Transcript Arithmetic 2

Other Arithmetic Functions
Section 4-5
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 }
2
Computing the Product
• Let us work out the multiplication
on the board:
3
Multiplier
4-Bit by 3-Bit Binary Multiplier
4
Contraction
Can implement other functions ( Like
increment, decrement ) by using basic
arithmetic circuits (e.g. Adder) And
removing unused portions
• This is called contraction
5
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 A0 + 001; Do not use C3 (why?)
Hiding Box
Hiding Box
Hiding Box
Hiding Box
Hiding Box
6
Customized adder
Reference
To increment A = { A2 A1 A0 }, do
A2 A1 A0 + 001; Ignore C3
C2
7
Eliminate Unneeded Gates
No need
for C3
0
C
C22
C2
Cell 2
C1
Cell 1
Cell 0
8
Multiplication by Constant
Case 1: Constant is a power of two
Case 2: Constant is NOT a power of
two
9
Multiplication by Power of 2
• Just wires
C=4*B
• Division by power of 2 similar
C=B/4
10
Multiplication by A Constant
Which Is Not A Power Of 2
AND gates removed
because A is a
constant
First adder removed
by contraction
11
Contraction Not Always Best
• Sometimes it pays to rethink the
function
♦ Example: study decrementer in book
(Example 4-6 page 167)
12
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 1111 1011
 1011 = -5
1111 1011 is also -5
13
We’ve Covered
• Adders
♦ Ripple carry
• Subtracting unsigned numbers
♦ New design for adder-subtractor
• Signed numbers
♦ Signed addition/subtraction
• Multiplication – just basic
• Modified Circuits
14