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

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

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

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

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

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

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

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

Octal Numbering System (Base 8) Allowable digits –0,1,2,3,4,5,6,7 Weighting factor of 8 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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

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

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

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

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

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

Comparison of Numbering Systems See Table 1-4 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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

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

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

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

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

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