For every position, the host computer reads data for that ***ff*** position and copy data into each supercard memory. **********f1*** There are two mail boxes used betwwen host and each *********ff*** supercard to provide synchronization ************************f

for(section=O;section<SECTION_SIZE;section++){for(position=O;position<NPOINT_APER/SECTION_SIZE;position++){printf("»»>processing section %d pos %d «««<\n",section,position);

1*** Read in data for a position *********************************/fread(data,sizeof(data),1,inputfile);

for(j=O;j<NO_SUPERCARD;j++){

1*** Copy data in each supercard memory **************************ffor(i=O;i<NPOINT_DATA;i++)

{cs[jJ->data_tempti]=data[i];}

f*** Send mail to supercard to inform data is ready **************fxlnwxmt_(cs[jJ,&data_ready,&one);}

f*** Wait until data is consumed by supercards *******************ffor(j=O;j <NO_SUPERCARD;j++)

{xlwtrec_(cs[jJ,&data_consumed,&msg);}

}}

f*** Check for supercards to finish all processing ******************ffor(j=O;j<NO_SUPERCARD;j++)

{xldone_(cs[jJ);}

AppendixB

69

Appendix B

1*** Save resulting radar image *************************************/for(j=O;j<BOX_BASE_START;j++){printf("»>Prepad zero to output file \n");fwrite(pixeIJero,sizeof(pixel_zero), 1,outputfile);}

printf("»>Save image to output file \n");for(j=O;j<NO_SUPERCARD;j++){fwrite(cs[j]->pixel,sizeof(float)*BOX_SIZE_AZIMUTH*NPOINT_RADIAL*NO_KBOX]ROCESS/

NO_SUPERCARD,l,outputfile);

}for(j=BOX_BASE_START+NO_KBOX_PROCESS;j<NBOX_AZIMUTH;j++){printf("»>Postpad zero to output file \n");fwrite(pixel_zero,sizeof(pixel_zero),1 ,outputfile);

}

fclose(inputfile);fclose(outputfile);

/*** Closing all supercards ****************************************/for(j=NO_SUPERCARD-l;j>=O;j--){xlclos_(cs[jJ);}

} /* end main */

B-2. Main Program for Multiple Array Processors (sc.c)/*******************************************************************

Program: sC.c

Description: This is the main program to be executed by each ofthe CSPI i860 array processor.

*******************************************************************/#include "focus.h"#include "focus.var"struct cstype *cs;

70

AppendixB

void sc(dummy)long *dummy;{void focus_varinitO;void filter_initO;void haminitO;void hamwtO;void eraseO;void interO;void fix_data"'pointerO;void polyO;void bpO;void xwtrec_();void xnwxmt_();void xvmov_0;void xvclr_();

float *dataJlointer; /* pointer to actual data */long section,position,box,coefc_order,kbox,ibox;long data_ready=1; /* maibox to indicate data ready host ---> SC */long data_consumed=2; /* mailbox to indicate data consumed host <--- SC */long msg; /* message read from mailbox */long one=1; /* nonzero message to put in mailbox */float fJlosition;long boxbase;

/*** initialize supercard pointer ***********************************/cs=O;

/*** initialize image with zeros ************************************/xvclr_(cs->pixel,&cs->pix_size,&cs->i_one);

/* * * initialize filter *** ***** ** ** ******** ***** *** ********* ** * ** * ***/filter initO;

/** * initialize hamming weight coefficients **** *** ***** *** ***** ** ***/haminit(cs->ham_coef,NPOINT_APER);

/*** Start forming image ********************************************1

boxbase=BOX_BASE_START+cs->supercard_id*NO_KBOX_PROCESS/NO_SUPERCARD;

for(section=O;section<SECTION_SIZE;section++){for(position=O;position<NPOINT_APER/SECTION_SIZE;positionH){

71

Appendix B

/* •• Wait for new data from host computer /

xwtrec_(&data_ready,&msg);

/* •• Copy data to working buffer •••••••••••••••••••••••••••••••••••/

xvmov_(cs->data,cs->data_temp,&cs->n_data,&cs->i_one,&cs->i_one);

/* •• Signal host computer that data are consumed /xnwxmt_(&data_consumed,&one);

fyosition=(float)position;

/••• Hamming weight data if needed ..·······························/if(cs->ham_flag==l){hamwt(cs->data,cs->n_data,section 'NPOINT_APER/SECTION_SIZE+position,

cs->ham_coef);}

/••• zeros last portion of data for circular convolution /erase(cs->data,cs->n_data,32);

/* '" '" Interpolate data"''' '" **'" '" III *' * '" '" '" *' '" '" * '" * '" * *' '" *' '" '" '" '" * '" '" '" *' * '" III ** '" '" * '" '" * '" */inter(cs->data,cs->n_data,cs->data_inter,

cs->n_data_ inter,cs->filter_fft);

fix_datayointer(&datayointer,&cs->data_inter[O],cS->lstart_err[NPOINT_APER/SECTION_SIZE'section+position],cs->ts,8);

for(kbox=boxbase;kbox<(boxbase+NO_ KBOX]ROCESS/NO_SUPERCARD);kbox++){for(ibox=IBOX_START;ibox«IBOX_START+NO_IBOX_PROCESS);ibox++){box=kbox'NBOX_RADIAL+ibox;

/* •• Generate coefficients for back projection /forecoefc_order=O;coefc_order<COEF_SIZE;coefc_order++){poly(&fyosition,&cs->coefc[coefc_order],l,

&cs->coefa[section][box][coefc_orderJ[OJ,cs->deg[coefc_orderJ);

}1* coefc_order ./

/*** Perform back projection **************************************/bp(cs->pixel,datayointer,cs->n_data)nter,cs->coefc,

72

AppendixB

boxbase,kbox,ibox,cs->k--pix_size,cs->i--pix_size,cs->i_box_size);

} 1* i_box *1} 1* k_box *1

} 1* position *1} 1* section *1

} 1* end supercard program *1

B-3. Routines Used to Interpolate (inter.c)

#include <math.h>#include "focus.h"#include "focus.var"1***************************************************** ***********************

Subroutine: haminit.cDescription:

this subroutine initialize array of hamming weighting coeffsaccross the aperture.

Input: float ham_coef[NPOINT_APER]long npoint Number of points for hamming coeffs

Output: float ham_coef[NPOINT_APER] is initialized****************************************************************************/

haminit(ham_coef,npoint)float *ham_coef;long npoint;{long i;for(i=O;i<npoint;i++)

{ham_coef[i]=.54-.46*cos(2. *3.1415927*(float)i/(float)npoint);}

}

/****************************************************************************

Subroutine: hamwt.c

Description:this subroutine takes the radar signal at a position andmultiplies the entire signal array with the hamming coef atthat position.

73

Appendix B

Input:float data[NPOINT_DATA]long npoint_datalong positionfloat ham_coef[];

Output:float data[NPOINT_DATA]

****************************************************************************/

hamwt(data,npoint_data,position,ham_coe!)float *data,*ham_coef;long npoint_data,position;{long i;for(i=O;i<npoint_data;i++)

{datali]=data[i] *ham_coef(position];

}}

/****************************************************************************

Subroutine: inter.cDescription:

This routine performs FIR interpolation by using convolutionin the frequency domain. The input buffer contains 2K of dataand is interpolated into 16K of data. The FIR filter hasbreakpoints at .95 Ghz and 1.05 Ghz

# of taps = 255name of filter coeff. file : filter.dat

Input: data(NPOINT_DATA]complex filter_fft[NPOINT_DATA_INTER/2]float nyquist_filter

Output: data_inter(NPOINT_DATA_INTER]****************************************************************************/

interO{void xvclr_0;void xfrU);void xcvmls_0;void xfrU);

74

Appendix B

extern struct cstype "cs;int i;

j" clear data_inter buffer "jxvclr_(cs->data_inter,&cs->n_data_inter,&cs->i_one);

j" interleave data into data-inter buffer" j

for(i=O;i<NPOINT_DATA;i++){cs->data_inter[i"8j=cs->data[ij;}

j" FFT of interleaved data" j

xfrf_(cs->data_inter_fft,&cs->nyquist_data,cs->data_inter,&cs->n data inter half);- - -

j" FFT of interpolated data (multiply with filter in freq. domain)"jxcvmls_(cs->data)nter_fft,&cs->f_one,cs->data_inter_fft,

cs->filterjft,&cs->n_data_ inter_half);cs->nyquist_data=cs->nyquist_data"cs->nyquistjilter;

j" inverse FFT to get interpolated data "jxfri_(cs->data_inter,&cs->nyquist_data,cs->da ta_inter_fft,

cs->data)nter_fft,&cs->n_data_inter_half);

}

/*************************************************************************

Subroutine: fix_dataJlointer

Description: This subroutine fixes the pointer to the start ofdata)nter buffer. The pointer needed to be adjusteddue to the following reasons:

(a) The linear FIR interpolation filter has phaseshift and the actual data point starts at bin 127of the data_inter buffer ( FIR has 255 coeffs )

(b) The actual time to start data acquisition has tobe rounded of to 1 nsec resolution for the scopeDSA602. However the backprojection algorithm usesthe exact time since the indeces have to be smoothfor curve fit.

75

Appendix B

Input:float 'data inter ( address of data_inter buffer)float tstart err ( tstart-tstart_trunc )float ts ( radar sampling period )long inter_factor (interpolation factor )

Output:float 'datayointer ( actual start pointer for data)

**************************************************************************/

fix_datayointer(pointer,data_inter,tstart_err,ts,inter_factor)float "pointer;float 'data_inter;float tstart_err;float ts;long interjactor;{

long i;double floatl,float2;unsigned long index_ofset;float1=modf(tstart_err'inter_factorfts,&float2);if(float1>=.5) float2=float2+ 1.;index_ofset=(unsigned long)127+(unsigned long)float2;'pointer=data)nter+index_ofset;

f' zeros fill first 128 points of data_inter since phase shift from linear filter 'ffor(i=0;i<128;i++){data_inter[i)=O.;

}

f' zeros fill the last 128 points of data_inter plus 8 points for tstart_err 'ffor(i=0;i<136;i++){data_inter[NPOINT_DATA_INTER+i)=O.;

}

/* first point and last point of actual data array are zeros for backprojection 'f, 'pointer=O.;'('pointer+NPOINT_DATA_INTER-1)=0.;

}

1***************************************************** ********************

76

AppendixB

Subroutine: erase

Description: Zero out the last nzero points of input buffer

***************************************************************************/

erase(buffer,ndata,nzero)float *buffer;long ndata,nzero;{long i;for(i=ndata-nzero;i<ndata;i++){buffer[i]=O.;}

}

B-4. Initialize Interpolation Filter (filtinit.c)/***************************************************************************

This subroutine performs the following:1- Read filter coefficients from file to filter buffer2- Zerro pad the filter buffer to the size of interpolated data

buffer size3- Perform Real Forward FFT of filter buffer to prepare for

interpolation

/***************************************************************************/

#include <stdio.h>#include "focus.h"#include "focus.var"

extern struct cstype *cs;

void filter initO{void xfrU);long i;

1* zero pad filter coeff *1for(i=FILTER_LENGTH;i<NPOINT_DATA_INTER;i++){cs->fiIter_coef[i]=O.;

77

Appendix B

}

1* take fft of filter coef for each supercard *1for(i=O;i<NO_SUPERCARD;i++)

{xfrf_(cs->filterjft,&cs->nyquistjilter,

cs->fi1ter_coef,&cs->n_data_inter_half);}

}

B-S. Main Back Projection Focusing Routing (bp.c)

#include <math.h>/****************************************************************************

Subroutine: poly.cDEscription:

Calculates values of polynomialsInput: x[] input array

n number of elementscoeff(] coefficients of polydeg degree of poly

Output: y[] output array****************************************************************************/

poly(x,y,n,coeff,deg)float *x,*y,*coeff;long n,deg;{long i,j;for(i=O;i<n;i++){y[i]=coeff[O];for(j=l;j«deg+ 1);j++){y[i ]=y[i]+coeff[j]*pow(x[i],(float)j);}

}}

/***************************************************************************

Subroutine: poly2.cDescription: calculate y=cO+cl *x+c2*x A 2

where x=[O,1,2,...,n-l]

78

Appendix B

Input:float c(3) coefficientslong n (max: 1000) number of points

Output:float y[] output vector

***************************************************************************/

#define MAX ELEMENT 1000

void poly2(y,n,c)float y[],c[);long n;{void xvrmp_O;void xdintg_O;float temp;float yl[MAX_ELEMENT);float yl_init,yl_inc;y1_init=c(1]-c[2);yl_inc=2. 'c(2);xvrmp_(y 1,&temp,&y l_init,&yl_inc,&n);yl[O)=O.;xdintg_(y,&c[O),yl,&temp,&n);}

/****************************************************************************

Two-dimensional back projection subroutineusing fast algorithm by NUllalto calculates the index to dataarray for each pixel on the ground in i_box,k_box positioned by i,kk: azimuth ( 2nd order)i: range (ist order)The equation for index calculation is

index= cO+c1'k+c2'k'2 + (c3+c4'k+cS'k'2)'i= dO + dl'i

where c(i) (i=O,S) is a set of coefficients for each box andeach position in the aperture

Input:float 'pixel pointer to first point of the image

2-dimensional areafloat 'data pointer to data array

Important: data[O)=O.Odata[dat_size-l)=O.O

long dat_size size of data buffer

79

AppendixB

float *c pointer to six coefficients forindex computation

long kJ>ix_size size of the patch (pixels) in k axislong iJ>ix_size size of the patch (pixels) in i axislong k_box patch number in k axislong i_box patch number in i axislong i_box_size number of patches in i axis

Output:index to data array is calculated and pixel is updated

********.*.********************.*.*****************************************'

#define MAX_K]IX_SIZE 1000#define MAX_I]IX_SIZE 1000

void bp(pixel,data,dat_size,c,k_box_base,k_box,i_box,kJ>ix_size,iJ>ix_size,i_box_size)

float *pixel; f* array of pixels of the whole Z-dimentional area *ffloat *data; f* data array *flong dat_size; /* size of data array *ffloat *c; f* pointer to six coefficients for index computation *flong k_box_base; f* 0 for 1st SC, i*NBOX_AZIMUTH/NO_SUPERCARO for ith SC *flong k_box; f* patch number in k axis *flong i_box; f* patch number in i axis *jlong kJ>ix_size; f* size of the patch (pixels) in k axis *flong iJ>ix_size; /* size of the patch (pixels) in i axis *flong i_box_size; /* number of patches in i axis *f

{void xVclip_0;void vclip_0;void xvfx4_0;void fix4_0;void xvrmp_O;void vindex_0;void vramp_O;void vgathr_();void vadd_O;void vtabU);

float findex[MAX_I]IX_SIZE]; /* data index value *jlong lindex[MAX_I]IX_SIZE]; /* data index value (round to integer) *ffloat pix_temp[MAX_I]IX_SlZE]; 1* temp buffer for back projection *ffloat dO[MAX_K]IX_SIZE]; f* dO= cO+c1*k+cZ*kA Z *ffloat dl[MAX_K]IX_SIZE]; f* dl= c3+c4*k+c5*kA Z *f

80

float *pix_index; /* pointer to current pixel */long pix_indexjnc; /* each time k is incremented, pixel pointer is jumped */long k; /* for k and i loops control */float maxindex; /* maximum value for data index */float zero=O.O;float one=l.O;long i_one=l;float temp;

maxindex=(float)(dat_size-I);

pix_ index=pixel+(k_box-k_box_base)*k.J'ix_size*i_box_size*i.J'ix_size+i_box*i.J'ix_size;

/* index of first point of the patch *//* with respect to pixel */

pix_index_inc=i.J'ix_size*i_box_size; /* index increment along k axis */

Appendix B

poly2(dO,k.J'ix_size,c); /* generate array of dOpoly2(d1,k.J'ix_size,c+3); /* generate array of d1

for(k=O;k<k.J'ix_size;k++){

/* generate floating point index vector */vramp_(&dO[k],&d1[k],findex,&i_one,&i.J'ix_size);

*/*/

1* table look up and interpolate *//* this function replaces xvcliPO,xvfx40,vgathrO *//*vtabi_(findex,&i_one,&one,&zero,data,pix_temp,&i_one,&dat_size,&i.J'ix_size);*/

/* clip index */vclip_(findex,&i_one,&zero,&maxindex,findex,&tone,&i.J'ix_size);

/* convert to integer index */fix4_(findex,&i_one,lindex,&i_one,&i.J'ix_size);

/* gather data into temporary buffer */vgathr_(data,Jindex,&i_one,pix_temp,&i_one,&i.J'ix_size);

/* update pixel array with data from temporary buffer */vadd_(pix_index,&i_one,pix_temp,&i_one,pix_index,&i_one,&i.J'ix_size);

/* pix_index jumps along k axis */pix_index=pix~index+pix_index_inc;

81

Appendix B

} /* k loop */

} /* subroutine */

B-6. Declare All Global Constants (Focus.h)#define NO SUPERCARD 3 /* number of supercards */#define NPOINT_AZIMUTH 600 /* number of bearing lines */#define NPOINT_RADIAL 4095 /* number of radial lines */#define NPOINT_APER 2304 /* # of positions in aperture *f#define NPOINT DATA 2048 f* number of original data points *f#define NPOINT_DATA_INTER NPOINT_DATA*8/* number of data points after

interpolated */#define FILTER_LENGTH 255 /* length of FIR filter */#define PI 3.141592654#define RADAR HEIGHT 60. /* radar height in feet *f#define A_OFSET -15.0 /* offset angle from center line */#define A_SPAN 54.0 f* spanning angle of the patch */#define RMINCENTER 550. /* range from center position to

reference point on tha patch */#define RCENTER_REF 802.0 f* range (ft) of ref. point from center pos. */#define THETA_CENTER_REF -15.0 /* angle (deg)ofref. point from center pos. */#define SAMPLING_RATE 2.0e09 f* sampling frequency of signal */#define SAMPLING]ERIOD 5.0e-1O /* ts=1/fs *f

#define BOX_SIZE_RADIAL 195#define BOX_SIZE_AZIMUTH 100#define NBOX_RADIAL NPOINT_RADIALlBOX_SIZE_RADIAL#define NBOX_AZIMUTH NPOINT_AZIMUTH/BOX_SIZE_AZIMUTH#define BOX_BASE_START 2#define NO_KBOX]ROCESS 3#define !BOX_START 10#define NO_IBOX]ROCESS 1

#define SECTION SIZE 4 /* # of sections for one aperture */#define COEF_SIZE 6 /* # of coeffs for curve fit *f#define MAXDEG 3 /* maximum degree for poly. fit */#define NPOINT_BOXSAMPLE_RADIAL 5 f* # of samples in a box in radial

direction *f#define NPOINT_BOXSAMPLE_AZIMUTH 5 /* # of samples in azimuth direction

for every box */#define NPOINT_BOXSAMPLE

82

Appendix B

NPOINT_BOXSAMPLE_RADIAL*NPOINT_BOXSAMPLE_AZIMUTIi/* # of samples in a box for curve fit */

B-7. Declare All Global Variables (Focus.var)struct complex { float r,i;};

struct cstype {

/* define a complex type */

/* Data Variables */float pixel[NPOINT_AZIMUTIi/NO_SUPERCARD][NPOINT_RADIAL]; /* array of image*/float data_temp[NPOINT_DATA]; /* temp buffer for data */float data[NPOINT_DATA]; /* original data bufer */float filter_coef[NPOINT_DATA_INTER]; /* filter coefficients */float data_inter[NPOINT_DATA_INTER+136]; /* interpolated data buffer */float tstart_err[NPOINT_APER]; /* tstart.tstarUrunc for each position */struct complex data)nterjft[NPOINT_DATA_INTERl2]; /* Real->Complex Forward FFT

of data)nter */struct complex filter_fft[NPOINT_DATA_INTER/2]; /* Real·>Complex Forward FFT

of filter_coef */float nyquist_filter; /* value of FFT of filter at Nyquist point */float nyquist_data; /* value of FFT of data at Nyquist Point */long ham_flag; /* to indicate wheter to use hamming or not */float ham_coef[NPOINT_APER]; /* hamming weight coefficients */

/* Geometry Variables */float a_ofset; /* offset angle from the center line */float a_span; /* spanning angle of the patch */float d_theta; /* delta angle */float x,theta; /* coordinate of a pixel */float c_theta,s_theta; /* cosine and sine of theha */float d; /* distance from radar to center position

positive to the right, neg. to the left */float rcenter,rmincenter,rmaxeenter; /* range from center position */float reenter_ref; /* range from center pos. to ref point */float theta_center_ref; /* angle of ref. point from center position */float r_ref; /* range from any pos. to ref. point */float dJcenter; /* sampling distance from center position */float dr8; /* (sampling distance)/8 */float r,rmin,rmax; /* range from any position */float aper)ength; /* length of aperture */float radar_height; /* height ofradar */float radar_height2; /* square the height of radar */

83

Appendix B

f* Radar Variables *ffloat c; f* speed of wave *ffloatts; f* sampling period of signal *ffloat fs; f* sampling frequency of signal *f

long n_azimuth;long nJadial;long n_aper;long pix_size;long n_data;long n_data)nter;long n_data)nter_half;long n_boxsample;long n_coef;long n_section;long k.Jlix_size;long i.Jlix_size;long k_box_size;long i_box_size;

f* # of points in azimuth direction *ff* # of points in radial direction •ff* # of positions in aperture *ff* # of points in pixel array *ff* number of original data points *f

f* number of interpolated data points *ff* 1/2 # of interpolated data points */f* # of samples in a box for curve fit */

/* # of coefficients curve fit *f/* # of sections for one aperture *//* # of pixels of a box in azimuth direction*/f* # of pixels of a box in radial dirction */

1* # of boxes in azimuth direction *ff* # of boxes in radial direction */

*f

*/

*f

/* to identify board numberf* floating point zero *//* floating point 1 *f/* integer 1 *ff* stride for supercard ssl *f

f* error message from lib call *f1* just a dummy argument pass to SC program *f

/* SSL VAriables *flong supercard_id;float f_zero;float Cone;long i_one;long stride;long ierr;long dummy;f* working variables */long n.Jl0sition;float e_index,a_index,error,maxerror,minerror;long ierr_max,jerr_max,ierr_min,jerr_min;long column_max,row_max,column_min,row_min;long ia,ib;long ipixel,jpixel;float coefc[COEF_SIZE]; f* c coefficientsfloat

coefa[SECTION_SIZEJ(NBOX_RADIAL*NBOX_AZIMUTHJ(COEF_SIZE][MAXDEG+1];/*a coef. for each c *flong deg[COEF_SIZE]; f* degree for each coefclong mat_c_stride,findex_c_stride,coeCc_stride;float tstart_diff[NPOINT_APER]; f* time difference between quantized and *f

1* unquantized values from trigger to *f1* start data acq. at each position *f

84

Appendix B

} ;

B-8. Initialize All Global Variables on Both Processors(varinit.c)

;***************************************************************************./

/* This subroutine performs the following; *//* Initialize both host and Supercard Variables *//* Output: *//* All variables *//****************************************************************************/

#include "focus.h"#include "focus.var"

extern struct cstype *cs[NO_SUPERCARD];

focus_varinitO{long j;

for(j=O;j<NO_SUPERCARD;j++){cs[j]->n_azimuth=NPOINT_AZIMUTH; /* # of points in azi. direction */cs[j]->n_radial=NPOINT_RADIAL; /* # of points in rad. direction */cs[j]->pix_size=cs[j]->n_azimuth*cs[j]->nJadial/NO_SUPERCARD;

/* Pixel array size */cs[j]->n_data=NPOINT_DATA; /* # of orig. data points */cs[j]->n_dataJnter=NPOINT_DATA_lNTER;1* # of interpolated. data points*/cs[j]->n_dataJnter_half=NPOINT_DATA_INTER/2;

/* 1/2 # of inter. data points */cs[j]->n_aper=NPOINT_APER; /* # of points in the aperture */cs[j]->n_coef=COEF_SIZE; /* # of coeffs for curve fit */cs[j]->n_section=SECTION_SlZE; /* # of sections for one aperture*/cs[j]->iyix_size=BOX_SIZE_RADIAL; /* # of pixels of a box in rad. dir.*/cs[j]->kyix_size=BOX_SIZE_AZlMUTH; /* # of pixels of a box in az. dir. */cs[j]->i_box_size=NBOX_RADIAL; /* # of boxes in radial direction *1cs[j]->k_box_size=NBOX_AZlMUTH; /* # of boxes in azimuth direction */

/* Radar Variables */cs[j]->fs=SAMPLING_RATE; /* Sampling Frequency */cs[j]->ts=SAMPLING]ERIOD; /* Sampling period */cs[j]->c=3.e8; /* Speed of wave */

85

Appendix B

/* SSL Variables */cs[j)->f_zero=O.; 1* floating point zero */cs[j)->stride=l; /* stride for ssl */cs[j)->f_one=1.0; /* floating point 1 */cs[j)->i_one=l; /* integer 1 */cs[j)_>mat_c_stride=cs[j)->n_boxsample; /* column stride for mat */cs[j)->findex_c_stride=l; /* column stride for findex */cs[j)->coef_c_stride=l; /* column stride for coeff */cs[j)->deg[O]=3; /* degree for first coefc */cs[j)->deg[1]=3;cs[j)->deg[2]=3;cs[j)->deg[3]=3;cs[j)->deg[4]=2;cs[j)->deg[5]=2;

}} /* end focus_varinit */

86

AdmnstrDefns Techl Info CtrAttn DTlC-DDA (2 copies)Cameron Sta Bldg 5Alexandria VA 22304-6145

Defns Advncd Rsrch Proj AgcyAttn ASTO D Giglio3701 N Fairfax DrArlington VA 22203-1714

Defns Intllgnc AgcyAttn CMO-4 G AndreievAttn E ThompsonCentral MASINT OfcWashington DC 20340-5100

Defns Spply AgcyAttn P Tomlinsen1110 N Glebe Rd Ste 400Arlington VA 22201

DoDAttn J B Dobsa9800 Savage RdIT Meade MD 20755-6000

DirectorDoD-BTlAttn Dr J R Transue1901 N Beauregard Ste 380Alexandria VA 22311

Inst for Defns AnlysAttn B CraneAttn J Ralston1801 N Beauregard StretAlexandria VA 22311

Dept Army NVESDAttn AMSEL-RD-NV-R A TarbellAttn AMSEL-RD-NV-RPPO K WilleyIT Monmouth NJ 07703

Distribution

Belvoir RDE CtrDept Army NVESDAttn SATBE-N-T B BernardAttn SATBE-N-T D FranklinAttn SATBE-N-T T Broach10101 Gridley Rd Ste 104IT Belvoir VA 22060-5806

Dept Army NVESDAttn AMSEL-RD-NV-GSID T Watts10221 Burbeck Rd Ste 430IT Belvoir VA 22060-5806

HdqtrsDept of the ArmyArmy Spc ProgAttn SARDSO C Keisar103 PentagonWashington DC 20310-0103

ECACAttn M Aasen 11TRl/DQTAttn C MadisonAttn D Quasny185 Admiral Cochrane DrAnnapolis MD 21401

Sp & Terrestrial Comm Dirctrt CECOMRDEC

Attn AMSEL-RD-ST-SE-S J DeewallIT Monmouth NJ 07703-5208

US Army CECOM IntllgnclElect WarfareDirctrt

Attn AMSEL-RD-IEW-SSP B BennettAttn AMSEL-RD-IEW-SSP J CumminsAttn AMSEL-RD-IEW-SSP R ScruppaVint Hill Farm StaWarrenton VA 22186-5100

87

Distribution (cont'd)

CommanderUS Army MICOMAttn AMSMI-RD-WS-UB D HolderAttn AMSMI-RD-AS-RA C KrolingerAttn AMSMI-RD-AS-RA J LoomisAttn AMSMI-RD-AS-RA R R SmithRedstone Arsenal AL 35898-8000

US Army Rsrch OfcAttn J HarveyPO Box 12211Research Triangle Park NJ 27709-2211

Nav Air Warfare CtrAttn Code 2154 M FrishbeeAttn Code C27731A M HendersonAttn Code C29504 D Schriener1 Administrative CircleChina Lake CA 93555-6001

Nav Postgraduate SchlAttn Code EC/GL G S GillAttn ECE Dept Code EC M Morgan833 Dryer Rd Rm 437Monterey CA 93942-5121

Nav Rsrch LabAttn Code 5348 P HansensAttn Code 5340 E L MokoleAttn Code 5340 M Steiner4555 Overlook Ave SWWashington DC 20375-5336

Naval Rsrch LabAttn M Sletten7204 13th PITakoma Park MD 20912

Naval Rsrch LabAttn S N SamaddarWashington DC 22375

CmdrNav Surfc Weapons CtrAttn Code F42 S LeongDahlgren VA 22448-5000

88

NRAD Nav Cmand Cntrl & OceanSurveillance Ctr

Attn B DingerAttn M PollacAttn TTiceSan Diego CA 92152

CmdrOfcrNSWCAttn G GaunaurdAttn P Winters10901 New Hampshire AveSilver Spring MD 20904

CmdrOfcrNSWCAttn F40 J CavenaughAttn F40 D KirkpatrickDahlgrin VA 22448

Ofc of Nav RsrchAttn G D RoyArlington VA 22217

Air Force Inst of TechlgyAttn D DunnAttn A J TerzuoliPO Box 3402Dayton OH 45401-3402

Air Force Inst of TechlgyAttn LT D J WolstenholmeBox 43198 (ENG) 2950 P StretWright Patterson AFB OH 45433-7765

Armstrong LabAttn P G Petropoulos AL-OES8103 15th StretBrooks AFB TX 78235

CMTCOAttn M DouglasAttn C GardnerAttn MAJ K ReichlAttn COL S Williams1030 S Hwy AlAPatrick AFB OH 32925-3002

Distribution (cont'd)

Phillips LabAttn PL/WSR C BaumAttn PL/WSR J BergerAttn PL/WSR S A BlocherAttn T S BowenAttn C J BuchenauerAttn R K MarekAttn S C Mason3550 Aberdeen Ave SEAlbuquerque NM 87109

Rome LabAttn R A Shore RUERASAttn B Tomasic RLIERAHanscom AFB MA 01731-3010

Rome LabAttn F WelkerAttn MWicks26 Electronic PkyGriffis AFB NY 13441-4514

Wright LabAttn WUMNMF K MinAttn A Sullivan101 W Eglin BlvdEglin AFB FL 32542-6810

Wright Rsrch Dev CtrAttn WL/AARM R KoeselAttn WL/AARM K McCoinAttn WL/AARM D MukiAttn WL/AARM J RomanAttn WL/AARM G Urban2690 C Stret Bldg 22 Ste 1Wright Patterson AFB OH 45422-7408

DirectorMarine Corps Intllgnc ActvtyAttn MCIAD M Howard2033 Barnet AveQuantico VA 22134-5011

Sandia Nat! LabAttn B BrockAttn D CressRadar Antenna Dev Code 2343Albuquerque NM 87185-5800

Sandia Nat! LabAttn Code 2345 R HurleyPO Box 5800Albuquerque NM 87185

Sandia Nat! LabAttn M Buttram1153 Sandia Nat! LabAlbuquerque NM 87185-5800

Sandia Nat! Lab High Power Electromag DeptAttn G M Loubriel MS 1153PO Box 5800Albuquerque NM 87185-1153

NASA-Goddard.Attn R Wallace Code 480Greenbelt MD 20771

Nat! Inst of Stand & TechlgyAttn J HorstBldg 220 Rm B124Gaithersburg MD 20399

Nat! Inst of Stand & TechlgyAttn M KandaAttn 81307 A Ondrejka325 BroadwayBoulder CO 80303

Boston Univ Dept of ECS EngAttn P R Kotiuga44 Cummington StretBoston MA 02215

Catholic Univ Physics DepartAttn H UberallWashington DC 20064

89

Distribution (cont'd)

FAMU/FSU Colg of EngrgAttn F GrossPO Box 2175Tallahassee FL 32316-21785

Univ ofFLElect Commctn LabAttn M BarlettAttn J BevingtonPO Box 140245Gainesville FL 32614-0245

GA Institute of TechAttn GTRI/ESL/CAD H EnglerAtlanta GA 30332

GA TechAttn M P Kesler CRB-600Attn J G Maloney Centenial Rsrch Bldg/STLAtlanta GA 30332

GA Tech Schl of MathematicsAttn a P BrunoAtlanta GA 30332-0160

Howard UnivAttn P Alakananda2300 Sixth St Dept EEWashington DC 20059

MI State Univ Dept of Electrd EngrgAttn Kun-Mu ChenEast Lansing MI 48824

Northeastern Univ ECE DeptAttn S McKnightAttn D J McLaughlinBoston MA 02115

OH State UnivAttn R MosesAttn J Young2015 Neil AveColumbus OH 43210-1272

90

OH State Univ OSU/ESLAttn E K Walton1320 Kinnear RdColumbus OH 43212-1191

Polytech UnivAttn L Carin6 Metrotech CenterBrooklyn NY 11201

Polytechnic Univ Electrd Engrg Dept/WR1Attn D M BolleAttn S U Pillai6 Metrotech CenterBrooklyn NY 11201

Polytechnic Univ WRI & Dept of PhysicsAttn Mok-Ming Leung6 Metrotech CenterBrooklyn NY 11201

Purdue UnivAttn M Melloch1285 Electrical Eng BldgWest Lafayette IN 47907-1285

TX A&M Univ Dept of Electrd EngrgAttn J C GoswamiCollege Station TX 77843-3128

Univ of AZ Dept of Electrd & Cmptr EngrgAttn D G DudleyAttn S L DvorakTucson AZ 85721

Univ of CA at Santa Barbara Electrd & CmptrEngrg Dept

Attn M RodwellSanta Barbara CA 93106

Univ of CA Electrd Engrg DeptAttn T !toh405 Hilgard Ave 56-125B Engr IV BldgLos Angeles CA 90024

Distribution (cont'd)

Univ of FL Elect Commctn LabAttn M BartlettAttn J BevingtonAttn J KurtzPO Box 140245Gainesville FL 32614-0245

Univ ofILAttn WCChew1406 W Green StretUrbana IL 61801

Univ of MD Dept of Electrcl EngrgAttn E FunkAttn CHLeeCollege Park MD 20742

Univ of MI Ctr for Ultrafast Optic Sci 2200Attn J WhitakerBonisteel Blvd Rm 1006Ann Arbor MI 48109-2099

Univ of NM CHTMAttn KAgiEECE Bldg Rm 125Albuquerque NM 87131

Univ of PA Electrcl Engrg DepartAttn R BoseAttn D CarlsonAttn B D Steinberg200 S 33rd Stret Rm 202Moore Philadelphia PA 19104

Univ ofTXAttn W NunnelyPO Box 19380Arlington TX 76019

Univ of TX at Austin Dept of Electrcl &CmptrEngrg

Attn H LingAustin TX 78712-1084

Univ of VT Dept of Cmptr Sci & Elec EngrgAttn K E OughstonAttn P SmithBurlington VT 05405-0156

Univ of Wash Dept ofElectrcl Engrg FT-IOAttn A IshimaruSeattle WA 98195

ANRO Engrg IncAttn G FRoss1800 Second Stret Ste 878Sarasota FL 34236

AT&T Bell LabsAttn H Lenzing40 New StretColts Neck NJ 07722

Barth Elect IncAttn J E Barth1300 Wyomiqg StretBoulder City NY 89005

BattellAttnG Ruck505 King AveColumbus OH 43201

BattelleAttn MSIN K2-31 D SheenPO Box 999Richland WA 99352

BDMFedAttn G R Salo1801 Randolph Rd SEAlbuquerque NM 87106

Boeing Defns & Sp GrpAttn D IversonAttn B MerlePO Box 3999 MS 82-36Seattle WA 98124-2499

91

ConsulantAttn V Van Lint1032 Skylark DrLa Jolla CA 92037

Dynamics Techlgy IncAttn S Borchrdt1555 Wilson Blvd #320Arlington VA 22209-2405

ESysAttn CBN 177 K RobinettPO Box 6065Greenville TX 75403-6056

Electromag Capability Anlys CtrAttn J WidnerAttn J Word185 Admiral Cochrane DrAnnapolis MD 21401

EMAAttn R ParellaAttn D Steffen7655 W Mississippi Ave #300Lakewood CA 80226-4332

Ensco IncAttn J Faist5400 Port Royal RdSpringfield VA 22151

Ensco IncAttn G BeasleyAttn R Kemerait445 Pineda CtMelbourne FL 32940

ERlMAttn P Lembo1101 Wilson Blvd #1100Arlington VA 22209-2043

92

Distribution (cont'd)

ERIMAttn D AushermanAttn D HerrickAttn D SheenAttn J GormanAttn R HeimillerAttn S DegrafAttn T LewisPO Box 134001Ann Arbor MI 48113-4001

Farr RsrchAttn E Farr614 Paseo Delmar NEAlbuquerque NM 87123

GORCAAttn KAbendAttn B FellPO Box 2325Cherry Hill NJ 08034-0181

GroupAttn D SegoAttn B MereleBox 3999 MS 82-36Seattle WA 98124-2499

Grumman Aircraft SysAttn S BolesAttn B29-035 D CerrnignianBethpage NY 11714-3582

HARCAttn A BlanchardAttn J GlaznerAttn B KrenekAttn B Williams4800 Research Forrest DrThe Woodlands TX 77381

Distribution (cont'd)

HP-Hewlett PackardAttn K Pastel29 Burlington Mall RdBurlington MA 01803

HP-Hewlett PackardAttn K McMenamin3701 Koppers StretBaltimore MD 21227

HTB Intrnt!Attn J W Luguire1840 Wilson Blvd #206Arlington VA 22201

Hughes Mis CompAttn G CooperAttn L FrazierAET Ctr Box 1973Rancho Cucamonga CA 91729-1973

Hughes Aircraft CoAttn A Wang 600/CISIPO Box 3310Los Angeles CA

IBM RsrchAttn A RuehliPO Box 218Yorktown Heights NY 10598

ITT Rsrch InstAttn H Riggins1875 Admiral Cochrane DrAnnapolis MD 21401

Kaman Sci CorpAttn J Demarest Dikewood Sect6400 Uptown Blvd NE Ste 300EAlbuquerque NM 87110

Lincoln LabAttn S AyasliAttn D BlejerAttn T BryantAttn C FrostAttn CLeeAttn TGrossAttn A K McIntoshAttn M MirkinAttn G MorseAttn LNovak244 Wood StretLexington MA 02173

Lockheed Arntel Sys CompAttn H B LorberDept 73-E6 Bldg L-886 South Cobb DrMarietta GA 30063-0649

Loral Defens,e SysAttn B WoodyPO Box 85Litchfield Park AZ 85340

Soft Machine Resources IncAttn M Rofheart2140 L Stret NW Ste 1007Washington DC 20037

Mirage SysAttn D BarrickAttn J ErwinAttn G MousallyAttn M RosowskiAttn P FailerAttn D Venters425 Lakeside DrSunnyvale CA 94086

93

Distribution (cont'd)

Modem Technologies CorpAttn B Williams494 Sycamore AveShrewsbury NJ 07702

Mssn Rsrch CorpAttn E K English3975 Research BlvdDayton OH 45430

Phys RPIAttn Xi-Cheng ZhangTroy NY 12180

Power Spectra IncAttn M GambleAttn J GrantAttn JOides919 Hermosa CtSunnyvale CA 94086-4103

Novel Technologies & ConceptsAttn S DavisPO Box 726Almeda CA 94501

Sci Applctn Intrntl CorpAttn L Manning1710 Goodridge Dr MS 1-5McLean VA 22102

Schlumberger-Doll RsrchAttn R GaylorAttn D R MarianiOld Quarry RdRidgefield CT 06877

Southwest Rsrch InstAttn B DuffAttn K Stevens6220 Culebra Rd PO Drawer 28510San Antonio TX 78228-0510

94

SRI IntrntlAttn M BaronAttn D BenderAttn D BuseckAttn K A DreyerAttn R Vickers333 Ravenswood AveMenlo Park CA 94025-3493

TechnisaAttn P Nelson1840 Wilson Blvd #206Arlington VA 22201

Techlgy Svc CorpAttn R LeFeure2950 31st StretSanta Monica CA 09405-3093

Texas InstrAttn G BurnhamAttn R VosPO Box 405 MIS 3451Lewisville TX 75067

The Boeing CompAttn B H Merle MS-8721Attn D IversonAttn D SegoPO Box 3999 MS 82-36Seattle WA 98124-2499

CSPIAttn J Warther40 Linnell CircleBillerica MA 01821

Thermo Trex CorpAttn C C Phillips9550 Distribution BlvdSan Diego CA 92121-2306

Distribution (cont'd)

ToyonAttn M Vanblaricum75 Aero Camino Ste AGoleta CA 93117-3139

Unisys Corp Gov Sys GrpAttn A KerdockAttn K S Lee MIS H16365 Lakeville RdGreat Neck NY 11020-1696

United Technologies Norden SysAttn M GreenspanNorden PI Box 5300Norwalk CT 06856

Army Rsrch LabAttn AMSRL-EP-ED W R BuchwaldAttn M WeinerFT Monmouth NJ 07703

Army Rsrch LabAttn J ArthurAttn AMSRL-SL-EP J B HendersonWhite Sands Missile Range NM 88002-5513

Army Rsrch Lab/EPSDAttn AMSRL-EP-EC A A KimAttn E LenzingFT Monmouth NJ 07703

Army Rsrch LabAttn AMSRL-EP-EC L KingsleyPulse Power CenterFT Monmouth NJ 07703-5601

US Army Rsrch LabAttn AMSRL-EP-MC A PaolellaFT Monmouth NJ 07703-5601

US Army Rsrch LabAttn AMSRL-OP-SD-TA Mail & Records

MgmtAttn AMSRL-OP-SD-TL Tech Library

(3 copies)Attn AMSRL-OP-SD-TP Tech PubAttn AMSRL-SS J SattlerAttn AMSRL-SS V DeMonteAttn AMSRL-SS-FG R TobinAttn AMSRL-SS-IB E AdlerAttn AMSRL-SS-S J MillerAttn AMSRL-SS-SA M BarnesAttn AMSRL-SS-SA F H LeAttn AMSRL-SS-SD C N TranAttn AMSRL-SS-SG P AlexanderAttn AMSRL-SS-SG A A BencivengaAttn AMSRL-SS-SG M BennettAttn AMSRL-SS-SG L HappAttn AMSRL-SS-SG R InnocentiAttn AMSRL,SS-SG K KappraAttn AMSRL-SS-SG H KliatriAttn AMSRL-SS-SG L Nguyen (10 copies)Attn AMSRL-SS-SG M ResslerAttn AMSRL-SS-SG V SabioAttn AMSRL-SS-SG J Sichina (10 copies)Attn AMSRL-SS-SG T TonAttn AMSRL-SS-SJ G H StolovyAttn AMSRL-SS-SM R KapoorAttn AMSRL-WT-NF L JasperAttn AMSRL-WT-NH L LibeloAttn AMSRL-SS-SG J McCorkle

(30 copies)

95~ u.s. GOVERNMENT PRINTING OFFiCE: 1994··379·161

; , poe, - k'1i VV'.'" mil ..-... /\ I J010 {L7

~)( '51.fJ.7

, '.Quality Assurance and OPSEC Review

This form is an approval record for AAL generated information to be presented or disseminated external to AAL Note: Submit al/ manuscripts

In electronic format or camera ready copy. See attached Instructlons. If mora space is needed, use reverse ollorm (include block numbers).

A. GenerallnformatJon 1. Present Date (rnmJdd/yyyy) 2. Unclassified Title: ~),<L '/,('-3,:,5

ol/29/20 10 Focusing of Dispersive Targets Using Synthetic Aperture Radar

3. Author(s): (Last, First, MI): 4. Office Symbol(s): 5. Telephone Nr(s):

John McCorkle, Lam Nguyen RDRL-SER-U (301) 394 - 0847

(LAM rJGVlc~)

6. Contractor generated c><;J No [J Yes 7. Type: ~Report 'LJAbstract "0 Publication [] Presentation (speech, briefing, video

Ifyes, enter Contract No. and ARL COR clip, poster, etc) []Book U Book Chapter OWeb

8. Key Words: UWB, Synthetic Paerture Radar (SAR), Back Projection

9. Distribution Statement (reqUired) is manuscript Check appropriate letter and number(see instructions for statement text 10. Security

DNo DYes A B C D E F X , 2 3 4 5 6 7 6 9 '0 11 Classificationsubject to export control?

~OOOOOO DO 0 00000 0 0 0 UNCLASSIFIE

B. Reports 11. Series 12. Type 13. No. of pages 14. Project No. 15. Period Covered 16. Sponsor

--r1ZJc. Publications

17. Is MS an invited paper? ~NO LJ Yes 19. Material will be submitted for publication in

18. Publication Is a refereed journal? ,Change DistdJl:Wion to Public Release ARLTR

~NO DYes Journal Country

D. Presentations 20. Conference Name I Location 21. Sponsor

ARLTR ARL

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DNo [J Ves (Ifyes, list countries and identify InternationalOral presented only [J Oral presented and pUblished inAgreement{s))

[J Don't know

(If published, complete block 18 and 19, Section C).

E. Authors Statement: 27. All authors have concurred In the technical content and the sequence of authors. All authors have made a

substantial contribution to the manuscript and all authors who have made a substantial contribution are identified in Block 3.

o //~/4..--- J! 'Z '1!U)OARL Lead ~thor r COR Date (mm/ddlyyyy)

F. Approvals

28. First line Supervisor of Senior ARL Author 29. Revlewer(s) C!!:..Chn~~ditori81/ NA)O'COlltfff "ylzy;; .. N~#e Ol/d<jPtJ/OName (Las, lr. t, MI) Date (mm/dd/yyyy Date (mm/dd/yyyy)

30. limited distribution information for release to 31. Classified Informationforeign nationals.

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ARL Form 1October 2003

OPSEC REVIEW CHECKLISTOPSEC poc: Complete and explain any positive responses in block 9.

Note: ARL must be the proponent of the proposed information for release.

1. Does this material contain Sensitive Information? DYES iI NO 8. Does this material contain:

2. Does this information contain state-of-the~art,a. Any contract proposals, bids, and/or proprietary

breakthrough technology? DYES iI NOinformation? DYES ilNO

3. Does the United States hold a significant leadb. Any information on inventions/patent application

time in this technology? DYES iI NOfor which patent secrecy orders have been issued? DYES ilNO

4. Does this information reveal aspects of reversec. Any weapon systems/component test results? DYES ilNO

engineering? DYES iI NO d. Any ARl-originated studies or after actlon reportscontaining advice and recommendations? DYES ilNO

5. Does this material reveal any security practicesor procedures? DYES iI NO e. Weakness andlor vulnerability information? DYES iINO

6. Does this information reveal any securityf. Any information on countermeasures? DYES ilNO

practices or procedures? DYES iI NO g. Any fleldingltest schedule information? DYES ilNO

7. Would release of this information be of economich. Any Force Protection, Homeland Defense

benefit to a foreign entity, adversary, or alloW for the(security) information? DYES ilNO

development of countermeasures to the system ori. Information on subjects of potential controversytechnology? DYES iI NO among military services or other federal agencies? DYES ilNO

j. Information on military applications in space,nuclear chemical or biological efforts: high energy

DYES iINOlaser information; particle beam technology; etc?

k. Contain information with foreign policy or foreignrelations implications? DYES iINO

OPSEC Approval StatementI, the undersigned, am aware of the adversary's interest in 000 publications and in the subject matter of this material and that. to the best of myknowledge, the net benefit of this release out weighJtthe potential damage to the essential security of all ARL. AMC. Army, or other DOD programsof which I am aware. h .

'/5ld"'"John Costanza • J I-~p£ ... A ")

/ ;1PSEC Reviewer (Printed nam~signature) Date

(I 9. Sr.l«for exnl.na'ions/continualinns/nPSEC reviewenmmen'"

Since this report~as published 16 years ago, the technology described herein has become routine and is no longer criticaltechnology.

Final Release Clearances32. Public/Limited release informationa. Material has been reviewed for OPSEC policy.

0. J i. A u~ A A2..) . }£-f Ie?j ,~

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b.The Inrorma:i A ~~:ned ,;;;er,al Is

[2' I is not D approved for public release/ has received appropriate tech/editorial review.

C) m", 10p v \ Division CIT A"'~ (.}h..r ~' Date

c. This inform~ion is accepted for public release. ~

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ARL Form 1October 2003

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