# Basics of Digital Logic Design

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1 CSE 675.2: Introduction to Computer Architecture Basics of Digital Logic Design Presentation D Study: B., B2, B.3 Slides by Gojko Babi From transistors to chips Chips from the bottom up: Basic building block: the transistor = on/off switch Digital signals voltage levels high/low Transistors are used to build logic gates Logic gates make up functional and control units Microprocessors contain several functional and control units This section provides an introduction into digital logic Combinatorial and sequential logic Boolean algebra and truth tables Basic logic circuits: Decoders, multiplexers, latches, flip-flops Simple register design Presentation D 2

2 Logic operation Signals, Logic Operations and Gates Rather than referring to voltage levels of signals, we shall consider signals that are logically or (or asserted or de-asserted). NOT A A AND A B A and B OR A B A or B XOR A B A xor B Gates Output is iff: Input is Both inputs are s At least one input is Inputs are not equal Gates are simplest digital logic circuits, and they implement basic logic operations (functions). Gates are designed using transistors. Gates are used to build more complex circuits that implement more complex logic functions. 3 Classification of Logic Functions/Circuits Combinational logic functions (circuits): any number of inputs and outputs outputs y i depend only on current values of inputs x i Logic equations may be used to define a logic function. Example: A logic function with 4 inputs and 2 outputs y = (x + (x 2 *x 3 )) + ((x 3 *x 4 )*x ) y 2 = (x + (x 2 *x 4 )) + ((x *x 2 )*x 3 ) * used for and, + used for or For sequential functions (circuits): outputs depend on current values of inputs and some internal states. Any logic function (circuit) can be realized using only and, or and not operations (gates). nand and nor operations (gates) are universal. g. babic Presentation D 4 2

3 Basic Laws of Boolean Algebra Identity laws: A + = A A * = A Zero and one laws: A + = A * = Commutative laws: A + B = B+A A * B = B * A Inverse laws: A + A = A * A = Associative laws: A + (B + C) = (A + B) + C A * (B * C) = (A * B) * C Distributive laws : A * (B + C) = (A * B) + (A * C) A + (B * C) = (A + B) * (A + C) DeMorgan s laws: (A + B) = A * B (A * B) = A + B g. babic Presentation D 5 Simple Circuit Design: Example Given logic equations, it is easy to design a corresponding circuit y = (x + (x 2 *x 3 )) + ((x 3 *x 4 )*x ) = x + (x 2 *x 3 ) + (x 3 *x 4 *x ) y 2 = (x + (x 2 *x 4 )) + ((x *x 2 )*x 3 ) = x + (x 2 *x 4 ) + (x *x 2 *x 3 ) g. babic Presentation D 6 3

4 Truth Tables Another way (in addition to logic equations) to define functionality Problem: their sizes grow exponentially with number of inputs. inputs outputs x x 2 x 3 y y 2 What are logic equations corresponding to this table? y = x + x 2 + x 3 y 2 = x * x 2 * x 3 Design corresponding circuit. g. babic Presentation D 7 Logic Equations in Sum of Products Form Systematic way to obtain logic equations from a given truth table. inputs outputs x x 2 x 3 y y 2 A product term is included for each row where y i has value A product term includes all input variables. At the end, all product terms are ored y = x *x 2 *x 3 + x *x 2 *x 3 + x *x 2 *x + x 3 *x 2 *x 3 y 2 = x *x 2 *x 3 + x *x 2 *x 3 + x *x 2 *x 3 + x *x 2 *x 3 + x *x 2 *x 3 g. babic Presentation D 8 4

5 Programmable Logic Array - PLA PLA structured logic implementation g. babic Presentation D 9 Circuit Logic Equation Truth Table For the given logic circuit find its logic equation and truth table. x x2 x3 x x 2 x 3 y y = x *x 2 + x 2 *x 3 y Note that y column above is identical to y column Slide 8. Thus, the given logic function may be defined with different logic equations and then designed by different circuits. g. babic Presentation D 5

6 Minimization Using Boolean Laws Consider one of previous logic equations: y = x *x 2 *x 3 + x *x 2 *x 3 + x *x 2 *x 3 + x *x 2 *x 3 = x *x 2 *(x 3 + x 3 ) + x 2 *x 3 *(x + x ) = x *x 2 + x 2 *x 3 But if we start grouping in some other way we may not end up with the minimal equation. g. babic Presentation D Gray codes a.k.a. reflected code binary numeral system in which two successive values differ by only one digit 4-bit Gray code 2-bit Gray code 3-bit Gray code Presentation D 2 6

7 Minimization Using Karnaugh Maps (/4) Provides more formal way to minimization Includes 3 steps. Form Karnaugh maps from the given truth table. There is one Karnaugh map for each output variable. 2. Group all s into as few groups as possible with groups as large as possible. 3. each group makes one term of a minimal logic equation for the given output variable. Forming Karnaugh maps using Gray code The key idea in forming the map is that horizontally and vertically adjacent squares correspond to input variables that differ in one variable only. Thus, a value for the first column (row) can be arbitrary, but labeling of adjacent columns (rows) should be such that those values differ in the value of only one variable. g. babic Presentation D 3 Minimization Using Karnaugh Maps (2/4) Grouping (This step is critical) When two adjacent squares contain s, they indicate the possibility of an algebraic simplification and they may be combined in one group of two. Similarly, two adjacent pairs of s may be combined to form a group of four, then two adjacent groups of four can be combined to form a group of eight, and so on. In general, the number of squares in any valid group must be equal to 2 k. Note that one can be a member of more than one group and keep in mind that you should end up with as few groups as possible, which are as large as possible. Finding Product Terms The product term that corresponds to a given group is the product of variables whose values are constant in the group. If the value of input variable x i is for the group, then x i is entered in the product, while if x i has value for the group, then x i is entered in the product. g. babic Presentation D 4 7

8 Minimization Using Karnaugh Maps (3/4) Example : Given truth table, find minimal circuit x x 2 x 3 y x x2 x3 y = x *x 2 + x 2 *x 3 g. babic Presentation D 5 Minimization Using Karnaugh Maps (4/4) Example 2: Example 3: x x 2 x x 2 x 3 x 4 y = x *x 3 + x 2 x 3 Example 4: y = x *x 2 *x 3 + x *x 2 *x 4 + x 2 *x 3 *x 4 x x 2 x 3 x 4 y = x *x 4 + x 2 *x 3 *x 4 + x *x 2 *x 3 *x 4 Presentation D 6 8

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