Homework #3 Computer Organization
1. Draw the circuit (using AND, OR, and NOT gates) to implement an 8-input to 1-output multiplexer (MUX).
Your MUX should have 3 control wires (c
2
, c
1
, c
0
) to select which input is switched to the single output.
2. Recall that the sum-of-products (SOP) Boolean formula for the carry-out (c
i+1
) of a n-bit adder was :
1-bit adder: c
i+1
= x
i
y
i
+ x
i
c
i
+ y
i
c
i
.
2-bit adder: c
i+1
= x
i
y
i
+ x
i
(x
i-1
y
i-1
+ x
i-1
c
i-1
+ y
i-1
c
i-1
) + y
i
(x
i-1
y
i-1
+ x
i-1
c
i-1
+ y
i-1
c
i-1
)
= x
i
y
i
+ x
i
x
i-1
y
i-1
+ x
i
x
i-1
c
i-1
+ x
i
y
i-1
c
i-1
+ y
i
x
i-1
y
i-1
+ y
i
x
i-1
c
i-1
+ y
i
y
i-1
c
i-1
3-bit adder: c
i+1
= x
i
y
i
+ x
i
( 7 product terms of 2-bit adder ) + y
i
( 7 product terms of 2-bit adder )
= x
i
y
i
+ x
i
x
i-1
y
i-1
+ ... ( 12 product terms omitted ) + y
i
y
i-1
y
i-2
c
i-2
4-bit adder: c
i+1
= x
i
y
i
+ x
i
( 15 product terms of 3-bit adder ) + y
i
( 15 product terms of 3-bit adder )
= x
i
y
i
+ x
i
x
i-1
y
i-1
+ ... ( 28 product terms omitted ) + y
i
y
i-1
y
i-2
y
i-3
c
i-2
a) Complete the following table showing the gate delays for different types of adders assuming a 9-input limit
into any gate.
101,0239-bit
95118-bit
82557-bit
71276-bit
6635-bit
8 x 3 = 243215314-bit
10 x 3 + 2 = 323214153-bit
16 x 2 = 32211372-bit
32 x 2 = 64211231-bit
Gate Delays for
32-bit ripple adder
using adders of this
type
Gate
Delay per
Adder
# gate delays
due to
sum/OR
# gate delays due
to product/AND
terms
Most # of inputs
in any of the
product terms
# of product terms
in SOP expression
to be OR’ed
Type
of
Adder
b) Consider the SOP Boolean formula for the carry-out (c
i+1
) of a 9-bit adder:
x x xx xxy y yy y
c
c
y
i i-1 i-6i-5 i-8i-7i i-1
i+1
i-6i-5 i-8
i-8
i-7
. . .
Give an example of each of the following:
i) a product term with only two terms
ii) a product term with 10 terms
Name:____________________
HW #3 Page 1
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