Multipliers in Digital Circuits

Till now we have seen classification as well as some examples of multipliers like Braun, Booth, Wallace Tree and Dadda Multiplier. This article will present a comparative analysis of the multipliers. The comparison will be between the the two types of multipliers i.e. Array and Tree. Each one has its own merits and demerits. We will be diving in the details. Parameter Array Multiplier Tree Multiplier Operation Speed Low Speed High Speed Time Delay More Delay Most Delay Area More Area (More Adders) Medium Area Complexity Less Complex More Complex Power Consumption Very High High                        Table 1: Analysis of different types of multipliers Design Summary Array Multiplier Tree Multiplier No. of slice flip-flops

 Array multiplier has more power consumption and propagation delay compared to other multipliers. Wallace multiplier has huge area wastage issue because of the irregularity in the structure. Dadda multiplier is proposed to overcome these disadvantages in the existing multipliers. Dadda multiplier is a fast parallel multiplier. It was presented by Luigi Dadda in 1965. It is a refinement of Wallace multiplier so its algorithm also has three general stages. The procedure of the three stages is same for Dadda multiplier as that for Wallace multiplier. But in Dadda multiplier, the row reduction processed by placing adders at maximum heights of matrix in the most optimal manner possible. Thus it requires less adders as compared to Wallace multiplier. Fig. 1: 4x4 Dadda Multiplier The algorithm of Dadda multiplier is  1. Let d1 = 2 and di+1 = |15.di|.Di is the height of the matrix for the ith stage. Repeat until the maximum ith stage is reached in which the original matrix of height N contains

A Wallace Tree is one of the efficient ways of implementing a multiplier circuit for two integers. It was derived by one of an Australian Computer scientists Chris Wallace in 1964. The Wallace tree uses the carry save reduction method.  In this multiplier technique firstly, the multiplication takes place according to the multiplication rules i.e. using AND gate basic multiplication done by usual multiplication rule. Then after getting the partial products take the 3 rows at a time for the addition of the partial product. For the addition of the 2-bit, half adder is used and for the successive addition of the 3-bit full adder is used. Because of the use of adder, we get 2 rows as a result for the respective 3 rows of addition. After getting two rows, add the one next row with these resultant two rows. Now again we have 3 rows and our aim is to reduce them into 2 rows as mentioned earlier. This reduction and addition with the next row process continues till the last row arrives. After th

This multiplier comes under the class of array multiplier. Booth multiplier is a powerful algorithm for multiplication of signed numbers, which treats both positive and negative numbers uniformly. Booth’s Multiplication Algorithm is an algorithm  that use to multiply two signed binary numbers by the method of  two’s complement for binary. Andrew Donald Booth invented the algorithm in 1950. This invention takes place when the Andrew booth doing research on crystallography in London. To check the speed Andrew used the desk calculators which not fast for adding purpose compare to the shifting purpose  from where algorithm is generated. Each multiplier bit generates one multiple which is added to the partial product for the standard operation add-shift,. If we take a large multiplier, then addition takes place in the large number of multiplicand. Basically the number of additions gives us the delay of the multiplier. But if we reduce the number of additions in the steps then the performanc

In Braun multiplier, the partial products are first computed in parallel then collected through a cascade of different types of adders. It consists of an array of AND gates and adders arranged in an iterative structure which doesn’t require any logic registers. So it is a type of parallel array multiplier. 4-Bit Braun Multiplier: 4 bit Braun multiplier contains two inputs of 4-bit i.e., A0-A3 and B0-B3 and the output is 8-bit which is P0-P8. Fig 1 gives the circuit of 4-bit Braun multiplier. For layout, to make the connections easier, an AND gate and a Full adder is taken as one unit. 9 of such units are placed accordingly in the layout. The top layer consists of 4 AND gates, high is taken as one instance. The remaining 3 and gates to the extreme left  is  taken as one more instance. Fig. 1: 4x4 Braun Multiplier   8-Bit Braun Multiplier: The circuit used in the 4-bit Braun multiplier is taken as the instance for the schematic implementation of the 8-bit Braun multiplier. Here there a

1. Array Multiplier An array multiplier is a digital  combinational circuit . It is used to multiply two binary numbers by using an array of full adders and half adders. This array is used for the addition of the various product terms involved at the same time. To form the various product terms, an array of AND gates is used prior to the Adder array. Checking the bits of the multiplier one at a time and forming partial products is a sequential operation that requires a combination of add and shift micro-operations. The multiplication of two binary numbers can be executed with one micro operation with the help of a combinational circuit that forms the product bits at the same time. This is a rapid way of multiplying two numbers because all it takes is the time for the signals to travel through the gates that form the multiplication array. However, it requires many gates. This is the reason it was not economical until the development of integrated circuits. Fig. 1: Array Multiplier  2. T