Introduction to Digital Combinational Logic and Systems Design So far we have been discussing the generation transmission and processing of signals whose amplitude voltage current varies continuously in time and can in principle take any value At a certain instant of time we may represent a signal by displaying its amplitude in an analog form or in a digital format The graphics below demonstrate the familiar representation of the two forms Both displays are asked to display the number 4 7 In this case the digital display on the left has the required resolution to represent the number exactly If we try to display the number 4 76 this digital display with its ability to display only 2 digits will have to round off the number representing it either as 4 7 or 4 8 depending on how the system processes the numerics Our reading of the number off the analog display requires some interpolation of the value but in principle the resolution is only limited by our ability to identify the position of the measuring needle In general the characteristics of the digital display correspond to the digital signal which is required to generate the digital display in the first place If we now consider an analog signal which varies continuously in time see Figure 2a then if we sample the signal at discrete times 2 3 we will obtain the values indicated by the solid circles on Figure 2b Furthermore if we consider the quantization of the signal at these discrete sampling times we obtain the signal indicated on Figure 2c which is a digital signal The analog signal is also shown on Figure 2c to emphasize the relationship between it and its digital representation 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 Signal V Signal V 8 8 6 6 4 2 4 2 0 2 3 4 5 6 7 a Time Signal V Level Level Level Level Level Level Level 0 2 3 4 5 6 7 b Time 7 6 5 4 3 2 1 0 2 3 4 5 6 7 Time c Figure 2 Schematic representation of analog and digital signals This review should have motivated us to ask a few questions about these signals and in particular about the digital signal shown on Figure 2c Some of these questions are 1 How is the information embodied by the digital signal represented 2 How is the signal generated a How is the sampling frequency selected and how is it related to the quality of signal representation b How is the amplitude quantization achieved 3 What are the advantages and disadvantages of generating the digital signal For example how does it perform in a Accuracy b Transmission c Noise immunity d Information storage e Computation In the next few classes we will answer these questions and explore the fundamental issues associated with the design of digital circuits 6 071 22 071 Spring 2006 Chaniotakis and Cory 2 Numbering Systems Binary Code In digital electronics the signals are formed with only two voltage values HI and LOW or level 1 and level 0 and it is called binary digital signal 1 Therefore the information contained in the digital signal is represented by the numbers 1 and 0 In most digital systems the state 1 corresponds to a voltage range from 2V to 5V while the state 0 corresponds to a voltage range from a fraction of a volt to 1 volts Digital operations are performed by creating and operating on binary numbers Binary numbers are comprised of the digits 0 and 1 and are based on powers of 2 Each digit of a binary number 0 or 1 is called a bit an abbreviation for binary digit Four bits together is a nibble 8 bits is called a byte 8 16 32 64 bit arrangements are also called words The rightmost bit is called the Least Significant Bit LSB while the leftmost bit is called the Most Significant Bit MSB The schematic below illustrates the general structure of a binary number and the associated labels MSB 1010 N 1101 0110 1010 nibble byte word LSB 1 In addition to binary digital systems and its associated binary logic multivalued logic also exists but we will not consider it in our discussion 6 071 22 071 Spring 2006 Chaniotakis and Cory 3 Binary to Decimal Conversion The conversion of a binary number to a decimal number may be accomplished by taking the successive powers of 2 and summing for the result For example let s consider the four bit binary number 0101 The conversion to a decimal number base 10 is illustrated below 0 1 0 1 3 0x2 N 2 1x2 N 1 0x2 N 0 1x2 N 0 4 0 1 510 For this four bit binary number the range of powers of 2 goes from 0 corresponding to the LSB to 3 corresponding to the MSB The number 5 is shown as 510 to indicate that it is a decimal number power of 10 The signal represented on Figure 2c has a value of 5 V at time 6 The binary representation of that value is 0101 and it is shown on Figure 3 replacing Level 4 We will see more of this later when we consider the fundamentals of the device which converts the analog signal to a digital signal Signal V Level 7 Level 6 Level 5 0101 Level 3 Level 2 Level 1 0 2 3 4 5 6 7 Time Figure 3 In the next few examples we will use the subscript 2 to indicate a binary number but the subscripts will be omitted after that 6 071 22 071 Spring 2006 Chaniotakis and Cory 4 Examples Verify the Binary to Decimal conversion 11112 1510 1111 00002 24010 1111 11112 25510 1101 10112 21910 0001 0101 10112 34710 1001 0101 10112 239510 Decimal to Binary Conversion The conversion of a decimal number to a binary number is accomplished by successively dividing the decimal number by 2 and recording the remainder as 0 or 1 Here is an example of the conversion of decimal number 125 to binary 125 2 62 2 31 2 15 2 7 2 3 2 1 2 62 1 LSB 31 0 15 1 7 1 0111 1101 3 1 1 1 0 1 MSB Practice number conversion by verifying the conversions from decimal to binary Decimal 69 299 756 Binary 0100 0101 0001 0010 1011 0010 1111 0100 6 071 22 071 Spring 2006 Chaniotakis and Cory 5 Representation of fractions and signed numbers A fractional number may be represented as a binary fraction by simply extending the procedure used in representing integer numbers For example 13 7510 1101 11002 The procedure is clearly visualized by considering the following mapping 23 8 22 4 21 20 2 1 1 1 0 1 13 2 1 2 2 2 3 2 4 0 5 0 25 0 125 0 0625 1 75 1 0 0 Signed binary numbers may be represented by assigning the MSB to indicate the sign A 0 is used to indicate a positive number and a 1 is used …