Embedded Systems and Computer Architecture

Graham R Wilson

• Publisher: Newnes
• Year: 2001
• ISBN:
  9780080477558
• Topic: Computer Science
• Subtopic: Hardware
• Index: Computer Science

The author has taught the design and use of microprocessor systems to undergraduate and technician level students for over 25 years.

• A core text for academic modules on microprocessors, embedded systems and computer architecture
• A practical design-orientated approach

Part 1

The Building Blocks

1

Binary numbers

1.1 Numbers within a computing machine

Our study of computing machines begins by looking at the basic components from which a machine might be constructed. We begin by asking how numbers may be represented in a machine.

The simplest numbers that we want to represent in the machine are the unsigned integers. These are whole numbers without a sign, for example, 0, 1, 2, 3, … The mechanical calculators of yesteryear and the car mileage meter of today both store unsigned integers on what are effectively cogs having ten numbered teeth¹. Thus a simple two-digit calculator capable of addition and subtraction will comprise two cogs, one indicating units, the other indicating tens, Figure 1.1.

Figure 1.1 Two cogs representing the two-digit number, 63

A simple device moves the tens cog one position every time the units cog completes a rotation and passes from 9 back to 0. Thus, if the tens cog currently indicates 4 and the units cog indicates 9, when the units cog is moved forward one position, so adding 1, the cogs correctly display the result 50. The ‘carry’ from the units cog to the tens cog is thus automatic².

Decimal³ numbers are represented by using the ten digits 0, 1, 2,. 9 in such a way that each digit is interpreted according to its position in the number. That is, a three-digit number represented on the cogs as <d₂, d₁, d₀> is interpreted as

100.d₂ + 10.d₁ + 1.d₀

or 10².d₂ + 10¹.d₁ + 10⁰.d₀

e.g. 406 is said to represent the number four hundreds, no tens, and six units.

The digits are weighted according to their position in the number; in general, digit d_J has a weight of 10^J.

Now consider four imaginary cogs having just two teeth, labelled 0 and 1. Again, a simple device moves the cog to the left one position every time a cog completes a rotation and passes from 1 back to 0. When advanced one step at a time the cogs will display the sequence shown in Figure 1.2.

Figure 1.2 Successive positions of four cogs, each having two teeth

The cogs now have weights of 8, 4, 2 and 1 and when they indicate <b₃, b₂, b₁, b₀>, the value of the number, in decimal, is obtained from:

8.b₃ + 4.b₂ + 2.b₁ + 1.b₀

or 2³.b₃ + 2².b₂ + 2¹.b₁ + 2⁰.b₀

e.g. the number 1101 is interpreted as one 8, one 4, no 2s, and 1 unit or 1.8 + 1.4 + 0.2 + 1.1 = 13, in decimal notation.

This method of representing numbers is called pure binary notation⁴. We use this notation to represent unsigned integers. The digits 0 and 1 are called binary digits, or bits. The four-cog mechanism represents numbers using 4 bits and so can represent only the 16 numbers, 0000 to 1111. If we use 5 bits to represent a number, the extra bit allows us to represent twice as many numbers, 0000 to 11111. In general, if we use N bits to represent a number, we have 2^N different numbers. An N-bit number <b_N–1 b_N–2 … b₂b₁b₀> has the decimal value 2^N–1.b_N–1 + 2^N–2.b_N–2 + … + 2².b₂ + 2¹.b₁ + 2⁰.b₀.

1.2 Adding binary integers

The familiar rules for adding two decimal digits are shown, in part, in Figure 1.3(a). Note that the addition of two one-digit numbers results in a two-digit number; we call the right-most digit the sum digit and the left-most digit the carry digit. Decimal addition proceeds as shown in the example Figure 1.3(b). Starting with the right-most pair of digits, the sum digit is written down and the carry digit is carried into the column to the left. The sum of this column of digits therefore requires that we add three digits. Note that, if the number mechanism holds only two-digit numbers, the sum in t...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• Notation used in the text
• Part 1: The Building Blocks
• Part 2: Computing Machines
• Part 3: Larger Computer
• Appendix A: G80 instruction set
• Appendix B: ASCII character codes
• Appendix C: Specifications of the input and output devices
• Appendix D: The GDS assembler and linker
• Index