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In today’s digital era, developing digital circuits is bounded by the research towards investigating various nano devices. This paper provides the design of compact Baugh-Wooley multiplier using reversible logic. Even though various researches have been done for designing reversible multiplier, this work is the first in the literature to use Baugh-Wooley algorithm using reversible logic. In this work, a new 5 × 5 reversible multiplier cell is proposed which will be useful in designing Baugh-Wooley multiplier. The proposed single multiplier cell is able to perform addition of a 1 × 1 product with the sum and carry from the previous cell. This reversible multiplier cell is useful in building up regularity in the array multipliers. The Toffoli gate synthesis of the proposed reversible multiplier cell is also given.

In today’s digital era, developing digital circuits is bounded by the research towards investigating various nano devices to provide substitution for CMOS technology. As the nano devices are developed, the density of digital chips is being increased naturally seeking the solution for the power consumption and the heat dissipation developed by this power consumption. This scenario motivates the study of reversible computing field. The origin of the reversible computing is the research work done by R. Landauer in early 1960’s stating that irrespective of the realization technique, the irreversible hardware computation results energy dissipation due to information loss [

Hence reversible logic is being applied in various research domains like DNA computing, nanotechnology, Low Power CMOS design and quantum computing. The quantum circuits can be constructed only with reversible logic gates. Besides, synthesizing reversible logic circuits is much difficult than conventional irreversible logic circuits due to the constraints. In the reversible logic circuit design, fan-out and feedback are not permitted [

In the recent years various reversible multiplier designs have been proposed [

The organization of the paper is as follows. Section 2 is an overview of basic reversible gates. Section 3 is an overview of Baugh-Wooley multiplier. A detailed representation and explanation is done in this section. The proposed reversible multiplier design and its functions are discussed in section 4. The results and discussions of the proposed reversible Baugh-Wooley multiplier are presented in section 5. Conclusion is in section 6.

A circuit will be known as reversible if it can bring back the inputs from the outputs. Also the relationship between the inputs and outputs should be maintained as one-to-one and unique. This constraint forces the number of inputs to be equal to the number of outputs. This section deals with the preliminary reversible gates available in the literature.

When we are performing the signed n × n multiplication there will be no difference if the result has the same bit-width as the inputs. For Eg: the multiplication of two numbers “−2” and “3” results in “−6”. Its binary representation is (1110)_{2} × (0011)_{2} = (1010)_{2}. But what happens if we want the result to be in “2n” bits. Either we need to use sign extension or “2n × 2n” array multiplier. One of the efficient algorithms to handle such situation is the Baugh-Wooley multiplication. This design method has been established in order to design structured multipliers, appropriate for 2’s complement numbers [

where the bits in A and B are denoted as a_{i}’s and b_{i}’s, respectively, and a_{n}_{−1} and b_{n}_{−1} are the sign bits. The below equation gives the n * n product, P = A × B:

The final product could be generated by subtracting the last two positive terms from the first two terms. Instead of doing subtraction operation as in the normal multipliers, it is possible to obtain the two’s complement of the last two terms and add all the terms to deliver the final product. The last two terms will be of n − 1 bits length where each term has the binary extension from position 2^{n}^{−1} up to 2^{2n−3}. In contrast the final product is 2^{n} bits which extend the binary weight from 2^{0} up to 2^{2n−1}. As a first step pad each of the last two terms in the product P with zeros to obtain a 2n-bit number to aid adding it with the other terms. Later the padded terms extend in binary weight from 2^{0} up to 2^{2n−1}. Let X be one of the last two terms that can represent it with zero padding as

The final n * n product, P = A × B is given by:

Let two 4-bit binary numbers be A and B, then the product, P = A × B will be 8 bit long and is

The block diagram representation of 4 bit Baugh-Wooley multiplier is shown in

In the block diagram shown in

The black cells are representing the multiplier cell used for 2’s complement numbers. The grey cells represent the multiplier cell. Each of the multiplier cell receives four inputs namely, the multiplier input (horizontal-green line), multiplicand input (vertical-red line), carry from previous cells (vertical-black line) and sum from previous cells (diagonal-black line). They produce two outputs namely sum output (diagonal-black line) and carry output (vertical black line).

In this work we are proposing two reversible multiplier cells representing black and grey cells. Since each cell is having four inputs and two outputs, the reversible multiplier cell, in order to maintain the reversible constraints it is developed as a cell having five inputs and five outputs. Out of this, three outputs are maintained as garbage outputs. Garbage outputs are by definition don’t care outputs and thus can be left unspecified leading to an incompletely specified function [

The Toffoli gate representation of the proposed Reversible Multiplier Cell is given in

The Reversible Complement Multiplier Cell (CMC) is shown in

These proposed multiplier cells are having one constant input. These cells will function as a multiplier circuit when the input “E = 0”. The input A is the multiplier bit. The input B is the multiplicand bit. The input C is the carry input from the previous cells. The input D is the sum input from the previous cells. For the first level the inputs C and S will be “0”. The outputs P, Q and R are considered as garbage outputs. Since this is an incompletely specified reversible logic gates the functions Q and R are not specified.

The reversible multiplier designs available in the literature are for the array multipliers. There is no any specific application of any algorithm except [

Measuring the reversible logic design in terms of number of gates is one of the major factors. In [

One of the major factors in the design of a reversible logic circuit is the number of constant inputs. The input used as a control input by connecting to either logical low or logical high to get the required function at the output is called garbage/constant input. The proposed reversible Baugh-Wooley multiplier design requires 16 constant inputs, but the design in [

The number of output of the reversible gate that is not making useful functions is referred as garbage output. Other constraint in designing reversible logic circuit is optimizing garbage outputs. The proposed reversible Baugh- Wooley multiplier design produces 48 garbage outputs, but the design in [

The conclusion of the above discussion is that, it is evident that the proposed reversible Baugh-Wooley multiplier circuit design is better than the existing designs with respect to gate counts, garbage inputs and garbage outputs.

The proposed reversible Baugh-Wooley multiplier circuit is more efficient compared to the existing circuits presented by [

In this work, the design of 5 × 5 reversible Multiplier Cell (MC) and reversible Complement Multiplier Cell is proposed. These are the first 5 × 5 reversible multiplier cells proposed in the literature. These reversible multiplier cells are targeted for reversible Baugh-Wooley multiplier design. The proposed reversible multiplier cells are capable of multiplying 2 bits in the current array and add the result with the sum and carry outputs of previous array. The Toffoli gate synthesis of the proposed reversible multiplier cell is also given. The functionality of the multiplier cell was verified with RC viewer. This design is useful in the multiplier design with reduced number of gates and constant inputs. Even the proposed design is having moderate garbage outputs; we can conclude that this design is better in terms of number of gates and constant inputs. The number of gates, constant inputs and garbage outputs

Work | Gate count | Constant inputs | Garbage outputs |
---|---|---|---|

[ | 40 | 52 | 52 |

[ | 44 | 44 | 52 |

[ | 32 | 40 | 40 |

[ | 42 | 42 | 49 |

Proposed | 20 | 16 | 32 |

are analyzed. It is comprehended that the number of gates, the constant inputs and garbage outputs values are fewer in number in the proposed design compared to the existing approaches. The design can be enhanced to construct n × n reversible Baugh-Wooley multiplier circuit.

V. Rajmohan,O. Uma Maheswari, (2016) Design of Compact Baugh-Wooley Multiplier Using Reversible Logic. Circuits and Systems,07,1522-1529. doi: 10.4236/cs.2016.78133