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Combinational Logic: Other Gate Types

Explore the fundamental types of combinational logic gates, including primitive and complex gates such as NAND, NOR, XOR, and XNOR. Learn how these gates can express any Boolean function and their applications in parity bit generation and error detection. Familiarize yourself with truth tables, XOR/XNOR symbols, and the construction of odd and even parity circuits. This overview will guide you through designing combinational circuits using various logic gate types, enhancing your digital logic design skills.

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Combinational Logic: Other Gate Types

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  1. Combinational Logic: Other Gate Types

  2. Gate classifications • Primitive gate - a gate that can be described using a single primitive operation type (AND or OR) plus an optional inversion(s). • Complex gate - a gate that requires more than one primitive operation type for its description

  3. primitive gates • NAND • NOR

  4. NAND Gates

  5. NOR • NOT OR • Also common

  6. NAND is Universal • Universal gate : Can express any Boolean Function using only this type of gate • Equivalents below

  7. Sum of Products with NAND Easy to think of bubbles as canceling

  8. AND-OR Circuit Easy to Convert

  9. NOR Also Universal • Dual of NAND

  10. Buffer • No inversion • No change, except in power or voltage • Used to enable driving more inputs

  11. Parity Function • How does parity work ? • Given 7- bit ASCII code for A (100 0001) • What is the ASCII code for A with even parity ? • Write truth table for two input even parity generator • What needs to be generated for parity bit? • What function of two inputs gives you this? • This is called: Exclusive OR function

  12. Example Complex Digital Logic Gates: Exclusive OR/ Exclusive NOR Å = + X Y X Y X Y Å = + X Y X Y X Y • The Exclusive OR (XOR) function is defined as: • The eXclusive NOR (XNOR) function, otherwise known as equivalence is:

  13. Symbols For XOR and XNOR • XOR symbol: • XNOR symbol: • Symbols exist only for two inputs

  14. Truth Tables for XOR/XNOR Å Å X Y X Y X Y (X Y) º or X Y 0 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1 1 1 • Operator Rules: XOR XNOR • The XOR function means: X OR Y, but NOT BOTH • Why is the XNOR function also known as the equivalence function, denoted by the operator ?

  15. XOR Implementations Å = + X Y X Y X Y Y Y X X • The simple SOP implementation uses the following structure: • A NAND only implementation is: X Y X Y

  16. XOR/XNOR (Continued) X Y Z X Y Z X Y Z X Y Z X Y Z Å X 0 X X 1 X Å = = Å X X 0 X X 1 Å = X Y Y X Å Å ) = = Å Å Å Å Å Å ( X Y Z X ( Y Z ) X Y Z • The XOR function can be extended to 3 or more variables. For more than 2 variables, it is called an odd function or modulo 2 sum (Mod 2 sum), not an XOR: • The complement of the odd function is the even function. • The XOR identities: Å Å = + + + = =

  17. Question C 1 1 1 1 B 1 1 A 1 1 D • Draw the K-map of a 4 variable odd function

  18. Example: Odd Function Implementation + + • Design a 3-input odd function F = X Y Zwith 2-input XOR gates

  19. Example: Odd Function Implementation + + + + • Design a 3-input odd function F = X Y Zwith 2-input XOR gates • Factoring, F = (X Y) Z

  20. Example: Odd Function Implementation X Y F Z + + + + • Design a 3-input odd function F = X Y Zwith 2-input XOR gates • Factoring, F = (X Y) Z • The circuit:

  21. Example: Odd Function Implementation X Y F Z + + + + • Design a 3-input odd function F = X Y Zwith 2-input XOR gates • Factoring, F = (X Y) Z • The circuit: • Based on the above, given (X,Y,Z,F), then F would be the even parity bit for the three bits X,Y,Z. Hence, the circuit is an even parity generator.

  22. Even Parity Generators and Checkers X Y P Z X Y E Z P • An even parity bit could be added to n-bit code to produce an n + 1 bit code: • Use an odd function to produce codes with even parity • Use odd function circuit to check code words with even parity • Example: n = 3. Generate even parity code words of length 4 withan odd function circuit (parity generator): • Check even parity code words of length 4 with odd function circuit • Operation: (X,Y,Z) = (0,0,1) gives(X,Y,Z,P) = (0,0,1,1) and E = 0.If Y changes from 0 to 1 betweengenerator and checker, then E = 1 indicates an error.

  23. Odd Parity Generators and Checkers Similarly, an odd parity bit could be added to n-bit code to produce an n + 1 bit code • Use an even function to produce codes with odd parity • Use even function circuit to check code words with odd parity

  24. Tri-State • Output w/ 3 states: H, L, and Hi-Z • High impedance • Behaves like no output connection if in Hi-Z (Hi Impedance) state • Allows connecting multiple outputs

  25. Data Selector (2 to 1 Multiplexer) IN0 OL IN1 s Data Selection • If s = 0, OL = IN0, else OL = IN1

  26. Data Selection Function Implementation with 3-State Logic IN0 OL EN0 S IN1 EN1 • Data Selection Function: If s = 0, OL = IN0, else OL = IN1 • Performing data selection with 3-state buffers: • Since EN0 = S and EN1 = S, one of the two buffer outputs is always Hi-Z plus the last row of the table never occurs.

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