EECS 150  Components and Design Techniques for Digital Systems Lec 18 – Arithmetic II (Multiplication). David Culler Electrical Engineering and Computer Sciences University of California, Berkeley http://www.eecs.berkeley.edu/~culler http://wwwinst.eecs.berkeley.edu/~cs150. Review.
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David Culler
Electrical Engineering and Computer Sciences
University of California, Berkeley
http://www.eecs.berkeley.edu/~culler
http://wwwinst.eecs.berkeley.edu/~cs150
8 + 5 Systems
Computer Number Systems1
+0
2
1111
0000
+1
1110
0001
3
+2
1101
0010
4
1100
+3
0011
5
1011
0100
+4
1010
6
0101
+5
1001
0110
7
+6
1000
0111
8
+7
How can you tell an overflow occurred?
Add two positive numbers to get a negative number
or two negative numbers to get a positive number
1
1
+0
+0
2
2
1111
0000
+1
1111
0000
+1
1110
1110
0001
0001
3
3
+2
+2
1101
1101
0010
0010
4
4
1100
+3
1100
+3
0011
0011
5
5
1011
1011
0100
+4
0100
+4
1010
1010
6
6
0101
0101
+5
+5
1001
1001
0110
0110
7
7
+6
+6
1000
0111
1000
0111
8
8
+7
+7
7  2 = +7!
5 + 3 = 8!
0 1 1 1
0 1 0 1
0 0 1 1
1 0 0 0
1 0 0 0
1 0 0 1
1 1 0 0
1 0 1 1 1
5
3
8
7
2
7
Overflow
Overflow
0 0 0 0
0 1 0 1
0 0 1 0
0 1 1 1
1 1 1 1
1 1 0 1
1 0 1 1
1 1 0 0 0
5
2
7
3
5
8
No overflow
No overflow
Overflow occurs when carry in to sign does not equal carry out
A  B = A + (B) = A + B + 1
Dedicated carry logic provides fast arithmetic carry capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.
The arithmetic logic includes an XOR gate and AND gate that allows a 2bit full adder to be implemented within a slice.
Cin to Cout delay = 0.1ns, versus 0.4ns for F to X delay.
Adders on the Xilinx VirtexHow do we map a 2bit adder to one slice?
a b c capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.i ci+1 s
Carry Lookahead Addersdelay log(n)
carry “kill”
ki = ai’ bi’
carry “propagate”
pi = ai bi
carry “generate”
gi = ai bi
ci+1 = gi + pici
si = pi ci
c capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.in
pi
pi+1
pi+k
gi
gi+1
gi+k
Carry Lookahead Adders – in blocksP = pi pi+1 … pi+k
G = gi+k + pi+kgi+k1 + … + (pi+1pi+2 … pi+k)gi
cout
Cout = G + PCin
c capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.0
Carry Lookahead Adders9bit Example of hierarchically generated P and G signals:
a0
Pa
b0
a1
a
b1
a2
Ga
P = PaPbPc
b2
c3 = Ga + Pac0
a3
Pb
b3
a4
b
b4
a5
Gb
b5
c6 = Gb + Pbc3
a6
G = Gc + PcGb + PbPcGa
Pc
b6
a7
c
b7
a8
c9 = G + Pc0
Gc
b8
x capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.
BA
30
74
30
74
30
30
BA
Ax
10
BAx
54
54
32
10
76
54
32
76
10
30
10
54
6
0
2
4
0
4
2
1
6
5
3
7
76
54
32
10
74
64
30
20
70
60
50
40
Parallel Prefix (generalizing CLA)70
B
A
c capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.0
cin
P = PaPb
G = Gb + GaPb
Cout = G + cinP
Pa,Ga
P,G
Pb,Gb
cout
c0
a0
ci
p = a b
g = ab
s = p ci
ci+1 = g + cip
p,g
b0
ai
s0
P,G
bi
p,g
c1
c0
a1
si
b1
s1
ci+1
c2
a2
b2
s2
c3
c0
a3
b3
8bit Carry Lookahead Adder
s3
P,G
c4
a4
b4
s4
c8
c5
c0
a5
b5
s5
c6
a6
b6
s6
c7
a7
b7
s7
Addition of 2 nbit numbers: capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.
takes n clock cycles,
uses 1 FF, 1 FA cell, plus registers
the bit streams may come from or go to other circuits, therefore the registers may be optional.
Requires controller
What does the FSM look like? Implemented?
Final carry out?
A, B, and R held in shiftregisters. Shift right once per clock cycle.
Reset is asserted by controller.
Bitserial Adderlsb
Partial products capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.
10001111
(143)
Basic concept of multiplicationmultiplicand
multiplier
1101 (13)
1011 (11)
1101
*
1101
0000
1101
A1
B1
A1 B0
A0 B1
A0
B0
A0 B0
A3
B3
A2 B0
A2 B1
A1 B2
A0 B3
A2
B2
A2 B0
A1 B1
A0 B2
A3 B1
A2 B2
A1 B3
A3 B3
A3 B2
A2 B3
S7
S6
S4
S5
S3
S2
S1
S0
Generates all n partial products simultaneously.
Each row: nbit adder with AND gates
What is the critical path?
Sums each partial product, one at a time. capability for highspeed arithmetic functions. The VirtexE CLB supports two separate carry chains, one per Slice. The height of the carry chains is two bits per CLB.
In binary, each partial product is shifted versions of A or 0.
Control Algorithm:
1. P 0, A multiplicand,
B multiplier
2. If LSB of B==1 then add A to P
else add 0
3. Shift [P][B] right 1
4. Repeat steps 2 and 3 n1 times.
5. [P][B] has product.
“Shift and Add” MultiplierSpeeding up multiplication is a matter of speeding up the summing of the partial products.
“Carrysave” addition can help.
Carrysave addition passes (saves) the carries to the output, rather than propagating them.
Example: sum three numbers,
310 = 0011, 210 = 0010, 310 = 0011
310 0011
+ 210 0010
c 0100 = 410
s 0001 = 110
310 0011
c 0010 = 210
s 0110 = 610
1000 = 810
Carrysave Additioncarrysave add
carrysave add
carrypropagate add
When adding sets of numbers, carrysave can be used on all but the final sum.
Standard adder (carry propagate) is used for final sum.
Carrysave CircuitsFast carrypropagate adder
Add but the final sum.
CPA
Another Representation (from book)Building block: full adder + and
4 x 4 array of building blocks
CSA is associative and communitive. For example:
(((X0 + X1)+X2 )+X3 ) = ((X0 + X1)+(X2 +X3 ))
Signed Multiplication:
Remember for 2’s complement numbers MSB has negative weight:
ex: 6 = 110102 = 0•20 + 1•21 + 0•22 + 1•23  1•24
= 0 + 2 + 0 + 8  16 = 6
a) subtract final partial product
b) signextend partial products
a) adder/subtractor
b) signextender on P shifter register
multiplicand
multiplier
1101 (3)
1011 (5)
1101
*
1111
Note: 2s complement
Sign extension
+
+(3)
111
+(6)
+
11010
+
00
000000

(24)
1
1101000
(15)
00001111
Implicit but the final sum.
Sign
extension




Signed Array Multiplierb3 0
b2 0
b1 0
b0 0
a0
0
P0
a1
0
P1
a2
0
P2
a3
0
P3
P7 P6 P5 P4
Signed extend partial product at each stage but the final sum.
Final step is a subtract
“Shift and Add” Signed Multiplier