Binary Adders

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Binary Adders

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n-bit Addition

• Ripple Carry Adder
• Conditional Sum Adder
• (Carry Lookahead Adder)

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Ripple Carry Adder

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder

• Main principle pre-computing upper sums for the
  cases ck1 and ck0
• Assume n is power of 2

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Conditional Sum Adder
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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Conditional Sum Adder

• full adder implements adder for n1 CSA(1)
  FA !!!!

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Carry Lookahead Adder

• Main principle combine carry computations of
  substrings
• Carry into bit position i ci is computed from
  bits ai-10,bi-10, cin
• condition

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Carry Lookahead Adder

• Main principle combine carry computations of
  substrings
• Carry into bit position i ci is computed from
  bits ai-10,bi-10, cin
• condition
• Definition propagate signals
• Definition generate signals

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Carry Lookahead Adder

• Definition propagate signals
• Definition generate signals
• for igt0
• for i0
• relation to carries
• Computation of the signals ,
  
• for igt0
• for i0

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Carry Lookahead Adder

• Combination of carry and propagate signals

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Carry Lookahead Adder

• Combination of carry and propagate signals

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Carry Lookahead Adder

• Combination of carry and propagate signals

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Carry Lookahead Adder

• Combination of carry and propagate signals
• Given two such pairs of generate and propagate
  signals and
• we define the operation

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Carry Lookahead Adder

• Carry computation based on computation of
  generate and propagate signals

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Carry Lookahead Adder

• Carry computation based on computation of
  generate and propagate signals

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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder

• Structure
• Implementation using Parallel Prefix Computation

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Parallel Prefix Computation

• For associative operation o, we define
• Parallel Prefix Operation
• n-bit INPUT
• n-bit OUTPUT
• Compute the operation o on all prefixes of
  Xn1

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Carry Lookahead Adder

• Parallel Prefix Computation
• n1 n2k

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Carry Lookahead Adder

• Parallel Prefix Computation

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Carry Lookahead Adder

• Parallel Prefix Computation

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Carry Lookahead Adder

• Parallel Prefix Computation

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Carry Lookahead Adder

• Parallel Prefix Computation

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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder

• cost

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Carry Lookahead Adder

• delay

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Carry Lookahead Adder

• Example for n8
• design
• computation of 18343

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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Carry Lookahead Adder
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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)

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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)

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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)

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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)

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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)

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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)
• assumption add/sub result is representable with
  n bits

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Addition/Substraction

• Addition and Subtration for n-bit twos
  complement inputs
• bit op signals the case of subtraction (op1)
• assumption add/sub result is representable with
  n bits
• Detection of the case that the result can not be
  represented with n bits(overflow)

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Overflow

• Overflow result too large for finite computer
  word
• e.g., adding two n-bit numbers does not yield an
  n-bit number 0111  0001 note that overflow
  term is somewhat misleading, 1000 it does not
  mean a carry overflowed
• Detecting Overflow
• No overflow when adding a positive and a negative
  number
• No overflow when signs are the same for
  subtraction
• Overflow occurs when the value affects the sign
• overflow when adding two positives yields a
  negative
• or, adding two negatives gives a positive
• or, subtract a negative from a positive and get a
  negative
• or, subtract a positive from a negative and get a
  positive
• Consider the operations A B, and A B
• Can overflow occur if B is 0 ?
• Can overflow occur if A is 0 ?

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Effects of Overflow

• An exception (interrupt) occurs
• Control jumps to predefined address for exception
• Interrupted address is saved for possible
  resumption
• Details based on software system / language
• example flight control vs. homework assignment
• correct sign has to be computed even for
  overflows
• branches should also work correctly after
  overflows
• neg signal

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Overflow detection

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Overflow Sign

• Sign condition

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Overflow Sign

• Sign condition
• consider sign extended sum

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Overflow Sign

• Sign condition
• consider sign extended sum
• sign computation

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Overflow Sign

• Sign condition
• consider sign extended sum
• sign computation

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Overflow Sign

• Sign condition
• consider sign extended sum
• sign computation

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Overflow Sign

• Sign condition
• consider sign extended sum
• sign computation

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Design of an Add/Subtract Unit

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Design of an Add/Subtract Unit

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Design of an Add/Subtract Unit

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Design of an Add/Subtract Unit

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Design of an Add/Subtract Unit

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Design of an Add/Subtract Unit