Modified
Booth
Encoding
Radix-4 8-bit
Multiplier
Final Project Report
Da Huang, Afsaneh Nassery
Table of Contents
Table of Contents ...................................................................................................................... 2
Abstract: .................................................................................................................................... 3
System Diagram:............................................................................................................... 3
Performance Estimation: ................................................................................................... 5
Transistor number and Area estimation .................................................................... 5
Time Delay:............................................................................................................... 6
Power Consumption: ................................................................................................. 6
Project Floor plane: ........................................................................................................... 7
Sub circuit Design ..................................................................................................................... 8
Encoder Circuit ............................................................... Error! Bookmark not defined.
Decoder Circuit ............................................................... Error! Bookmark not defined.
12 Bit Adder circuit......................................................................................................... 12
Full System Design ................................................................................................................. 13
Test simulations………………………………………………………………………… 16
Summary ................................................................................. Error! Bookmark not defined.
Reference ................................................................................................................................ 20
Page 2 of 20 Abstract:
In this project, we are building up a Modified Booth Encoding Radix-4 8-bit Multiplier using 0.5um
CMOS technology. Booth multiplication allows for smaller, faster multiplication circuits through encoding
the signed numbers to 2’s complement, which is also a standard technique used in chip design, and
provides significant improvements by reducing the number of partial product to half over “long
multiplication” techniques. In this project, we demonstrate an extendable system diagram for 8-bit radix-4
MBE algorithm. Encoder, decoder and Carry Look Ahead Adder (CLA) are presented in this system.
System Diagram:
Y[7:0]
8
X[7:0]
8
8
8
8
“0”
X[-1]
X[0]
X[1]
Encode
X[1]
X[2]
X[3]
Encode
3
9 bit DECODE
3
7
9 bit DECODE
2
9
X[3]
X[4]
X[5]
Encode
3
9 bit DECODE
12-BIT ADDER
9
X[5]
X[6]
X[7]
Encode
3
7
9 bit DECODE
12-BIT ADDER
9
2
7
2
12-BIT ADDER
9
OUTPUT:
P[15]p[14]P[13]P[12]P[11]P[10]P[9]P[8]P[7]P[6]
P[5]P[4] P[3]P[2] P[1]P[0]
System Architecture
We applied three basic unit cells in this design: Encoder, Decoder, and 12-bit Adder.
Each unit schematic is shown below:
For 8-bit Multiplicand, we have the following structure:
Page 3 of 20 Y[7:0]
X[2i-1]
X[2i]
X[2i+1]
ENCODER
DECODER
DECODER
DECODER
DECODER
DECODER
DECODER
DECODER
DECODER
12-bit Adder
We use 3 4-bit CLA units to build up our 12bit adder circuit. The diagram is shown below.
4-bit CLA Adder:
Page 4 of 20 Performance Estimation:
Transistor number and Area estimation
T
Num of
Number Transisto
Area for
Area
needed
Gate(um
Est.
ype of
Transisto
Gate
r
And2
6
134
804
595
79730
OR2
6
62
372
577.5
35805
NOR2
4
32
128
315
10080
XNOR
8
32
256
1147.5
36720
2
r Est.
2)
XOR2
8
30
240
885
26550
OR3
8
18
144
612.5
11025
AND3
8
21217.
30
240
707.25
Total
2184
Total
0.265
Area
mm2
Transist
5
or
Page 5 of 20 Time Delay:
From the system diagram, we can see the critical path is as below:
For the 12-bit adder, each adder will give a 10 gate delay time.
Including the encoder, decoder and 3 adder, we have 38gate delay.
Power Consumption:
Ptotal = Pstatic + Pdynamic = 1.887 mW / MHz + 0.205mW = 0.2092mW ( f = 100MHz )
Page 6 of 20 Project Floor plane:
Page 7 of 20 Sub circuit Design
Encoder Block
The encoder block generates the selector signals for each 3 bits of multiplicand. This is the logic for
the encoder block:
M i = X1 ;
X=X 0 ⊕ X -1 ;
X 2 = X1X 0 X -1 +X1 X 0 X -1
Here the schematic and the layout of the encoder.
(a)
(b)
Decoder Block
The decoder block generates the partial product from the selector signals that they are generated in
encoder block.
This is the logic for the 1bit decoder that generates 1 bit of the partial product:
PPij =(YjXX 2 +Yj-1 XX 2 ) ⊕ M i
Fig.1 shows the schematic and the layout of 1 bit decoder block.
We design the 9 bit decoder using 9 blocks of 1bit decoder that their selector signals are the same and
9 blocks of half adder is used to make the partial product in the 2’s complement form in the negative cases.
Fig.2 and 3 shows the schematic and the layout of 9 bit decoder block.
Page 8 of 20 Fig.1 1bit decoder block schematic and layout
Fig.3 9bit Decoder Block schematic
Fig.3 9bit Decoder Block layout
Page 9 of 20 12 Bit Adder circuit
Logic
We use three 4bit Carry Look Ahead (CLA) Adder to make up the 12bit adder.
The 4bit CLA contains 4 PFA units and one CLL unit, which will increase the computation time.
Block Diagram:
Schematic
4bit Look Ahead ADDER
Page 10 of 20 12bit Look ahead ADDER
Page 11 of 20 Layout
4bit Look Ahead ADDER layout
12bit Look ahead ADDER
Page 12 of 20 Full System Design
Logic
Here we will show how the signed multiplier works, and how to extend the sign bit.
We get the 9bit output from Decoder, which is the 2’s complement result of partial products.
Here is the sign extension for 16 bit x 16 bit.
Here shift two bit for each partial product, and also we extend 9bit result to 12bit. Except the highest
bit set to be 0. There are 3 or 2 bit sign extension for each partial product.
Ei = Mi ⊕ PPi [8]
This sign bit extension is different from the book and reference value, we apply the highest partial
product bit rather than the multiplicand bit. This difference maybe come from we take a different approach
to add sign bit. The book and reference paper all take the same way to put sign bit add at adder part. We
finish this target at the generation partial product. Our 9bit decoder schematic carefully shows how this
works.
Page 13 of 20 Here is the schematic and layout of the multiplier:
Schematic
Page 14 of 20 Encoder circuit:
Decoder and bit shift, sign extension:
We have sign extension on each partial product before they goes into 12 bit adder.
Page 15 of 20 Connection between adder blocks.
For each output of 12 bit adder, the highest 10 bits go to next adder, and the lowest 2 bit goes to
output directly.
Page 16 of 20 Multiplier Layout
Page 17 of 20 Here is our DRC and LVS check results:
Page 18 of 20 Test Simulations:
Digital Simulation Result:
In order to make sure about the multiplication procedure we wrote the verilog code for all the blocks
and check our multiplier with digital simulation. The table shows our simulation result:
A[7:0]
B[7:0]
Output
00000001
00000001
0000000000000001
00001111
00000010
0000000000011110
00001111
00000100
0000000000111100
00001111
00001111
0000000000110001
10000001
00000100
1111111000000100
10110111
01011010
1110011001010110
Eldo Simulation Result:
12 bit CLA adder Eldo Simulation Result:
Page 19 of 20 Reference
[1]: CMOS VLSI Design: A Circuits and Systems Perspective, Thrid Edition, Neil H.E. Weste, David
Harris
[2]:G. Bewick, Fast Multiplication: Algorithm and Implementation PhD. Thesis, Stanford University,
1994;
[3]: Leonardo L.de Oliveira, Eduardo. C. Sergio B. Array Hybrid Multiplier versus Modified Booth
Multiplier: Comparing Area and Power Consumption of Layout Implementations of Signed Radix-4
Architectures.
[5]: Vojin G. Oklobdziji. High-Speed VLSI Arithmetic Units: Adders and Multipliers
[4 ]:Hsin-Lei Lin, Robert C. Chang, Ming-Tsai Chan. Design of a Novel Radix-4 Booth Multiplier
Page 20 of 20