Binary Multipliers

1
Binary Multipliers
The key trick of multiplication is memorizing a
digit-to-digit table Everything else was just
adding
0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
9 0 9 18 27 36 45 54 63 72 81
0 1
0 0 0
1 0 1
Youve got to be kidding It cant be that easy
Reading Study Chapter 3.1-3.4 (including Booth
coding)
2
Binary Multiplication
Binary multiplication is implemented using the
same basic longhand algorithm that you learned in
grade school.
A0
A1
A2
A3
B0
B1
B2
x
B3
A0B0
A1B0
A2B0
A3B0
AjBi is a partial product
A0B1
A1B1
A2B1
A3B1
A0B2
A1B2
A2B2
A3B2
A0B3

A1B3
A2B3
A3B3
Multiplying N-digit number by M-digit number
gives (NM)-digit result
Easy part forming partial products (just an AND
gate since BI is either 0 or 1) Hard part adding
M, N-bit partial products
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Multiplication Implementation
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Second Version
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Example for second version
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Final Version
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The trick is to use the lower half of the
productto hold the multiplier during the
operation.
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What about the sign?

• Positive numbers are easy.
• How about negative numbers?
• Please read Booth coding in textbook

8
Faster Multiply
A1 B
A0 B
A2 B
A3 B
A31 B
P1
P2
P0
P32-P63
P31
9
Simple Combinational Multiplier
tPD (2(N-1) N) tPD,FA
Components N HA N(N-1) FA
NB this circuit only works for nonnegative
operands
10
Carry-Save Combinational Multiplier
Observation Rather than propagating the sums
across each row, the carries can instead be
forwarded onto the next column of the following
row
tPD 8 tPD,FA
tPD (NN) tPD,FA
Components N HA N2 FA
11
Division
Start
1. Subtract Divisor from the Remainder leave the
result in the Remainder
gt0
lt0
Test Remainder
Restore Remainder by adding DivisorShift
Quotient to the leftset its rightmost bit 0
Shift Quotient to the leftset its rightmost bit
1
Shift Divisor Register right 1 bit
Repeat 33times