Digital Logic Fundamental

Digital Logic

All digital systems are founded on logic design. Logic design transforms algorithms and processes conceived by people into computing machines. A grasp of digital logic is crucial to the understanding of other basic elements of digital systems, including microprocessors. This chapter addresses vital topics ranging from Boolean algebra to synchronous logic to timing analysis with the goal of providing a working set of knowledge that is the prerequisite for learning how to design and implement an unbounded range of digital systems.

Boolean algebra is the mathematical basis for logic design and establishes the means by which a

task’s defining rules are represented digitally. The topic is introduced in stages starting with basic logical operations and progressing through the design and manipulation of logic equations. Binary and hexadecimal numbering and arithmetic are discussed to explain how logic elements accomplish significant and practical tasks. With an understanding of how basic logical relationships are established and implemented, the discussion moves on to explain flip-flops and synchronous logic design. Synchronous logic complements Boolean algebra, because it allows logic operations to store and manipulate data over time. Digital systems would be impossible without a deterministic means of advancing through an algoithm’s sequential steps. Boolean algebra defines algorithmic steps, and the progression between steps is enabled by synchronous logic. Synchronous logic brings time into play along with the associated issue of how fast a circuit can reliably operate. Logic elements are constructed using real electrical components, each of which has physical requirements that must be satisfied for proper operation. Timing analysis is discussed as a basic part of logic design, because it quantifies the requirements of real components and thereby establishes a digital circuit’s practical operating conditions.

The chapter concludes with a presentation of higher-level logic constructs that are built up from

the basic logic elements already discussed. These elements, including multiplexers, tri-state buffers, and shift registers, are considered to be fundamental building blocks in digital system design. The remainder of this book, and digital engineering as a discipline, builds on and makes frequent reference to the fundamental items included in this discussion.

1.1 BOOLEAN LOGIC

Machines of all types, including computers, are designed to perform specific tasks in exact well defined manners. Some machine components are purely physical in nature, because their composition and behavior are strictly regulated by chemical, thermodynamic, and physical properties. For example, an engine is designed to transform the energy released by the combustion of gasoline and oxygen into rotating a crankshaft. Other machine components are algorithmic in nature, because their designs primarily follow constraints necessary to implement a set of logical functions as defined by human beings rather than the laws of physics. A traffic light’s behavior is predominantly defined by human beings rather than by natural physical laws. This book is concerned with the design of digital systems that are suited to the algorithmic requirements of their particular range of applications. Digital logic and arithmetic are critical building blocks in constructing such systems. An algorithm is a procedure for solving a problem through a series of finite and specific steps. It can be represented as a set of mathematical formulas, lists of sequential operations, or any combination thereof. Each of these finite steps can be represented by a Boolean logic equation. Boolean logic is a branch of mathematics that was discovered in the nineteenth century by an English mathematician named George Boole. The basic theory is that logical relationships can be modeled by algebraic equations. Rather than using arithmetic operations such as addition and subtraction, Boolean algera employs logical operations including AND, OR, and NOT. Boolean variables have two enumerated values: true and false, represented numerically as 1 and 0, respectively.

The AND operation is mathematically defined as the product of two Boolean values, denoted A

and B for reference.

Truth tables are often used to illustrate logical relationships as shown for the

AND operation in Table 1.1. A truth table provides a direct mapping between the possible inputs and outputs. A basic AND operation has two inputs with four possible combinations, because each input can be either 1 or 0 — true or false. Mathematical rules apply to Boolean algebra, resulting in a nonzero product only when both inputs are 1. Summation is represented by the OR operation in Boolean algebra as shown in Table 1.2. Only one combination of inputs to the OR operation result in a zero sum: 0 + 0 = 0.

AND and OR are referred to as binary operators, because they require two operands. NOT is a

unary operator , meaning that it requires only one operand. The NOT operator returns the complement of the input: 1 becomes 0, and 0 becomes 1. When a variable is passed through a NOT operator, it is said to be inverted .

TABLE

1.1 AND Operation Truth Table

AND and OR are referred to as binary operators, because they require two operands. NOT is a

unary operator , meaning that it requires only one operand. The NOT operator returns the complement of the input: 1 becomes 0, and 0 becomes 1. When a variable is passed through a NOT operator, it is said to be inverted.

Boolean variables may not seem too interesting on their own. It is what they can be made to represent that leads to useful constructs. A rather contrived example can be made from the following logical statement:

“If today is Saturday or Sunday and it is warm, then put on shorts.”

Three Boolean inputs can be inferred from this statement: Saturday, Sunday, and warm. One Boolean output can be inferred: shorts. These four variables can be assembled into a single logic equation that computes the desired result, shorts = (Saturday OR Sunday) AND warm. While this is a simple example, it is representative of the fact that any logical relationship can be expressed algebraically with products and sums by combining the basic logic functions AND, OR, and NOT.

Several other logic functions are regarded as elemental, even though they can be broken down

into AND, OR, and NOT functions. These are not–AND (NAND), not–OR (NOR), exclusive–OR (XOR), and exclusive–NOR (XNOR). Table 1.3 presents the logical definitions of these other basic functions. XOR is an interesting function, because it implements a sum that is distinct from OR by taking into account that 1 + 1 does not equal 1. As will be seen later, XOR plays a key role in arithmetic for this reason.

All binary operators can be chained together to implement a wide function of any number of inputs. For example, the truth table for a ten-input AND function would result in a 1 output only when all inputs are 1. Similarly, the truth table for a seven-input OR function would result in a 1 output if any of the seven inputs are 1. A four-input XOR, however, will only result in a 1 output if there are an odd number of ones at the inputs. This is because of the logical daisy chaining of multiple binary XOR operations. As shown in Table 1.3, an even number of 1s presented to an XOR function cancel each other out.

It quickly grows unwieldy to write out the names of logical operators. Concise algebraic expressions are written by using the graphical representations shown in Table 1.4. Note that each operation has multiple symbolic representations. The choice of representation is a matter of style when handwritten and is predetermined when programming a computer by the syntactical requirements of each computer programming language.

A common means of representing the output of a generic logical function is with the variable Y.

Therefore, the AND function of two variables, A and B, can be written as Y = A & B or Y = A*B. As with normal mathematical notation, products can also be written by placing terms right next to each other, such as Y = AB. Notation for the inverted functions, NAND, NOR, and XNOR, is achieved by

inverting the base function. Two equally valid ways of representing NAND are Y = A & B and Y =!(AB). Similarly, an XNOR might be written as Y = A (+)B. When logical functions are converted into circuits, graphical representations of the seven basic operators are commonly used. In circuit terminology, the logical operators are called gates. Figure 1.1 shows how the basic logic gates are drawn on a circuit diagram. Naming the inputs of each gate A and B and the output Y is for reference only; any name can be chosen for convenience. A small bubble is drawn at a gate’s output to indicate a logical inversion. More complex Boolean functions are created by combining Boolean operators in the same way that arithmetic operators are combined in normal mathematics. Parentheses are useful to explicitly convey precedence information so that there is no ambiguity over how two variables should be treated. A Boolean function might be written as

Y=(AB+C+ )D&E ? F

This same equation could be represented graphically in a circuit diagram, also called a schematic

Diagram , as shown in Fig. 1.2. This representation uses only two-input logic gates. As already mentioned, binary operators can be chained together to implement functions of more than two variables.

An alternative graphical representation would use a three-input OR gate by collapsing the two-input OR gates into a single entity.

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