Lo’ai Tawalbeh
Lecture #2
Standard combinational modules:
decoders, encoders and
Multiplexers
1/3/2005
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Decoders
• General decoder structure
• Typically n inputs, 2n outputs
– 2-to-4, 3-to-8, 4-to-16, etc.
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Binary 2-to-4 decoder
Note: “x” = (don’t care) cases.
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Decoder Use – Operation Decoding
•Microprocessor instruction decoding
opcode field
instruction
other fields
1
4-input binary
En decoder
0 1 2 ………. 15
jump
load
store
add
decoded instructions
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2-to-4-decoder logic diagram
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3-input Binary Decoder
E
1
1
1
1
1
1
1
1
0
x2
0
0
0
0
1
1
1
1
-
x1
0
0
1
1
0
0
1
1
-
x0
0
1
0
1
0
1
0
1
-
x
0
1
2
3
4
5
6
7
-
y7
0
0
0
0
0
0
0
1
0
y6
0
0
0
0
0
0
1
0
0
y5
0
0
0
0
0
1
0
0
0
y4
0
0
0
0
1
0
0
0
0
y3
0
0
0
1
0
0
0
0
0
y2
0
0
1
0
0
0
0
0
0
y1
0
1
0
0
0
0
0
0
0
y0
1
0
0
0
0
0
0
0
0
Inputs:
x = (x2, x1, x0), with xi in {0,1} and E in {0,1}
Outputs:
y = (y7,y6,y5,…,y1,y0) with yi in {0,1}
Function: yi = E. mi(x), i = 0,1,…,7
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Implementing functions using a
Binary Decoder and OR Gates
Function
x1
0
0
1
1
0
0
1
1
x0
0
1
0
1
0
1
0
1
z2
0
1
0
0
0
1
0
1
z1
1
0
0
1
0
0
1
0
z0
0
0
1
0
1
1
0
0
x2
x1
x0
2
1
0
Binary
Decoder
x2
0
0
0
0
1
1
1
1
z2
0
1
2
3
4
5
6
7
z1
z0
E
1
Remember that any function can be represented as a sum of minterms
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Binary Encoders
• Only one input Ij has value 1 at any given time
• Output Y corresponds to the binary code of j when Ij = 1
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8-to-3 Binary Encoders
Y0 = I1 + I3 + I5 + I7
(odd input indices)
Y1 = I2 + I3 + I6 + I7
Y2 = I4 + I5 + I6 + I7
(indices > 3)
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Multiplexers
data
inputs
1
0
0
1
1
MUX
0
2
1
3
1
0
1
0
1
0
selection inputs
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4-input Multiplexer
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Typical Multiplexer use
selection between multiple paths to a functional unit
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Multiplexers as universal modules
• Universal module: using only this module
you are able to implement ANY logic
function.
• NAND and NOR gates for example are
universal gates.
• Question: how do you assign inputs for the
multiplexer in order to implement a given
function?
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Exercise
Implement the following function using:
a) 8-input multiplexer.
b) 4-input multiplexer.
F=x,y,z(1,2,6,7)
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Lo’ai Tawalbeh
Lecture #2
Signed and Unsigned Numbers
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4-bit Unsigned Numbers
Range of values for n-bit vector is:
0 ≤ x ≤ (2^n-1) cpe 252: Computer Organization
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Representation of Signed
Integers and Basic Operations
• Two common representations
– Sign and Magnitude (SM)
– True and Complement (TC)
• In both cases there is a mapping from
signed values to positive values.
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Sign and Magnitude
• x represented by a pair (s,m) where
– s is the sign: 0 for positive and 1 for negative
– m is the magnitude
– example: (-23)10 = -(10111) = (1,10111)
• Range of values for n-bit vector (n-1 bits
for m)
–
- (2n-1 – 1) ≤ x ≤ (2n-1 – 1)
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• Two representations
for
zero
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2’s complement
• No separation between sign and magnitude
• Signed integer x represented by positive integer
xR such that:
when x  0
x
xR   n
 2  x otherwise
Example: n=4, 2^4=16. To represent x = -7; xR = 9
• Range of values for n-bit vector ( 2’s comlement)
–
- (2n-1 ) ≤ x ≤ (2n-1 – 1)
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4-bit Two’s Complement
Numbers
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Change of Sign
1. complement each bit of x
2. add 1
Example: n = 4 x = (0011)2= 3
x’ =
1100
+1
1101  representation of -3
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Positive integer
addition/subtraction
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Two’s complement
addition/subtraction
• Addition: same as positive integer addition
– just discard carry out
• Subtraction: x - y
– step 1: change the sign of y to obtain -y
– step 2: add x and -y
• Example: x = 8, y = 5, 5-bit vectors
y = 00101
-y = 11011 <<< change sign
x - y = 01000
11011 +
00011
<< carry out discarded
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