Multiplication Circuits (part I)

1
Multiplication Circuits(part I)

• Lecture 11

2
Single Digit Multipliers in Decimal
1) Write the Multiplicand and Multiplier
2) Write the intermediate terms using the table.
3) Add the intermediate terms to form the Product.
3
Single Digit Multipliers in Binary
1) Write the Multiplicand and Multiplier
2) Write the intermediate terms using the table.
3) Add the intermediate terms to form the Product.
4
Intermediate Terms in Binary
The Circuit to form the intermediate terms in
binary is simply a column of AND gates.
5
Multi-Digit Multipliers in Decimal
We shall assume that forming products from single
digit multipliers is easy.
6
Accumulating Partial Products
7
Hardware for Unsigned Binary Multiplication
This Circuit performs the Right-Shifting Algorithm
We need to add Control signals to coordinate
the registers.

• The Multiplicand Register
• Hold, Load

• The Multiplier Register
• Hold, Load Shift Right

• The Product Register
• Hold, Load, Clear Shift Right

8
Combining the Registers
At the beginning of the calculation the
Multiplier Register is full of valid bits and the
lower half of the Product Register does not
contain any valid data. After each Add / Shift
stage of the calculation the Multiplier Register
contains one less valid bit while the Lower half
of the Product Register gains a Valid Bit.
We can save ourselves a register by combining the
Multiplier Register with the Lower half of the
Product Register.
9
Signed Multiplications
So far we have developed TWO algorithms for
multiplying Unsigned Binary Integers. We must now
ask ourselves, How do we multiply Signed Binary
Numbers? How do we multiply Fractional Binary
Numbers?
If the numbers are coded as Sign/Magnitude then
the solution is simple. We simply compute the
sign of the product using an XOR gate and compute
the magnitude as before.
10
2s Complement Multiplications
Recall that 2s Complement coded numbers have a
radix vector identical to that of unsigned binary
EXCEPT for one difference. The MSB has a negative
weight.
As we form the intermediate terms we must be
careful when we extend the length of the term in
preparation for the additions. The terms must be
sign extended. Each intermediate term will be
correct except for the last one. Since the MSB of
the multiplier has a negative weight, the last
intermediate term must be SUBTRACTED instead of
added. Subtraction is handled in the usual
manner. Simply negate the operand and then ADD.
The operand is negated by 1s complement and then
adding 1 by injection into the carry. The
final result is given in the next slide.
11
2s Complement Right-Shift Multiplication
Notice that AddSub is a Control Signal.