Chapter 3 Digital Logic Structures

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3-2 Building Functions from Logic Gates We've already seen how to implement truth tables using AND, OR, and NOT -- an example of combinational logic. Combinational Logic Circuit output depends only on the current inputs stateless Sequential Logic Circuit output depends on the sequence of inputs (past and present) stores information (state) from past inputs We'll first look at some useful combinational circuits, then show how to use sequential circuits to store information.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Full Adder 3-3 Consider computing 7+6=13: Now, consider one column of this addition: A combinational logic design

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1-bit Full Adder 3-4 Truth table for a 1-bit adder: Formulate a circuit for each output

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The Majority Circuit for CarryOut 3-5

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Odd-Parity Circuit for the Sum 3-6

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3-7 Putting It All Together: Full Adder Add two bits and carry-in, produce one-bit sum and carry-out. ABC in SC out

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3-8 Four-bit Adder

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multiplexer (MUX) A device with multiple inputs and 1 output Could be used to allocate a resource to one of multiple clients: 3-9

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. MUX A 2 n -to-1 multiplexer (MUX) sends one of 2 n input lines to a single output line A MUX has two sets of inputs: 2 n data input lines n select lines used to pick one of the 2 n data inputs Simplest example is a 2-to-1 MUX 3-10

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-to-1 MUX 3-11

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Multiplexer (MUX) n-bit selector and 2 n inputs, one output output equals one of the inputs, depending on selector 4-to-1 MUX

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4-to-1 MUX from Two 2-to-1 MUXs 3-13 S0 S1

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Decoder General example: Assume that some information is encoded in n bits For each encoding, we want to activate the (one) correct output line The general idea: given an n-bit input Detect which of the 2 n combinations is represented Produce 2 n output, only one of which is “1” A n-to-2 n decoder takes an n-bit input and produces 2 n outputs. The n inputs represent a binary number that determines which one of the 2 n outputs is “true” (i.e., 1). 3-14

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-to4 Decoder 3-15 This circuit decodes a binary input into one of four possible choices, or codes

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Decoder n inputs, 2 n outputs exactly one output is 1 for each possible input pattern 2-bit decoder

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Representing Multi-bit Values Number bits from right (0) to left (n-1) just a convention -- could be left to right, but must be consistent Use brackets to denote range: D[l:r] denotes bit l to bit r, from left to right May also see A, especially in hardware block diagrams. A = A[2:0] = 101 A[14:9] =

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multibit Values in Circuit Diagrams A 4-to-1 mux, selecting one byte out of a 32-bit value

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. MUX example -- what does this circuit do? 3-19

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Combinational vs. Sequential Combinational Circuit always gives the same output for a given set of inputs  ex: adder always generates sum and carry, regardless of previous inputs Sequential Circuit stores information output depends on stored information (state) plus input  so a given input might produce different outputs, depending on the stored information example: ticket counter  advances when you push the button  output depends on previous state useful for building “memory” elements and “state machines”