CASASENT, D.

PSALTIS, D.

Infoscience - École polytechnique fédérale de Lausanne

Volume 27, number 3 OPTICS COMMUNICATIONS December 1978 DECIMAL/RF_.SIDUE/DECIMAL OPTICAL CONVERTERS David CASASENT and Demetri PSALTIS Department of Electrical Engineering, Pittsburgh, PA 15213, USA Received 12 June 1978 Revised manuscript received 21 August 1978 An optical numerical processor using residue arithmetic sdescribed. Specific designs and experimental demonstrations of decimal-to-residue andresidue-to-decimal converters u ing a coherent optical processor are presented. 1. Introduction Residue arithmetic [1-3] appears to be an attrac- tive scheme for realizing a high speed numerical pro- cessor. Huang [4,5 ] was one of the first to suggest fa- brication of an optical residue arithmetic processor. One recent optical approach [6] to such a system uses ~he phase or polarization of a spatial point as the cy- clic variable. Another approach [7] uses pulse-position modulation to represent numbers and achieves the ne. cessary cyclic permutation by use of maps. Many pos- sible realizations of optical residue arithmetic systems using integrated optical switches, planar waveguides, etc. have been suggested [7]. In this paper we address the design of a decimal/residue and residue/decimal converter. We present acorrelation formulation of the required operations rather than expressing them by maps [7]. This approach leads directly to 2-D optical processor designs that can utilize the large available space bandwidth product of optical systems. In residue notation, we represent a decimal num- ber X by its N residues Rmi moduli m i (where i = 1, . . . .  N). All computations in residue arithmetic an thus be divided into sub-computations i  each of N moduli. Since there are no carries in residue addition, subtraction and multiplication, all subcomputations can be performed in parallel. Such systems are thus di- rectly implementable in a parallel multichannel opti- cal processor. Furthermore, the maximum decimal number that can be represented in residue notation equals the prod- uct of the moduli. However, since a given computa- tion is divided into sub-computations i  the N moduli mi, the dynamic range of each sub-computation is greatly reduced with an associated reduction in the ac- curacy required. Since the accuracy and dynamic range of an optical processor are often questioned, an optical residue arithmetic system is an attractive way to reduce the required ynamic range and improve the accuracy of the overall system. In residue arithmetic, this is achieved at an increased system space band- width (SBW). However, since the available SBW of an optical processor can rarely be fully utilized, this re- presents no limitation in practice. For these many rea- sons optical residue arithmetic systems have recently received considerable attention [4-7]. 2. Residue arithmetic For completeness, webriefly review residue arith- metic [1-3].  An integer X is represented in a residue number system by the N-tuple set of residues (R m t' • .. RmN) with respect to N different moduli m i. The residues Rmi are the least positive integer emainders of the quotient of X and m i. To represent 13 in a resi- due system with moduli (2,5,7,9), we divide 13 by the first modulus 2 and obtain 6 and a remainder or resi- due of 1. We proceed similarly to divide 13 by the other moduli and find 13 = (1,3,6,4). Similarly 59 = (1,4,3,5) in the same moduli. To add these two numbers in residue, we add each residue number with respect to its respective modulus. We retain only the residue of the sum with respect to 317 Volume 27, number 3 OPTICS COMMUNICATIONS December 1978 to the corresponding modulus. For example 13 + 59 is mi= (2,5,7,9) 13 = (1,3,6,4) 59 = (1,4,3,5) 72 = (0,2,2,0) where for the far right residue number, 5 + 4 --- 9 or in modulo 9 this is 1 with a residue or remainder of 0. The decimal sum 13 + 59 = 72 is thus found to be (0,2,2,0) in residue notation with moduli (2,5,7,9). To subtract wo numbers in residue, we invert the subtrahend by complementing each residue number with respect o its modulus m i. We then add these two numbers as before. I 'o multiply two numbers in resi- due notation, we multiply each residue number and retain only the residue with respect o the correspond- ing modulus. As seen in all cases, no carries occur, each sub-computation is of vastly reduced dynamic range, and all sub-computations can be performed in parallel. Thus, the motivation for considering an opti- cal residue arithmetic processor are apparent. To perform computations optically in residue nota- tion, we choose to denote decimal and residue num- bers in pulse-position otation by the horizontal posi- tion of a pulse of light as Huang [4,5] and others [7] have chosen to do. Alternate representations involve dertoting a number by the phase or polarization of a spot of light [6]. The conversion scheme we propose is based on a correlation formulation of residue arith- metic and is thus more directly realizable in other technologies besides coherent optics. We describe spe- cific residue arithmetic architectures using coherent optical systems because such systems are readily real- izable with present echnology and because of the par- allel processing and real-time high throughput of an optical processor. , ,G .T  GRATINGS ~ad p, i ,¢  L,s ",R L,c L,s P, " Fig. 1. Schematic diagram of a decimal-to-residue optical con- verter. 3. Decimal-to-residue conversion We can represent any decimal integer X by X = nm i+ Rmi , where n is the number of times X is divisible by m i and R is the remainder or residue for the modu- lus m i. Solving for the residue, we obtain Rmi = X-nm i. To realize this conversion from decimal-to- residue, m i must be subtracted n times from X until the remainder satisfies 0 ~< Rmi <~ m i - 1. Multiplying these last two expressions by Ax (unit distance in decimal and residue space) we obtain Rmi Ax  = (X - nmi )Ax  , (1) 0 <~ Rmi Ax  <~ (m i - l )  Ax. (2) Refer to the decimal-to-residue optical converter of fig. 1. Lenses L 1 and L 2 Fourier transform horizontal- ly and image vertically. The input to this system (on one vertical channel) is the decimal number X which, in pulse-position otation, is g(xo) = 6 (x 0 - XAx) .  (3) We describe the system's impulse response for channel /by  h (Xo) = ~ ~ (x 0 -nm i Ax).  (4) ?/ This is easily realized by a grating at P1. The system's output is then g ® h = ~f6(xo-XAx  ) 6 (xo ,nmi+xz)dx  0 n - -  = ~ 6 Ix 2 - (X -nmi )  Ax l .  (5) n From (5), we see that at P2 of fig. 1 we find delta functions at x 2 = (X -nmi )  Ax. The locations of these delta functions correspond to the various residues pos- sible as described by (1). If we place a rectangular aperture properly centered at P2 of fig. 1 and of length miAx,  we can select the desired minimum positive x 2 value satisfying 0 <~ x 2 ~< (m i - 1) Ax, which is identi- cal to (2). Thus the output within this P2 aperture will con- sist of a peak of light whose location (in one of m i possible positions) within the aperture denotes the desired residue number Rmi modulo m i. As depicted in fig. 1, to convert X to N residue numbers Rmi 318 Volume 27, number 3 OPTICS COMMUNICATIONS December 1978 (where N moduli m i are assumed), the input at P0 is a slit, N gratings at frequencies u~ satisfying miAx = UgiXfL (6) are recorded on N channels at PI, and N channels ap- pear at P2 with the position of the peak of light on each channel (within the P2 aperture) denoting the residue of X for the modulus m i of the particular channel. 4. Experimental demonstration tween channels. This input pattern represents he dec- imal numbers 0 to 20 on the 21 channels from bottom to top in fig. 2a in pulse position coding. At P1 of fig. 1 we place a grating of 10 cy/mm spatial frequency. With X = 633 nm, fLe = 300 mm and fLs = 762 mm fo- cal length lenses, (6) is satisfied for a modulus m i = 7. The output P2 plane of fig. 1 is shown in fig. 2b. It contains 21 channels. The location of the output peak of light (in one of 7 possible locations) on each of the 21 channels represents he residue number Rmi mod- ulo 7 for the corresponding decimal input number. As the first experimental demonstration f optical residue arithmetic, we consider the conversion of the 21 decimal numbers 0 to 20 to residue with modulo 7. We represent the input P0 in fig. 1 as shown in fig. 2a by 21 channels with a single pulse on each channel and an equal horizontal separation Ax = 0.7 mm be- Fig. 2. Experimental demonstration f the decimal-to-residue conversion ofthe decimal numbers 0 to 21 to residue modulo 7 using the optical system of fig. 1. (a) Input plane Po pattern, (b) Output plane P2 pattern. 5. Residue4o-dedmal conversion The system of fig. 1 can also be used to perform residue-to-decimal conversion. In this case, the input PO pattern consists of N channels with a pulse of light on each channel at a horizontal location correspond- ing to the residue number Rmi with Rrn i for different moduli recorded on different channels. At P1, we place N gratings on N lines with the grating frequen- cies corresponding to the N moduli m i as in (6). At P2, we again find N channels of delta functions. The hori- zontal ocations of the delta functions on each chan- nel now denote the various decimal numbers to which the input residue number could correspond. At one horizontal location in P2, all channels will have out- put peaks of light. This horizontal location denotes the decimal equivalent of the input set of residue num- bers. The output detection is easily accomplished by an N input NOR gate fed by a set of N horizontally oriented linear self scanned photodiode CCD arrays. 6. Discussion The pulse-position modulation scheme used re- quires an input and output space bandwidth of M for the decimal/residue and residue/decimal converters e- spectively. With M limited to 1000 for most practical spatial light modulators, the system will be severely limited. Several solutions to this problem exist. First- ly, the use of input data in binary rather than decimal representation would accomodate an input dynamic range of 64 000 for 16 bits. Secondly, by raster e- cording the input data, M values of 64 000 could be ac- comodated by allowing 64 lines of 1000 points each 319 Volume 27, number 3 OPTICS COMMUNICATIONS December 1978 to represent an input number. Thirdly, the decimal in- put can be used to control an oscillator's frequency which is raster ecorded. In the 2-D Fourier transform of such a signal, a folded spectrum results with coarse and fine frequency axes. The location of the output peak along the free frequency axis is the desired resi- due modulo the input horizontal line scan rate. In this case, a set of gratings is not required. Which of the ap- proaches is best in the design of an optical residue arith- metic processor can best be determined by application to a specific problem. Acknowledgement References [1] H.L. Garner, The residue number system, IRE Trans. Electron. Comput., EC-8 (1959) 140. [2] N.S. Szabo and R.I. Tanaka, Residue arithmetic and its applications to computer technology (New York, McGraw- Hill, 1967). [3] A. Svoboda nd M. Valaeh, in: Digital information proces- sor, ed. W. Hoffman (John Wiley and Sons, Inc., NY, 1962). [4] A. Huang, IOCC Conf., Washington, D.C., (April 1975) p. 14. [5] A. Huang, Proc. EOSD, Anaheim, (Oct. 1977) p. 208. [6] S. Collins, SPIE 128 (1977) 313. [7] A. Huang, Y. Tsunoda nd J.W. Goodman, Appl. Opt., submitted (1978). The financial support of the National Science Foundation on Contract ENG 77-20038 for the work reported herein is gratefully acknowledged. 320 

Decimal-Residue Decimal Optical Converters

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Decimal-Residue Decimal Optical Converters

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