MODULE I S.NO TOPICS PAGE NO 1 INTRODUCTION 1 2 REVIEW OF NUMBER SYSTEMS 3 NUMBER BASE CONVERSIONS 5 3.1 Any Number System to Decimal Number System 5 3.2 Decimal to any Number System 5 3.3 Binary number to Hexadecimal number 6 3.4 Hexadecimal number to Binary number 6 COMPLEMENTS 6 4.1 1’s complement 7 4.2 2’s complement 7 SIGNED BINARY NUMBER REPRESENTATIONS 8 3 4 5 BINARY ARITHMETIC 6 7 8 8 6.1 Addition 8 6.2 Subtraction 9 6.3 Multiplication 10 6.4 Division 10 BOOLEAN ALGEBRA 10 7.1 Theorems of Boolean Algebra 11 7.2 Minimization of Boolean Functions using Boolean theorems 12 DIGITAL LOGIC GATES 13
1 1. INTRODUCTION A signal is defined as a variation of electrical quantity (usually a voltage or a current) with time. There are two basic ways to represent the signals. 1. Analog signal 2. Digital signal Analog Signal: Analog signal is a continuously varying signal within a given range. It can take any value at any instant of defined time. Digital Signal: Digital signal is a signal that represents a sequence of discrete values. Analog versus Digital Representation: Analog representation is to express the numerical value of the quantity as a continuous range of values, between the two expected extreme values. For example, consider the temperature of an oven settable anywhere from 0 to 100 °C. It can be measured as 65 °C or 64.96 °C or 64.958 °C or even 64.9579 °C and so on. It depends up on the accuracy of the measuring instrument. Similarly, voltage across a certain component in an electronic circuit may be measured as 6.5 V or 6.49 V or 6.487 V or 6.4869 V. In this mode of representation the variation in the numerical value of the quantity is continuous and can take any possible values between the two extremes. This is an example of an analog representation of a number. Digital representation is to express the numerical value of the quantity in steps of discrete values. For example, digital voltmeter displays the voltage directly in the numerical form as 6.5 V or 6.6 V and so on. To summarize, while an analog representation gives a continuous output, a digital representation produces a discrete output.
2 Analog systems contain devices that process or work on various physical quantities represented in analog form. Digital systems contain devices that process the physical quantities represented in digital form. It means the digital representation is not necessarily more accurate than analog. Just because a clock is digital doesn‟t mean that it will always read time more accurately than an analog clock, it just means that the interpretation of its display is less ambiguous. Difference between the analog and digital representation can be illuminated by their representations. Example: The analog sound signal is converted into digital signal through an Analog to Digital converter (ADC). Now the analog sound signal can be stored and also can be retrieved using Digital to Analog converter (DAC). Advantages of Digital system: 1. Easier to design by duplicating similar circuits. 2. Noise immunity. 3. Easy to transmit and store the data. 4. Ease of fabrication in integrated circuit form, leading to availability of more complex functions in a smaller size.
3 Limitation of Digital system: 1. Any physical quantity should be converted from analog to digital signal for processing. For conversion, extra circuitry (Analog to Digital Convertor - ADC) is needed. 2. When an analog signal is converted into a digital form, the digital one is definitely not a 100% reproduction of the analog signal. Some information may be lost in the digital form. For the reproduction of processed digital signal into analog signal, digital to analog conversion (DAC) is required. Still it may not be a 100% reproduction of the analog signal. Note: An Analog to Digital Convertor (ADC) is a circuit which is used to convert analog signals into digital signals. A Digital to analog convertor (DAC) is a circuit which is used to convert digital signals into analog signals. 2. REVIEW OF NUMBER SYSTEMS Number system is the way to represent a number in different forms. There are four basic number systems as follows i. Decimal Number System: It is the number system with base or radix (r) value 10. It has 10 digits. The digits are (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). ii. Binary Number System: It is the number system with base or radix (r) value 2.It has only two binary digits (bits). The digits are (0, 1). iii. Hexadecimal Number System: It is the number system with base or radix (r) value 16. It has 16 digits. The digits are (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F) alphabets represents number starting from 10 where (A=10, B=11, C=12, D=13, E=14, F=15) iv. Octal Number System: It is the number system with base or radix (r) value 8.It has 8 digits. The digits are (0, 1, 2, 3, 4, 5, 6, 7). For example, the decimal number system (base 10) has a radix of 10. Decimal uses different combinations of 10 symbols to represent any value. (i.e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9).