Number Systems and Codes Chapter 1 Number Systems and Codes Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Digital Versus Analog Digital Analog See Figure 1-1 ON and OFF 0 and 1 continuously varying temperature, pressure, velocity See Figure 1-1 Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Figure 1-1 Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Digital Representations of Analog Quantities Audio Recording CD or DAT Conversions digital-to-analog analog voltage to 8-bit Digital equivalent See Figure 1-2 and 1-3 Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Figure 1-2 Figure 1-3 Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Figure 1–4 Adding unwanted electrostatic noise to: (a) an analog waveform; (b) a digital waveform. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Decimal Numbering System (Base 10) 10 different possible digits Least significant position rightmost Most significant digit leftmost Weighting factor of 10 Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Binary Numbering System (Base 2) Only 0 and 1 Weighting factor of 2 Conversion techniques digit times weighting factor successive division Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Table 1–1 Powers-of-2 Binary Weighting Factors William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Figure 1–5 Successive division by 2 to develop fractional binary weighting factors and show that 20 is equal to 1. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Decimal-to-Binary Conversion Subtracting weighting factors Successive division Least Significant Bit (LSB) Most Significant Bit (MSB) Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Octal Numbering System (Base 8) Allowable digits 0,1,2,3,4,5,6,7 Number after 7 is 10 Weighting factor of 8 Used by computers that utilize 3-bit codes to indicate instructions or operations Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Table 1–2 Octal Numbering System William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Octal Conversions Binary to Octal Octal to Binary Octal to Decimal group binary positions in groups of three write the octal equivalent Octal to Binary reverse the process Octal to Decimal multiply by weighting factors Decimal to Octal successive division Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Hexadecimal Numbering System (Base 16) 4-bit groupings Allowable digits 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Number after F is 10 Two hex digits are used to represent 8 bits a byte 4 bits are a nibble Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Hexadecimal Conversions Binary to Hexadecimal group the binary in groups of four write the equivalent hex digit Hexadecimal to Binary reverse the process Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Hexadecimal Conversions Hexadecimal to Decimal multiply by weighting factors Decimal to Hexadecimal successive division Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Binary-Coded-Decimal System BCD Each of the 10 decimal digits has a 4-bit binary code Conversion convert each decimal digit to its 4-bit binary code BCD to Decimal - reverse the process Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

The ASCII Code American Standard Code for Information Interchange Represents alphanumeric data Uses 7 bits 128 different code combinations See Table 1-5 3-bit group is most significant 4-bit group is least significant Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Summary Numerical quantities occur in analog form but must be converted to digital form to be used by computers or digital circuitry. The binary numbering system is used in digital systems because the 1’s and 0’s are easily represented by ON or OFF transistors, which output 0V for 0 and 5V for 1. Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Summary Any number system can be converted to decimal by multiplying each digit by its weighting factor. The weighting factor for the least significant digit in any number system is always 1. Binary numbers can be converted to octal by forming groups of 3 bits and to hexadecimal by forming groups of 4 bits. Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version

Summary The successive division procedure can be used to convert from decimal to binary, octal or hexadecimal The binary-coded-decimal system uses groups of 4 bits to drive decimal displays such as those in a calculator. ASCII is used by computers to represent all letters, numbers and symbols in digital form. Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. William Kleitz Digital Electronics with VHDL, Quartus® II Version