EE241 1 UC Berkeley EE241 B. Nikolic EE241 - Spring 2000 Advanced Digital Integrated Circuits Lecture 16 Adders UC Berkeley EE241 B. Nikolic Carry-Skip Adder FA FA FA FA P0 G1 P0 G1 P2 G2 P3 G3 Co,3 Co,2 Co,1 Co,0 Ci,0 FA FA FA FA P0 G1 P0 G1 P2 G2 P3 G3 Co,2 Co,1 Co,0 Ci,0 Co,3 Multiplexer BP=P oP1P2P3 Idea: If (P0 and P1 and P2 and P3 = 1) then C o3 = C 0 , else “kill” or “generate”. Bypass MacSorley, Proc IRE 1/61 Lehman, Burla, IRE Trans on Comp, 12/61
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EE241
1
UC Berkeley EE241 B. Nikoli c
EE241 - Spring 2000Advanced Digital Integrated Circuits
Lecture 16
Adders
UC Berkeley EE241 B. Nikoli c
Carry-Skip Adder
FA FA FA FA
P0 G1 P0 G1 P2 G2 P3 G3
Co,3Co,2Co,1Co,0Ci,0
FA FA FA FA
P0 G1 P0 G1 P2 G2 P3 G3
Co,2Co,1Co,0Ci,0
Co,3
Mul
tiple
xer
BP=PoP1P2P3
Idea: If (P0 and P1 and P2 and P3 = 1)then Co3 = C0, else “kill” or “generate”.
Bypass
MacSorley, Proc IRE 1/61Lehman, Burla, IRE Trans on Comp, 12/61
EE241
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UC Berkeley EE241 B. Nikoli c
Carry-Skip Adder
Setup
CarryPropagation
Sum
Setup
CarryPropagation
Sum
Setup
CarryPropagation
Sum
Setup
CarryPropagation
Sum
Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15
C i,0
Critical Path
( ) ( ) RCASKIPRCAd tktkN
tkt 121 −+
−+−=
For N-bit adder with k-bit groups
UC Berkeley EE241 B. Nikoli c
Carry-Skip Adder
( ) SKIPRCAd tkN
tkt
−+−= 212
Critical path delay with constant groups
N
tp
ripple adder
bypass adder
4..8
EE241
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UC Berkeley EE241 B. Nikoli c
Carry-Skip Adder
Variable Group Length
Oklobdzija, Barnes, Arith’85
321 cNcctd ++=
UC Berkeley EE241 B. Nikoli c
Carry-Skip Adder
Variable Block Lengths
EE241
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UC Berkeley EE241 B. Nikoli c
Manchester Carry Chain
P0
Ci,0
P1
G0
P2
G1
P3
G2
P4
G3 G4
φ
φ
VDD
Kilburn, et al, IEE Proc, 1959.
•Implement P with pass-transistors•Implement G with pull-up, kill (delete) with pull-down•Use dynamic logic to reduce the complexity and speed up
UC Berkeley EE241 B. Nikoli c
Sizing Manchester Carry Chain
R1
C1
R2
C2
R3
C3
R4
C4
R5
C5
R6
C6
Out
M0 M1 M2 M3 M4MC
Discharge Transistor
1 2 3 4 5 6
tp 0.69 Ci Rjj 1=
i∑
i 1=
N∑=
1 1.5 2.0 2.5 3.0k
5
10
15
20
25
Spe
ed
1 1.5 2.0 2.5 3.0k
0
100
200
300
400
Are
a
Speed (normalized by 0.69RC) Area (in minimum size devices)
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UC Berkeley EE241 B. Nikoli c
Manchester Chain with Carry-Skip
P0
Ci,0
P1
G0
P2
G1
P3
G2
BP
G3
BP
Co,3
Delay model:
UC Berkeley EE241 B. Nikoli c
PTL with SA-F/F Implementation
Matsui,JSSC 12/94
EE241
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UC Berkeley EE241 B. Nikoli c
Propagate and Generate Signals
Define 3 new variable which ONLY depend on A, B
Generate (G) = AB
Propagate (P) = A ⊕ B
Delete = A B
Can also derive expressions for S and Co based on D and P
UC Berkeley EE241 B. Nikoli c
A0,B 0 A1,B 1 AN-1 ,BN-1...
Ci,0 P0 Ci,1 P1Ci,N-1 PN-1
...
Carry Lookahead Adder
Weinberger, Smith, 1958.
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UC Berkeley EE241 B. Nikoli c
Lookahead Adder
1−+= iiii cpgc
Looakahead Equations
( )1111
111
111
−+++
−++
+++
++=
++=+=
iiiiii
iiiii
iiii
cppgpg
cpgpgcpgc
Position i:
Position i + 1:
Carry exists if:- generated in stage i + 1- generated in stage i and propagated through i + 1- propagated through both i and i + 1
UC Berkeley EE241 B. Nikoli c
Lookahead Adder
• Unrolling of carry recurrence can be continued• If unrolled to level k, resulting in two-level AND-OR
structure• AND Fan-In = k + 1, OR Fan-In = k + 1• k + 1 transistors in the MOS stack• Limits k to 3 -4