1. Other Arithmetic Functions Section 4-5

2. Combinational Multiplier
Example:
4-Bit Multiplicand
B = {B3 B2 B1 B0}
3-Bit
Multiplier A = {A2 A1 A0}
Product is a 7-Bit number,
C = {C6 C5 C4 C3 C2 C1 C0 }

3. Computing the Product
Let us work out the multiplication on the board:

4. Multiplier
4-Bit by 3-Bit Binary Multiplier

5. Contraction
Can implement other functions ( Like increment, decrement ) by using basic arithmetic circuits (e.g. Adder) And removing unused portions
This is called contraction

6. To increment A = { A2 A1 A0 }, do
Incrementing
Very common
Computation of memory address for the next instruction to execute by CPU
Use a customized adder
To increment A = { A2 A1 A0 }, do
A2 A1 A ; Do not use C3 (why?)
Hiding Box

7. Customized adder To increment A = { A2 A1 A0 }, do
Reference
Customized adder
To increment A = { A2 A1 A0 }, do
A2 A1 A ; Ignore C3 C2

8. Eliminate Unneeded Gates
No need for C3 C2
Cell 0 C1 C2
For the right cell, the output of gate 1 becomes A0’ so it can be replaced by an inverter.
The output of gate 2 becomes A0, so it can be replaced by a wire connected to A0. Applying A0 and 0 to gate 3, it can be replaced by a wire, connecting A0 to the output S0. The output of gate 4 is 0, so it can be replaced with a 0 value. Applying this 0 and A0 from gate 2 to gate 5, gate 5 can be replaced by a wire connecting A0 to C1. The resulting circuit is shown as the right cell in Figure 5-12(b).
Apply the same technique to the typical cell 1
Cell 2
Cell 1

9. Multiplication by Constant
Case 1: Constant is a power of two
Case 2: Constant is NOT a power of two

10. Multiplication by Power of 2
Just wires
Division by power of 2 similar
C = 4 * B
C = B / 4

11. Multiplication by A Constant Which Is Not A Power Of 2
AND gates removed because A is a constant
First adder removed by contraction

12. Contraction Not Always Best
Sometimes it pays to rethink the function
Example: study decrementer in book (Example 4-6 page 167)

13. Sign Extension
Changing the number of bits used to store a number is common
Example:
-5 in stored in 4-bits (1011)
to store -5 in 8 bits, extend the “Sign Bit” in the 4 left most bits
1011 is changed to
1011 = -5
is also -5

14. We’ve Covered Adders Subtracting unsigned numbers Signed numbers
Ripple carry
Subtracting unsigned numbers
New design for adder-subtractor
Signed numbers
Signed addition/subtraction
Multiplication – just basic
Modified Circuits