Arithmetic Circuits

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Arithmetic Circuits
2
Outline

• Adders
• Multipliers
• ALU Design

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Adders

• Half adder without carry in
• Two inputs
• Two outputs sum and carry out
• Full adder with carry in
• Three inputs include carry in
• Two outputs sum and carry out

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Half Adder

• S A?B
• Cout AB

A B S Cout
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
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Full Adder

• S A?B?Cin
• Cout ABBCin ACin

A B Cin S Cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
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Full Adder
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4-bit Parallel Adder
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4-Bit Adder-Subtractor
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4-Bit Adder-Subtractor
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Carry Look-Ahead Logic
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Carry Look-Ahead Logic

• Pi Ai ? Bi (carry Propagate)
• Gi AiBi (carry generate)
• Si Ai ? Bi ? Ci Pi ? Ci
• Ci1 AiBi BiCi AiCi
• Gi (Ai Bi)Ci
• Gi (Ai ? Bi)Ci
• Gi PiCi

Cin A B Cout S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
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Carry Look-Ahead Logic

• C1 G0 P0C0
• C2 G1 P1C1 G1 P1 (G0 P0C0) G1 P1G0
  P1P0C0
• C3 G2 P2C2 G2 P2G1 P2P1G0 P2P1P0C0
• C4 G3 P3C3 G3 P3G2 P3P2G1 P3P2P1G0
  P3P2P1P0C0
• S0 A0 ? B0 ? C0
• S1 A1 ? B1 ? C1
• S2 A2 ? B2 ? C2
• S3 A3 ? B3 ? C3

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Carry Look-Ahead Logic
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BCD Addition

• A BCD adder is a circuit that adds two BCD digits
  in parallel and produces a sum digit also in BCD.
  Consider the sum in BCD and binary
  representations in 5 bits

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Decimal Cout B8 B4 B2 B1 Cout S8 S4 S2 S1
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 1 0 0 0 0 1
2 0 0 0 1 0 0 0 0 1 0
3 0 0 0 1 1 0 0 0 1 1
4 0 0 1 0 0 0 0 1 0 0
5 0 0 1 0 1 0 0 1 0 1
6 0 0 1 1 0 0 0 1 1 0
7 0 0 1 1 1 0 0 1 1 1
8 0 0 0 0 0 0 1 0 0 0
9 0 1 0 0 1 0 1 0 0 1
10 0 1 0 1 0 1 0 0 0 0
11 0 1 0 1 1 1 0 0 0 1
12 0 1 1 0 0 1 0 0 1 0
13 0 1 1 0 1 1 0 0 1 1
14 0 1 1 1 0 1 0 1 0 0
15 0 1 1 1 1 1 0 1 0 1
16 1 0 0 0 0 1 0 1 1 0
17 1 0 0 0 1 1 0 1 1 1
18 1 0 0 1 0 1 1 0 0 0
19 1 0 0 1 1 1 1 0 0 1
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BCD Addition
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BCD Addition
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Magnitude Comparator

• 1-bit comparator
• F AiBi Ai' Bi' (Ai ? Bi)'
• Flt Ai'Bi
• Fgt AiBi'

A B F Flt Fgt
0 0 1 0 0
0 1 0 1 0
1 0 0 0 1
1 1 1 0 0
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2-bit Comparator

• Suppose N1 A1A0, N0 B1B0
• Case 1 N1 N0 -- A1 B1 and A0 B0
• F (A1 ? B1)' (A0 ? B0)'
• Case 2 N1 lt N0 -- A1lt B1 or A1 B1 and A0 lt B0
• Flt A1'B1 (A1 ? B1)' A0'B0
• Case 3 N1 lt N0 -- A1gt B1 or A1 B1 and A0 gt B0
• Fgt A1B1' (A1 ? B1)' A0B0'

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Multiplier

• Partial product accumulation
• 1001 (9)
• 1101 (13)
• ----
• 1001
• 0000
• 1001
• 1001
• -------
• 1110101 (117)
• 64321641 117

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2-Bit Multiplier

• A1 A0
• B1 B0
• -------
• A1B0 A0B0
• A1B1 A0B1
• ---------------
• S3 S2 S1 S0

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4-Bit Multiplier

• A3 A2 A1 A0
• B3 B2 B1 B0
• ---------
• A3B0 A2B0 A1B0 A0B0
• A3B1 A2B1 A1B1 A0B1
• A3B2 A2B2 A1B2 A0B2
• A3B3 A2B3 A1B3 A0B3
• -----------------------------------
• S7 S6 S5 S4 S3 S2 S1 S0

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Arithmetic Logic Unit (ALU) Design
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ALU
S1 S0 Cin Yi F
0 0 0 0 F A
0 0 1 0 F A 1
0 1 0 B F A B
0 1 1 B F A B 1
1 0 0 B' F A B'
1 0 1 B' F A B' 1
1 1 0 1 F A -- 1
1 1 1 1 F A
S1 S0 Yi
0 0 0
0 1 B
1 0 B'
1 1 1
Xi Ai
Yi s0Bi s1Bi'
Cin Cin
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Example

• Design one bit slice for an ALU unit using a full
  adder block to perform the following

M S1 S0 Function Name F Xi Yi Cin
0 0 0 Complement A' Ai' 0 0
0 0 1 AND A AND B Ai AND Bi 0 0
0 1 0 Identity A Ai 0 0
0 1 1 OR A OR B Ai OR Bi 0 0
1 0 0 Decrement A - 1 Ai 1 0
1 0 1 Add A B Ai Bi 0
1 1 0 Subtract A B' Ai Bi' 1
1 1 1 Increment A 1 Ai 1 1
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Example
Determine Yi
M S1 S0 Yi
1 0 0 1
1 0 1 Bi
1 1 0 Bi'
1 1 1 0
Yi MS1'Bi MS0'Bi'
Determine Xi
M S1 S0 Xi
0 0 0 Ai'
0 0 1 Ai Bi
0 1 0 Ai
0 1 1 Ai Bi
1 X X Ai
Xi M'S1'S0'Ai' M'S1S0Bi S0AiBi S1Ai
MAi Cin MS1
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Example

• Design one bit slice for an ALU unit using a full
  adder block to perform the following

M S1 S0 Function Name F Xi Yi Cin
0 0 0 Add A B Ai Bi 0
0 0 1 Subtract A - B Ai Bi' 1
0 1 0 Increment A 1 Ai 0 1
0 1 1 Decrement A - 1 Ai 1 0
1 0 0 AND A AND B Ai AND Bi 0 0
1 0 1 OR A OR B Ai OR Bi 0 0
1 1 0 Complement A' Ai' 0 0
1 1 1 XOR A ? B Ai ?Bi 0 0
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Example
Determine Yi
M S1 S0 Yi
0 0 0 Bi
0 0 1 Bi'
0 1 0 0
0 1 1 1
Yi M'S1S0 M'S1S0Bi M'S0'Bi'
Determine Xi
M S1 S0 Xi
1 0 0 Ai Bi
1 0 1 Ai Bi
1 1 0 Ai'
1 1 1 Ai ?Bi
0 X X Ai
Xi MS1S0'Ai' MS0Ai'Bi MS1'AiBi MS0AiBi'
M'Ai Cin M' (S1 ?S0)
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Exercises

• p. 325 4.32, 4.33, 4.34, 4.36, 4.37, 4.39, 4.40,
  4.43, 4.44