All-Optical Conversion of Binary Number to Quaternary Signed Digit (QSD) Number Tanay Chattopadhyay and Jitendra Nath Roy Department of Physics, College of Engineering & Management, Kolaghat KTPP Township. Midnapur (east). 721171, W.B., India. [email protected]

Abstract. A suitable number system and an efficient encoding/decoding scheme for handling the data are very much essential to achieve the parallelism in computation. Binary number is accepted as the best representing number system in almost all types of existing electronic computers. But, binary number (0 and 1) is insufficient in respect to the demand of the coming generation. Multi-valued logic (with Radix>2) can be viewed as an alternative approach to solve many problems in transmission, storage and processing of large amount of information in digital signal processing. A new method for conversion from binary number (2’s compliment representation) to quaternary signed digit (QSD) is reported. For the first time to our knowledge, the principle and possibilities of design of an all-optical binary to QSD converter circuit is proposed and described with the help of Terahertz Optical Asymmetric Demultiplexer (TOAD) based interferometric switch.

1

Introduction

In high-speed arithmetical calculation, carry free adders improve the operational performance. Signed digit representation of a number is very much essential in this regard. A higher radix (quaternary, radix=4) based representation of binarysigned digit numbers (2’s compliment) is not only allows carry-free arithmetical operation but also, offers other important advantages such as simplicity in logical operation and higher storage density [1]. In quaternary (radix=4) system, the positive integer set {0, 1, 2, 3} is called ordinary quaternary digit (OQD) and the set of both positive and negative integer {¯3, ¯2, ¯1, 0, 1, 2, 3} is called quaternary signed digit (QSD). Where ¯3 = −3, ¯2 = −2, ¯1 = −1. In signed digit representation QSD number can be written as [2,5]:

X=

n−1 i=0

xi 4i

where xi ∈ {¯3, ¯2, ¯1, 0, 1, 2, 3}

(1)

In electronics, efforts have been made to design ordinary quaternary encoder (ENC) and decoder (DEC) [3-4] and QSD arithmetic [5-6]. But in optics a little A. Ghosh and D. Choudhury (Eds.): IConTOP 2009 c Department of Applied Optics and Photonics, University of Calcutta 2009 

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effort has been given in this regard. The new generation of communication networks is moving towards terabit per second data rates. Such data rates can be achieved if the traditional carrier of information, electron, are replaced by photon for devices based on switching and logic. In our earlier paper we proposed an all-optical circuit for conversion of binary to quaternary number and vice versa that incorporate only the ordinary quaternary number [7]. But in this paper we proposed and described the conversion of binary (2’s compliment) to QSD, with the help of Terahertz Optical Asymmetric Demultiplexer (TOAD) based interferometric switchs [8-9]. For the quaternary data processing in optics, the quaternary signed digit (¯3, ¯2, ¯1, 0, 1, 2, 3) can be represented by four discrete polarized state of light as mentioned below: ¯3 ¯2 ¯1 0 1 2 3

2

= = = = = = =

Plane polarized light 45◦ to the +x axis () Plane polarized light 45◦ to the -x axis () Left circularly polarized light () No light (Ø) Vertically polarized light () Horizontally polarized light (•) Partially polarized light ( •)

TOAD-based optical tree-net architecture

Tree architecture is a multiplying system of single straight path into several distributed branches and sub-branch paths [10-14]. In our recently published paper we have explained and described TOAD-based switching system for designing of optical tree architecture (OTA)[10,11,14]. For this purpose, seven TOAD-based optical switches S1 to S7 are to be set as shown in fig-1. These switches control the light in such a manner that, in the absence of control signal, the incoming light signal emerges from lower channel of the switch. In the presence of control signal, the incoming light signal emerges from upper channel. Now, let us consider there is a constant unpolarized light source (CLS) which may be a laser source. The light signal that comes from CLS can be taken as the incoming signal. The incoming light signal incidents on switch S1 first. Now we can get the light in different desired branches or sub-branches by proper placing of control signals. Control signals are also the light signals. Here we have three control signals A, B and C. They can take two binary values 1 and 0. Presence of light beam is considered to be as one (1) state and absence of light beam is zero (0) state. The output obtained is shown in the table-1. Case-1: When A=0, B=0, C=0 output T-1 only receives light where as the other seven output terminals do not receive any light. Hence T-1 is in one state and others are in zero state, when A=B=C=0. Case 2: When A=0, B=0, C =1. Light from CLS reaches output T-2. Case 3: When A=0, B=1, C=0. Light beam reaches output T-3.

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Case 4: When A=0, B=1, C=1. Here the light signal reaches at output T-4.

Fig 1. TOAD-based optical switch in tree architecture Table 1. State of different output terminals for different input variables (three input) in OTA

Input A 0 0 0 0 1 1 1 1

Case Case Case Case

3 3.1

5: 6: 7: 8:

B 0 0 1 1 0 0 1 1

When When When When

State of different output terminals C 0 1 0 1 0 1 0 1

T-1 T-2 T-3 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

A=1, A=1, A=1, A=1,

B=0, B=0, B=1, B=1,

C=0. C=1. C=0. C=1.

T-4 0 0 0 1 0 0 0 0

T-5 0 0 0 0 1 0 0 0

T-6 0 0 0 0 0 1 0 0

T-7 0 0 0 0 0 0 1 0

T-8 0 0 0 0 0 0 0 1

Light from CLS reaches output T-5. Light reaches output T-6. Output T-7 receives the light. Light reaches at output T-8.

Binary to quaternary conversion All-optical 1-digit QSD converter unit

1-digit QSD can be represented by 3-bit binary equivalent (2’s complement form) as follows [5]:

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¯3 = 101 ¯2 = 110 ¯1 = 111 0 = 000 1 = 001 2 = 010 3 = 011 All-optical circuit TOAD based OTA explained in section-2 can successfully used to convert 3-bit binary data to one digit QSD data. The circuit is shown in the fig-2. In this circuit output will not be taken from T-1 & T-5. Here a polarizing beam splitter (PBS) is placed in the optical path T-4, such that it split into two component vertically polarized light () and horizontally polarized light (•). They are indicated by O1 and O2 respectively in fig-2(a). Five polarizer P C1 (linear polarizer with azimuth angle of transmission axis is 0◦ ), P C2 (linear polarizer with azimuth angle of transmission axis is 90◦ ), P C¯3 (linear polarizer with azimuth angle of transmission axis is 45◦ ), P C¯2 (linear polarizer with azimuth angle of transmission axis is −45◦ ) & P C¯1 (nonlinear right circular polarizer) are placed at the optical path T-2, T-3, T-6, T-7 and T-8 respectively. Now X1 , X2 , X3 , X4 and X5 combined by beam splitter (BS) to get the final output ‘Y’. The principle of operation of this circuit can be understand if we give some examples:

Fig 2. OTA based 1-digit QSD converter (bQ-block) 1) When A=B=C=0, then light (unpolarized) comes out at T-1 (according to the table-1). As T-1 is not connected to the output ‘Y’, hence ‘Y’ = 0 (no light or Ø). 2) When A=B=C=1, then light from CLS comes out only at T-8 (according to the table-1) & falls on P C¯1 . So unpolarized light after passing through this right circular polarizer, it becomes right circularly polarized light (). Hence

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the final output ‘Y’=¯1 or (). 3) When A=0 and B=C=1, Then we find light only at port T-4. It then passed through PBS. Hence O1 = 1() & O1 = 1(•). Hence the final output ‘Y’ is the mixer of vertically ()i.e. pertially polarized light ( •) or logical state ‘3’. Other cases can be obtained as the same way. This result is shown in the table-2.

Table 2. Truth table of 1-digit QSD converter unit Input A 0 0 0 0 1 1 1

3.2

B 0 0 1 1 0 1 1

Output C 0 1 0 1 1 0 1

Y 0 (Ø) 1 ( ) 2 ( •) 3 ( •) ¯3 () ¯2 () ¯1 ()

Conversion of n-bit binary data to QSD data

Technique From the previous section we see that 3-bit binary number generates 1-digit QSD. So to convert n-bit binary data to its equivalent q-digit QSD data, we have to convert the n-bit input binary data into 3q-bit binary data. To achieve this target, we have to split the 3rd, 5th, 7th . . . bit of the given binary data. i.e. odd bit (from the LSB to MSB) into two portion. But we don’t split the MSB. If the odd bit is 1 then, it is split into 1 & 0 and if it is 0 then, it is split into 0 & 0. An example makes it clear, the splitting technique of a binary data (1101101)2 is shown below:

So, we have to split the binary data for (q − 1) times (as example, for conversion of 2-digit quaternary number, the splitting is for 1 time; for converting 3-digit quaternary number the split is 2-times and so on). In each such splitting one extra bit is generated. So, the required binary bits for conversion to it’s QSD equivalent (n) = (Total numbers of bits generated after divisions) - (extra bit generated due to splitting) i.e. n = 3q − {1 × (q − 1)} = (2q + 1)

(2)

So, number of bits of the binary number should be 3, 5, 7, 9 . . . etc for converting it to its equivalent QSD number. Now every 3-bit binary number can be convert

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to its equivalent QSD (discused in the subsection-4.1).

1. Let (49)10 = (110001)2 should be converted to its equivalent QSD by the above techniques. It is 6-bit binary data, but according to the previous discussion we can say that, it should be 7-bit. Hence we can write this number as (0110001)2 . Now it is 7-bit binary data, so from the subsection-(4.1) we can say that, its QSD equivalent is of 3-bit. The techniques is shown below:

So the QSD equivalent of (110001)2 is (301)4 . 2. Let (−155)10 = (101100101)2 should be converted to its equivalent QSD by the above techniques is shown below, It is 9-bit binary data. So from the subsection-(4.1) we can say that, its QSD equivalent is of 4-bit.

So the QSD equivalent of (101100101)2 is (¯3211)4 .

All-optical circuit The above techniques can be easily used for designing nbit binary to q-bit QSD data (mq mq−1 . . . m2 m1 m0 )4 with the help of single bit QSD converter unit (bQ-block). The corresponding circuit is shown in the fig-4. Here binary bits an , an−1 and an−2 are put into three inputs ’A’, ’B’ and ’C’ respectively of bQn -block. The output of this circuit is mq . Again, an−3 and an−4 binary bits are put into ’B’ and ’C’ inputs of next bQn−1 -block. ’A’ input of this block is kept open i.e. fed to no light (Ø). The output of this block is mq−1 . Other bits are put into other bQ-blocks (viz. bQn−1 , . . . , bQ1 , bQ0 respectively) as the same way of bQn−1 -block is shown in the fig-4.

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Fig 4. n-bit binary data to q-digit QSD data conversion circuit

4

Discussion and Conclusion

In this paper we have reported a new and easy method for conversion from binary number (2’s compliment representation) to quaternary-signed digit (QSD). Some important issues as follows are discussed. (a). This conversion technique is very simple. (b). MSB of this converted QSD number is only signed bit. So, if we look only the MSB then, we can easily understand (with out calculation) whether it is positive or negative. 3, ¯2, ¯1 then, the data is negative. Otherwise it is If the MSB of any QSD-data is one of ¯ positive. (c). From this conversion scheme we see that (+3)10 & (−3)10 is expressed as 1-digit QSD number. But it is not the same for higher bit QSD-data. As an example: (+45)10 is expressed as 2-digit QSD number, but (−45)10 is expressed as 3-digit QSD number. The largest q-digit QSD number is (333 . . . q terms)4 , the decimal equivalent q−1 i of this data is (Dmax )q = i=0 3 · 4 (from the equation-1). Similarly the smallest ¯ q-digit QSD number is (300 . . . {q − 1 } terms)4 , the decimal equivalent of this data is Dmin )q = −3 · 4q−1 . So the range (R) of the QSD value for a particular q-digit is from (Dmax )q−1 to (Dmax )q in positive side and from (Dmin )q−1 to (Dmin )q in negative side. i.e. (R+ )q = (Dmax )q − (Dmax )q−1 = 3 · 4q−1 and (R− )q = (Dmin )q − (Dmin )q−1 = −3 · 4q−1 + 3 · 4q−2 , where q > 1. So (R+ )q = |(R− )q | for q > 1.

References 1. S.L.Hurst, “Multiple-Valued Logic-Its Status and Future”, IEEE Transactions computers. C-33(12) (1984) 1160-1179.

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2. A.K.Datta , A. Basuray and S.Mukhopadhyay, “Arithmetic operation using modified ternary number system for use in optical communication”, Opt.Lett. 14(5) (1989) 426-427. 3. H.G.Kerkhoff and M.L.Tervoert, “Multiple-Valued logic Charge-Coupled Devices”, IEEE Transactions computers. C-30(9) (1981) 644-652. 4. J.L.Mangin and K.W.Current, “Characteristics of prototype CMOS quaternary logic Encoder-Decoder circuits”, IEEE Transactions on Computers. C-35(2) (1986) 157-161. 5. A.A.S.Awwal and J.U.Ahmed, “Fast Carry free Adder Design Using QSD Number System”, IEEE. CH3306-8/93/0000-1085 (1993) 1085-1088. 6. M.Kameyama and T.Higuchi, “Signed digit arithmetic circuits based upon multiple valued logic and its application”, in Proc. IEEE Int. Symp Multiple Valued Logic. (1981) 41-53. 7. T.Chattopadhyay and J.N.Roy, “All-optical conversion scheme: binary to quaternary and quaternary to binary number”, Optics & Laser Technology, 41 (2009) 289-294. 8. J.P.Sokoloff, P.R.Prucnal , I.Glesk and M.Kane, “A terahertz optical asymmetric demultiplexer (TOAD)”, IEEE Photon. Technol. Lett. 5(7) (1993) 787-790. 9. Y.J.Jung, S.Lee, N.Park, “All-optical 4-bit gray code to binary coded decimal converter”, Proc. of SPIE 6890, 68900S (2008) 1-10. 10. J.N.Roy and D.K.Gayen, “Integrated all-optical logic and arithmetic operations with the help of TOAD based interferometer device - alternative approach”, Appl Opt. 46(22) (2007) 5304-5310. 11. D.K.Gayen and J. N. Roy, “All-Optical Arithmetic Unit with the help of Terahertz Optical Asymmetric Demultiplexer (TOAD) based Tree Architecture”, Appl Opt. 47(7) (2008) 933-943. 12. Z.Y.Shen and L.L.Wu, “Reconfigurable optical logic unit with a terahertz optical asymmertic demultiplexer and electro-optic switches”, Appl Opt. 47(21) (2008) 3737-3742. 13. A.K.Maiti, J.N.Roy and S.Mukhopadhyay, “All-optical conversion scheme from binary to its MTN form with the help of nonlinear material based tree-net architecture”, Chin Opt letters. 5(8) (2007) 480-483. 14. J.N.Roy, G.K.Maity, D.K.Gayen and T.Chattopadhyay , “Terahertz optical asymmetric demultiplexer based tree-net architecture for all-optical conversion scheme from binary to its other 2n radix based form”, Chin Opt letters. 6(7) (2008) 536540.