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Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 2
Gates and Boolean Algebra
The symbols and functional behavior for the five basic gates. Boolean function of n variable is completely described by
table with 2n rows, known as truth table
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 3
Majority Function
(a) Truth table for majority function of three variables.
(a) A circuit for (a).
M=ABC+ABC+ABC
+ABC
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 4
Circuit Equivalence
Construction of (a) NOT, (b) AND, and (c) OR gates using only NAND gates or only NOR gates.
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 5
Circuit Equivalence (Cont…)
To reduce complexity designers try to reduce no. of gates
Circuit computing same function as original with fewer gates is preferred
Boolean algebra can be valuable
Using Distributive law:
AB + AC = A (B+C)
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 6
Boolean Algebra Main Identities
To Circuit designers start with a Boolean function and then apply the laws of Boolean algebra to it
In an attempt to find a simpler but equivalent one
From the final function, circuit can be constructed
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 7
Alternate Symbols for NAND, NOR, AND, OR
De Morgan’s law suggests an alternative notation, (a) an OR gate is equivalent to the NAND gate with inverted inputs
(b) NOR gate can be drawn with an AND gate with inverted inputs By negating both forms we arrive at (c) and (d) respectively
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 8
XOR, Conversion to NAND Gate Only
(a) The truth table for the XOR function
(b-d) Three circuits for computing it
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 9
Circuit Equivalence (Cont…)
Same physical gate can compute different functions, depending on the convention used
(a) Electrical characteristics of a device.
(b) Positive logic,
(c) Negative logic,
AND function
OR function
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 12
Multiplexers
(a) An MSI multiplexer.
(b) The same multiplexer wired to compute the majority function.
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 15
Programmable Logic Arrays A 12-input, 6-output programmable logic
array
The little squares represent fuses that can be burned out
.
.
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 17
Half Adder
(a) A truth table for 1-bit addition
(b) A circuit for a half adder
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 18
Full Adder
(a) Truth table for a full adder
(b) Circuit for a full adder
Lecture # 6, DIGITAL LOGIC LEVEL DE-32-CE 20
Arithmetic Logic Units
Eight 1-bit ALU slices connected to make an 8-bit ALU The enables and invert signals are not shown for simplicity