CS 121 Digital Logic Design. Chapter 1. Teacher Assistant. Hadeel Al-Ateeq
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1 CS 121 Digital Logic Design Chapter 1 Teacher Assistant Hadeel Al-Ateeq
2 Announcement DON T forgot to SIGN your schedule OR you will not be allowed to attend next lecture.
3 Communication Office hours (8 hours per Week): Saturday(10-11). Sunday(10-12). Monday (10-11). Tuesday (10-12). Wednesday (10-11) & (12:30-1:30). Blog:
4 Textbook DIGITAL DESIGN, by Mano M. Morris, 4th edition, Prentice- Hall.
5 Grades Evaluation Type Midterm 1 Midterm 2 Quiz 1 Including Date Grade Chapter 1 + chapter 2 Chapter 3 + chapter 4 Chapter 1 + part of chapter 2 Sunday AH (Week # 7) Sunday AH (Week # 12) Quiz 2 Chapter 3 Week # 10 40% Week # 5 5% 4 HW Every chapter % Project After week 15 10% Final All 5 chapters After week 15 40% Total 100%
6 Topics Topics to be covered Digital Systems and Binary numbers Boolean Algebra and Logic Gates Gate Level Minimization Combinational Logic Synchronous Sequential Logic
7 Outline 1.1 Digital Systems 1.2 Binary Numbers 1.3 Number-Base Conversions 1.4 Octal and Hexadecimal Numbers 1.5 Complements 1.6 Signed Binary Numbers 1.7 Binary Codes
8 1.1 Digital Systems (1-2) Digital system is a system that uses discrete values such as numbers and letters. The signal in most digital systems use two values : 0 and 1 which called a bit. Discrete elements of information are represented with a group of bits called binary codes. Thus, Digital system is a system that manipulates discrete elements of information represented internally in binary form.
9 1.1 Digital Systems (2-2) Stores programs, inputs, outputs & intermediate data Data and program prepared by user Input Unit Central Processing Unit Memory Unit Arithmetic & data processing operations Output Unit Receives results of computations and prints them to user Block diagram of a digital Computer
10 1.2 Binary Numbers (1-6) In general, a number expressed in a base-r system has coefficients multiplied by powers of r: a n.r n + a n-1.r n a 2.r 2 + a 1.r + a 0 + a -1.r -1 + a- 2.r a- m.r- m r is called base or radix. a j ranges in thr value from 0 to r-1
11 1.2 Binary Numbers (2-6) Example: (127.4)8 = 1 x x x x 8-1 = (87.5) 10 (1101) 2 = 1 x x x x 2 0 = (13) 10
12 1.2 Binary Numbers (3-6) Numbering System Base Symbols Decimal 10 0, 1, 3, 4, 5, 6, 7, 8, 9 Binary 2 0, 1 Octal 8 0, 1, 3, 4, 5, 6, 7 Hexadecimal 16 0, 1, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
13 1.2 Binary Numbers (4-6) Arithmetic Operations Follow the same rules of as for decimal numbers Addition Carries Augend Addend Sum Carries Augend Addend Sum
14 1.2 Binary Numbers (5-6) Arithmetic Operations Subtraction Borrows Minuend Subtrahend Difference Borrows Minuend Subtrahend Difference Borrows Minuend Subtrahend Difference
15 1.2 Binary Numbers (6-6) Arithmetic Operations Multiplication Multiplicand 1011 Multiplier X Product
16 1.3 Number-Base Conversions (1-6) Base r to Decimal Conversion Rule: a n.r n + a n-1.r n a 2.r 2 + a 1.r + a 0 + a -1.r -1 + a- 2.r a- m.r- m Integral part a n.r n + a n-1.r n a 2.r 2 + a 1.r + a 0 Fractional part a -1.r -1 + a- 2.r a- m.r- m Result Integral part Fractional part
17 1.3 Number-Base Conversions (2-6) Base r to Decimal Conversion Example: Find the decimal equivalent ofthe binary number ( ) 2. Integral part Fractional part Result 1001 = 1 x x x x 2 0 = = = 0 x x x x 2-4 = = (9.3125) 10
18 1.3 Number-Base Conversions (3-6) Base r to Decimal Conversion Example: ( ) 2 = =(10.375) 10 (127.4) 8 = 1 x x x x 8-1 = (87.5) 10 (1101) 2 = 1 x x x x 2 0 = (13) 10
19 1.3 Number-Base Conversions (4-6) Decimal to Base r Conversion Rule: Convert each part differently. Integral part Fractional part Divide number & its quotients by r. Accumulate reminders. Multiply number & its quotients by r. Accumulate integers. Result Integral part Fractional part
20 1.3 Number-Base Conversions (5-6) Decimal to Base r Conversion Example: Find the binary equivalent of the decimal number (41) 10. Integer 41 Reminder Divide by Result (101001) 2
21 1.3 Number-Base Conversions (6-6) Decimal to Base r Conversion Example: Find the binary equivalent of the decimal number (0.6875) 10. Integer Fraction Multiply by Result (0.1011) 2
22 1.4 Octal and Hexadecimal Numbers (1-17)
23 1.4 Octal and Hexadecimal Numbers (2-17) Decimal-to-Octal Conversion Example: Find the octal equivalent of the decimal number ( ) 10. Divide by 8 Integer Reminder Multiply by 8 Integer Fraction Result ( ) 8
24 1.4 Octal and Hexadecimal Numbers (3-17) Octal-to-Decimal Conversion Example: Find the decimal equivalent ofthe octal number (137.21) 8. Integral part Fractional part Result 137= 1 x x x 8 0 = =95.21 = 2 x x 8-2 =0.265 (95.265) 10
25 1.4 Octal and Hexadecimal Numbers (4-17) Decimal-to-Hexadecimal Conversion Example: Find the hexadecimal equivalent of the decimal number (82.25) 10. Divide by 16 Integer Reminder 82 Multiply by 16 Integer Fraction Result (52.4) 16
26 1.4 Octal and Hexadecimal Numbers (5-17) Hexadecimal-to-Decimal Conversion Example: Find the decimal equivalent ofthe hexadecimal number (1E0.2A) 16. Integral part Fractional part Result 1E0= 1 x x x 16 0 = =480.2A = 2 x x 16-2 =0.164 ( ) 10
27 1.4 Octal and Hexadecimal Numbers (6-17) Binary Octal and Octal Binary Conversions Binary Octal Rule: Divide integral part into three bits starting from right integral bit. Divide fractional part into three bits starting from left fractional bit. Adding zero s if necessary.
28 1.4 Octal and Hexadecimal Numbers (7-17) Binary Octal and Octal Binary Conversions Example: Find the octal equivalent ofthe binary number ( ) ( ) 8
29 1.4 Octal and Hexadecimal Numbers (8-17) Binary Octal and Octal Binary Conversions Octal Binary Rule: BOTH integral and fractional parts: Convert each digit seperately into binary of three bits
30 1.4 Octal and Hexadecimal Numbers (9-17) Binary Octal and Octal Binary Conversions Example: Find the binary equivalent ofthe octal number (374.26) ( ) 2
31 1.4 Octal and Hexadecimal Numbers (10-17) Binary Hex and Hex Binary Conversions Binary Hex Rule: Divide integral part into four bits starting from right integral bit. Divide fractional part into four bits starting from left fractional bit. Adding zero s if necessary.
32 1.4 Octal and Hexadecimal Numbers (11-17) Binary Hex and Hex Binary Conversions Example: Find the hexdecimal equivalent ofthe binary number ( ) (2 C E. 6 E 8) 16
33 1.4 Octal and Hexadecimal Numbers (12-17) Binary Hex and Hex Binary Conversions Hex Binary Rule: BOTH integral and fractional parts: Convert each digit seperately into binary of four bits
34 1.4 Octal and Hexadecimal Numbers (13-17) Binary Hex and Hex Binary Conversions Example: Find the binary equivalent ofthe hexadecimal number (17E.F6) E. F ( ) 2
35 1.4 Octal and Hexadecimal Numbers (14-17) Hex Octal and Octal Hex Conversions Hex Octal Rule: Convert Hex number into binary. Convert the result binary number into octal. OR Convert Hex number into decimal. Convert the result decimal number into octal.
36 1.4 Octal and Hexadecimal Numbers (15-17) Hex Octal and Octal Hex Conversions Example: Find the Hex equivalent ofthe octal number ( ) Binary ( ) 2 Hex (1 F ) 16
37 1.4 Octal and Hexadecimal Numbers (16-17) Hex Octal and Octal Hex Conversions Octal Hex Rule: Convert Octal number into binary. Convert the result binary number into Hex. OR Convert Octal number into decimal. Convert the result decimal number into Hex.
38 1.4 Octal and Hexadecimal Numbers (17-17) Hex Octal and Octal Hex Conversions Example: Find the octal equivalent ofthe Hex number (2F.C4) F. C 4 Binary ( ) 2 Octal ( ) 8 ( ) 8
39 1.5 Complements (1-9) Complement s Types Radix Complement (r s complement) Diminished radix Complement ((r-1) s complement)
40 1.5 Complements (1-9) Diminished Radix Complement ((r-1) s complement) Rule o o Given a number N in base r having n digits, the (r- 1) s complement of N is defined as (r n - 1) -N. Examples o 9 s complement of is = o 1 s complement of is Note The (r-1) s complement of octal or hexadecimal numbers is obtained by subtracting each digit from 7 or F (decimal 15), respectively.
41 1.5 Complements (2-9) Radix Complement Rule o OR o Given a number N in base r having n digit, the r s complement of N is defined as (r n -N) for N 0 and as 0 for N =0. Adding 1 to (r-1) s complement. Examples o The 10 s complement of is o The 10 s complement of is o The 2 s complement of is
42 1.5 Complements (2-9) Note: The complement of the complement restores the number to its origunal value.
43 1.5 Complements (3-9) Subtraction with Complement The subtraction of two n-digit unsigned numbers M N in base r can be done as follows: o M + (r n - N), note that (r n - N) is r s complement of N. o If (M N), the sum will produce an end carry x, which can be discarded; what is left is the result M- N. o If (M < N), the sum does not produce an end carry and is (N - M). Take the r x complement of the sum and place a negative sign in front.
44 1.5 Complements (4-9) Example: Using 10 s complement subtract M = s complement of N = sum = Discard end carry. answer: 69282
45 1.5 Complements (5-9) Example: Using 10 s complement subtract M = s complement of N = sum = The answer is (10 s complement of 30718) =
46 1.5 Complements (6-9) Example: Using 2 s complement subtract M = N = , 2 s complement of N = = Discard end carry. answer:
47 1.5 Complements (7-9) Example: Using 2 s complement subtract M = N = , 2 s complement of N = = The answer is (2 s complement of ) =
48 1.5 Complements (8-9) Example: Using 1 s complement subtract M = N = , 1 s complement of N = = end-around carry = + 1 The answer is:
49 1.5 Complements (9-9) Example: Using 1 s complement subtract M = N = , 1 s complement of N = = No end-around carry Take the 1 s complement of the result with negetive sign. The answer is:
50 1.6 Signed Binary Numbers (1-10) Sign and Magnitude representation Signed complement representation 1 s complement representation 2 s complement representation
51 1.6 Signed Binary Numbers (2-10) Notes: o o o The previous representation (unsigned numbers) are the same for positive numbers and different for negative numbers. For a unsigned binary number the most significant bit of the number. For a signed binary number the most significant bit is used for representing the sign of the number and the rest of the bits represent the number. We use 0 for positive numbers and 1 for negative numbers.
52 1.6 Signed Binary Numbers (3-10) Example: Represent +76 (76) 10 = ( ) 2 Unsigned number (+76) 10 = ( ) 2 (+76) 10 = ( ) 2 (+76) 10 = ( ) 2 Sign & Magnitude 1 s Complement 2 s Complement
53 1.6 Signed Binary Numbers (4-10) Negative numbers: 1 Complement obtained by flipping all bits of the positive binary number 2 Complement obtained by adding 1 to the 1 s Complement. OR: by flipping bits of the positive binary number after the first one from the right
54 1.6 Signed Binary Numbers (5-10) Negative numbers: Sign & Magnitude obtained by changing the left most bit of the positive binary number to 1. Example: Represent -76 (-76) 10 = ( ) 2 (-76) 10 = ( ) 2 (-76) 10 = ( ) 2 Sign & Magnitude 1 s Complement 2 s Complement
55 1.6 Signed Binary Numbers (6-10) Arithmetic Addition: With Comprison The addition of two numbers in the signed & magnitude system follows the rules of ordinary arithmetic: If the signed are the same, we add the two magnitudes and give the sum the common sign. If the signed are different, we subtract the smaller magnitude from the larger and give the difference the sign of the larger magnitude. EX. (+25) + (-38) = -(38-25) = -13
56 1.6 Signed Binary Numbers (7-10) Arithmetic Addition: Without Comprison The addition of two numbers in the signed 2 s complement form is obtained from the addition of the two numbers, including their signed bits. A carry out of the signed bit position is discarded (note that the 4th case of the example). If the result is negetive (left most bot is one), then the must be complemented to extract its value.
57 1.6 Signed Binary Numbers (8-10) Arithmetic Addition: Without Comprison
58 1.6 Signed Binary Numbers (9-10) Arithmetic Subtraction: Rule Change the artithmetic operation into addition. Change the sign of the subtrahend (positive to negative or negative to positive). (+/-) A (+B)= (+/-) A + (-B) (+/-) A (-B)= (+/-) A + (+B)
59 1.6 Signed Binary Numbers (10-10) Arithmetic Subtraction: Example (-6) (-13)= +7 In binary: ( )= ( ) = After removing the carry out the result will be :
60 1.7 Binary Codes (1-24) Any discrete element of information that is distinct among a group of quantities can be represented with a binary code (i.e. a pettern of 0's and 1's). An n-bit binary code is a group of n bits that assumes up to 2" distinct combinations of 1's and 0'r. Each combination representing one element of the set that is being coded. Example A set of four elements can be coded with two bits, with each element assigned one of the following bit combinations:
61 1.7 Binary Codes (2-24) The bit combination of an n-bit code is determined from the count in binary from 0 to 2" - 1. Each element must be assigned a unique binary bit combination and no two elements can have the same value; otherwise. the code assignment will be ambiguous. Minimum number of bits required to code 2" distinct quantities is n. NO Maximum number of bits that may be used for a binary code.
62 1.7 Binary Codes (3-24) BCD Stands for binary-coded decimal. Represents the 10 decimal digits. Each digit is represented in 4 bits. Makes 16 possible combinations. 6 are unassigned. A number with k decimal digits will require 4k bits in BCD. Weight (8,4,2,1.)
63 1.7 Binary Codes (4-24) BCD Example Represent (945) 10 in BCD 3 decimal digits will require (3*4) = 12 bits. Each decimal digit is represented in 4 bits Result ( ) BCD
64 1.7 Binary Codes (5-24) BCD Addition Notes Same as adding two decimal numbers. Sum cannot be greater than 9, such as: 9+9+1=19 with 1 is being a previous carry. If the result value was greater than (1001) 2 =(9) or if the result was more than 4 digits. Then it is an invalid BCD digit. Add (0110) 2 =(6) to convert it to correct digit. A carry will be produced Result more than 4 digit is greater than 9(1001)
65 1.7 Binary Codes (6-24) BCD Addition Example Evaluate (3+4) in BCD System 3 BCD BCD Decimal 0111
66 1.7 Binary Codes (7-24) BCD Addition Example Evaluate (4+8) in BCD System BCD BCD ERROR We must add 6 (0110) to the result
67 1.7 Binary Codes (8-24) BCD Addition Example Evaluate ( ) in BCD System 184 BCD BCD
68 1.7 Binary Codes (9-24) Decimal Arithmetic Signed numbers Signed decimal numbers in BCD similar to signed numbers in binary. Can be signed-magnitude system or signedcomplement system (9 s or 10 s complement) Sign represented with four bits in BCD. + -> 0000 (four 0 s). - -> 1001 (nine).
69 1.7 Binary Codes (10-24) Decimal Arithmetic Addition for signed numbers Apply 10 s complement to the negative number only. Addition is done by summing all digits, including the sign digit,and discarding the end carry. Example (+375) + (- 240) (+0 375) + (9 760) = ( 1135) (end carry is discarded) = (0 135) in BCD
70 1.7 Binary Codes (11-24) Decimal Arithmetic Subtraction For BOTH signed and unsigned numbers: Apply 10 s complement to the subtrahend and apply addition (same as binary case)
71 1.7 Binary Codes (12-24) Other Decimal codes
72 1.7 Binary Codes (13-24) Excess -3 (ex-3) Another system to represent a decimal number. Like (BCD) in the way of representing number (each digit is represented in 4 bits) Except that : each digit is firstly incremented by three
73 1.7 Binary Codes (14-24) Excess -3 (ex-3) Example Represent (945) 10 in ex Result ( ) ex-3
74 1.7 Binary Codes (15-24) Gray Code Output data of many physical systems are quantities that are continuous. Such data must be converted into digital form before they are applied to a digital system using analog-todigital converter. Such converted data represented using Gray code. Used in applications in which the normal sequence of binary numbers may produce an error or ambiguity during the transition from one number to the next. Because only 1bit in the code group changes in going from one number to the next.
75 1.7 Binary Codes (16-24) Gray Code
76 1.7 Binary Codes (17-24) Gray Code
77 1.7 Binary Codes (18-24) ASCII Character Code ASCII : American Standard Code for Information Interchange Used to represent characters, Symbols, Consists of 7-bits (to represent 128 character). A bit can be added to produce a total of eight bits (byte). Used for other purposes.
78 1.7 Binary Codes (19-24) ASCII Character Code # ASCII Ch A B Z a b Upper case Letters are represented by ASCII (65 : 90) Lower case Letters are represented by ASCII (97 : 122) z
79 1.7 Binary Codes (20-24) ASCII Character Code
80 1.7 Binary Codes (21-24) ASCII Character Code
81 1.7 Binary Codes (22-24) ASCII Character Code Error Detecting Code Detecting errors in data communication and processing. Eight bit added to ASCII code. A parity bit is an extra- bit included with a message lo make the total number of 1's either even or odd. Example: even parity odd parity ASCII (A) ASCII (T)
82 1.7 Binary Codes (23-24) ASCII Character Code Error Detecting Code
83 1.7 Binary Codes (24-24) ASCII Character Code Error Detecting Code The eight-bit character that include parity bits are transmitted to their destination. The parity of each character is then checked at the receiving end. If the parity of the received character is not even. then at least one bit has changed value during transmission. Detects one, three or any odd combination of errors in each character that is transmitted. Other detection codes may be needed to take care of that possibility
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