Digital Techniques

1. Digital Techniques Instructor: Dr. André Deutz Office: Room 116 Office Phone: 071 - 527 – 7071 Email Address: deutz@liacs.nl Office Hours: Mondays 16:30-18:00

2. Digital Techniques • Lab Assistants • Gerben van der Lubbe (email: spoofedexistence@gmail.com ) • Drs. Dmitry Nadezhkin (email: dmitry@liacs.nl; phone: 071-527-5775) • Sjoerd Henstra (email: shenstra@liacs.nl), BSc in Computer Science • Simon Zaaijer (email: szaaijer@liacs.nl ) Fall 2007 Digital Techniques by André Deutz, Leiden University

3. Digital Techniques Text Book: M.Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals, 4th Ed., 2008, Pearson Education ; isbn-13: 978-0-13-198926-9; isbn-10: 0-13-198926-X Useful Link: The URL of the Companion Website: http://www.prenhall.com/mano Tip: see also the errata Fall 2007 Digital Techniques by André Deutz, Leiden University

4. Digital Techniques References Snyder, L., Fluency with Information Technology, second edition, 2005, Addison-Wesley (third edition due in October of 2007) (NB this textbook is not required!) The site of Great Principles of Computing: http://cs.gmu.edu/cne/pjd/GP/ Fall 2007 Digital Techniques by André Deutz, Leiden University

5. Digital Techniques Tentative Course Outline Exam (tentamen): Test, Monday, 22 October; Friday, Jan. 11, 2008; retake 1, Wednesday, Mar. 26, 2008; retake 2, Friday, Aug. 8, 2008 Fall 2007 Digital Techniques by André Deutz, Leiden University

6. Digital Techniques Test: 14:00 – 17:00; Monday, October 22, 2007 Exam: 14:00-17:00; Friday, Jan. 11, 2008; retake 1: 14:00-17:00; Wednesday, Mar. 26, 2008; retake 2: 10:00-13:00; Friday, Aug. 8, 2008

7. Digital Techniques Assignments There are two kinds of assignments: 1) pencil-and-paper assignments and 2) processor-project assignments.

8. Digital Techniques Collaboration The pencil-and-paper assignments are done individually. You may work on the processor-project assignments in teams of two. You may consult any source for design and implementation ideas, as long as the synthesis and implementation of these ideas is your own work. It goes without saying that you need to credit any source you are using in your work. The above remarks pertain to both kinds of assignments. If in doubt on how to proceed in this matter, consult the instructor of the course.

9. Digital Techniques Deadlines Please submit the assignments by the stated deadlines. Pencil-and-paper are due 7 days after availability on the web or by email . Therefore you are obliged to check your email and/or web daily (as already stated above). The deadlines for the processor-project assignments will be stated separately. Deadlines are hard deadlines because the answers to the assignments will be posted on the web shortly after the deadline. It goes without saying that we will deal appropriately with exceptionally harsh circumstances beyond your control.

10. Digital Techniques Grade The grade for the course will be determined by considering the grade for the exam (tentamen), the assignments, and the practicum project. The weights for computing the course grade are as follows: Exam and Test: 50% Pencil-and-Paper Assignments: 10% Processor-Project Assignments : 40% Moreover the grade for the Exam as well as the grade for the Processor-Project Assignment should exceed 6..

11. Digital Techniques Give a definition of the term Computer Science (informatica).

12. Digital Techniques Give a definition of the term Computer Science (informatica). Possible answers: To make the digital world (faster), more efficient, and above all more intelligent. The study of phenomena with respect to (digital) computers. The study of representation, analysis, transformation of information van Dale: leer van de mechanische verzameling en verwerking van informatie Van Dale (paar jaar geleden): leer van het verzamelen en de verwerking van gegevens m.b.v. computers Computing is the study of information processes, natural and artificial.

13. Digital Techniques Goal of the Course: Our goal is to gain more insight into natural and man-made/artificial information processes with a focus on man-made/artificial information processing. This goal is, of course, too ambitious and encompasses more than one (if not all) course(s) of our BSc in Computer Science program. A slightly more modest (and perhaps more feasible) goal is to learn to understand information processing à la von Neumann.

14. Digital Techniques Objective Build a simple computer based on the von Neumann model of computation (which in the man-made world is still the ubiquitous model of computation).

15. Digital Techniques Description In this course we will learn how to build a simple computer based on the von Neumann model of computation. To that end we need to study the following topics: Digital Systems and Information, Number Systems, Binary Arithmetic Operations, Decimal and Alphanumeric Codes, Boolean Algebra, Combinational Logic Circuits, Logic Functions and Circuits, Arithmetic Functions and Circuits, Sequential Circuits, Memory Basics, Registers and Register Transfers, Computer Design Basics. As simulation tool we will use Digital Works 3.04.The key for Digital Works: 8290-0018-0300-0087

16. Number Systems

17. Overview • Number Systems • Positional Number Systems (decimal, binary, octal, and hexadecimal) • Number Conversions (r-to-decimal, decimal-to-r, other conversions) • Representations of Numbers in Digital Computers • Integer Numbers (unsigned and signed representations) • Arithmetic Operations • General Remarks • Unsigned, Signed, • Decimal Codes • BCD code, Alphanumeric Codes • ASCII and Unicode Fall 2007 Digital Techniques by André Deutz, Leiden University

18. Digital Systems: General Remarks • A Digital System manipulates discrete elements/quantities of information • Discrete quantities of information emerge from: • the nature of the data being processed • the data may be purposely quantized from continues values • Early computer systems were used mostly for numeric computations: the discrete elements used were the digits, hence the term digital computer/system emerged. • In general, any system uses an alphabet (set of symbols) to represent information • The English language system uses an alphabet of 26 symbols (letters) • The decimal number system uses an alphabet of 10 symbols (digits) • What about the alphabet of the Digital Systems? Fall 2007 Digital Techniques by André Deutz, Leiden University

19. The Digital Systems’ Alphabet is Binary • Digital Systems use only one alphabet with two symbols (digits) ‘0’ and ‘1’ (hence binary ). • A binary digit is called a bit • Information is represented by groups of bits • Why is a binary alphabet used? • Digital systems have a basic building block called a switch, that can only be “on” or “off”, i.e., two discrete values ‘0’ and ‘1’ can be distinguished. • An electric device, called a transistor, physically implements the switch. • The two discrete values are physically represented by ranges of voltage values called HIGH and LOW. • “on” (closed) switch corresponds to bit value ‘0’ and is represented by LOW voltage value (between 0.0 and 1.0 Volt). • “off” (open) switch corresponds to bit value ‘1’ and is represented by HIGH voltage value (between 4.0 and 5.0 Volts). • More information will be given later in another lecture. Fall 2007 Digital Techniques by André Deutz, Leiden University

20. Information Representation • All information in Digital Systems is represented in binary form. An Aside: Wine merchants in England (13th century or earlier) 2 gills = 1 chopin 2 chopins = 1 pint 2 pints = 1 quart 2 quarts = 1 pottle 2 pottles = 1 peck 2 pecks = 1 demibushel 2 demibushels = 1 bushel or firkin 2 firkins = 1 kilderkins 2 kilderkins = 1 barrel 2 barrels = 1 hogshead 2 hogshead = 1 pipe 2 pipes = 1 tun Fall 2007 Digital Techniques by André Deutz, Leiden University

21. Information Representation All information that is not binary is converted to binary before processed by a Digital Systems. Decimal numbers are expressed in the binary number system or by means of a binary code. How is this done? That is not too difficult, once we understand how all number systems, not only the decimalone (0-9), have a similar formal representation and how a number in one number system can be converted into another. Let us look into number systems and conversions. Fall 2007 Digital Techniques by André Deutz, Leiden University

22. Number Systems Number Systems are employed in arithmetic to represent numbers by strings of digits. There are two types of number systems: • Positional number systems • The meaning of each digit depends on its position in the number. • Example: • Decimal number system (we know it very well and use it in everyday arithmetic). • 585.5 is a decimal number in positional code – “5 hundreds plus 8 tens plus 5 units plus 5 tenths”. The hundreds, tens, units, and tenths are powers of 10 implied by the position of the digits. • Decimal number system is said to be of base or radix 10 because it uses 10 distinct digits (0 – 9) and the digits are multiplied by power of 10: 585.5 = 5x102 + 8x101 + 5x100 + 5x10-1 • Non-positional number systems • Old Roman numbers: for example, XIX equals to 19 Fall 2007 Digital Techniques by André Deutz, Leiden University

23. Positional Number Systems We can represent numbers in any number system with base r • Number in positional code • (An-1An-2…A1A0.A-1A-2…A-m+1A-m)r • r is the base (radix) of the system, r  {2, 3, …, I}. • every digit Ai  {0, 1, 2, …, r-1}, where {0, 1, 2, …, r-1} is the digit set. • “.” is called the radix point. • An-1 is referred to as the most significant digit. • A-m is referred to as the least significant digit. • Number in base r expressed as power series of r • An-1 rn-1 +An-2 rn-2 +…+ A1 r1 +A0 r0 + A-1 r-1 +A-2 r-2 +…+ A-m+1 r–m+1 +A-m r-m • Example: a number in number systems with base 5 • (132.4)5 = 1x52 + 3x51 + 2x50 + 4x5-1 = 25 + 15 + 2 + 0.8 = (42.8)10 Fall 2007 Digital Techniques by André Deutz, Leiden University

24. Binary Number System This is the system used for arithmetic in all digital computers • Number in positional code • (bn-1bn-2…b1b0.b-1b-2…b-m+1b-m)r • r = 2 is the base of the binary system. • every digit bi  {0, 1} • the digits bi in a binary number are called bits • bn-1 is referred to as the most significant bit (MSB). • b-m is referred to as the least significant bit (LSB). • Number in base 2 expressed as power series of 2 • bn-1 2n-1 +bn-2 2n-2 +…+ b1 21 +b0 20 + b-1 2-1 +b-2 2-2 +…+ b-m+1 2–m+1+b-m 2-m • Example: a number in the binary number system • (1011.01)2 = 1x23 + 0x22 + 1x21 + 1x20 + 0x2-1 + 1x2-2 = 8 + 2 + 1 + 0.25 = (11.25)10 Fall 2007 Digital Techniques by André Deutz, Leiden University

25. is 1K (kilo) is 1M (mega) is 1G (giga) Memorize this table by heart!!!!!!!! Power of Two Fall 2007 Digital Techniques by André Deutz, Leiden University

26. Other Useful Number System Apart from the ordinary binary number system, the octal (base-8) and the hexadecimal (base-16) number systems are useful for representing binary quantities indirectly because their bases are powers of two. These systems have a more compact representation of binary quantities. • Octal number system • (on-1on-2…o1o0.o-1o-2…o-m+1o-m)8 • every digit oi  {0, 1, 2, 3, 4, 5, 6, 7}. • on-1 8n-1 +on-2 8n-2 +…+ o1 81 +o0 80 + o-1 8-1 +o-2 8-2 +…+ o-m+1 8–m+1 +o-m 8-m • (127.4)8 = 1x82 + 2x81 + 7x80 + 4x8-1 = (87.5)10 = (001 010 111.100)2 • Hexadecimal number system • (hn-1hn-2…h1h0.h-1h-2…h-m+1h-m)16 • every digit hi  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. • hn-1 16n-1 +hn-2 16n-2 +…+ h1 161 +h0 160 + h-1 16-1 +h-2 16-2 +…+ h-m+1 16–m+1 +h-m 16–m • (B6F.4)16 = 11x162 + 6x161 + 15x160 + 4x16-1 = (2927.25)10 = (1011 0110 1111.0100)2 Fall 2007 Digital Techniques by André Deutz, Leiden University

27. Another Important Table Fall 2007 Digital Techniques by André Deutz, Leiden University

28. Conversion from base r to Decimal The conversion of a number in base r to decimal number (base 10) is done by expanding the number in power series and adding all the terms as shown below: (An-1An-2…A1A0.A-1A-2…A-m+1A-m)r = An-1 rn-1 +An-2 rn-2 +…+ A1 r1 +A0 r0 + A-1 r-1 +A-2 r-2 +…+ A-m+1 r–m+1 +A-m r-m • Example of converting Binary (base 2) to Decimal (base 10): (1011.01)2= 1x23 + 0x22 + 1x21 + 1x20 + 0x2-1 + 1x2-2 = 8 + 2 + 1 + 0.25 = (11.25)10 • Example of converting number in base 5 to Decimal (base 10): (132.4)5= 1x52 + 3x51 + 2x50 + 4x5-1 = 25 + 15 + 2 + 0.8 = (42.8)10 • Example of converting Octal (base 8) to Decimal (base 10): (127.4)8= 1x82 + 2x81 + 7x80 + 4x8-1 = (87.5)10 • Example of converting Hexadecimal (base 16) to Decimal (base 10): (B6F.4)16 = 11x162 + 6x161 + 15x160 + 4x16-1 = (2927.25)10 Fall 2007 Digital Techniques by André Deutz, Leiden University

29. Converting the integer part (41)10: 41/2 = 20 + 1/2 Remainder = 1LSB 20/2 = 10 + 0/2 0 10/2 = 5 + 0/2 0 5/2 = 2 + 1/2 1 2/2 = 1 + 0/2 0 1/2 = 0 + 1/2 1 MSB (41)10 = (101001)2 Converting the fraction part (0.6875)10: 0.6875 x 2 = 1.3750 Integer = 1MSB 0.3750 x 2 = 0.7500 0 0.7500 x 2 = 1.5000 1 0.5000 x 2 = 1.0000 1 LSB (0.6875)10 = (0.1011)2 (41.6875)10 = (101001.1011)2 Conversion from Decimal to base r • The conversion is done as follows: • If the number has a radix point then separate the number into an integer part and a fraction part, since the two parts must be converted differently. • The conversion of a decimal integer part to a number in base r is done by dividing the integer part and all successive quotients by r and accumulating the remainders. • The conversion of a decimal fraction part to a number in base r is done by multiplying the fractional parts by r and accumulating integers. • Example of converting Decimal (base 10) to Binary (base 2):(41.6875)10 Fall 2007 Digital Techniques by André Deutz, Leiden University

30. Other Conversions • Binary to Octal or Hexadecimal: grouping bits starting from the radix point • (1101010.01)2 to Octal (groups of 3): (001|101|010.010|)2 = (152.2)8 • (1101010.01)2 to Hex (groups of 4): (0110|1010.0100|)2 = (6A.4)16 • Octal to Binary: convert each digit to binary using 3 bits • (475.2)8= (100 111 101. 010)2 • Hexadecimal to Binary: convert each digit to binary using 4 bits • (7A5F.C)16= (0111 1010 0101 1111. 1100)2= (111101001011111.11)2 • Hexadecimal to Octal • Hexadecimal  Binary  Octal • Octal to Hexadecimal • Octal  Binary Hexadecimal Fall 2007 Digital Techniques by André Deutz, Leiden University

31. Number Ranges • The range of numbers in base (radix) r depends on the number of digits used to represent the numbers. • Assume the number (An-1An-2…A1A0.A-1A-2…A-m+1A-m)r represented by n digits for the integer part and m digits for the fraction part. • The smallest integer number is 0 and the largest is (r-1)rn-1 +(r-1)rn-2 + …+ (r-1)r1 +(r-1)r0 = rn-1,i.e., the range is from 0 torn-1 • The smallest fraction number is 0.0 and the largest is (r-1)r-1 +(r-1)r-2 + …+ (r-1)r–m+1 +(r-1)r–m = 1- r–m,i.e., the range is from 0.0 to1- r–m • The range of numbers is from 0.0 to rn - r–m • Examples: • Largest 3-digit integer decimal (base 10) number is 103-1 = 1000 - 1 = 999 • Largest 8-digit integer binary (base 2) number is (11111111)2 ,i.e., 28-1 = 255 • Largest 5-digit decimal (base 10) fraction is 1-10-5 = 1 – 0.00001 = 0.99999 • Largest 16-digit binary (base 2) fraction is 1-2-16 = 0.9999847412 • What about the range of negative numbers? Fall 2007 Digital Techniques by André Deutz, Leiden University

32. Representations of Numbers in Digital Computers (1) • Numbers are represented in binary format as strings of bits • Bit is the smallest binary quantity with a value of 0 or 1. • Byte is a string (sequence) of eight bits. • Word is a string (sequence) of n bits (n > 8). In most cases n is a power of 2 (n = 24 = 16, n = 25 = 32, n = 26 = 64, etc…). • Examples  bit: 1 byte: 01101111 16-bit word: 11110100 10001010 • Positive Integer Numbers • Positive integers and the number zero can be represented as unsigned binary numbers using a byte or an n-bit word. • Magnitude representation – number N in binary having n bits. • Example: 00001001 ( represents integer number 9 using 8 bits ). • Radix complement ( r’s complement ) representation in our case 2’s complement – Given number N in binary having n bits, the 2’s complement of N is defined as 2n – N . • Example: 11110111 ( 2’s complement of integer number 9 ). • Diminished radix complement ( (r-1)’s complement ) representation in our case 1’s complement - Given number N in binary having n bits, the 1’s complement of N is defined as (2n – 1) – N . • Example: 11110110 ( 1’s complement of integer number 9 ). Fall 2007 Digital Techniques by André Deutz, Leiden University

33. Representations of Numbers in Digital Computers (2) • Positive and Negative Integer Numbers • Positive and negative integers and the number zero can be represented as signed binary numbers using a byte or an n-bit word where the most significant bit is interpreted as a sign bit. The convention is to make the sign bit 0 for positive numbers and 1 for negative numbers. • Signed-Magnitude representation • Example: 0|0001001 ( represents integer number +9 using 8 bits ). • Example: 1|0001001 ( represents integer number -9 using 8 bits ). • Signed-Radix complement ( signed-r’s complement ) representation in our case signed-2’s complement (n=8 in the example below) • Example: 0|0001001 ( signed-2’s complement of integer number +9 ). • Example: 1|1110111 ( signed-2’s complement of integer number -9 ). • How do we get the signed complement? Fall 2007 Digital Techniques by André Deutz, Leiden University

34. Multiplicand: Multiplier: Product: 1010 101 1010 0000 1010 110010 Carries: Augend: Addend: Sum: 111 10110 01110 100100 x Borrows: Minuend: Subtrahend: Difference: 00110 11101 11 11101 00110 -10111 + - + - Arithmetic Operations • Arithmetic operations with numbers in base r follow the same rules as for decimal numbers. • Examples: addition, subtraction, and multiplication in base-2. • For more information study pages 18-20 in the text book. • In Digital Computers arithmetic operations are done with the binary number system (base-2) - Binary Arithmetic. • In many cases binary subtraction is done in a special way by binary addition. Why? • It is much more simple to do it that way. • One simple building block called adder can be implemented and used for both binary addition and subtraction. Fall 2007 Digital Techniques by André Deutz, Leiden University

35. 0101 0001 0110 (5) (2’s compl. of 15) (the sum) 1111 1011 1 1010 (15) (2’s compl. of 5) (the sum) + + 5 15 -10 15 5 10 - - correction is needed no carry discard carry ( “-” 2’s compl. of sum ) -1010 Unsigned Binary Subtraction • Binary subtraction by 2’s complement addition • Assume two n-bit unsigned numbers M and N, M - N can be done as follows: • Add the 2’s complement of N to M. This performs the sum M + (2n - N) == M – N + 2n. • If M ≥ N, the sum produces an end carry, 2n. We can discard it, leaving the correct result M – N. • If M < N, the sum does not produces an end carry since it is equal to 2n – (N - M), which is the 2’s complement of N – M. To obtain the correct result take the 2’s complement of the sum, i.e., 2n – (2n – (N - M)) = (N - M) and place a minus sign in front. • Examples: • What about binary subtraction by 1’s complement addition? • I leave this for you as a home work Fall 2007 Digital Techniques by André Deutz, Leiden University

36. (+ 9) (- 5) (+ 4) 0|1001 1|1011 10|0100 (- 9) (+ 5) (- 4) 1|0111 0|0101 1|1100 (- 9) (- 5) (-14) 1|0111 1|1011 11|0010 + + + 1|0111 0|0101 1|1100 1|0111 1|1011 11|0010 (- 9) (- 5) (- 4) 1|0111 1|1011 (- 9) (+ 5) (-14) 1|0111 0|0101 + + - - discard discard discard Signed Binary Addition/Subtraction • Signed binary addition using 2’s complement representation • Assume two n-bit signed numbers M and N represented in signed-2’s complement format. The sum M + N can be obtained as follows: • Add M to N, including their sign bits to get the correct sum in signed-2’s complement format. A carry out of the sign bit position is discarded. • Examples: • Signed binary subtraction using 2’s complement representation • Assume two n-bit signed numbers M and N represented in signed-2’s complement format. The difference M - N can be obtained as follows: • Take the 2’s complement of N (including the sign bit) and add it to M (including the sign bit). A carry out of the sign bit position is discarded. • Examples: (+ 9) (+ 5) (+14) 0|1001 0|0101 0|1110 + Fall 2007 Digital Techniques by André Deutz, Leiden University

37. Decimal Codes • The binary number system is the most natural system for a digital computer, but people are accustomed to the decimal system. • How to resolve this difference? • Convert decimal numbers to binary, perform all arithmetic calculations in binary, and then convert the binary result back to decimal. • You already know how to do this. • Digital computers can do this as well, but: • We have to store the decimal numbers in the computer in a way that they can be converted to binary. • Since the computer can accept only binary values, we must represent the decimal digits by a code that contains 1’s and 0’s. Fall 2007 Digital Techniques by André Deutz, Leiden University

38. Binary Coded Decimals (BCD) (1) • The BCD code is the most commonly used code. Each decimal digit (0, 1, 2, …, 9) is coded by a 4-bit string (half a byte) called BCD digit. • A decimal number is converted to a BCD number by replacing each decimal digit of the number with the corresponding BCD digit code. • Example: (369)10 = ( 0011 0110 1001 )BCD= (101110001)2 3 6 9 • A BCD number needs more bits than its equivalent binary value. However, the advantages of using BCD are: • Computer input and output data are handled by people who use the decimal system. • BCD numbers are decimal numbers (not binary numbers) even though they are represented in bits. • Computers can store decimal numbers using BCD, convert the BCD numbers to binary, perform binary operations, and convert the result back to BCD. Note: the binary combinations 1010 through 1111 are not used and have no meaning in the BCD code. Fall 2007 Digital Techniques by André Deutz, Leiden University

39. Convert the number (11001)2 by dividing it to (1010)2 = (10)10 (11001)2 / (1010)2 = (0010)2 and Remainder =(0101)2Least significant BCD digit (0010)2 / (1010)2 = (0000)2 and Remainder =(0010)2Most significant BCD digit (11001)2= ( 00100101 )BCD = (25)10 Binary Coded Decimals (BCD) (2) • Converting a BCD number to a binary number (25)10 = (0010 0101)BCD= (0010)2 x 101 + (0101)2 x 100 = = (0010)2 x (1010)2 + (0101)2 x (0001)2 = = (10100)+(0101) = (11001)2 • Converting a binary number to a BCD number • BCD Arithmetic • Digital computers can performed arithmetic operations directly with decimal numbers when they are stored in BDC format. • How is this done? (study the text book or go to internet for information). Fall 2007 Digital Techniques by André Deutz, Leiden University

40. Other Useful Decimal Codes:Seven-Segment Code • Used to display numeric info on seven-segment displays. • Seven-segment display: • 7 LEDs (light emitting diodes), each one controlled by an input • 1 means “on”, 0 means “off” • Display digit “3”? • Set a, b, c, d, g to 1 • Set e, f to 0 a f b g e c d Fall 2007 Digital Techniques by André Deutz, Leiden University

41. Alphanumeric Codes • Many applications of digital computers require the handling of data consisting not only of numbers, but also of letters. • Alphanumeric character set of English includes: • The 10 decimal digits • The 26 letters of the alphabet (uppercase and lowercase letters) • Several (more than three) special characters • We need to code these symbols • The code must be binary – computers can handle only 0’s and 1’s • We need binary code of at least seven bits (27 = 128 symbols) • American Standard Code for Information Interchange (ASCII) • ASCII is a 7-bit standard code for representing the symbols of the English language. • Unicode • A 16-bit standard code for representing the symbols and ideographs for the world’s languages. Fall 2007 Digital Techniques by André Deutz, Leiden University

42. ASCII Code Table Fall 2007 Digital Techniques by André Deutz, Leiden University

43. ASCII Character Code • • ASCII is a 7-bit code, commonly stored in 8-bit bytes. • • “A” is at 4116. To convert upper case letters to lower case letters, add 2016. Thus “a” is at 4116 + 2016 = 6116. • • The character “5” at position 3516 is different than the number 5. To convert character-numbers into number-numbers, subtract 3016: 3516 - 3016 = 5. Fall 2007 Digital Techniques by André Deutz, Leiden University

44. ASCII Code Table Fall 2007 Digital Techniques by André Deutz, Leiden University

45. Gray Codes Fall 2007 Digital Techniques by André Deutz, Leiden University

46. What is What? How do you know whether binary bit string 00110001 represents an ASCII code with odd parity for the character ‘1’ , the positive number 49, or the positive number 31?