Introduction to Computer architecture

Digital Circuits

The digital computer is a digital system that performs various computational tasks. The word digital implies that the information in the computer is represented by variables that take a limited number of discrete values. These values are processed internally by components that can maintain a limited number of discrete states. The decimal digits 0, 1, 2, . . . , 9, for example, provide 10 discrete values. The first electronic digital computers, developed in the late 1940s, were used primarily for numerical computations. In this case the discrete elements are the digits. From this application the term digital computer has emerged. In practice, digital computers function more reliably if only two states are used. Because of the physical restriction of components, and because human logic tends to be binary (i.e., true-or-false, yes-or-no statements), digital components that are constrained to take discrete values are further constrained to take only two values and are said to be binary.

Digital computers use the binary number system, which has two digits: 0 and 1 A binary digit is called a bit. Information is represented in digital computers in groups of bits. By using various coding techniques, groups of bits can be made to represent not only binary numbers but also other discrete symbols, such as decimal digits or letters of the alphabet. By judicious use of binary arrangements and by using various coding techniques, the groups of bits are used to develop complete sets of instructions for performing various types of computations.

In contrast to the common decimal numbers that employ the base 10 system, binary numbers use a base 2 system with two digits: 0 and 1. The decimal equivalent of a binary number can be found by expanding it into a power series with a base of 2. For example, the binary number 1001011 represents a quantity that can be converted to a decimal number by multiplying each bit by the base 2 raised to an integer power as follows:

\(1 \times 26 \times 0 \times 25 \times 0 \times 24 \times 1 \times 23 \times 0 \times 22 \times 1 \times 21 \times 1 \times 20 = 75\)

Two basic types of computer architectures are von Neumann architecture and Harvard architecture. von Neumann architecture describes a general framework, or structure, that a computer’s hardware, programming, and data should follow. Although other structures for computing have been devised and implemented, the vast majority of computers in use today operate according to the von

Neumann architecture. Von Neumann envisioned the structure of a computer system as being composed of the following components:

the central arithmetic unit, which today is called the arithmetic-logic unit (ALU). This unit performs the computer’s computational and logical functions;
memory; more specifically, the computer’s main, or fast, memory, such as random access memory (RAM);
a control unit that directs other components of the computer to perform certain actions, such as directing the fetching of data or instructions from memory to be processed by the ALU; and
man-machine interfaces; i.e., input and output devices, such as a keyboard for input and display monitor for output,

Two methods of simplifying Boolean algebraic expressions are the map method and the tabular method. The map method is used for functions upto six variables. To manipulate functions of a large number of variables, the tabular method also known as the Quine-McCluskey method, is used. If a function to be minimized is not in a canonical form, it must first be converted into canonical form before applying Quine-McCluskey tabular procedure. Another tabular method, known as the iterative consensus method, begins the simplification process even if the function is not in a canonical form. The map method provides a simple, straightforward procedure for simplifying Boolean expressions. This method may be regarded as a pictorial arrangement of the truth table which allows an easy interpretation for choosing the minimum number of terms needed to express the function algebraically. The map method is also known as the Karnaugh map or K-map.

Logic Gates

Binary information is represented in digital computers by physical quantities called signals. Electrical signals such as voltages exist throughout the computer in either one of two recognizable states. The two states represent a binary variable that can be equal to 1 or 0. For example, a particular digital computer may employ a signal of 3 volts to represent binary 1 and 0.5 volt to represent binary 0. The input terminals of digital circuits accept binary signals of 3 and 0.5 volts and the circuits respond at the output terminals with signals of 3 and 0.5 volts to represent binary input and output corresponding to 1 and 0, respectively.

The basic logic gates are AND and inclusive OR with multiple inputs and NOT with a single input. Each gate with more than one input is sensitive to either logic 0 or logic 1 input at any one of its inputs, generating the output according to its function. For example, a multi-input AND gate is sensitive to logic 0 on any one of its inputs, irrespective of any values at other inputs.

Boolean Algebra

The two-element Boolean algebra \(B_2\) among all other \(B_i\) , where \(i \ge 2\), defined as switching algebra, is the most useful. Switching algebra consists of two elements represented by 1 and 0 as the largest number and the smallest number respectively.

*Boolean algebra is a switching algebra that deals with binary variables and logic operations. The variables are designated by letters such as \(A\), \(B\), \(x\), and \(y\). The three basic logic operations are AND, OR, and complement. A Boolean function can be expressed algebraically with binary variables, the logic operation symbols, parentheses, and equal sign. For a given value of the variables, the Boolean function can be either 1 or 0. Consider, for example, the Boolean function

\(F = x \times y`z\)

The function \(F\) is equal to 1 if \(x\) is 1 or if both \(y`\) and z are equal to 1; \(F\) is equal to 0 otherwise. But saying that \(y`\) 1 is equivalent to saying that \(y=0\) since \(y`\) is the complement of \(y\). Therefore, we may say that \(F\) is equal to 1 if \(x = 1\) or if \(yz =01\). The relationship between a function and its binary variables can be represented in a truth table.

DeMorgan's Theorm

DeMorgan’s theorem is very important in dealing with NOR and NAND gates. It states that a NOR gate that performs the \((x \times y)`\) function is equivalent to the function xy. Similarly, a NAND function can be expressed by either \((xy)`\) or \((x` \times y`)\). For this reason the NOR and NAND gates have two distinct graphic symbols.

Map Simplification

The complexity of the logic diagram that implements a Boolean function is related directly to the complexity of the algebraic expression from which the function is implemented. The truth table representation of a function is unique, but the function can appear in many different forms when expressed algebraically. The expression may be simplified using the basic relations of Boolean algebra. However, this procedure is sometimes difficult because it lacks specific rules for predicting each succeeding step in the manipulative process. Two methods of simplifying Boolean algebraic expressions are the map method and the tabular method. The map method is used for functions upto six variables. To manipulate functions of a large number of variables, the tabular method also known as the Quine-McCluskey method, is used. If a function to be minimized is not in a canonical form, it must first be converted into canonical form before applying Quine-McCluskey tabular

procedure. Another tabular method, known as the iterative consensus method, begins the simplification process even if the function is not in a canonical form. The map method provides a simple, straightforward procedure for simplifying Boolean expressions. This method may be regarded as a pictorial arrangement of the truth table which allows an easy interpretation for choosing the minimum number of terms needed to express the function algebraically. The map method is also known as the Karnaugh map or K-map.

K-Map

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TODO Combinational Circuits

Digital logic circuits are basically categorized into two types:

Combinational circuits in which there are no feedback paths from outputs to inputs and there is no memory.
Sequential circuits in which feedback paths exist from outputs to inputs, and they have memory.