174 IRE TRANSACTIONS ON MILITARY ELECTRONICS April

Optical Processing of Simulated IF Pulse-Doppler Signals*W. G. HOEFERt, MEMBER, IRE

Summary-The potentially wide bandwidth and large storage PLANEcapacity of optical recording and processing systems leads naturally WAVEto consideration of direct IF recording. A method of recording and SOURCEZprocessing IF pulse-Doppler signals is presented and it is shown how ..sambiguities are reduced, compared to video processing, in the man- - |ner expected from sampling theory. Processing of a linearly varyingDoppler frequency is also discussed. Experimental results with sim- Fulated constant frequency and linearly varying frequency recordings SLITSLIT LI ~~Y PLANE Z PLANEare shown.

INTRODUCTION Fig. 1-Elementary optical processing system.T_ HE THEORY of optical data processing has been

discussed very comprehensively by Cutrona, et al.1 the modulated Y-plane wave as an infinitesimal pointThis introduction presents only a brief review of the source of light, emitting spherical (or, in the one-dimen-

basic principles which were employed in the work described sional case, cylindrical) waves. The expression for aiiy onehere. of these infinitestimal waves can be writtenThe light wave in an optical processing system is analo-

gous to the "carrier frequency" of conventional electronic dA = Ka(y) exp [j-(ct-r)] dy, (2)systems, and the signal information is modulated upon it. LA jIt is not modulated in time, however, but in space. Re-ferring to Fig. 1, if a point source of light is placed at the where r is now the distance from the point source in thefocal point of a lens L1, the spherical light waves emanating Y plane to the point where the wave-disturbance A is beingfrom the point source are converted by the lens into plane measured, a(y) is the amplitude of the modulated wave,waves, i.e., the electromagnetic wave oscillation is in phase and K is a factor which may be assumed constant if theor coherent through any given plane perpendicular to the diffraction is limited to small angles.direction of propagation. This plane wave can be rep- The amplitude of the wave at any point in the Z planeresented by the expression is given by the vector sum of all the infinitesimal wavelets

arriving from the Y plane, orA = aexp j - (ct- x),(1 27r 1A = aexP[i-

-(ct-x)j,A 2(z) = K a(y) exp j-(ct-r) dy (3)where J L XA =instantaneous amplitude of wave disturbance at from Fig. 1,

time t and distance xa=peak amplitude of wave r = [F2 + (y -Z)2 11/2 = F[1 + (y - Z)2 2F2 + *c=wavelength of light using the binomial series expansion. Therefore, neglecting

the terms of higher order, (3) becomesNow, by introducing in the plane Y, a film or other opticalmedium having a transmittance which varies as some func- A2(z) = K U a(y)tion of y, one can perform a spatial modulation of thisplane wave. One of the interesting features of optical r 2r / Y2

-

2zy +z2processing is that operations can be carried out in two *exp - (\ct -F - dy.dimensions, but for simplicity, the discussion here will be LX 2Fconfined to one-dimensional processing.Thnfinedtofundam ient sisofnopticalprocessing. is At this point, the term in the exponent containing ct canThe fundamental basis of optical processing iS the fact 1 --- - '4

thatonecanproucein noter paneZ aligt apliude be dropped, because it represents the "carrier frequency"and contains no information. The two other terms notpattrn hichis n itegrl tansfrm f te pater in containing y as a factor can also be dropped, because they

the plne. odmontrat ths, cnsier ach oin in contribute nothing but phase shift to the final result. So* Received by the PGMIL, November 30, 1961. the expression becomest Electronics Laboratory, General Electric Company, Syracuse, N. Y.1 L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, 2lY2 Z\"Optical data processing and filtering systems," IRE TRANS. ON IN: A 2(Z) =K f a(y) explj-lldy(4

FORMATION THEORY, VOl. IT-6, pp. 386-400; June, 1960._X\2

1962 Hoefer: Optical Processing of Simulated Doppler Signals 175

Now, let z/F=6, the diffraction angle, and let F becomevery large; the term y2/2F becomes negligible and

A2(O/X) = Kf a(y) exp [ji- y]dy. (5)In optics, diffraction patterns obtained from (4) are

called Fresnel diffraction patterns and those obtained from(5) are called Fraunhofer patterns. The latter have thedesired Fourier-transform relation to the modulating func-tion a(y). The limits of these integrals are over the entireY-plane aperture. Fraunhofer patterns are also obtained ifthe Y and Z planes are in the two focal planes of a converg-ing lens, and this is the method which is commonly used.Now, if the modulating film in the Y plane is a recording

of a time function g(l), the following analogies hold:Fig. 2

-P'ulse-Doppler processing system.y t-

a(y) -g(t) aAf1O/X

176 IRE TRANSACTIONS ON MILITARY ELECTRONICS April

where The diffraction pattern may be obtained in the same way= d/XF. as before, by expressing the samples as pulses of sine wave:

Following optical terminology, let us call the three terms / y - md\ 1in this expression the zero-order, positive first-order, and Am = ao + a, cos 2?r (k + mfdTrtnegative first-order diffraction patterns, respectively. Fig. \ d / i4 shows a plot of (8) in which the three terms are plotted md < y < (m +1)d, (9)separately for clarity. The dashed line is the function sin7r//3z/iij3z, which is the envelope of the first-order terms as where k is the number of cycles of IF per sample (notfd varies. The zero-order pattern is independent of Doppler necessarily an integer). Substituting in (7) and evaluatingfrequency; the principal maxima of the first-order patterns the integrals gives the resultmove outward from the center as Doppler frequency in-creases. When the Doppler frequency reaches one-half the sin irNozpulse repetition frequency fr, the two principal maxima A2(Z) =aooverlap. This simply illustrates a limitation predicted bysampling theory, which says that without quadrature a1 sin ir(k - 3z) sin 7rN(fdTr - /z)phase information one can determine the Doppler fre- + - Nsin_r(__T_-As)quency unambiguously only over a range of 1/2 fr. This z w(k - fz) N sin r(fdTr -3z)range is still further limited by the presence of the zero- a1 sin ir(k + Az) sin lrN(fdTr + #3z)order light, which is normally much stronger than the first- + - ____(to)+ _____sn ____+ _

order.~~ ~ ~~~~92 ir(k-I-f3z) N sinrr(fjcT+ Oz) (0order.W Eq. (10) is identical with (8) except in the "envelope"

factors which now have their maxima at z= k/fl instead ofz=0. Fig. 6 shows a plot of the three terms for k=2. Asthe Doppler frequency varies from - 1/2 fr to + 1/2 fr,

+Z b0 ORDER the principal first-order line moves between the limitsshown. In this same region, there is also a much weaker

,_- _ t line from the negative order which moves in the oppositedirection. Therefore, if the Z-plane field stop is located so as

+Z I_ S-IORDER

So C/0 SWCEP

+Z - /41 ORDER , ,Aif41I/)

Fig. 4-Spectrum of video pulse-Doppler signal. ' 46

IF RECORDING OF PULSE-DOPPLER DATAIt is well known that by filtering at IF rather than

video, one can determine Doppler frequencies both as tomagnitude and polarity over a range of- 12fr to + 112fr,or double the range possible with video processing. Thesame result can be achieved in optical processing by record- SCALEing the data in a somewhat different way. One method of tdhrIF recording is shown in Fig. 5. The signal input to the -'/2 o 1X2recorder is at IF rather than at video. The recording sweepproceeds across the film in a zigzag pattern, not in a uni-form straight line, as it was assumed to do in video record- . Oing. That is, the sweep moves longitudinally for a period oftime equal to one range element (or approximately theradar pulse length) then jumps back and records the , ^ 1'- REnext range element alongside the preceding one. Thus the ; ~ ~ ^ ~- frange elements lie in their usual order along the transversedirection, which corresponds to range, but each individual

-

element is recorded longitudinally as several cycles of IF. A --' + , ORDERconstant Doppler shift causes the phase of the IF to changein equal increments from pulse to pulse. Fig. 6-Spectrum of IF pulse-Doppler signal.

1962 Hoefer: Optical Processing of Simulated Doppler Signals 177

to exclude the remaining lines, it is possible to determine For convenience, the three integrals in this form willD)oppler frequency over the range - 1/2 Jr to + 1/2 f, ex- again be referred to as zero-order, and positive andcept for the small ambiguity represented by the negative- negative first-order terms, respectively. Note that iforder line. As k increases, the envelopes of the first-order F=Nd1/2AfT,, the y2 term in the second integral van-patterns move farther apart. ishes, leaving only a first-order term. This results in aAssuming in each case that the maximum information Fraunhofer diffraction pattern at this one particular value

density of the recording medium is utilized, IF processing of F, which is the focal length of the FM recording. Simi-requires more recording space than video processing. How larly, the third term exhibits a virtual focus at the negativemuch more depends, as shown by the above analysis, upon of this value. Fig. 7 is a geometric representation of thethe tolerable ambiguity level. An approximate measure of three diffraction patterns. The useful information is con-the improvement gained by IF processing can be expressed tained in the positive first-order term; its focal length is aby the ratio of the peaks of the positive- and negative- measure of the rate of change of Doppler frequency, andorder envelopes in the vicinity of the principal line (Fig. 6), its position on the Zaxis is a measure of the initial valuefi.which is 2k7r. The separation between this term and the other two, which

are generally unwanted, is proportional to k.PROCESSING A SIGNAI WITII LINEARLY VARYING

DOPPLER FREQUENCYA second case of interest is one in which the Doppler . i

frequency of the signal varies linearly with time. TheFresnel-integral transform (4) turns out to have propertiesuseful in processing this type of signal. Letfibe the Dop- 1 ORDERpler frequency at the beginning of the interval being proc- +1 ORDERessed andfi+Af the Doppler frequency at the end of the VIRTL STOPinterval. To avoid considering ambiguities, assume both of FOCthese frequencies to be less than the prf. The IF samples Fig. 7-Geometrical representation of linear FM signal spectrum.can be written

/kyAr= ao + a, cos 2r -+ m. md < y < (m + l)d (II)d EXPERIMENTAL RESULTS WITII SIMULATEDwliere IIF RECORDINGS

m=(f,+ mAf/.N)mT,. Some IF recordings of the type described above weresimulated by a photographic method and processed on an

Now the Fresnel-integral transform of this sampled wave- optical bench. The apparatus used for recording is shownform is very tedious to evaluate, and the purpose of this in Fig. 8. The spatial modulation of an IF pulse wasdiscussion will be served just as well if the sampled func- simulated by a Ronchi ruling (black and white lines oftion is approximated by a continuous one. This can be done equal width) on transparent film. The pulse length wasby letting m=y/d. Then, using (4), the Z-plane amplitude controlled by means of an aperture slit of variable width inis front of the ruling. The phase of the simulated pulse could

Nd-AfT, be varied by moving the ruling relative to the aperture byA(3) =f ao+ a, cos 2r (k + f1T)y/d + y2 means of a micrometer screw.

J o Nd2 _ The Ronchi ruling had 2.5 lines per centimeter, and theF 2r 2 camera distance D1 was chosen to produce an image mag-

X exp _- (Y--yz)] dy nification of 1/100; so the ruling was recorded on the film_i AF \2 with a line density of 25 lines per millimeter. The camera

fNd r 2r /y2 was mounted on a pivoted plate, and could be rotated=IJ exp - yz)Idy through precise angular increments by means of a long0 LAF 2 lever arm and a dial gauge. Therefore, by taking a series of

r Nd .-lAfT, _ l Xy2 exposures and rotating the camera and varying the pulse

+ J exl)21r - phase by the proper amount between exposures, a train ofpulses simulating a Doppler-shifted signal could be re-

zz k +fir 1 corded on the film.+AIS d~--y-) _ dy Fig. 9 shows portions of some of the recordings obtained

by this method. They are simulated pulse-Doppler record-+ r expj2[AJTr+ y2 ings with k= 2.5, and various (constant) values of fd, as

+O L\expr +AF indicated. The actual size of the recordings was approxi-k T mately 4 mm by 1 mm.z(

- I) yI dy. (12) The result of processing these recordings is shown in Fig.\AF d / 10. An optical system similar to that of Fig. 2 was used,

178 IRE TRANSACTIONS ON MILITARY ELECTRONICS April

PRECISION/DIAL GAUGE ROTATING PLATFORMROCIULN

CAMERA

__~~~~~~~T10= w~~~~~~~~~~~~~~~IHNOT DRAWN TO SCALEFig. 8-Apparatus for simulating IF recordings. Fig. 10- --Spectra of simulated pIulse-Dopper recordings.Fig. 11- Spectra of simulatedi linear FMI recordings.

Fig. 9-Simulated pulsc-Dopp)ler recor(lings (enlarged 50X).

except that since the recordings simulated only one range the various orders in the diffraction spectrum is propor-element, the cylindrical lens was omitted. The shift of the tional to k, the number of IF cycles per sample. The effectprincipal first-order line proportional to Doppler frequency of varying k is illustrated in parts A, B, and C of Fig. 11.is clearly shown. The dark line at the right is the focused, positive first-

It was also possible, with the same equipment, to order line. At k=0.5, this coincides with the edge of thesimulate recordings with linearly varying Doppler fre- zero-order light. However, if the Y-plane aperture is re-quency. The results of processing some of these are shown duced slightly, it can be isolated, as shown by Fig. 11 1);in Fig. 11. In all cases, fd was varied between the limits so useful recordings may still be obtained with k as- 1/2f, to + 1/2fr. As noted above, the separation between low as 0.5.