Introductory Concepts This section of the course introduces the concept of digital circuits and devices, that are building block of digital circuits. Generally, there are two class of signals which we can apply to a circuit: analogue or digital. An analogue signal is the one where the quantities that we are passing to circuit are continuous range of values, for example sending sound signal to amplifier circuit. An digital signal is the one where the quantities that we are passing to circuit are Discrete values, for example Switch in a electrical circuit can pass discrete signal (on , off) to circuit.

Introduction to Logic Circuits Based on the information (digital signal) that pass to a circuit, we can design systems which can use binary logic In binary logic quantities are described by 0 or 1, YES or NO, TRUE or FALSE, etc. Logic circuits are the basis of digital computers. In this course, we deal with binary logic circuits, which have a dominant role in digital technology. The goal is to introduce and provide an understanding of how these circuits work, how they are represented in mathematical notation, and how they are designed using modern automated design technique

Variables and Functions In binary circuits, signals can have only one of two possible values. The simplest binary element is a switch that has two states. The switch can be used in a simple application where it turns a small light bulb on or off. The state of the switch determines whether or not current will flow. When the switch is open (state 0) no current can flow, when it is closed (state 1) current can flow. State 0 (x =0) State 1 (x =1)

x = 0 x = 1 The input to this circuit which causes the change in the behavior, is the switch, that is represented as ‘x’. The output of the circuit is the state of the light “L”. If the light is on, we will say that L=1. If the light is off, we will say that L=0. We can describe the state of the light L as a function of the input variable x. L = f(x) = x This simple logic expression describes the output as a function of the input. We say that f(x) = x is a logic function with the variable x as the input variable.

(state of light) L = f ( x 1 , x 2 ) = x 1 . x 2 Now consider using two switches to control the state of the light. The switches can be connected either in series or in parallel. Using a series connection, the light will be turned on only if both switches are closed. If either switch is open, the light will be off. (state of light) L = f ( x 1 , x 2 ) = x 1 . x 2 where L = 1 if x1 = 1, and x2 = 1 otherwise L = 0 The . symbol is called the AND operator, and the above circuit is said to implement a logical AND function. x1 x2

If the switches are connected in parallel, the light will be on if one, or both, of the switches are closed. The light will be off only if both switches are open. L = f ( x 1 , x 2 ) = x 1 + x 2 If either x1 = 1 or x2 = 1 then L = 1 (light is on ) If both x1 = x2 = 1 then L = 1 (light is on ) If both x1 = x2 = 0 then L = 0 (light is off ) The “+ “symbol is called the OR operator, and the above circuit is said to implement a logical OR function. x1 x2

Finally, consider a circuit which will generate an output when a switch is opened. In this case, the light will be turned on when the switch opens. Conversely, when the switch is closed, it will short-circuit the light and prevent any current from flowing through it. L = f ( x ) = x where L = 1 if x = 0; L = 0 if x = 1 We say that L is the inverse or complement of x. Another frequently used term is the NOT operation. There are several commonly used notations for indicating the complement: x = x’ = ! x = NOT x x= 0

The AND and OR functions can be extended to n variables. The AND, OR, and NOT functions are the three most important logic functions we will use. They can be used as building blocks for the implementation of all logic circuits. The AND and OR functions can be extended to n variables. An AND function of variables x1, x2, …., xn has the value of 1 only if all n variables are equal to 1. An OR function of variables x1, x2, …., xn has the value of 1 if one, or more of the variables are equal to 1.

Truth Tables The operations of a logic circuit can be defined in a tabular form, called a truth table. A truth table lists all the possible combinations of the input variables and shows the relationship between the input variables and the resulting output. Truth tables grow exponentially in size with the number of variables. A truth table with three input variables has eight rows,23 since there are eight possible valuations of these variables. For four-input variables the truth table has 16 rows, 24 ,and so on. x1 x2 x1 x2 x1+x2 1

Logic Gates Each logic operation can be implemented by electronic device called transistor (you will learn about the physical property of transistors and how they work on Physics301 or CS232 ). Each of these transistor which in this course we call them logic gates , are element and building block of digital circuits. A logic gate has one or more inputs and one output AND Gate OR Gate NOT Gate A larger circuit is implemented by a network of gates. The complexity of a given network has a direct impact on its cost. The more gate we use on the circuit, the more costly is the circuit. Thus it is important to simplify circuit as much as possible x1 . x2 x1 x2 x1 x2 x1+x2 x’ x

Tasks on logical circuits and logical functions Analysis is the task of determining the logical function that a logic network represent. Synthesis is the reverse task . It means having logical function find out what logical circuit network is. In general, any logic function can be implemented with a variety of different networks, and each likely having a different cost. (next two examples will show this fact) So how does one find the best implementation for a given function? In order to find the best implementation (less costly circuit), we need to know basic algebraic manipulation of logic function

Example1: Determine the following circuit's functional behavior. Consider what happens if we apply all possible input signals to it. x2 x1 A B f(x1,x2) x1 x2 A=x1 B=x1 x2 f(x1,x2) 1

Example2: Analyze of the following network Note that in the end, g changes in exactly the same way as f in the previous example. Thus, g(x1, x2) = f(x1, x2), which indicates that two networks are functionally equivalent. Since both networks represent the same function, it makes sense to use the simpler one, which is less costly to implement. x2 x1 g(x1,x2) A B x1 x2 A B g(x1,x2)

Boolean Algebra In order to simplify a function and design a less costly circuit we need to know the rule of Boolean Algebra G. Boole (1849) published a model for the algebraic description of processes that involved in logical thought and reasoning .This model became known as Boolean algebra. C. Shannon (1938) showed that Boolean algebra provides an effective means of describing logic circuits. Boolean algebra is based upon a small number of assumptions referred to as axioms, that can be applied to logical functions in order to simplify them and design a less costly digital circuit.

Axioms of Boolean Algebra Assume that Boolean algebra involves elements that can take on only one of two values, 0 or 1. Then the following theorems are true 1a) 0 .0 = 0 1b) 1 + 1 = 1 2a) 1 .1 = 1 2b) 0 + 0 = 0 3a) 0 .1 = 1 .0 = 0 3b) 1 + 0 = 0 + 1 = 1 4a) If x = 0 , then x = 1 4b) If x = 1 , then x = 0

This call The principal of duality . Assume that in Boolean algebra we are dealing with Single or more variables, then the following terms are true 1a) x + 0 = x 1b) x . 1 = x 2a) x + x’ = 1 2b) x . x’ = 0 3a) x + x = x 3b) x . x = x 4a) x + 1 = 1 4b) x . 0 = 0 5a) (x’)’ = x Commutative: a) x + y = y + x b) xy = yx Associative: a) x + (y + z) = (x + y) + z b) x(yz) = (xy)z Distributive: a) x (y + z) = xy + xz b) x + yz = (x+y) . (x+z) DeMorgan: a) (x + y)’ = x’ . y’ b) (xy)’ = x’ + y’ Absorption: a) x + xy = x b) x (x+y) = x The dual of an expression is obtained by replacing all “+”operators with “ .” operators, and vice versa, and by replacing all 0s with 1s, and vice versa. (DeMorgan law) This call The principal of duality .

Example Verify DeMorgan law (a) by truth table. DeMorgan's theorem can also be extended to three or more variables: (x1 + x2 + x3 + … + xn)’ = x1’ . x2’ . x3’ . … . xn’ (x1 . x2 . x3 . … . xn)’ = x1’ + x2’ + x3’ + … + xn’ x y x’ y’ (x+y) (x+y)’ x’ y’ 0 0 1 1 0 1 0 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 0

Example Prove of theorem 3a: x + x = x x + x = (x + x) . 1 by theorem 1b = (x + x) . (x + x’) by theorem 2a = x + xx’ by distributive (b) = x + 0 by theorem 2b = x

Example Prove of theorem 3b: x . x = x x . x = (x . x) + 0 by theorem 1a = (x . x) + (x . x’) by theorem 2b = x (x + x’) by distributive (a) = x . 1 by theorem 2a = x