Chapter 1 Number Systems and Codes William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle

Chapter 1

Number Systems and Codes

William KleitzDigital Electronics with VHDL, Quartus® II Version

Copyright ©2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Digital Versus Analog

• Digital– ON and OFF– 0 and 1

• Analog– continuously varying– temperature, pressure, velocity

• See Figure 1-1




Figure 1-1




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




Figure 1-2

Figure 1-3




Decimal Numbering System (Base 10)

• 10 different possible digits

• Least significant position– rightmost

• Most significant digit– leftmost

• Weighting factor of 10




Binary Numbering System (Base 2)

• Only 0 and 1


• Conversion techniques– digit times weighting factor– successive division




Decimal-to-Binary Conversion

• Subtracting weighting factors

• Successive division

• Least Significant Bit (LSB)

• Most Significant Bit (MSB)




Octal Numbering System(Base 8)

• Allowable digits– 0,1,2,3,4,5,6,7





Octal Conversions• Binary to Octal

– 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




Hexadecimal Numbering System(Base 16)

• 4-bit groupings

• See Table 1-3

• Two hex digits are used to represent 8 bits– a byte– 4 bits are a nibble







Hexadecimal Conversions

• Binary to Hexadecimal– group the binary in groups of four– write the equivalent hex digit

• Hexadecimal to Binary– reverse the process




Hexadecimal Conversions

• Hexadecimal to Decimal– multiply by weighting factors

• Decimal to Hexadecimal– successive division




Binary-Coded-Decimal SystemBCD

• 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




Comparison of Numbering Systems

• See Table 1-4







The ASCII Code

• 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







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.




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.




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.




Documents

Chapter 1 Number Systems and Codes William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle