Loai Tawalbeh Lecture

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Loai Tawalbeh Lecture 2
Digital Logic Review

• 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 (dont care) cases.
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Decoder Use Operation Decoding

• Microprocessor instruction decoding

opcode field
instruction
other fields
4-input binary decoder
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En
0 1 2 . 15
jump
load
decoded instructions
store
add
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2-to-4-decoder logic diagram
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3-input Binary Decoder
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
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 gt 3)
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Multiplexers
data inputs
0 1 2 3
MUX
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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Loai Tawalbeh Lecture 2
Digital Logic Review

• Signed and Unsigned Numbers

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4-bit Unsigned Numbers
Range of values for n-bit vector is
0 x (2n-1)
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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)
• Two representations for zero

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2s complement

• No separation between sign and magnitude
• Signed integer x represented by positive integer
  xR such that
• Example n4, 2416. To represent x -7 xR 9
• Range of values for n-bit vector ( 2s comlement)
• - (2n-1 ) x (2n-1 1)

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4-bit Twos Complement Numbers
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Change of Sign

• complement each bit of x
• 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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Twos 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 ltltlt change
sign x - y 01000 11011
00011 ltlt carry out discarded