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Multioperand Addition

Behrooz Parhami,

Computer Arithmetic: Algorithms and Hardware Design

Chapter 8, Multioperand Addition

Applications of multioperand addition

Inner product

Multiplication

n-1

n-1

p=a·x

s = x(i) y(i)=

p(i)

i=0

i=0

# of bits of S = log2 (Smax + 1)

=

= log2 (n (2k-1) + 1) log2 n 2k =

= k + log2 n

Number of bits of the result

n-1

S = x(i)

x(i) [0..2k-1]

i=0

Smin = 0

Smax = n (2k-1)

Ripple-Carry Carry Propagate Adder (CPA)

a2

b2

a1

b1

a0

b0

an-1

bn-1

c0

c3

c2

c1

cn

cn-1

. . .

FA

FA

FA

FA

s2

s1

s0

sn-1

cn-1

c2

c1

c0

bn-1

b2

b1

b0

an-1

a2

a1

a0

. . .

FA

FA

FA

FA

c3

c2

s2

s3

c1

s1

s0

sn-1

cn

cn-1

Operation of a Carry Save Adder (CSA)

Example

20

22

23

24

21

x

y

z

0 1 0 1 0

1 1 0 1 1

1 0 1 1 1

s

c

0 0 1 1 0

1 1 011

x+y+z = s + c

Carry propagate and carry-save adders

in dot notation

Specifying full- and half-adder blocks

in dot notation

x3 x2 x1 x0

y3 y2 y1 y0

z3 z2 z1 z0

w3 w2 w1 w0

s3 s2 s1 s0

c4 c3 c2 c1

w3 w2 w1 w0

c4 s3 s2 s1 s0

c4 c3 c2 c1

’

’

’

’

’

’

’

’

S5 S4 S3 S2 S1 S0

Carry-save adder for four operands

Tree of carry save adders reducing

seven numbers to two

Number of Inputs and Tree Height

Parameters of tree carry-save adders (1)

Latency

LatencyCSA = h(n) TFA + LatencyCPA(k, n)

Tree height

for n operands

Component Adders

Widths

typically

close to k bits

k .. k + log2 n

CSA

k + log2 n

CPA

3

3

2

2

2

Parameters of tree carry-save adders (2)

Maximum number of inputs that can be reduced

to two by an h-level tree, n(h)

n(0) = 2

n(h) = n(h-1)

n(1) = 3

n(2) = 4

n(3) = 6

n(4) = 9

n(5) = 13

n(6) = 19

2 ( )h-1 < n(h) 2 ()h

Parameters of tree carry-save adders (3)

Smallest height of the tree carry save adder

for n operands, h(n)

h(n) = 1 + h( )

2

n

3

h(2) = 0

h(n) log ( )

n

3

2

2

Wallace trees

- Reduce the size of the final Carry Propagate Adder (CPA)
- Optimum from the point of view of speed

Dadda trees

- Reduce the cost of the carry save tree
- Optimum (among the CSA trees) from the point of
- view of area

- Wallace reduces number of operands at earliest opportunity
- Goal of this is to have smallest number of bits for CPA adder
- However, sometimes having a few bits longer CPA adder does not affect the propagation delay significantly (i.e. carry-lookahead)

- Dadda seeks to reduce the number of FA and HA units
- May be at the cost of a slightly larger final CPA

22

23

21

20

24

a

b

c

d

e

0 1 0 1 0

1 1 0 1 1

1 0 1 1 1

1 0 1 1 1

1 1 1 1 1

s0

s1

s2

0 1 1 1 0

0 1 1 0 0

1 0 0 1 1

a+b+c+d+e = s0+s1+s2

Implementation of 1-bit of 5-to-3 parallel counter

using single CLB slice of a Virtex FPGA

d

e

a

b

w

w

w

w

w

PC

s1

s2

s0

CSA

CPA

w

y=a+b+c+d+e mod 2w

Carry Save Adder vs. 5-to-3 Parallel Counter

a

b

c

d

e

w

w

w

w

w

CSA

CSA

CSA

CPA

w

y=a+b+c+d+e mod 2w

Fig. 8.17 Dot notation for a (5, 5; 4)-counter and the use of such counters for reducing five numbers to two numbers.

Unequal columns

Generalized Parallel CountersMulticolumn reduction

(5, 5; 4)-counter

Generalized parallel counter =

Parallel compressor

(2, 3; 3)-counter

Drop of such counters for reducing five numbers to two numbers.

Invert

(a) m = 2k

(b) m = 2k – 1

(c) m = 2k + 1

8.6 Modular Multioperand AddersFig. 8.20 Modular carry-save addition with special moduli.

Computer Arithmetic, Addition/Subtraction

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