AN OPTICAL MULTIPLICATION BASED ON A QUADRATIC RESIDUE NUMBER SYSTEM Bo Liu, Liren Liu, and Guoqiang Li Shanghai Institute of Optics and Fine Mechanics Academia Sinica

Shanghai 201800, China P.O. BOX 800-21 1

KEY TERMS Optical computing, quadratic residue number system, matrix multiplica- tion.

ABSTRACT Based on the quadratic residue number system {QRhS), a novel optical matrix-uector multiplication is proposed. Complev number multiplication has higher parallelism than other algorithms. If the residue number is binary encoded, this multiplication can be easily implemented with a multichannel incoherent optical cornlator. Q 19% John Wley & Sons, Inc.

1. INTRODUCTION Matrix-vector multiplication is a basic mathematics operation widely used in signal-processing fields such as a linear pro- gramming, linear equation solving, and image processing. It is well known that the propagation properties of light can be used for signal processing. Because of its innate characteris- tics of high parallelism and huge free-space interconnection, optical matrix operation is of vital concern. Many efforts have been made to perform matrix-matrix or matrix-vector multi- plication with optics. A wide variety of analog optical systems [l , 21 have been developed for this purpose in which matrix and vector element information has been encoded using conventional analog representation. They have high paral- lelism and speed, but the accuracy of the analog system is limited to approximately 10 bits under the best conditions. However, methods that retain the high degree of parallelism obtainable from optics but offer improved accuracy and dy- namics range are of great interest. Many digital optical com- puting systems, such as the residue number system (RNS) [3-61, binary representation [7-lo], redundant representation [ll-131, and symbolic substitution [14, 151, have been devel- oped to increase the limited accuracy. During these represen- tations, the RNS, which is a trade-off between the analog and digital optical computing, has for a long time been an attrac- tive subject studied by many researchers, because it has two merits well matched to optical systems: First, there is no carry mechanism needed in residue arithmetic, which allows all the computations to run in parallel, and second, the RNS decom- poses a calculation into many subcalculations of smaller computational complexity.

In many fields, such as sonar, adaptive phase array radar, and image pattern recognition, complex matrix multiplication should be a concern. So it is very important to consider how to solve complex matrix multiplication effectively. Conven- tional complex multiplication algorithms of two complex numbers requires the formulation of real and imaginary channels and the special handling of cross-product terms. It needs four multiplications and four additions or subtractions, including two cross multiplications of the real and imaginary parts of the complex numbers. This lowers the parallelism of the operation seriously. One approach [16], that uses a bi-

imaginary number system with the base i f i has been pro- posed, but this method needs many more channels.

Recently it has been shown that certain types of modular number systems [6, 171 admit an unusual representation of complex data, which leads to a complete decoupling of the real and imaginary channels, thereby simplifying complex multiplication. However, this method [61 is performed by an optical systolic structure that decreases the parallelism and is difficult to perform, for it is based on look-up truth tables.

In this article, the QRNS is applied to optical matrix multiplication based on a common multichannel incoherent optical correlator [ 181. The residue has binary representation, so it is easy to implement by optics in parallel.

2. QUADRATIC RESIDUE NUMBER SYSTEM

A. Introduction to Finite Rings. In the RNS we deal with rings that are used for the actual implementation and rings that are isomorphic to direct products of implementation rings or extensions of them. We define the ring of computation in a residue number system by

where the symbols @ and 8 correspond to module mk addition and multiplication, respectively.

B. Introduction to RNS. It has been shown theoretically that residue arithmetic is the fastest way to perform addition, subtraction, multiplication, and polynomial transform. A number X is represented by an N-tuple of residue:

x= (XI? x2,. . . I X k ) ,

where xi = (XI,,,, is the ith residue and m, is the ith module. Closed computations (addition, multiplication) over the implementation rings map to the direct product ring via the Chinese remainder theorem (CRT):

A g B - {(al Q bl)ml , ( 0 , @ b2Im2, . . . , ( a N 8 bNImn) ,

A E B cJ { ( a , 8 bl )mi , (a2 8 b ~ ) ~ 2 , . . . , @ b ~ ) ~ ~ ) ,

with A , B, R(M); ui, bi, R(mk), and R ( M ) = n R ( m k ) (di- rect product).

The final mapping is found from the CRT:

with mk = M / m k , x E R ( M ) , and xk E R(mk). (a)-' is the multiplicative inverse operator.

The isomorphic (=) between R ( M ) and the direct prod- uct of (R(rnk)) means that calculations over R ( M ) can be effectively performed, over each R(mk), independently and in parallel. A final mapping (i.e., the CRT) to R ( M ) is carried out at the end of a chain of calculations. Therefore, we have broken down a calculation set in a large dynamic range M to a set of N calculations set in small dynamic ranges given by {mi}. This is the main advantage of using the RNS over a

274 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 11, No. 5, April 5 1996

conventional weighted-value number system. Therefore, opti- cal computing by RNS is a very good trade-off between analog optical computing and binary-coded optical comput- ing.

C. Quadratic Residue Number System. Recall that the ordi- nary complex number system results because the equation X 2 = - 1 does not have a solution among the real numbers. Similarly, to form a QRNS structure, it is necessary to deter- mine if (1 ) has a solution in R(m) = (1,2,. . . , m - 1):

X 2 = (-1)modm. (1)

If there exists a solution j to (l), j E R(m), then -1 is a quadratic residue mod m.

THEOREM. The number - 1 is a quadratic residue mod p of all primes of the form p = 4k + 1 and a quadratic non- residue modp of all primes of the form p = 4k + 3.

By the theorem, we select a particular set of modular m,, m2, . . . , mN with m, = 4k + 1, so that (1) has a solution j E R(Mk) , and - 1 is a quadratic residue mod m k .

Now we define the quadratic ring (QR) and the direct producf QR as follows:

quadratic ring: QR(mk) = {s: @ , 81;

S = ( x , x * } ,

direct product quadratic ring:

QK(M2) = {S:@-,@; M = n m , .

with A , A* E R(m,) ,

N

k = 1 s = { ( X 1 , x y ) , ( X 2 , X ; ) , . . . I ( x N , x ; ) l ;

Under these conditions, the ring R ( M 2 ) can be mapped onto a quadratic ring QR(MZ) that is isomorphic to R(m2):

where

and

It is essential that j E R(Mk) , so that both x and x* are real numbers that form an ordered pair ( x , x * ) to properly repre- sent ( x , x*> in the quadratic ring QR(m2).

Addition and multiplication within QR(m2) are defined by

( x , x * > o ( y , y * ) = ( z , z * ) , ( 5 )

where

z = ( x 0 y ) mod ink,

z* = ( x * oy*)modm,, (6 )

and where the circle denotes either addition or multiplica-

tion. In the quadratic residue number system, addition and multiplication have the same simple form as addition in conventional complex arithmetic. Both are computed by inde- pendent operations in the uncoupled channels. The elimina- tion of cross-product terms in the complex multiply suggests high parallclism, high speed, and reduced hardware complex- ity.

For example, we are given two complex numbers A = A, + jA, and B = B, + jB,, and a set of modular m,, m2,. . . , mN. To get the product C of A and B, we do the following steps.

1. Get the RNS representations of A and B with respect to ( m l , in2,, , . , mN):

where A,, = A, , =

Convert the residue representations of A and B to the QRNS:

A * I ( q , a T ) , ( u z , a ; > , . .., ( a N , a ; ) } ,

B ( ( b , , b T > , ( b , , b ~ ) , ..., ( b N , b s ) } ,

where = (Ark +jkALk)ml and a% = (Ark - j kAIJmk with j , can be obtained from Eq. (1). Perform the two complex number calculations with only component wise multiplication by (6)

C = A B ! ( c I , c y > , ( c 2 , c ~ > ,..., ( c N , c $ ) } ,

where c k = (Ukbk)mod m,; C z = (a2b:)mod m,. Now rcverse the QRNS decomposition, operating on the elements of the QRNS representations of C by (3)

C = {(Crl + jCl1) , (Cr2 + jCc2) , . . . , (CrN + j C t N ) l .

We decode C by CRT:

C = (C, + jC,).

3. OPTICAL IMPLEMENTATION

For demonstration, we select rn, = 5, m2 = 13 ( N = 2), and the matrix

vector

From Eq. (11, we get

Now perform the QRNS map on the real and imaginary parts

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 11 No. 5, April 5 1996 275

incohcnnt optical codator I

Figure 1 Optics-electronics matrix multiplication configuration

of the elements of A and B, respectively, as dictated by (4):

and now encode the QRNS representations by binary to get

(0010,0010) (0011,0001) (0001,0010) (0011,0100)

(0001,0011~ (0100, oooo> (1001,1100) (0110,1001)

(0000,0001~

(1000,1011)

! !

A, , =

AIBB =

An incoherent optical correlator, which is shown in Fig- ure 1 and can easily perform convolution and additions, is used to implement the complex matrix-vector operation.

In this configuration, two masks feed the elements of matrix A and vector B, respectively. To implement the component wise additions and multiplications by the correla- tor, the data arrangements in the two masks are shown in Figure 2, where the closed circles represent 0 and the open circles represent 1. If d, is the interval between the two adjacent coded bits of the two masks A and B, dr is the distance between the correlation intensity points of scatter, L is the distance between A and B, and f is the focal length of the lens. then

df = d , f / L .

By such a configuration, all the operations can be carried out in parallel.

Figure 3 shows the theoretical intensity distributions of the detector and the experimental results are shown in Fig-

3 2 1 0 3 2 1 0 3 2 4 0

I I I

Mask A Vector B Figure 2 Data arrangement of two masks

654321 0 654321 0

Figure 3

ure 4. In the

Theoretical intensity distribution of the scatter

Figure 4

experiment, a photodetector has been used to receive the output intensity, which is a fixed binary represen- tation. A PC does the preprocessing and postprocessing. The diameter of each point of the masks is 1 mm, d, is 1.5 mm, and f is 240 mm.

4. CONCLUSION

This article introduced a new code method for optical matrix multiplication, especially for complex numbers. The method is a good trade-off between analog and the digital optical computing. It effectively eliminates the cross product of the real and imaginary parts in the multiplication of complex numbers. Optical implementation is also presented.

ACKNOWLEDGMENT

The authors appreciate the support of the National Natural Science Foundation of China.

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Received 9-5-95; recised 10-28-95

Microwave and Optical Technology Letters, 11/5, 274-277 0 1996 John Wiley & Sons, lnc. CCC 0895-2477/96

EXPERIMENTAL RESULTS ON THE FOCUSING PROPERTIES OF TAPERED FREQUENCY-SELECTIVE SURFACES Sathish Chandran Department of Electrical and Electronic Engineering University of Nottingham Nottingham NG7 2RD United Kingdom

John C. Vardaxoglou Department of Electronic and Electrical Engineering Loughborough University of Technology Loughborough, Leicestershire LEI 1 3TU United Kingdom

KEY TERMS Frequency-selectwe surfaces, focusing, tapering

ABSTRACT The properties of frequency-selectiue surfaces (FSS) exploited in antenna systems are well known. The purpose of this article is to report some experimental obsemations made on the focusing properties of FSS by tapering the lengths of the array elements. Two FSSs with different tapers are used to study this effect. Both surfaces show increased gain oLw a range offrequencies. Mass reduction in the lens structure is thus obtained here by uurying the element size of the FSS. 0 1996 John Wiley & Sons, Inc.

INTRODUCTION

Antenna systems that use lenses play an important role among the different types of antennas that have been devel- oped for use at various frequency bands [l, 21 . Lenses have been used in conjunction with microwave horn antennas for a considerable period [3]. An attempt is made here to use passive filters like frequency-selective surfaces (FSS) by taper- ing the lengths of its array elements. An advantage of these FSSs compared to the existing form of lenses, such as zoned lenses, is that the physical size of the lens structure can be reduced. An FSS can be made to act as a diverger by appropriately selecting the element geometry. Certain struc- tures help to changc thc direction of an incoming electromag- netic wave to other directions by proper choice of element geometry [4]. In our study two FSSs with different tapers are used for the experimental evaluation.

FOCUSING ACTION OF AN FSS

The focusing action that uses an FSS is discussed in this section. It has been shown that regular arrangements of periodic elements can be viewed as artificial dielectrics or phase-shifting structures [S]. A plurality of elements or an FSS can be used to introduce a phase shift in the transmittcd portion of the radiation incident on the array. The phase shift can be made stepped or progressive, by altering the geometry or dimensions of the array elements used. Therefore, if an array with phase shifting properties varying spatially across its surface is used, incident radiation can be influenced in such a way to beam form, beam direct, filter, or converge the inci- dent wave.

The basic principle involved in creating a newly formed wave front is to introducc symmctrical or antisymmetrical phase advances, incrcasing in relation to thc spatial distance from the axis (center) or from the perimeter of the array. The amount of beam shaping near the lossless region of the band depends on the variation of the transmission coefficient (am- plitude and phase) to the local angles of incidence that are present on the structure.

The lengths of the slot arms are varied to obtain the phase change to produce a focused beam. The lengths of the slot arms are varied, as shown in Figure 1. The distance between the centers of adjacent clcmcnts rcmains thc samc. Thc lengths of the slot arms can be increased or decreased according to the requirements. To produce a converging lens, the lengths of thc slot arms are dccrcascd as they movc from thc ccntcr to thc cdgcs and vicc vcrsa.

EXPERIMENTAL RESULTS

Figure l(a) shows a symmetrically spatially varying slotted crossed dipole array. The choice of the crosscd dipolc gcomc- try is justified by the flexibility in tapering in both the horizontal and vertical directions. There is taper in both the vertical and horizontal arm lengths, from the left to the right as well as from the top to the bottom of the surface. The taper rate is chosen as 0.5 mm. The dimensions were as follows: LHI = LVI = 4.15 mm, LHR = 3 mm, LHF = 2 mm, and LVF = 3.6 mm. There was a 33% reduction in the vertical arm. The periodicity between elements is 4.6 mm and the width of each slot arm is 0.2 mm.

A different type of symmetrical arrangement of the array elements can be seen in Figure l(b). This formation of elements is axisymmetric, where square-loop zones increase

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 1 1 , No. 5, April 5 1996 277