Synthetic optical processor for generalized logic realization problems

Optics and ¸asers in Engineering 29 (1998) 485—497( 1998 Elsevier Science Ltd. All rights reserved

Printed in Northern Ireland0143—8166/98/$19.00

PII: S0143–8166(97)00041–9

Synthetic Optical Processorfor Generalized Logic Realization Problems

Farid Ahmed,a,* Abdul A. S. Awwalb & Mohammad A. Karimc

aElectrical Engineering, Pennsylvania State University, The Behrend College, Erie, PA, USAbWright State University, Computer Science and Engineering Department, Dayton, OH, USA

cUniversity of Dayton, Center for Electro-Optics, Dayton, OH, USA

(Received 23 December 1996; accepted 22 May 1997)

ABS¹RAC¹

A synthetic joint Fourier transform (SJF¹) correlator is proposed for therealization of generalized logic problems. As a case study, a multi-channelSJF¹ correlator is employed to realize a multi-output logic unit. For a givencorrelator set-up, this particular scheme ensures the maximal utilization ofspace bandwidth product. ¹his work establishes a pragmatic approach forimplementing real-time programmable content addressable memory for in-formation processing. ( 1998 Elsevier Science ¸td. All rights reserved.

1 INTRODUCTION

Due to the free-space communication and parallel processing capabilities1 ofoptics, optical computing has attracted much attention lately. Different typesof optical techniques have been proposed so far for the realization of arith-metic and logical operations. In particular, truth table look-up employingholographic optical content addressable memory (CAM) is one of severalpromising approaches for optical logic realization.2,3 Its real-time implemen-tation, however, is limited by the number of holograms needed for the system.In addition, polarization-encoded optical shadow-casting (POSC) system hasbeen used for designing binary multiprocessor.4 However, it requires polariz-ation encoding of the inputs, which is not necessarily too trivial to achieve.

Joint Fourier transform (JFT) correlation5 is widely used for real-timeoptical matching of two spatial domain signals. The JFT technique results ina flexible optical set-up that can be easily programmed.6 In addition, different

*Author to whom correspondence should be addressed.

485

types of nonlinear processing 7—11 have also been proposed to make a JFTcorrelator more robust. Recent reported works on JFT correlator for logicimplementation,12,13 however, suffer from suboptimal utilization of thespace-bandwidth product and not-so-attractive correlation discrimination.Here, multiple minterms1 corresponding to the logic function serve as thereference image while the input minterm is loaded as the input image of theJFT correlator. The output correlation peaks when thresholded determine thelogical value (1 or 0) of the Boolean function. The available space-bandwidthproduct in a JFT correlator is normally shared by the input image, referenceimage and a safety band to ensure proper separation of the correlation terms.In this paper, we show how a substantial improvement of space-bandwidthutilization can be obtained using a superposition scheme for generating thereference image and a simple dual-rail code for encoding the minterms. Thecorrelation discrimination measure, on the other hand, is improved by the useof a synthetic reference constructed from the corresponding phase-only filter(POF)14 or the amplitude-modulated phase-only filter (AMPOF).15 Thesmaller space-bandwidth requirement of the resulting system, in effect, leadsto the realization of a multi-channel JFT correlation for multi-output logicimplementation. One obvious concern is the utility of this approach whenthere are very sophisticated VLSI fabrication techniques available for digitalrealization of logic. The answer is that the result obtained here is expected tohave far-reaching implications in a promising optical multi-processor archi-tecture. It is not the intention here to show a performance comparison withdigital realization.

2 THE MULTI-CHANNEL SYNTHETIC JFT MODEL

Let us consider an n-input m-output logic processor. In general, one startswith the truth table of the processor. For n binary inputs, the table thus has2n entries, each of them representing a binary combination. For each outputvariable, we first isolate the terms which produce an output of 1. These arealso called minterms.1 These minterms are then encoded using a simple dualrail scheme as shown in Fig. 1. The top subcells of each code are settransparent so that the resulting minterm representation is interpreted asa single image. Each coded minterm is thus of size 2]2n only. For the purposeof obtaining more discriminatory correlations, these coded minterms are thentransformed using some optimal filter formulations. For example, the follow-ing AMPOF formulation transforms F into H:

H"

Dej(DFD#A

(1)

486 F. Ahmed, A. A. S. Awwal, M. A. Karim

Fig. 1. Dual-rail code for logical 0, 1, and ‘don’t care’ outputs.

where D and A are constant parameters, F is the Fourier transform of thecorresponding coded minterms, and / is the phase of this transform. In otherwords, F"DFDej(. When amplitude modulation is not desired, the denomi-nator is set to 1 This, coupled with D"1, results in the corresponding POFformulation.

From all such coded minterms for each output a composite syntheticreference pattern is determined next using the point spread function (PSF) ofthese filter formulations. Let en

idenote the PSF of the filter formulation [using

eqn (1)] of the nth minterm which results in a logical 1 for the output i.Mathematically:

eni"F~1GDe j/n

i

DFniD#AH (2)

where F represents inverse the Fourier transform. The synthetic pattern asdescribed in eqn (2) is the inverse Fourier transform of a Hermitian matrix(because the AMPOF formulation results in a Hermitian matrix). Therefore, itis real pattern. The composite reference pattern c

ifor the output i is then found

using the following superposition:

ci"e1

i#e2

i#2#el

i(3)

where l is the number of logical 1’s for the output variable i in question, with0)l)2n. Therefore, in the reference half of input joint image of JFT correla-tor, we have m such composite patterns for m outputs. These m compositepatterns are then placed side by side in the reference half of the input jointimage. With the input minterm t (representing the coded inputs of the logicunit) placed next to it, the input joint image of the JFT is thus given by:

g(x, y)"t(x, y)#m+i/1

Ci(x!x

i, y!y

i) (4)

where t(x, y) and ci(x!x

i, y!y

i) represent the input minterm and the com-

posite reference image for output i, respectively. The joint Fourier transform

Generalized logic realization problems 487

of this input joint image results in:

G(u, v)"¹(u, v)#m+i/1

Ci(u, v) exp(!jux

i!jvy

i) (5)

where u and v are independent variables in the frequency domain scaled bya factor of 2n

jfj, j is the wavelength of the collimated light and f is the focal

length of the transforming lens. The joint power spectrum (JPS) is thus givenby

DG(u, v)D2"D¹(u, v)D2#+i

DCi(u, v)D2#¹ *(u, v)+

i

Ci(u, v) exp(!jux

i!jvy

i)

#¹(u, v)+i

C*i(u, v) exp( jux

i#jvy

i)

#

m+i/1

m+

k/1,kEi

Ci(u, v)C*

k(u, v) exp[!ju(x

i!x

k)!jv (y

i!y

k)] (6)

Now we will show that the term Ci(u, v)D2 is significantly smaller

compared with the term D¹(u, v)D2 Using the linearity property of Fouriertransform, eqns (2) and (3) can be combined to result in:

Ci(u, v)"+

n

De j/ni

DFniD#A

(7)

The parameter D can, in general, be used as a weighting coefficient of theindividual transform features. We here assume an equal contribution from allthe minterm features, thus considering an identical value of D"1. ParameterA can be considered as determining the relative weights of the high- andlow-frequency components of the transform features. A very large valueresults in POF feature, whereas A"0 results in inverse filter feature. In thesimulation, we have used a value such that A(0)1(maxDFn

iD), which gives more

weight to the high-frequency components of the features (thus exploiting theadvantages of AMPOF formulation) as well as resulting a smaller absolutevalue for the term of eqn (7). With a more rigorous mathematical analysis, theparameters D and A can be tuned to find a desired minimum value for thisterm. In addition, the last term, which represents the cross correlations amongthe composite patterns, contributes very little to JPS. Therefore, for allpractical purposes, eqn (6) can be approximated by:

DGD2KD¹D2#m+i/1

(Ci¹*#C*

i¹) (8)


The correlation output is obtained by first subtracting the zero-order termcontributed by the input scene (i.e. D¹D2) from the JPS and then taking theinverse Fourier transform of the difference.

Note that due to the use of superposition in constructing composite pat-terns, the minterms corresponding to the same logical output share the samespace in the reference half of the input joint image. Accordingly, for thisapplication, we need space for only one minterm instead of a maximum of 2n,thus improving the correlator space utilization by a factor of 2n. Now thequestion is how much space is sufficient to encode one minterm? It is observedthat in a JFT correlator, up to a certain limit, an increase in the size of theminterm code may actually improve the degree of correlation discrimination ifthe coded images are used directly in the reference half. However, due to thespatial synthesis of the minterms using, for example, POF or AMPOFformulation, a larger-sized code does not significantly improve the per-formance and it was found that a code size of 2]2n for an n-input logicprocessor suffices. For example, a three-input processor will require 12 pixelsfor encoding one minterm, whereas the approach used in the work by Alam12employed 72 pixels for the same. Accordingly, we gain an additional improve-ment in space-bandwidth utilization, by using smaller number of pixels for theminterm encoding.

One limitation of the proposed method, however, is the possible reductionin the discrimination ratio, which is defined as the ratio of the autocorrelationpeak intensity to the cross-correlation peak intensity. Consider a two-output( f

1, f

2) logic problem having l

1and l

2minterms with high logical outputs. For

example, the full-adder problem described in Table 1 has l1"4 and l

2"4 for

carry (C) and sum (S) outputs, respectively. In other words, these four min-terms are used in the code synthesis process which is also demonstrated in thenext section. Now, suppose a minterm that gives a high output for f

1and low

TABLE 1Truth Table for a Full Adder Circuit

Minterm A2

A1

A0

C S

M0

0 0 0 0 0M

10 0 1 0 1

M2

0 1 0 0 1M

30 1 1 1 0

M4

1 0 0 0 1M

51 0 1 1 0

M6

1 1 0 1 0M

71 1 1 1 1


output for f2

is introduced in the input joint image. Our goal, obviously, is toget satisfactorily higher correlation peak for f

1compared with the peak for f

2.

There will be one autocorrelation peak (P!) and l

1!1 cross-correlation peaks

(P#) for f

1, and l

2cross-correlation peaks for f

2. Auto-correlation here repres-

ents the correlation of the reference with an input which is considered in thereference synthesis. Considering average correlation peak values (which de-pend upon the Hamming distance of the patterns), the total correlation peakfor f

1is P

!#(l

1!1)]P

#and that for f

2is l

2]P

#. The discrimination ratio is

then given by:

R"

P!#(l

1!1)]P

#l2]P

#

(9)

Now defining the term R0"P

!/P

#, we have the following results

R"

R0#l

1!1

l2

(10)

This can be rewritten as:

R0"Rl

2!l

1#1 (11)

Note that R0

depends upon the reference image synthesis formulae given ineqn (1). Equation (11) describes one design criteria for the reference synthesisprocess. For a particular problem, l

1and l

2are known. Now, for a desired

discrimination ratio R, we get an estimate of R0. The parameters D and A can

then be tuned in the synthesis formulae given by eqn (1) to obtain this value.To be more specific, for the same problem described in Table 1, suppose thatthe desired discrimination ratio is R"2. This puts a constraint on thesynthesis process, R

0"5. As mentioned earlier, an exact analytical formu-

lation for this constraint is beyond the scope of the current work, but adjustingthe parameters of eqn (1) we can always get values closer to this result.

Equation (10) shows that if the number of minterms l2

(with logic 1 output)in the non-target class increases and then the discrimination ratio decreasesfor the target input. Another observation is that we can even have thediscrimination ratio R larger than R

0in the situation where l

1'R

0(l2!1)#1.

In one extreme case when l1"l

2"#1, then R"R

0, which represents the

case when there is only one pattern in the synthesis process. In our approach,however, the reduction in correlation discrimination ratio is partly offset bythe use of the synthesis formulation because R

0is higher than the discrimi-

nation ratio without any synthesis. Furthermore, we propose to use differentthreshold value for each output to alleviate this correlation discriminationproblem all together. Here, we record the output correlation for each outputby introducing all the coded minterms in the input half of the input joint


image. A threshold value for each output variable is then computed so thata correlation value larger than this threshold is treated as resulting in a logic 1,whereas a correlation value smaller than this is treated as resulting in a logic0. The threshold value, on the other hand, can be determined from the value ofR. Note that if it is desired to have a single threshold for a multi-functionsystem, the discrimination ratio should be redefined as the lowest autocorre-lation from any function divided by the highest cross-correlation from anyother function. In the present work though, we propose to use variablethreshold. Due to the use of composite reference pattern for each output, thelocations of the output peaks are exactly known a priori, which greatlyfacilitates the implementation of this variable threshold.

3 SIMULATION RESULTS

Let us consider the implementation of a full adder logic circuit. Table 1 showsthe truth table of this adder. Here, l

1"l, 1

2"4. The sum and carry outputs

are given by:

S"+(M1, M

2, M

4, M

7)

(12)C"+(M

3, M

5, M

6, M

7)

For encoding the minterms we use the dual-rail code. Each coded mintermin the truth table is represented by a 2]6 image, as shown in Fig. 2. The codedminterms are then transformed with the filter formulation according to eqn(1). The four transforms corresponding to S outputs are then added and,subsequently, the impulse response is calculated, which acts as the compositereference pattern for this S output. Similar composite pattern for the other

Fig. 2. Coding of the minterms for a three-input processor.


Fig. 3. Composite reference patterns for a full adder: (a) sum output; and (b) carry output.

Fig. 4. Output correlation of a full-adder logic circuit corresponding to input minterm 011.

output C is obtained as shown in Fig. 3. Figure 4 shows the output correlationwhen the input minterm is M

3(011). The C peak is larger than the S peak with

a discrimination ratio of 1)92. The corresponding logical outputs should be0 and 1, as seen from Table 1. Note that for the two outputs we get two peaksat specified locations. Therefore, we can implement a simple threshold logicfor each output to decode it as logical 1 or 0. Figure 5 shows this thresholdingoperation. Figure 5(a) shows the correlation output for C for all eight inputminterms. The threshold value set for output C is 23)5. Correlation peakslarger than this are decoded as logical 1 and those smaller are decoded aslogical 0. This results in correct logical 1 outputs for C as corresponding tominterms 3, 5, 6 and 7. For S outputs, the chosen threshold (20)5) properly


Fig. 5. Threshold determination of the full adder circuit: (a) carry output with AMPOFformulation; (b) sum output with AMPOF formulation; (c) carry output with POF formulation;

and (d) sum output with POF formulation.

TABLE 2Truth Table for a 2 Bit Multiplier Circuit

A B Product

A1

A0

B1

B0

P R O D

0 0 0 0 0 0 0 00 0 0 1 0 0 0 00 0 1 0 0 0 0 00 0 1 1 0 0 0 00 1 0 0 0 0 0 00 1 0 1 0 0 0 10 1 1 0 0 0 1 00 1 1 1 0 0 1 11 0 0 0 0 0 0 01 0 0 1 0 0 1 01 0 1 0 0 1 0 01 0 1 1 0 1 1 01 1 0 0 0 0 0 01 1 0 1 0 0 1 11 1 1 0 0 1 1 01 1 1 1 1 0 0 11 1 1 1 1 0 0 1


gives the sum output for minterms 1, 2, 4 and 7, as demonstrated by Fig. 5(b).Figure 5(c and d) represents the corresponding C and S outputs when POFformulation is used.

Next we consider a 2 bit multiplier whose truth table is shown in Table 2.Here, number of logical 1 outputs, for the output variables is not same. As seenfrom Table 2, the outputs P, R, O and D have minterms 1, 3, 6 and 4 resultingin logical 1 output. Figure 6 for example, shows the output correlation

Fig. 6. Output correlation for 2 bit multiplier for two high outputs with input minterm 0111.

Fig. 7. Output correlation for 2 bit multiplier for one high output with input minterm 0110.


corresponding to the minterm 0111. The output variables are labeled in theright side as P, R, O and D from top to bottom. Note from Table 2 that O andD here have logical 1 output, whereas P and R have logical 0 output.Correlation outputs from our model are found to implement this by givinghigher correlation values for O and D and lower values for P and R. Next,consider the correlation for input minterm 0110. Here, only O should givea logical 1 output. Figure 7 shows that O (third from the top on right half) hassignificantly higher correlation than the other three outputs. Figure 8 illus-trates the thresholding operation for the outputs P, R, O and D, respectively.Correlation peaks for all four outputs for the 16 minterm combinations arefirst registered. The threshold values for the four cases are then chosen as 7)2,16)5, 17 and 21, respectively. Correlation outputs larger than the threshold aredecoded as 1 and those smaller are decoded as 0. Everything works asexpected, except in one case. For the O output, the minterm 1111 resulted ina false output, i.e. 1 instead of 0 [Fig. 6(c)]. This could be attributed to the factthat this minterm has smaller Hamming distance with the six minterms ofO output which were used in the spatial synthesis. One approach to resolvethis problem is to use weighted sum of the output minterms in the spatialsynthesis process.

Fig. 8. Threshold determination for the 2 bit multiplier circuit: (a) the most significant bit P; (b)the second bit R; (c) the third bit O; and (d) the least significant bit (D).


4 CONCLUSION

The spatial synthesis of the reference image within a JFT framework allowedthe implementation of a reasonably larger sized truth table, which was notpossible in any previously reported work. With the chosen small-sized code,our system yielded good discrimination between the auto- and cross-corre-lation peaks. This also ensured better utilization of the space-bandwidthproduct. Although the simulation results did not show a higher value ofdiscrimination ratio (required for noise immunity) for the particular problemschosen, theoretical analysis has established that higher discrimination isachievable using the proposed technique. Finally, the small-sized code,coupled with the superposition of synthetic minterms resulted in the optimalrealization of multi-output logic processor with a significantly better utiliz-ation of space-bandwidth product. This will have a far-reaching impact on theoptical multiprocessor design.

ACKNOWLEDGEMENT

We would like to thank the reviewers for their constructive comments andsuggestions.

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Synthetic optical processor for generalized logic realization problems