CSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra


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1 CSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra Logic Gates Logic gatesare simple digital circuits that take one or more binary inputs and produce a binary output The relationship between the inputs and outputs can be described by a truth tableor a Boolean equation Each type of logic gate has its own corresponding symbol Inputs to a circuit are generally drawn at the left or top of a circuit diagramand the outputs on the right or bottom Usually we ll use letters A, B, C, for inputs and the letter Y for the output We will also see examples where subscripts are used to label inputs and outputs (e.g.,,,,,, ) 1 2 The NOT Gate Buffers Takes one input and produces one output The little circle is called a bubbleand indicates that negation is taking place The overline indicates negation Say it like this: Yequals NOT A Negation can also be denoted. I will use both notations in lecture notes, examples, quizzes and exams. A NOT gate is also called an inverter because it inverts its input In these truth tables, 0 indicates falseand 1 indicates true A buffertakes one input and reproduces that input on its output This seems pointless, but buffers can be used for useful tasks like changing voltage or amplifying a signal Could we implement a buffer using NOT gates? 3 4
2 The AND Gate The OR Gate The AND gate performs a logical AND operation over bits It produces a 1 only when both inputs are 1; otherwise, it produces a 0 Also denoted with a dot: Yequals AAND B (not Atimes B ) Note the order in which the values for A and Bare listed. This is the reverse order of what you typically see in a discrete mathematics class (e.g., CSE 215) If Aand Bare taken together to form a 2bit number, we are counting in binary from 0 to 3 The OR gate performs a logical OR operation over bits It produces a 1 when either input or both inputs are 1; otherwise, it produces a 0 Sometimes called inclusive ORbecause the gate produces a 1 when both inputs are 1 Also denoted with a plus sign: + Yequals AOR B (not Aplus B ) 5 6 The XOR Gate The NAND Gate The XOR gate performs a logical exclusiveor operator over bits It produces a 1 when only one of the inputs is 1; otherwise, it produces a 0 In general, an Ninput XOR gate produces a 1 if an odd number of its inputs are 1 The NAND gate produces a 0 only when both inputs are 1; otherwise, it produces a 1 In other words, it behaves like an AND gate with a NOT gate connected to the AND gate s output wire 7 8
3 The NOR Gate The XNOR Gate The NOR gate outputs a 1 if neither input is 1 Stated another way, a NOR gate produces a 1 only when both inputs are 0 It behaves like an OR gate with a NOT gate connected to the OR gate s output wire NAND and NOR gates are preferred in circuit manufacturing because each requires only 2 transistors, whereas AND and OR gates require 3 transistors each Fewer transistors smaller, cheaper circuit; more gates can fit on a silicon die The XNOR gate produces a 1 when both inputs are the same It behaves like an XOR gate with a NOT gate connected to the XOR gate s output wire 9 10 MultipleInput Logic Gates MultipleInput Logic Gates Multipleinput logic gates exist for most of the gates we have studied so far (NOT being an obvious exception) An Ninput AND gate produces a 1 if all the inputs are 1 An Ninput OR gate produces a 1 if at least one input is 1 What would be the behavior of a 3input NOR gate? Answer: produce a 1 only when all three inputs are 0s 11 12
4 Digital Circuit Design Digital Circuit Design A logic circuit is composed of: Inputs and outputs Functional specification (what does it do) Timing specification (how long does it take to do that) A circuit as a black box of inputs and outputs. Digital circuits can be combinational or sequential A combinational circuit s output depends only on its current inputs In contrast, sequential circuits have feedback the outputs are connected to the inputs Sequential circuits can be used to create memory Nodes(wires) Inputs: A, B, C Outputs: Y, Z Internal: n1 Circuit elements E1, E2, E3 Each is a circuit Specification vs. Implementation As is the case with algorithms, there could be different ways to implement a circuit s specification Below is a specification for a circuit ( CL indicates that it is implemented using combinational logic) Rules of Combinational Composition These rules tell us how we can build a large combinational circuit from smaller combinational circuits 1. Every element is combinational 2. Every node is either an input or connects to exactly one output 3. The circuit contains no cyclic paths It could be implemented in a variety of ways. Two possibilities are shown on the right
5 Rules of Combinational Composition Boolean Equations Which of these are combinational circuits and which ones aren t? A Boolean equation gives a formula that gives a functional specification of outputs in terms of inputs Example: =,, =,, Some Definitions Complement: variable with a bar over it,, Remember: can also be written. Literal: variable or its complement,,,,, Implicant: a product of literals,, Minterm: a product that includes all input variables For a 3input circuit (inputs: A, B, C),, and would be minterms, but and would not be Maxterm: a sum that includes all input variables For a 3input circuit, + + and + + would be maxterms, but + and + would not be SumofProducts (SOP) Form Each row in a truth table corresponds with a minterm Each mintermis TRUE for that row (and only that row) The rows are labeled 0 through 2 1where Nis the number of inputs The row number can be determined by taking the Ninputs as binary digits of an Nbit number 19 20
6 SumofProducts (SOP) Form Any equation can be written in sumofproducts canonical formas a sum (OR) of products (AND) Form the function by ORingonly those mintermsfor which the output is TRUE SumofProducts (SOP) Form SOP form can also be written in sigma notation For example, = + could also be written, = Σ, or, = Σ(1,3)or, = Σ(1,3) Note that you MUST list the input variables in the same order as given above: start with 0 and increment the count from there For the function given above, its SOP form is = ProductofSums (POS) Form ProductofSums (POS) Form Any equation can be written in productofsums canonical formas a product (AND) of sums (OR) Each row in a truth table corresponds with a maxterm Each maxtermis FALSE for that row (and only that row) Form the function by ANDingonly those maxtermsfor which the output is FALSE For the function given above, its POS form is = + + POS form can also be written in pi notation For example, = + + could also be written, = Π #,# or, = Π(0,2)or, = Π#(0,2) 23 24
7 SOP vs. POS The sumofproducts form of a Boolean equation and its corresponding productofsums form are dualsof each other: they are equivalent ways of expressing the same function Generally we use SOP form when a truth table has only a few rows where the output is TRUE. Why? Because it leads to a shorter equations. Likewise, we use POS form when a truth table has only a few rows where the output is FALSE The logic used in SOP and POS forms is called twolevel logicbecause it consists of two levels of gates: ANDOR or ORAND, depending on the form (SOP = ANDOR, POS = ORAND) SOP vs. POS (cont.) We use the generic word termto refer to the minimal unit of a Boolean expression Depending on the form of the function, a term is the firstlevel gate unit Examples: ()has one term, but + has two terms ( + +)has one term, but has two terms Boolean Equation Example You are going to the cafeteria for lunch You won t eat lunch (%) if it s not open (&)or if they only serve corndogs () Write a truth table for determining if you will eat lunch (%) and both the SOP and POS forms of the Boolean equation Boolean Algebra Boolean equations written in SOP and POS form lead to circuits that have many gates, which increases cost of the circuit and makes it slower than equivalent circuits that have simpler (but logically equivalent) Boolean expressions We can use axioms and theorems of Boolean algebra to simplify complex equations Boolean algebra is similar to regular algebra, but we have Boolean values (1 and 0) instead of numerical values Most of these axioms and theorems exhibit duality, meaning you can exchange AND for OR and also 0 for 1 and the theorem still holds 27 29
8 Boolean Algebra Axioms Axiom Dual = 0if 1 = 1if 0 0 = 1 1 = = = = = = 1 0 = = = 1 Boolean Algebra Theorems Theorem Dual Name 1 = 0 = 0 = = + 0 = Identity + 1 = 1 Null element + = Idempotency Involution = 0 + = 1 Complement Boolean algebra theorems allow us to simplify Boolean expressions, which simplify the corresponding digital circuits All of these theorems involve only one variable. We will see theorems for two or more variables shortly Boolean Algebra Theorems Boolean Algebra Theorems The identity theoremallows us to replace an AND or OR gate with a wire connected to the input The idempotency theorem allows us to replace an AND or ORgate with a wire connected to the input The null element theorem allows us to replace an AND or ORgate with a wire tied to high (1) or low (0) The involution theorem allows us to replace two inverters wired in series with a wire connected to the input 32 33
9 Boolean Algebra Theorems Boolean Algebra Theorems The complement theorem allows us to replace an AND or ORgate with a wire connected to high or low Theorem Dual Name = + = + Commutativity ( = ( () + + ( = + ( + () Associativity + ( = + ( + + ( = + ( Distributivity + = + = Absorption + = + + = Combining + ( + ( = + ( + + ( + ( = + + ( Consensus + = + + = No name = = De Morgan s Laws De Morgan s Laws as Circuits Simplifying Boolean Expressions De Morgan s Laws give us an interesting result that is helpful in implementing circuits: a NAND gate is equivalent to an OR gate with inverted inputs, and a NOR gate is equivalent to an AND gate with inverted inputs Since it is often advantageous to use exclusively NAND or NOR gates in a circuits, we find that de Morgan s laws will always let us replace AND gates with NOR gates, and OR gates with NAND gates First look for terms that you can group together and factor out elements that evaluate to 1. Example: + = + = (why?) Example: + = 1+ = (why?) Second, if you cannot find any terms to group together and simplify, then multiply elements by identities equivalent to 1 (but only using variables which don tappear in the term) Example of this on the next slide 36 37
10 Simplifying Boolean Expressions Simplifying Boolean Expressions Example: + + = identity = complement = distributivity = identity = (1 + ) distributivity = + null element This is a proof of the consensus theorem When all else fails, the best bet is to expand out all the terms into minterms, use the idempotency theorem to duplicate terms, and then regroup them Example: minimize + + = idempotency = + + ( + ) distributivity (used 2x) = complement (used 2x) = + identity (used 2x) It s not always obvious (i) that an expression is fully simplified, or (ii) if it isn t simplified, how to simplify it Later I will teach you a visual technique called Karnaugh maps that makes this process easier From Logic to Gates Example: From Logic to Gates To make circuit diagrams (schematics) as easy as possible to draw and read, we will adopt a few guidelines: Inputs are on the left (or top) Outputs are on right (or bottom) Gates flow from left to right Use straight wires only Wires always connect at a T junction A dot where wires cross indicates a connection between the wires Wires crossing withouta dot make no connection Let s draw a schematic for =
11 Example: From Logic to Gates Recall that we can simplify = + + to = + This means we can draw a much simpler diagram, resulting in a cheaper, faster circuit (hopefully!) Example: From Logic to Gates But wait, we can do even better! Both inputs to the top AND gate are complemented, and nowhere else do we use. This means we can use a NOR gate (drawn using AND) and eliminate one gate (an inverter) Example: Majority Circuit Example: Majority Circuit The majority function takes several 1bit inputs and outputs 1 if a majority of the inputs is 1 Consider the 3bit majority function (,, ) A B C (,,) Write Fin sumofproducts form and draw its circuit diagram
12 Example: Majority Circuit Example: Majority Circuit The majority function takes several 1bit inputs and outputs 1 if a majority of the inputs is 1 Consider the 3bit majority function (,, ) A B C (,,) Write Fin productofsums form and draw its circuit diagram MultipleOutput Circuits 4bit Priority Circuit Example: Priority circuit the output is asserted which corresponds to the most significant TRUE input Suppose we have four inputs:,,, and they are given in order of priority,,, (i.e., has highest priority and has lowest) We have four outputs,,, and corresponding to these four inputs, respectively is TRUE whenever asserted is TRUE if is asserted and is not is TRUE if is asserted and neither nor is is TRUE if is asserted and no other is 0 if no input is asserted How would we implement this circuit? First, derive Boolean expressions for,,, 55 56
13 4bit Priority Circuit 4bit Priority Circuit: Don t Cares In that truth table, we actually don t care what certain values are in many cases For example, if is asserted, it doesn t matter what the other three values are; they could be any combination of 1s and 0s and we don t care We write the letter X when we want to indicate that we don t care whata particular input or output is We will see in a later unit how to exploit don t cares to simplify circuits Example: p. 99, #2.18(a) Simplify =
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