Digital Logic: Basic Gates, Boolean Algebra, and Circuits, Study notes of Computer Architecture and Organization

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Digital logic Circuits

Basic Gates

• (^) NOT
• (^) AND
• (^) OR
• (^) BUFFER
• (^) XOR - ODD
• (^) XNOR - EVEN
• (^) NAND
• (^) NOR

Boolean Algebra Identities

Complement Laws x + ~x = 1 x · ~x = 0 Law of the Double Complement ~(~x) = x Idempotent Laws x + x = x x · x = x Identity Laws x + 0 = x x · 1 = x Dominance Laws x + 1 = 1 x · 0 = 0 Commutative Laws x + y = y + x x · y = y · x Associative Laws x + (y + z) = (x + y) + z x · (y · z) = (x · y) · z Distributive Laws x + (y · z) = (x + y) · (x + z) x · (y + z) = (x · y)+(x · z) DeMorgan's Laws ~(x · y) = ~x + ~y ~(x + y) = ~x · ~y Absorption Laws x · (x + y) = x x + (x · y) = x Simplification Laws

x · (~x + y) = x · y

x + (~x · y) = x + y

Combinational Circuits

Example – Line Follower

Example – Line follower

L C R O1 O

Half Adder

Full Adder

Implementation of Full Adder using two half adder S=x XOR y XOR z C=xy + (x XOR y) and z

Clocked SR Flip Flop

• (^) S
• (^) R
• (^) Q
• (^) C
• (^) Q

Clocked SR Flip Flop Truth Table

S R Qt+1 nQ clk 0 0 Qt nQt 1 0 1 0 1 1 1 0 1 0 1 1 1 indetermina te indetermina te

D Flip Flip Truth Table

D Qt+1 nQ clk 0 0 1 1 1 1 0 1

D Flip Flop Timing Diagram

T

T

T

JK Flip Flop Truth Table

J K Qt+1 nQ clk 0 0 Qt nQt 1 0 1 0 1 1 1 0 1 0 1 1 1 nQt Qt 1

JK Flip Flop Timing Diagram

T

T

T

T

T