Chapter 01 Basic Principles of Digital Systems
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This document provides an overview of digital systems fundamentals, including: - Analog signals have continuous values while digital signals can only have discrete values (0 or 1). - Digital electronics uses binary logic levels to represent information, with a high voltage representing 1 and a low voltage representing 0. - The binary number system uses positional notation to represent numbers using only the digits 0 and 1. - Digital circuits operate on binary inputs and outputs, with truth tables listing all possible input-output combinations for a logic gate or circuit.
Chapter 01 Basic Principles of Digital Systems
- 1. CHAPTER 1 Basic Principles of Digital Systems
- 2. ANALOG VS. DIGITAL Analog: A way of representing a physical quantity by a proportional continuous voltage or current. Digital: A way of representing a physical quantity in discrete voltage steps. 2
- 3. ANALOG ELECTRONICS Values are continuously variable between defined values. Can have any value within a defined range. 3
- 5. DIGITAL ELECTRONICS Values can vary only by distinct, or discrete, steps. Can only have two values. 5
- 6. DIGITAL LOGIC LEVELS Logic HIGH is the higher voltage and represented by binary digit ‘1’. Logic LOW is the lower voltage and represented by binary digit ‘0’. 6
- 8. BINARY NUMBER SYSTEM Uses two digits, 0 and 1. Represents any number using the positional notation. 8
- 9. POSITIONAL NOTATION The value of a digit depends on its placement within a number. In base 10, the positional values are (starting to the left of the decimal) – 1 (100), 10 (101), 100 (102), 1000 (103), etc. In base 2, the positional values are 1 (20), 2 (21), 4 (22), 8 (23), etc. 9
- 10. DECIMAL EQUIVALENCE OF BINARY NUMBERS 10 13 1048 1)(12)04)(18)(1 )2(1)2(0)2(1)2(11101 0123 (
- 11. BIT Shorthand for binary digit, a logic 0 or 1. The most significant bit (MSB) is the leftmost bit of a binary number. The least significant bit (LSB) is the rightmost bit of a binary number. 11
- 12. BINARY INPUTS Digital circuits operate by accepting logic levels (0,1) at their input(s). The corresponding output(s) logic level will change (0,1). 12
- 13. BINARY INPUTS 13
- 14. TRUTH TABLE A list of output logic levels corresponding to all possible input combinations. The number of input combinations is 2n, where n is the number of inputs. 14
- 15. INPUT COMBINATIONS A logic circuit with 3 inputs will have 23 or 8 possible input conditions. For this logic circuit there would also be 8 possible output conditions. 15
- 16. CONSTRUCTING A BINARY SEQUENCE FOR A TRUTH TABLE – 1 Two methods: Learn to count in binary Follow a simple repetitive pattern Memorize the binary numbers from 0000 to 1111 and their decimal equivalents (0 to 15). Use the weighted values of binary bits. 16
- 17. BINARY SEQUENCE FOR A TRUTH TABLE – 1 Logic Level Binary Value Decimal Equivalent A B C A B C L L L 0 0 0 0 L L H 0 0 1 1 L H L 0 1 0 2 L H H 0 1 1 3 H L L 1 0 0 4 H L H 1 0 1 5 H H L 1 1 0 6 H H H 1 1 1 7 17
- 18. FOLLOW A SIMPLE REPETITIVE PATTERN The LSB of any binary number alternates between 0 and 1 with every line. The next bit alternates every two lines. The next bit alternates every four lines, and so on. 18
- 19. 19 3 Input Truth Table Decimal Value Binary Value Base 10 22 21 20 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1
- 20. 4-INPUT DIGITAL CIRCUIT 24 = 16 possible input conditions. 20
- 22. A B C D Decimal A B C D Decimal (Cont) 0 0 0 0 0 1 0 0 0 8 0 0 0 1 1 1 0 0 1 9 0 0 1 0 2 1 0 1 0 10 0 0 1 1 3 1 0 1 1 11 0 1 0 0 4 1 1 0 0 12 0 1 0 1 5 1 1 0 1 13 0 1 1 0 6 1 1 1 0 14 0 1 1 1 7 1 1 1 1 15 22 4-Input Digital Circuit
- 23. BINARY WEIGHTS 23 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1
- 24. DECIMAL-TO-BINARY CONVERSION Two methods: Sum powers of 2 Repeated division by 2 24
- 25. SUM POWERS OF 2 Step 1: Determine the largest power of 2 less than or equal to the number to be converted. Place a 1 in that positional location. 25
- 26. SUM POWERS OF 2 Step 2: Subtract the number found in Step 1 from the number to be converted. For the new number, determine if the next lowest power of 2 is less than or equal to that number. 26
- 27. SUM POWERS OF 2 Step 3: If the new power of two from Step 2 is larger, place a 0 in that positional location. If the new value is less than or equal, place a 1 in that positional location. 27
- 28. SUM POWERS OF 2 Step 4: Repeat Steps 2 and 3 until there is nothing left to subtract. All remaining bits are set to 0. 28
- 29. REPEATED DIVISION BY 2 Step 1: Take the number to be converted, and divide it by 2. The remainder (0 or 1) is the LSB of the binary value. 29
- 30. REPEATED DIVISION BY 2 Step 2: Divide the quotient from Step 1 by 2. The remainder (0 or 1) is the next most significant bit. 30
- 31. REPEATED DIVISION BY 2 Step 3: Continue to execute Step 2 until the quotient is 0. The last remainder is the MSB. 31
- 32. FRACTIONAL BINARY NUMBERS Radix point: The generalized decimal point. The dividing line between positive and negative powers for positional multipliers. Binary point: The radix point for binary numbers. 32
- 33. FRACTIONAL BINARY VALUES The value immediately to the right of the binary point is 2–1 = 0.5. The next value to the right is 2–2 = 0.25. The next value to the right is 2–4 = 0.125, and so on. 33
- 34. FRACTIONAL BINARY WEIGHTS 34 2-1 2-2 2-3 2-4 ½ ¼ 1/8 1/16 0.5 0.25 0.125 0.0625
- 36. FRACTIONAL-DECIMAL-TO-FRACTIONAL- BINARY CONVERSION Step 1: Multiply the decimal fraction by 2. The integer part, 0 or 1, is the first bit to the right of the binary point. 36
- 37. FRACTIONAL-DECIMAL-TO-FRACTIONAL- BINARY CONVERSION Step 2: Discard the integer part from Step 1 and repeat Step 1 until the fraction repeats or terminates. 37
- 38. HEXADECIMAL NUMBERS Base 16 number system. Primarily used as a shorthand form of binary numbers. 38
- 39. COUNTING IN HEXADECIMAL Values range from 0 to F with the letters A to F used to represent the values 10 to 15 respectively. Positional multipliers are powers of 16: 160 = 1, 161 = 16, 162 = 256, etc. 39
- 40. HEXADECIMAL VS. DECIMAL NUMBERS 40 Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
- 41. Hex Decimal Binary Hex (Cont) Decimal Binary 0 0 0000 8 8 1000 1 1 0001 9 9 1001 2 2 0010 A 10 1010 3 3 0011 B 11 1011 4 4 0100 C 12 1100 5 5 0101 D 13 1101 6 6 0110 E 14 1110 7 7 0111 F 15 1111 41
- 43. DECIMAL-TO-HEXADECIMAL CONVERSION Two methods: Sum of weighted hexadecimal digits. Repeated division by 16. 43
- 44. HEXADECIMAL CONVERSION: METHOD ONE Convert 13510 to hexadecimal: Since 25610 = 162, the number will have 2 digits. (9 16) > 135 > (8 16) 135 – (8 16) = 135 – 128 = 7 (13510 = 8716) Verifying: 135 – ((8 16) + (7 1)) = 135 – 128 – 7 = 0 44
- 45. CONVERSION BETWEEN HEXADECIMAL AND BINARY Each hexadecimal digit represents 4 binary bits. 45
- 46. CONVERTING FD69H TO BINARY 46 HEX F D 6 9 BIN 1111 1101 0110 1001 DEC 15 13 6 9
- 47. PERIODIC DIGITAL WAVEFORMS A periodic digital waveform is a time-varying sequence of logic HIGHs and LOWs that repeat over some period of time. Period (T) is the time required for the pattern to repeat. 47
- 48. PERIODIC DIGITAL WAVEFORMS Frequency (f) is the number of times per second a signal repeats and is the reciprocal of period. f = 1/T 48
- 50. APERIODIC DIGITAL WAVEFORMS An aperiodic digital waveform is a time-varying sequence of logic HIGHs and LOWs that does not repeat. 50
- 52. WAVEFORM DEFINITIONS Time HIGH (th) is the time a logic signal is in its HIGH state. Time LOW (tl) is the time a logic signal is in its LOW state. Duty cycle is the ratio of the time a logic signal is HIGH (th) to the period (T). 52
- 53. DUTY CYCLE 53 th tl T Duty Cycle = th/T
- 54. PULSE WAVEFORMS A pulse is a momentary variation of voltage from one logic level to the opposite level and back again. Amplitude is the voltage magnitude of a pulse. Edge is the part of a pulse representing the transition from one logic level to the other. 54
- 56. PULSE WAVEFORM CHARACTERISTICS Rising edge is the transition from LOW to HIGH. Falling edge is the transition from HIGH to LOW. Leading edge is the earliest transition. Falling edge is the latest transition. 56
- 58. PULSE WAVEFORM TIMING Pulse width (tw) is the time from the 50% point of the leading edge to the 50% point of the trailing edge. Rise time is the time from 10% to 90% amplitude of the rising edge. Fall time is the time from 90% to 10% amplitude of the falling edge. 58