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CS 61C L22 Representations of Combinatorial Logic Circuits (1) Garcia, Spring 2004 © UCB
Lecturer PSOE Dan Garcia
www.cs.berkeley.edu/~ddgarcia
inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures
Lecture 22 – Representations of Combinatorial Logic Circuits
2004-10-20
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CS 61C L22 Representations of Combinatorial Logic Circuits (2) Garcia, Spring 2004 © UCB
Review…•We use feedback to maintain state•Register files used to build memories
•D-FlipFlops used for Register files
•Clocks usually tied to D-FlipFlop load•Setup and Hold times important
•Pipeline big-delay CL for faster clock•Finite State Machines extremely useful•You’ll see them again in 150, 152 & 164
CS 61C L22 Representations of Combinatorial Logic Circuits (3) Garcia, Spring 2004 © UCB
Representations of CL Circuits…•Truth Tables
•Logic Gates
•Boolean Algebra
CS 61C L22 Representations of Combinatorial Logic Circuits (4) Garcia, Spring 2004 © UCB
Truth Tables
0
CS 61C L22 Representations of Combinatorial Logic Circuits (5) Garcia, Spring 2004 © UCB
TT Example #1: 1 iff one (not both) a,b=1
a b y
0 0 0
0 1 1
1 0 1
1 1 0
CS 61C L22 Representations of Combinatorial Logic Circuits (6) Garcia, Spring 2004 © UCB
TT Example #2: 2-bit adder
HowManyRows?
CS 61C L22 Representations of Combinatorial Logic Circuits (7) Garcia, Spring 2004 © UCB
TT Example #3: 32-bit unsigned adder
HowManyRows?
CS 61C L22 Representations of Combinatorial Logic Circuits (8) Garcia, Spring 2004 © UCB
TT Example #3: 3-input majority circuit
CS 61C L22 Representations of Combinatorial Logic Circuits (9) Garcia, Spring 2004 © UCB
Logic Gates (1/2)
CS 61C L22 Representations of Combinatorial Logic Circuits (10) Garcia, Spring 2004 © UCB
And vs. Or review – Dan’s mnemonic
AND Gate
CA
B
Symbol
A B C0 0 00 1 01 0 01 1 1
Definition
AND
CS 61C L22 Representations of Combinatorial Logic Circuits (11) Garcia, Spring 2004 © UCB
Logic Gates (2/2)
CS 61C L22 Representations of Combinatorial Logic Circuits (12) Garcia, Spring 2004 © UCB
2-input gates extend to n-inputs
• N-input XOR is the only one which isn’t so obvious
• It’s simple: XOR is a 1 iff the # of 1s at its input is odd
CS 61C L22 Representations of Combinatorial Logic Circuits (13) Garcia, Spring 2004 © UCB
Truth Table Gates (e.g., majority circ.)
CS 61C L22 Representations of Combinatorial Logic Circuits (14) Garcia, Spring 2004 © UCB
Truth Table Gates (e.g., FSM circ.)
PS Input NS Output
00 0 00 0
00 1 01 0
01 0 00 0
01 1 10 0
10 0 00 010 1 00 1
or equivalently…
CS 61C L22 Representations of Combinatorial Logic Circuits (15) Garcia, Spring 2004 © UCB
Boolean Algebra
•George Boole, 19th Century mathematician
•Developed a mathematical system (algebra) involving logic• later known as “Boolean Algebra”
•Primitive functions: AND, OR and NOT
•The power of BA is there’s a one-to-one correspondence between circuits made up of AND, OR and NOT gates and equations in BA
+ means OR,• means AND, x means NOT
CS 61C L22 Representations of Combinatorial Logic Circuits (16) Garcia, Spring 2004 © UCB
Boolean Algebra (e.g., for majority fun.)
y = a • b + a • c + b • c
y = ab + ac + bc
CS 61C L22 Representations of Combinatorial Logic Circuits (17) Garcia, Spring 2004 © UCB
Boolean Algebra (e.g., for FSM)
PS Input NS Output
00 0 00 0
00 1 01 0
01 0 00 0
01 1 10 0
10 0 00 010 1 00 1
or equivalently…
y = PS1 • PS0 • INPUT
CS 61C L22 Representations of Combinatorial Logic Circuits (18) Garcia, Spring 2004 © UCB
BA: Circuit & Algebraic Simplification
BA also great for circuit verificationCirc X = Circ Y?use BA to prove!
CS 61C L22 Representations of Combinatorial Logic Circuits (19) Garcia, Spring 2004 © UCB
Laws of Boolean Algebra
CS 61C L22 Representations of Combinatorial Logic Circuits (20) Garcia, Spring 2004 © UCB
Boolean Algebraic Simplification Example
CS 61C L22 Representations of Combinatorial Logic Circuits (21) Garcia, Spring 2004 © UCB
Canonical forms (1/2)
Sum-of-products(ORs of ANDs)
CS 61C L22 Representations of Combinatorial Logic Circuits (22) Garcia, Spring 2004 © UCB
Canonical forms (2/2)
CS 61C L22 Representations of Combinatorial Logic Circuits (23) Garcia, Spring 2004 © UCB
Peer Instruction
A. (a+b)• (a+b) = b
B. N-input gates can be thought of cascaded 2-input gates. I.e., (a ∆ bc ∆ d ∆ e) = a ∆ (bc ∆ (d ∆ e))where ∆ is one of AND, OR, XOR, NAND
C. You can use NOR(s) with clever wiring to simulate AND, OR, & NOT
ABC1: FFF2: FFT3: FTF4: FTT5: TFF6: TFT7: TTF8: TTT