Paametes of spinning AM eticles onald G. Digges, Cal E. Halfod, and Glenn D. Boeman A new method of obtaining amplitude modulation (AM) fo detemining taget location with spinning eticles is pesented. The method is based on the use of gaded tansmission capabilities. The AM spinning eticles peviously pesented wee functions of thee paametes: amplitude vs angle, amplitude vs adius, and phase. This pape pesents these paametes along with thei capabilities and limitations and shows that multiple paametes can be integated into a single eticle. It is also shown that AM paametes can be combined with FM paametes in a single eticle. Also, a geneal equation is developed that elates the AM paametes to a eticle tansmission equation. Key wods: eticles, tacking devices, AM modulation. 1. Intoduction Amplitude modulation (AM) in eticles has been studied" 2 as a means of poviding taget position infomation to tacking systems. The poduction of an AM signal in eticles has been diffeent fom poduction of AM signals in othe systems. This diffeence esults in pat fom a lack of vaiable tansmission chaacteistics in infaed eticle mateials. This pape pesents a simple and diect appoach of obtaining an AM signal with eticles using paametes simila to the paametes involved in FM modulations Conside the histoical 2 case of amplitude modulation in spinning eticles as shown in Fig. 1(a). A taget is imaged onto the spinning eticle and the taget image is tansmitted though the tanspaent white sectos and blocked by the opaque dak sectos. Usually, the spatial integal of the light is collected by a single detecto and a tempoal signal fom the detecto is analyzed fo taget location infomation. If a cicula taget wee located at the cente of the spinning eticle (location A), the amount of image light tansmitting though the eticle does not change with the spinning motion of the eticle. Hence, the spatial integal of the light tansmitting though the eticle would be a dc value and the modulation (ac esponse) would be zeo. A dc signal would coespond to a taget located at the Glenn Boeman is with Univesity of Cental Floida, Depatment of Electical Engineeing-CEOL, Olando, Floida 32816; the othe authos ae with Memphis State Univesity, Depatment of Electical Engineeing, Memphis, Tennessee 38152. eceived 24 Apil 199. 3-6935/91/192675-1$5./. 1991 Optical Society of Ameica. eticle cente. As the taget is moved fom the cente of the eticle, the eticle begins to chop the taget image and the modulation gows as the taget is moved towads the eticle peiphey. It can be seen that once the taget passes beyond the adius at which it ovelaps one secto, the modulation is at a maximum and emains constant to the eticle peiphey. It is not obvious, howeve, that the modulation goes though peaks and toughs as the taget is moved fom the cente of the eticle to the location of the modulation peak. The peak occu when the taget image ovelaps an odd numbe of sectos and the toughs occu when the taget ovelaps an even numbe of sectos. Conside taget location B whee the taget image ovelaps an even numbe of sectos. As the eticle spins, a white secto is leaving the taget aea and a dak secto is enteing the taget aea esulting in a lage change in tansmitted image light. Now conside the taget location C whee the image size coesponds to an even secto ovelap. A white secto is leaving the taget aea and a white ba is enteing the taget aea, minimizing the change in tansmitted light (modulation). If the modulation wee plotted as a function of adial taget location, a cuve simila to the one shown in Fig. 1(b) would be geneated. It can be seen that the modulation cuve depends upon taget size. Also, fo small tagets, o point souces, the AM configuation is useless in geneating an eo signal fo taget location since the modulation peaks at a vey small distance fom the eticle cente. A method fo obtaining an AM signal fo any size taget is pesented in this pape. The ability to gade the tansmission chaacteistics of the eticle mateials is assumed, but the gading of tansmission is feasible even in infaed mateials using halftone techniques. 4 With this assumption, spinning AM eticles can be descibed using thee amplitude paametes 1 July 1991 / Vol. 3, No. 19 / APPLIED OPTICS 2675
Taget Modulation SPIJ at Taget Size is Position Less A Position B Than One Secto Modulation I 1.,Taget g I \. Modulation Position C adial aget Taget Position C Location a. b. T_= 'i " - = Fig. 1. Classical amplitude modulation in eticles. (simila to FM paametes 5 ): amplitude vs angle, f(o), amplitude vs adius, g(), and phase, p(). The following sections descibe these paametes in detail. Afte the paametes ae pesented, a combination of the paametes ae shown to povide an eo signal useful in tacking tagets. The last section shows how AM can be combined with FM in eticles to povide impoved eo signals without the constaints of pue AM signals. II. Amplitude vs Angle FF Fig. 2. Fan blade eticle. mission function as taveses fom - to 7. If a point souce wee imaged onto the eticle and the eticle wee spinning in the diection shown, the eticle would modulate the point souce light at a ate of thity cycles pe eticle otation time. This fequency can be consideed the caie fequency k fo all the eticles pesented in this pape. Amplitude modulation can be descibed by the equation (t) Phase evesial Phase evesal.9.7.6 The amplitude vs angle paamete encodes azimuth taget location. Conside the fan blade eticle shown in Fig. 2. Thee ae thity angula cycles in the tans-.4.3.2-.1 2 4 6 wt a. eticle one. b. Spinning impulse esponse fo eticle one. _ c. eticle two. I(t) A.9.8.7.6-8=.5.4 3.~~~~... Wt, 2.1 2 4 6 Wt d. Spinning impulse esponse fo eticle two. Fig. 3. Two angula AM eticles. 2676 APPLIED OPTICS / Vol. 3, No. 19 / 1 July 1991
-e=o -1 eticle a. 1 77-77- -e=o -77 7 8 eticle b. _=o Fig. 4. Vaious amplitude vs angle eticles. eticle c. -1 e S(O) = 1/2 + V[1 + mf(o)] cos(ko), (1) whee s(o) is the modulated signal, Vis a constant, m is the modulation index, and f(o) is the low fequency modulation signal. Fo the eticle shown in Fig. 2, m is 1, f(o) is a constant 1, and V is 1/4. The 1/2 dc tem in Eq. (1) allows an aveage eticle tansmission of 1/2 athe than zeo (i.e., no light passing the eticle). Fo symbol convention, let us define and as the spatial vaiables of the eticle tansmission function with anges of to and -i to 7, espectively. Also, let the eticle spin ate be w ad/s and let, O be the spatial coodinates of a point souce that is imaged onto the eticle. Usually, a single detecto collects all of the modulated light of the taget though the spinning eticle. This light collected is known as the spinning impulse esponse and can be found by J [ I(t) = ' T(,O)A[- o, - (wt - 6d~d, -fo whee the impulse function epesents the spatial extent of the point souce with a light bightness amplitude of A. The tansmission function of the eticle shown in Fig. 2 is T(,) = 1/2 + 1/2 cos(3). (3) Noting that Eq. (2) is a 1-D coelation function, it can be easily applied to Eq. (3). The spinning impulse esponse of the eticle shown in Fig. 2 is (2) 1 July 1991 / Vol. 3, No. 19 / APPLIED OPTICS 2677
g() 1 e=o a. eticle one. b. Amplitude vesus adius fo eticle one. g() 1i-- 77. -8= c. eticle two. d. Amplitude vesus adius fo eticle two. Fig. 5. Two adial AM eticles. I(t) = A + A cos[3(wt - )]. (4) 2 Note the signal is not dependent on and that thee ae thity possible taget locations in angle that povide identical spinning impulse esponse signals. Fo an AM eticle to povide angula taget location, a vaiation in amplitude must be imposed on the eticle as a function of angle. That is, an amplitude vs angle paamete is fomed. One must be caeful in imposing this vaiation since it is easy to mistake the eticle equiements fo angula vaiations as being simila to FM eticles. Conside the eticle tansmission function T(,O) = /2 + 1/2 f(o) cos(3), (5) with f(o) = coso. The eticle coesponding to the tansmission function is shown in Fig. 3(a) with a spinning impulse esponse fo a point souce located at = shown in Fig. 3(b). Thee appeas to be two peiods of modulation in one otation of the eticle. The modulation envelope of the caie is the magnitude of the low fequency cosine function as shown in Fig. 3(b). Note thee is a phase change (black becomes white and white becomes black) that occus at ct = 1I2 and cot = 37/2. The phase change occus when coso becomes negative causing the amplitude of the caie to evese. Unless the tacking system electonics can utilize the phase change fo angula taget location, the eticle is not useful as an AM tacking eticle. Tagets sepaated by give identical spinning impulse esponses. Fo this eason, Eq. (5) is not a useful tansmission equation fo amplitude modulation. If amplitude modulation in useful tacking eticles is stictly descibed by Eq. (1), we can place two simple constaints on the equation vaiables. The magnitude of V[1 + mf()] cannot be >1/2 since this would cause the tansmission function to have a value >1. Second, mf() cannot have a value <-1 since phase evesal of the caie occus at this point. If phase evesal is not a poblem fo tacke electonics, this constaint can be elaxed povided the magnitude of f() does not contain a peiod <27 as in the case of the eticle shown in Fig. 3(a). To illustate the esults of these constaints, conside the eticle shown in Fig. 3(c). The tansmission equation fo the eticle is T(O) = 1/2 + /4 [1 + cos(o)] cos(3), (6) whee V is 1/4 and m is 1. The equation satisfies the constaints descibed above. Hence, the amplitude envelope contains one peiod pe eticle otation and no phase evesal occus. Defining f() descibed in Eq. (1) as the amplitude vs angle paamete will allow insight into AM eticles though a few examples. Conside the eticles shown in Fig. 4. eticle 4a includes a linea incease in modulation as vaies fom - to. V, m and f() fo the eticle ae 1/4, 1, and Oh, espectively. The eticle shown in Fig. 4(b) has anf(o) that is popotional to the 2678 APPLIED OPTICS / Vol. 3, No. 19 / 1 July 1991
g()1 T = it1-9=o. eticle a. g()= e 1 -,T eticle b. g()=- 1-2 -&= Fig. 6. Vaious amplitude vs adius eticles. eticle c. squae of the eticle angle. V, m, and f(o) fo this eticle ae 1/8, 3, and (/7) 2, espectively. The last example shown, eticle 4c, has a Gaussian amplitude vs angle paamete. Since the ange off(o) is-1 to and m is 1, the value fo V is set at its maximum constaint value of 1/2. It should be noted that f(o) fo useful tacking eticles can be any function with the limitations descibed in this section. These ae the minimum equiements fo a useful amplitude vs angle paamete. Othe consideations include image size, eticle fabication, and tacke electonics. Ill. Amplitude vs adius adial taget location is povided by the amplitude vs adius paamete, g(). Conside the tansmission equation T(,O) = 1/2 + Vg()[I + mf(o)] cos(ko), (7) 1 July 1991 / Vol. 3, No. 19 / APPLIED OPTICS 2679
a. eticle one. b. Phase fo eticle one. -8= c. eticle two. d. Phase fo eticle two. Fig. 7. Two phase AM eticles. wheeg() has a -1 ange. The tansmission function is now a function of both and. Also, g() weights the eticle modulation as a function of adius. Since this section descibes the adial paamete, let mf() equal one and let V equal 1/4 fo the eticles in this section. Now, fo illustative puposes, a linea incease in modulation as a function of adius will be imposed on eticle Eq. (7). That is, we will let g() equal. With a caie fequency of thity cycles pe otation, the tansmission equation becomes T(,O) = 2 + I cos(3). (8) 2 2 The eticle coesponding to the tansmission equation is shown along with its g() in Figs. 5(a) and (b). The one-to-one linea mapping of to g() allows an incease in the amplitude modulated signal of a point souce as its adial location inceases. Hence, one signal amplitude coesponds to one adial taget location. This one-to-one mapping is equied since the function is nonpeiodic contay to the case of f(). A case whee two adial taget locations can give the same amplitude modulated signal is shown by the eticle and g() function in Figs. 5(c) and (d), espectively. The amplitude vs adius function g() is a Gaussian function centeed on /2. Fo evey adial taget location on one side of /2, thee is anothe adial taget location on the othe side of /2 that coesponds to an identical amplitude modulated signal. Theefoe, the eticle would not be useful as a adial taget locate when coupled to a single detecto. To povide moe insight to the amplitude vs adius paamete, a numbe of examples ae pesented in Fig. 6. The eticle shown in Fig. 6(a) has an amplitude vs adius function that depends on the squae of the eticle adius. The eticle shown in Fig. 6(b) has a negative exponential function fo amplitude vs adius. Theefoe, the modulation of a taget towads the eticle cente is lage than the modulation of a taget nea the eticle peiphey. The last eticle example, shown in Fig. 6(c), has an inceasing exponential amplitude vs adius function. The modulation inceases shaply fom the eticle cente towad the eticle peiphey. IV. Phase The final paamete associated with AM modulated eticles is the phase function. In FM eticles, this paamete is often efeed to as the spoke function since it shapes the geomety of the eticle spokes (bas). The eticle tansmission functions pesented in this pape so fa have been witten with espect to some phase efeence. This phase efeence has been the O = line on the eticle. The phase function can be consideed the line fom which the eticle tansmission function is efeenced. This line changes in angle as a function of adius. The phase, p(), is the line whee the aguments of the angula functions ae zeo and T(,O) = 1. 268 APPLIED OPTICS / Vol. 3, No. 19 / 1 July 1991
7/3 8 7_ -9= eticle a. 7/3 7-4(/) '=3 7 86 t7-8= eticle b. piecewise continuous 7- -8= Fig. 8. Vaious phase eticles. eticle c. The pupose of imposing a phase on a eticle is that it changes the eticle ba shapes in such a manne that the eticle can coelate to vaious images well and othe images pooly. Fo instance, a eticle can be designed so that it coelates with small tagets well and coelates with lage tagets pooly. This type of eticle would be useful in tacking small objects in cloud clutte. Conside the eticle shown in Fig. 7(a) along with its phase function shown in Fig. 7(b). The eticle has a cosine amplitude vs angle function with a linea phase. The cosine amplitude vs angle paamete was etained so that the vaiation in the angula modulation can be seen to follow the same cuves as that of the caie spokes. That is, the modulation in angle cuves aound in the same manne, as the eticle bas. To keep this in mind, all of the eticles pesented in the Phase Sec. will have a cosine amplitude vs angle pa- 1 July 1991 / Vol. 3, No. 19 / APPLIED OPTICS 2681
1 f(8) a. eticle. 1, weo -1.,..., 7 b. Apltude vesus angle. 9. () I c. At4ude vesus adius... n_ i. d. Phase. Fig. 9. eticle with two nonconstant AM paametes. amete. Since the amplitude vs adius paamete is not affected by phase, the paamete is not imposed on the eticle hee. To visualize the phase function of eticle 7a, conside the phase gaph of Fig. 7(b). When the eticle adius is zeo, the phase shift in angle is zeo and the eticle is the same as the eticle shown in Fig. 3(c). Now, when the adius is /2, the eticle tansmission function is shifted by /(12) and the bas cuve to meet the T(,O) = 1 o + p() = point at the adius. At a adius equal to, the tansmission function has cuved to 7/6. In compaison, the eticle shown in Fig. 7(c) has a squaed phase function as shown in Fig. 7(d). The squaed function effect can be easily seen since the bas towad the cente of the eticle do not cuve as much as towad the peiphey of the eticle. The phase paamete can be imposed on the eticle in the following equation: T(,O) = 1/2 + Vg()[1 + mf(a) cs(ka)ia=o+p(). (9) Fo the eticles shown in this section, V is 1/4, g() is 1, m is 1, and f() = coso. To illustate the impact of the phase function on the eticle, a few examples ae shown in Fig. 8. The fist eticle, Fig. 8(a), has a Gaussian phase function. A one-to-one mapping is often not of concen in phase functions since enhancement of the coelation of the spoke edge with taget objects and suppession of the coelation of the spoke edge with backgound objects ae the usual design goals. The eticle shown in Fig. 8(b) has a negative exponential phase function. This type of phase function may be useful in giving good coelation with staight line tagets nea the eticle peiphey while disciminating against staight line tagets nea the eticle cente. The eticle shown in Fig. 8(c) may be useful in filteing clouds, hoizons, and othe lage objects while maintaining a good coelation with small objects. All of the eticles pesented in this section have some type of phase paamete. With the insight into phase functions, it can now be shown that the thee amplitude paametes can be combined using Eq. (9) to constuct an AM eticle tansmission function. V. Combinations of AM Paametes Equation (9) can be used to constuct an AM eticle with multiple paametes. That is, a eticle tansmission function can be developed that detemines both angula and adial taget locations while coelating well with tagets and pooly with backgounds using amplitude modulation. The envelope function can be any signal desied with the simple constaints descibed in the pevious sections. Once the thee paametes ae known, the eticle tansmission function can be witten. Two examples ae pesented in this section that show combinations of AM eticle paametes. The fist example is the eticle shown in Fig. 9. The eticle has a cosine amplitude vs angle paamete and a linea amplitude vs adius paamete. The phase is held at a 2682 APPLIED OPTICS / Vol. 3, No. 19 / 1 July 1991
- 1 NO) -1-7 8 a. eticle. b. Ampitude vesus angle. 71-3 Fig. 1. eticle with thee nonconstant AM paametes. c. Ampitude vesus adius. d. Phase. constant zeo value. The tansmission function fo the eticle was found using Eq. (9) to be T(,O) = 2 + - [1 + cosol cos(3). (1) 2 4 In this case, taget location is found by detemining the phase offset of the cosine envelope fo angula location and the magnitude of the AM envelope detemines the adial taget location. Theefoe, the eticle is useful in detemining taget location. The detecto output due to a point souce located at (o,o) imaged onto a spinning eticle with the tansmission function descibed would be Eq. (9) with (,O) eplaced by (owt - Oo) The second example of AM paamete combinations is shown in Fig. 1. The eticle is a combination of thee nonconstant AM paametes. The amplitude vs angle paamete is linea, the amplitude vs adius paamete is one minus a negative exponential, and the phase is Gaussian as shown in Figs. 1(b), (c), and (d), espectively. The angula taget location is found by the phase offset of peiodic amp envelope and the adial taget location is found by the magnitude of the amp envelope. The phase decoelates the eticle with staight lines, especially nea the /2 adial location. The eticle tansmission function was found again using Eq. (9) to be T(,O) = + 1 1- exp(2 )] {1 + [ + p()} X cosj3[ + p()], (11) whee p() = 3 exp[-4 (- and b was15. Thee ae impotant consideations in designing an AM eticle that must be consideed befoe selecting the AM paametes. A few of these consideations ae taget size, backgound geometies, electonics bandwidths, amplitude nomalization, and othes. Howeve, once the AM paametes ae detemining using these consideations, the eticle tansmission function can be found diectly using Eq. (9). VI. Combination of FM and AM Paametes An altenative to the use of both angula and adial AM paametes involves a combination of AM and FM eticle paametes. Conside the case whee the amplitude vs angle paamete is etained and the amplitude vs adius paamete is eplaced with a fequency vs adius paamete. The phase paamete affects AM and FM eticles in the same manne. The use of this type of configuation allows the elimination of the magnitude of the AM envelope as a taget location device. A change in taget fequency (i.e., caie fequency) will coespond to a change in taget adial location. The angula taget location is still detemined by the phase of the AM envelope. Conside the eticle shown in Fig. 11. The caie 1 July 1991 / Vol. 3, No. 19 / APPLIED OPTICS 2683
give a numbe of ambiguous taget locations equal to the numbe of cycles pe eticle otation at the taget adius. Howeve, since an appoximate angula location has been detemined, the high esolution caie phase location used will coespond to the appoximate angula location. The angula location esolution fo the same electonic phase disciminato has inceased by the numbe of cycles pe otation at the taget adius. Fo example, fo a taget adius of /2 fo the eticle shown, the angula location esolution has inceased by a facto of 15. Fig. 11. eticle with AM amplitude vs angle paamete and FM fequency vs adius paamete. fequency inceases linealy with adial taget location and the amplitude vs angle paamete vaies as a cosine. The phase imposed on the eticle is a constant (zeo). The tansmission function fo the eticle is T(,O) = 2 4(1 + cos) Cos3 ). (12) The eticle appeas to be extemely useful in that an inceased accuacy can be obtained in the following manne. Once a point souce is imaged onto the eticle, the adial taget location can be detemined fom the value of the caie fequency. A ough angula taget location is then detemined by compaing the phase of the amplitude modulated envelope to the eticle efeence signal (choppe electonic signal o LED efeence). The esolution in this ough angula taget location is detemined by the accuacy of electonic phase disciminatos. Now, since the caie fequency is known, the phase of the taget caie fequency is compaed to the efeence signal. It is known that using the phase of the caie signal will VII. Conclusion It has been shown that AM spinning eticles can be descibed using the thee amplitude modulation paametes: amplitude vs angle, amplitude vs adius, and phase. The amplitude vs angle paamete, the amplitude vs adius paamete, and the phase paamete give angula taget location, adial taget location, and coelation, espectively. Once the paametes have been detemined, a eticle tansmission function can be developed using Eq. (9). Also, it has been shown that amplitude modulation can be combined with fequency modulation in eticles to povide a signal without amplitude nomalization while inceasing angula taget location esolution. efeences 1. T. Buttweile, "Optimum Modulation Chaacteistics fo Amplitude-Modulated and Fequency-Modulated Infaed Systems," J. Opt. Soc. Am. 51, 111-115 (1961). 2. L. M. Bibeman, eticles in Electo-optical Devices, (Pegamon, New Yok, 1968). 3.. G. Digges, C. E. Halfod, G.. Boeman, D. Lattman, and K. F. Williams, "Paametes of Spinning FM eticles," Appl. Opt. 3, 887-78 (1991). 4. J. Lee, "Devices fo Optical Signal Pocessing," Opt. News Vol 11 # 1 pg 22-26 (1985). 5. P. E. Menges and K. B. O'Bien, "Analysis of Eo esponse of Amplitude Modulated eticles," J. Opt. Soc. Am. 668-671 (1964). 2684 APPLIED OPTICS / Vol. 3, No. 19 / 1 July 1991