An 8-Bit Multiplier

PROJECT TITLE: An8-bit Multiplier Circuit.

COURSE NUMBER & NAME: EE- 211 Digital Logic Design.

GROUP MEMBERS:

Muhammad Asad Nawaz.

Syed Mobeen Riaz.

Muhammad Sharjeel Hashmi.

GROUP REGISTRATION#

11-EE-57

11-EE-153

11-EE-159

DUE DATE: 3rd January 2013

DATE HANDED IN: 13th December 2012

ABSTRACT

We will see how to apply the principles and components of arithmetic circuits

to implement a subsystem of moderate complexity. Our objective is to design a fast 8-

by-8 bit multiplier using 4-by-4 bit multipliers as building blocks, along with adders,

arithmetic logic, and carry look-ahead units.

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An 8-Bit Multiplier

INTRODUCTION

We can express an 8-bit product as a series of sums of 1-bit products, so-called Partial Product Accumulation. We can exploit the same principle to construct multipliers of wider bit widths using primitive 4-by-4 multiplier blocks.

First, we denote the two 8-bit magnitudes to be multiplied as A7-0 and B7-0 and the 16-bit product that results as P15-0. We can partition A and B into two 4-bit groups, A7-4, A3-0, B7-4, B3-0, and form their 16-bit product as a sum of several 8-bit products:

To see how this works, let's examine the multiplication of the 8-bit binary numbers 111100102 and 100011002. These correspond to the decimal numbers 242 and 140, respectively. As a check, we see that 242 * 140 = 33880, which is equal to 10000100010110002. 

The hardware implementation follows directly from this observation. It requires four 4-by-4 multipliers, plus logic to sum the four-bit wide slices of the partial products.

Let's call the four 8-bit partial products PP0, PP1, PP2, and PP3. Then the final product bits are computed as follows:

Of course, any carry-out of the calculation of P7-4 must be added to the sum for P11-8, and likewise for the carry-out of P11-8 to P15-12.

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An 8-Bit Multiplier

IMPLEMENTATION

The basic blocks of the implementation are (1) the calculation of partial

products, (2) the summing of the 4-bit product slices, and (3) the carry look-ahead

unit. We examine each of these in turn.

Calculation of Partial Products :

1st we want to use the 4 bit multiplier ICs to generate the partial products but

we haven’t find these ICs so we use gates instead of IC’s. A simple 4-bit multiplier

circuit is shown below. We have to use four such multipliers to get four partial

products, because each multiplier produced one partial product. The AND gates we

have used here are 7408

A 4x4 Multiplier Circuit

Calculation of Sums:

The low-order 4 bits of the final product, P3-0, are the same as PP03-0 and do

not participate in the sums. P7-4 and P11-8 are sums of three 4-bit quantities. How do

we compute these?

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An 8-Bit Multiplier

For this purpose, we need a 3 bit adder IC which was not available so we use 4-bit

adder ICs with two of its terminals shorted.

Calculating the Sum using 4-bit Adders instead of 3-bit

Adders

Note: In the above figure we can suppose these 3-bit adders to be 4-bit adder with

two terminals of each IC shorted.

The rightmost 74181 component and its two associated 74283s implement bit

slice P7-4. The logic is cascaded with an identical block of components to implement

bit slice P11-8.

The above figure also includes the implementation of slice P15-12. The final slice is

formed from the partial product PP37-4, plus any carry-outs from lower-order sums.

We implement this using a 74181 component configured as an adder, with the B data

inputs set to 0, the A inputs set to the partial product, and the carry-in coming from the

adjacent adder block.

Putting the Pieces Together: The last step in the design combines the

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An 8-Bit Multiplier

multiplier block with the accumulation block. To further improve the performance, the

carries between the 74181s can be replaced with a 74182 carry look-ahead unit.

Complete circuit diagram of an 8-bit Multiplier

Package Count and Performance:

 In terms of package count, the complete implementation uses four 4-bit

multipliers,(every multiplier requires 16 AND gates and three 4-bit adders) ,four 4-bit

adder IC’s 74283 packages, three 74181 arithmetic logic units, and one 74182 carry

look-ahead unit(optional). This is a total of 12 packages.

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An 8-Bit Multiplier

CONCLUSIONS

About The Current Circuit:

Here we looked at ways to use the 4-by-4 multiplier as a fundamental building

block to make 8 bit multiplier. The 8-by-8 multiplier we designed used a considerable

large amount of logic, much greater than if we had built the multiplier directly using

4-bit multiplier IC’s rather than using gates to build 4-bit multipliers building blocks.

Here we see that 4 bit multiplier, in conjunction with even more adders, can be used to

build multipliers of larger bit widths.

A much Easier Approach:

We could have done it quite easily if we just implement the 4-bit multiplier

circuit shown above to 8-bit multiplier. But in that case we would need 8-bit adders

IC’s which might be a problem so we have avoided that.

References

[1]http://www2.elo.utfsm.cl/~lsb/elo211/aplicaciones/katz/chapter5/

chapter05.doc6.html .

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An 8-Bit Multiplier

[2] http://www.datasheetarchive.com/5%20bit%20binary%20multiplier%20using

%20adders-datasheet.html .

[3] http://lap2.epfl.ch/courses/archord1/labs/A_8bit_Sequential_Multiplier.pdf .

[4] https://wiki.engr.illinois.edu/download/attachments/84770821/09-

AdditionMultiplication.pdf?version=1&modificationDate=1254112213000 .

Undertaking

We certify that project work titled “An 8-bit Multiplier” is our own work. Where

material has been used from other sources it has been properly acknowledged/referred.

Names Reg# Signatures

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An 8-Bit Multiplier

Muhammad Asad Nawaz 11-EE-57

Syed Mubeen Riaz 11-EE-153

Muhammad Sharjeel Hashmi 11-EE-159

_____________________________________________________________________

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