Download EE5410 Computer Arithmetic

Multi-Operand Addition
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Using 2-opr adder
Carry-save adder
Wallace Tree
Dadda Tree
Parallel Counters
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Two-Operand Adders
• X1+X2+ …+Xk with n-bit each
• Sum= n +log2(k) bits
• Serial Addition
 Tadd= (k–1) (TAdder + TReg )
X
Aadd= AAdder + AReg
i
Adder
Reg.
n+log 2 (k)
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Two-Operand Adders(2)
• Serial Addition: Cascading
k 1
 Tadd= i1T (n  log2 i )
k 1
Aadd=  AAdder=  (n  log2 i )bits adders
X3
...
Adder
X2
Adder
X1
Adder
i 1
Xk
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Two-Operand Adders(3)

(k / 2i )A(n  i  1)
i 1
log2 k 
i 1
X1
X2
X3
X4
/
n+1
/
n+2
...
/
Adder
T (n  i  1)
Adder
Aadd=

Adder
 Tadd=
log2 k 
Adder
• Parallel Addition: Binary tree
/
n+log2K
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Carry-Save Adder
• Add more than two numbers (says n)
• Carry not added (Carry save)
• 3 #  (3,2)  2#
0 1 0 1
1 0 0 1
0 1 1 1
FA FA FA FA
0
1
1
0
0
1
1
1
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Carry-Save Adder(2)
CPA
CSaveA
CSaveA
X1
X2
X3
CSaveA CsaveA
• CPA in the last step
• 3 #  (3,2)  2#;
• TCPA+log3/2kTFA
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Carry-Save Adder(3)
• 4#  (4,2)  2#;
• TCPA+ 2 log2kTFA
FA
FA
FA
FA
FA
FA
FA
FA
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Carry-Save Adder(4)
• 7#  (7,3)  3#;
• TCPA+ 2 log7/3kTFA
FA
FA
FA
FA
(7,3)-counter
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Carry-Save Adder(5-1)
• Applying (3,2) FA & (2,2) HA with dot
notation n=6&k=6
 12 FA in 1st level
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Carry-Save Adder(5-2)
• Applying (3,2) FA & (2,2) HA with dot
notation n=6&k=6
 5 FA & 2HA in 2nd level
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Carry-Save Adder(5-2-1)
• Applying (3,2) FA & (2,2) HA with dot
notation n=6&k=6
 12HA in 2nd level
 4 numbers (same)
Not so good
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Carry-Save Adder(5-3)
• Applying (3,2) FA & (2,2) HA with dot
notation n=6&k=6
 5 FA & 1HA in 3rd level
7-bit CPA in last level
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Wallace Tree
• k-input Wallace Tree reduces to two
(n+log2k –1)-bit outputs
 h(k)=1+h(2k/3)
h(k): the smallest height of an k-input
Wallace tree
h(k)= 0
k= 2
1
3
2
4
3
6
4
9
5 6
13 19
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Wallace Tree(2)
• 7-input Wallace Tree reduces to two
(n+log2k –1)=(n+2)-bit outputs

(n,1)(n-1,0)
csa
csa
csa
csa
csa
(1)
(0)
(n+1,2)
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Dadda Tree
• Reduce the number to the next lower
number
h(k)= 0
k= 2
1
3
2
4
h(k)= 7
k= 28
8
42
9
63
3
6
4
9
10
94
5 6
13 19
11 12 13
141 211 316
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Dadda Tree(2)
• Reduce the number to the next lower
number
• Ex1: k=8 8 (2CSA)6 (2CSA)4
(1CSA)3(1CSA)2CPA
• Ex2: k=12 12 (3CSA)9 6 4 2
h(k)= 0 1 2 3 4 5 6
k= 2 3 4 6 9 13 19
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Parallel Counters
• One column
(3,2) counter  at most 3 1’s 2 bits
(k,m) counter  at most k 1’s 
m=log2(k+1) bits
• How about multi-column?
(k,k,m) counter: at most k 3’s
m=log2(3k+1) bits
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Parallel Counters(2)
• Ex: (5,5,m); m= log2(3*5+1)= 4 bits
Overlapped  m/(# of col.)  = 2 bits
CPA at last stage too
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Parallel Counters(3)
• Ex: (5,5,5,m); m= log2(7*5+1)= 6 bits
Overlapped  m/(# of col.)  = 2 bits
CPA at last stage too
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