Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1

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1 Digital Systems Digital systems have such a prominent role in everyday life The digital age The technology around us is ubiquitous, that is we don t even notice it anymore Digital systems are used in: communication, business transactions, traffic control, spacecraft guidance, medical treatment, weather monitoring, the Internet, and many other commercial, industrial, and scientific enterprises We have: digital telephones, digital televisions, digital versatile discs, digital cameras, handheld devices, and, of course, digital computers, etc Roberto Muscedere Images 2013 Pearson Education Inc. 1

2 Digital Systems We enjoy music and video downloaded directly to our portable media devices (phones) which have very high resolution displays via complex digital communications networks (4G, LTE) These devices have advanced graphical user interfaces (GUIs), which enable them to execute commands that appear to the user to be simple, but which, in fact involve precise execution of a sequence of complex internal instructions Your phone s OS is over 3GB in size! These devices have a general purpose digital computer embedded within them Also contains specific components such as a radio, audio/video encoders/decoders, 3D graphics subsystems Roberto Muscedere Images 2013 Pearson Education Inc. 2

3 Digital Systems The digital computer can follow a sequence of instructions, called a program The user can specify and change the program or the data according to the specific need Because of this flexibility, the general purpose digital computer can perform a variety of information processing tasks that range over a wide spectrum of applications Digital systems manipulate discrete elements of information Any set that is restricted to a finite number of elements Examples: 10 decimal digits, 26 letters of the alphabet, 52 playing cards, 64 squares of a chessboard Roberto Muscedere Images 2013 Pearson Education Inc. 3

4 Digital Systems Early digital computers were used strictly for numeric computations (for which computer was derived) Discrete elements of information are represented in a digital system by physical quantities called signals which are usually represented as voltages and currents Digital systems today generally use just two discrete values and are therefore said to be binary A binary digit, called a bit, has two values: 0 and Roberto Muscedere Images 2013 Pearson Education Inc. 4

5 Digital Systems Discrete elements of information are represented with groups of bits called binary codes Decimal digits 0 through 9 could be represented in a digital system with four bits How a pattern of bits is interpreted as a number depends on the code system in which it resides Discrete quantities of information either emerge from the nature of the data being processed or may be quantized from a continuous process The quantization of a process can be performed automatically by an analog-to-digital converter, a device that forms a digital (discrete) representation of an analog (continuous) quantity Roberto Muscedere Images 2013 Pearson Education Inc. 5

6 Digital Systems The general purpose digital computer is the best known example of a digital system The major parts of a computer are a memory unit, a central processing unit, and input-output units Memory unit stores programs, input, output, and intermediate data Central processing unit performs arithmetic and data movement Program and data are transferred into memory by an input device An output device receives the results of the computations A digital computer can accommodate many input and output devices Roberto Muscedere Images 2013 Pearson Education Inc. 6

7 Digital Systems A digital computer is a powerful instrument that can perform not only arithmetic computations, but also logical operations It can be programmed to make decisions based on internal and external conditions By changing the program the same underlying hardware can be used for many different applications The cost of development to be spread across a wider customer base Advances in digital integrated circuit technology have reduced overall costs As the number of transistors that can be put on a piece of silicon increases to produce complex functions, the cost per unit decreases and digital devices can be bought at an increasingly reduced price Roberto Muscedere Images 2013 Pearson Education Inc. 7

8 Digital Systems Digital integrated circuits can perform at a speed of hundreds of millions of operations per second To improve reliability digital systems often utilize error correcting codes For example: Digital communications include additional bits to verify the proper transmission of information All digital storage includes additional bits of information which are used to not only verify but reconstruct data which has been damaged Roberto Muscedere Images 2013 Pearson Education Inc. 8

9 Digital Systems A digital system is an interconnection of digital modules It is necessary to have a basic knowledge of digital circuits and their logical function to understand the overall operation of each digital module Modern day digital design methodology uses hardware description languages (HDLs) to describe and simulate the functionality of a digital circuit Resembles a programming language and is suitable for describing digital circuits in textual form Used to simulate a digital system to verify its operation before hardware is built Also used with logic synthesis tools to automate the design process Roberto Muscedere Images 2013 Pearson Education Inc. 9

10 Digital Systems Digital systems manipulate discrete quantities of information that are represented in binary form Operands used for calculations may be expressed in the binary number system Other discrete elements, including the decimal digits and characters of the alphabet, are represented in binary codes Digital circuits, also referred to as logic circuits, process data by means of binary logic elements (logic gates) using binary signals Quantities are stored in binary (two-valued) storage elements (flip-flops) Roberto Muscedere Images 2013 Pearson Education Inc. 10

11 Number Representations Series of digits where the placement of the digits indicates its magnitude or significance a i is the digit, r is the base, i is the position or significance (lower is less), n is the number of digits For the decimal system, r = 10 Each digit ranges from 0 to 9 n 1 y a r i, 0 a r i i i Roberto Muscedere Images 2013 Pearson Education Inc. 11

12 Decimal Number System Example: r = 10 y = 7392 (10) a 0 = 2, the ones column a 1 = 9, the tens column a 2 = 3, the hundreds column a 3 = 7, the thousands column y a r a r a r a r y y " a a aa " Roberto Muscedere Images 2013 Pearson Education Inc. 12

13 Decimal Number System Example: r = 10 y = (10) a -2 = 5, the hundredths column a -1 = 7, the tenths column a 0 = 6, the ones column a 1 = 2, the tens column Radix point distinguishes where the positive powers switch to negative y a r a r a r a r y y " aa. a a " Roberto Muscedere Images 2013 Pearson Education Inc. 13

14 Binary Numbers Digital systems represent information in 0 s and 1 s Almost always built with transistors Circuits operate as switches ( 0 =off, 1 =on) Discrete levels compensate for electronic noise and power fluctuations E.g. 0 is from 0V to 2V, 1 is from 3V to 5V Roberto Muscedere Images 2013 Pearson Education Inc. 14

15 Binary Number System (r( = 2) Example: r = 2 y = (2) y (10) Always refer to bit positions from right to left (negative to positive) Roberto Muscedere Images 2013 Pearson Education Inc. 15

16 Other Number System Base 5: Digits between 0 and (5) (10) Base 8: Digits between 0 and (8) (10) Base 16: Digits between 0 and 9, A and F (10 to 15) B65F (16) (10) Roberto Muscedere Images 2013 Pearson Education Inc. 16

17 Binary Number System When converting from binary, you can skip 0 bits since they don t contribute: (2) (10) Roberto Muscedere Images 2013 Pearson Education Inc. 17

18 Binary Arithmetic Binary operations follow the same rules as decimal Remember to use the proper digits! (0 & 1) Roberto Muscedere Images 2013 Pearson Education Inc. 18

19 Number Base Conversions Convert 41 (10) to binary (base 2) Starting at position 0 Divide value by r Integer Remainder become coefficient Integer Quotient becomes next value Repeat until next value is 0 Solution is (2) Integer Quotient Integer Remainder Coefficient 41/ a 0 = 1 20/ a 1 = 0 10/2 5 0 a 2 = 0 5/2 2 1 a 3 = 1 2/2 1 0 a 4 = 0 1/2 0 1 a 5 = Roberto Muscedere Images 2013 Pearson Education Inc. 19

20 Number Base Conversions Convert 153 (10) to octal (base 8) 153/8 19/8 2/8 Integer Quotient Integer Remainder Coefficient a 0 = 1 a 1 = 3 a 2 = 2 Convert (10) to hexadecimal (base 16) Integer Quotient Integer Remainder Coefficient 41394/ a 0 = 2 = / a 1 = 11 = B 161/ a 2 = 1 = 1 10/ a 3 = 10 = A Roberto Muscedere Images 2013 Pearson Education Inc. 20

21 Number Base Conversions Convert (10) to binary (base 2) Starting at position -1 Multiple value by r Integer Product becomes coefficient Fraction becomes next value Repeat until next value is 0 Solution is (2) Integer Product Fraction Coefficient x a -1 = x a -2 = x a -3 = x a -4 = Roberto Muscedere Images 2013 Pearson Education Inc. 21

22 Number Base Conversions Convert (10) to octal (base 8) Integer Product Fraction Coefficient x a -1 = x a -2 = x a -3 = x a -4 = x a -5 = x a -6 = x Solution is (8) Roberto Muscedere Images 2013 Pearson Education Inc. 22

23 Number Base Conversions Full solution: (8) Not every base gives a compact answer Base 10 was certainly smaller! Roberto Muscedere Images 2013 Pearson Education Inc. 23

24 Number Base Conversions Conversion of both integer and fraction parts is done separately and then combining the two answers For example: (10) = (2) (10) = (8) Roberto Muscedere Images 2013 Pearson Education Inc. 24

25 Octal and Hexadecimal Numbers Octal: (2) (2) (8) (8) Hexadecimal: (2) (2) E. 3 2 (16) 665E.32 (16) Roberto Muscedere Images 2013 Pearson Education Inc. 25

26 Octal and Hexadecimal Numbers Roberto Muscedere Images 2013 Pearson Education Inc. 26

27 Complements of Numbers The larger a circuit becomes, generally the slower it becomes Complements are used to simplify subtraction Two types: Diminished Radix Complement (r-1) s complement (1 s complement in binary, 9 s complement in decimal) Radix Complement r s complement (2 s complement in binary, 10 s complement in decimal) Roberto Muscedere Images 2013 Pearson Education Inc. 27

28 Diminished Radix Complement Given N in base r having n digits (r-1) s complement of N is: n r 1 N For example (r=10,n=7): 9 s complement of is: = s complement of is: = Roberto Muscedere Images 2013 Pearson Education Inc. 28

29 Diminished Radix Complement For example (r=2,n=8): 1 s complement of is: = s complement of is: = Quick way: Invert the bits 1 s complement of is s complement of is For octal or hexadecimal, subtract each digit from 7 or F (15 decimal) respectively Roberto Muscedere Images 2013 Pearson Education Inc. 29

30 Radix Complement Given N in base r having n digits r s complement of N is: n r 0, N 0 N, N 0 Can also be obtained by adding 1 to the diminished radix complement since: n n r N r 1 N Roberto Muscedere Images 2013 Pearson Education Inc. 30

31 Radix Complement For example (r=10,n=7): 10 s complement of Add extra 0 s at front to make 7 digits 9 s complement of is s complement is = s complement of Add extra 0 s at front to make 7 digits 9 s complement of is s complement is = Roberto Muscedere Images 2013 Pearson Education Inc. 31

32 Radix Complement Quick way: Leave all least significant 0 s unchanged Subtract first nonzero least significant digit from 10 Subtract all higher significant digits from 9 10 s complement of Leave last 2 zeros unchanged to get 00 Subtract 7 from 10 to get 3 Subtract 0246 digits from 9 s to get 9753 Combine to get Roberto Muscedere Images 2013 Pearson Education Inc. 32

33 Radix Complement For example (r=2, and n=8): 2 s complement of Add extra 0 s at front to make 8 digits 1 s complement is = s complement is = s complement of Add extra 0 s at front to make 8 digits 1 s complement is = s complement is Roberto Muscedere Images 2013 Pearson Education Inc. 33

34 Radix Complement Quick way: Leave all least significant 0s and first 1 unchanged Invert remaining 2 s complement of s complement of Roberto Muscedere Images 2013 Pearson Education Inc. 34

35 Radix Complement When a radix point is present Remove point Perform complement Place point back in original place Essentially ignore it The complement of a complement restores the number to its original value Roberto Muscedere Images 2013 Pearson Education Inc. 35

36 Subtraction with Complements To subtract two n-digit unsigned numbers M - N in base r: 1. Add the minuend M to the r s complement of the subtrahend N: M + ( r n -N ) = M - N + r n 2. If there is a carry (r n ), discard it 3. If high digit is not zero, solution is in r s complement Take r s complement to get absolute value and place a negative sign in front to get normal form Roberto Muscedere Images 2013 Pearson Education Inc. 36

37 Subtraction with Radix Complements Solve: (r=10,n=6) M 10 s complement of N = = Sum = Discard end carry of 10 6 = Answer = Roberto Muscedere Images 2013 Pearson Education Inc. 37

38 Subtraction with Radix Complements Solve: (r=10,n=6) M 10 s complement of N Sum High digit not zero, do 10 s complement of Sum Place negative sign = = = = = Roberto Muscedere Images 2013 Pearson Education Inc. 38

39 Subtraction with Radix Complements r = 2, n = 8, X = , Y = Find: X - Y X 2 s complement of Y = = Sum = Discard end carry of 2 8 = Answer = Roberto Muscedere Images 2013 Pearson Education Inc. 39

40 Subtraction with Radix Complements r = 2, n = 8, X = , Y = Find: Y - X Y 2 s complement of X Sum High digit not zero, do 2 s complement of Sum Place negative sign = = = = = Roberto Muscedere Images 2013 Pearson Education Inc. 40

41 Subtraction with Diminished Radix Complements r = 2, n = 8, X = , Y = Find: X - Y X 1 s complement of Y Sum End-around carry Answer = = = = = Roberto Muscedere Images 2013 Pearson Education Inc. 41

42 Subtraction with Diminished Radix Complements r = 2, n = 8, X = , Y = Find: Y - X Y 1 s complement of X Sum High digit not zero, do 1 s complement of Sum Place negative sign = = = = = Roberto Muscedere Images 2013 Pearson Education Inc. 42

43 Signed Binary Numbers Range is (2 (n-1) ) to 2 (n-1) Roberto Muscedere Images 2013 Pearson Education Inc. 43

44 Arithmetic Addition (2 s s comp) Carry out is discarded Make sure we have enough bits or an overflow will be generated Roberto Muscedere Images 2013 Pearson Education Inc. 44

45 Arithmetic Subtraction (2 s s comp) Do the same as addition except perform subtraction Discard the borrow Beneficial since we don t need to make a decision on which operation to do based on the signs Binary adders and subtractors can be combined to obtain more savings in circuit implementation Roberto Muscedere Images 2013 Pearson Education Inc. 45

46 Binary Codes Digital systems use signals that have two distinct values and circuit elements that have two stable states Direct analogy among binary signals, binary circuit elements, and binary digits A binary number of n digits may be represented by n binary circuit elements; each a 0 or 1 Digital systems represent and manipulate not only binary numbers but also many other discrete elements of information Any discrete element of information that is distinct among a group of quantities can be represented with a binary code; a pattern of 0 s and 1 s Roberto Muscedere Images 2013 Pearson Education Inc. 46

47 Binary Codes Due to technology limitations, codes must be in binary Binary codes merely change the symbols and not the meaning of the elements of information that they represent Most bits of a digital system represent some type of coded information rather than just binary numbers An n-bit binary code is a group of n bits that assumes up to 2 n distinct combinations of 1 s and 0 s, with each combination representing one element of the set that is being coded Four elements can be coded with two bits: 00, 01, 10, 11 Eight elements requires a three bit code Sixteen elements requires a four bit code Roberto Muscedere Images 2013 Pearson Education Inc. 47

48 Binary Codes The bit combination of an n-bit code is determined from the count in binary from 0 to 2 n -1 Each element must be assigned a unique binary bit combination No two elements can have the same value or the code assignment will be ambiguous Minimum number of bits required to code 2 n distinct values is n No maximum number of bits that may be used For example, the 10 decimal digits can be coded with 10 bits, and each decimal digit can be assigned a bit combination of nine 0 s and a 1 (one hot) The bit combination represents Roberto Muscedere Images 2013 Pearson Education Inc. 48

49 Binary Coded Decimal r=bcd Perform computation directly on decimal information Avoid conversion between binary and decimal Can save time on slower systems Each digit takes 4 bits; 6 of 16 states unused Roberto Muscedere Images 2013 Pearson Education Inc. 49

50 BCD Addition If the addition of two digits exceeds 9, need to correct by adding 6 This may produce a new digit or add to an existing digit Carry propagation may continue onward to higher significant digits Roberto Muscedere Images 2013 Pearson Education Inc. 50

51 Solve: BCD Addition Carry is sent to next column Roberto Muscedere Images 2013 Pearson Education Inc. 51

52 Other Decimal Codes Roberto Muscedere Images 2013 Pearson Education Inc. 52

53 Other Decimal Codes BCD8421 code has weights 8, 4, 2, and 1 8x0 + 4x1 + 2x1 + 1x0 = 6 BCD2421 has weights 2, 4, 2, and 1 2x1 + 4x1 + 2x0 + 1x1 = 7 Have duplicate representations: 0100 & 1010 = 4, 0101 & 1011 = 5, etc. Excess-3 adds 3 to BCD8421 Excess-3 and BCD2421 are self complementing so 9 s complement is easily achieved by inverting bits assigns both positive and negative weights 8x0 + 4x1-2x1-1x0 = Roberto Muscedere Images 2013 Pearson Education Inc. 53

54 Gray Code Only one bit changes in going from one number to the next ie. 7 (0111) to 8 (1000) in binary requires 4 bit changes whereas 7 to 8 in gray code requires just 1 Traditionally used in applications where many bit changes could have produced errors Analog/asynchronous designs Not an issue today Roberto Muscedere Images 2013 Pearson Education Inc. 54

55 American Standard Code for Information Interchange (ASCII) Roberto Muscedere Images 2013 Pearson Education Inc. 55

56 American Standard Code for Information Interchange (ASCII) Roberto Muscedere Images 2013 Pearson Education Inc. 56

57 American Standard Code for Information Interchange (ASCII) ASCII is 7 bits Additional bit made each character 8 bits Bit used for more characters such as Greek letters or graphical symbols This has all been replaced by UNICODE which uses 16 or 32 bits per character Additional bit was also used for error detection using simple parity system: It can only detect 1, 3, 5, or 7 incorrect bits Simple but not used in modern technology Roberto Muscedere Images 2013 Pearson Education Inc. 57

58 Binary Storage and Registers Information needs to be stored in binary due to modern technology limitations A group of binary cells is called a register An n-bit register can store up to 2 n possibilities Modern technology groups information into 8-bits or a byte Advancements have transitioned from 1 to 2 to 4 to 8 bytes since it is easy to double up ASCII uses 1 byte UNICODE may use 2 or 4 Floating point may use 4 or 8 or more The amount of bits has to be determined for each application Roberto Muscedere Images 2013 Pearson Education Inc. 58

59 Register transfer is a basic operation which moves information from one register to another It may be a direct copy or may be processed before They essentially hold data Example: A keyboard sending data (with odd parity) to a control circuit which stores the information every time a key is pressed Pressed key is shifted 8 bits to the left and stored in memory Register Transfer Roberto Muscedere Images 2013 Pearson Education Inc. 59

60 Register Transfer Example: Memory unit holds millions of registers Processing unit takes 2 operands (R1, R2) and adds them and places them sum in (R3) The memory unit cannot perform any processing; it only stores information Transfer information from the memory unit to the processing unit, process it, and store it back The process is not immediate; it takes many steps Roberto Muscedere Images 2013 Pearson Education Inc. 60

61 Binary Logic Binary logic deals with variables that take on two discrete values and with operations that assume logical meaning Two values may be called by different names: true and false, yes and no, etc. Use the values 1 and 0 The binary logic is equivalent to Boolean algebra More formal presentation in Chapter 2 Binary logic consists of binary variables and a set of logic operations Variables are: A, B, C, x, y, z, etc. each having two distinct values (0 or 1) Three operations are AND, OR, and NOT Roberto Muscedere Images 2013 Pearson Education Inc. 61

62 Definition of Binary Logic AND: Represented by a dot or by the absence of an operator For example, z = x y or z = xy, is read z is equal to x AND y Interpreted to mean that z = 1 if and only if x = 1 and y = 1; otherwise z = 0 OR: Represented by a plus sign For example, z = x + y, is read z is equal to x OR y Interpreted to mean that z = 1 if x = 1 or if y = 1 or if both x = 1 and y = 1; otherwise z = 0 NOT: Represented by a prime after or an overbar or a slash before For example: z = x' or z = x or z = /x, is read z is equal to not x or z is not equal to x Interpreted to mean if x = 1, then z = 0, but if x = 0, then z = 1 Also referred to as the complement operation, since it changes a 1 to 0 and a 0 to Roberto Muscedere Images 2013 Pearson Education Inc. 62

63 Definition of Binary Logic Binary logic resembles binary arithmetic, and the operations AND and OR have similarities to multiplication and addition, respectively The symbols used for AND and OR are the same as those used for multiplication and addition Binary logic should not be confused with binary arithmetic Arithmetic variable designates a number that may consist of many digits A logic variable is always either 1 or 0 For example: Binary arithmetic: = 10 Binary logic: = Roberto Muscedere Images 2013 Pearson Education Inc. 63

64 Definition of Binary Logic For each combination of the values of x and y, there is a value of z specified by the definition of the logical operation Listed in a compact form called truth tables A table of all possible combinations of the variables, showing the relation between the values that the variables may take and the result of the operation The truth tables for the operations AND, OR, and NOT with variable(s) (x and y) are obtained by listing all possible values that the variables may have when combined in pairs: Roberto Muscedere Images 2013 Pearson Education Inc. 64

65 Logic Gates Logic gates are electronic circuits that operate on one or more input signals to produce an output signal Electrical signals exist as analog signals having values over a given continuous range but are interpreted to be either of two recognizable values, 0 or 1 for digital systems Modern day technology uses voltages to determine values; for example: Logic 0 is from 0 V to 1 V Logic 1 is from 2 V to 3 V Roberto Muscedere Images 2013 Pearson Education Inc. 65

66 Logic Gates The input terminals of digital circuits accept binary signals within the allowable range and respond at the output terminals with binary signals that fall within the specified range The intermediate region between the allowed regions is crossed only during a state transition Any information for computing or control can be operated on by passing binary signals through various combinations of logic gates Roberto Muscedere Images 2013 Pearson Education Inc. 66

67 Logic Gates Graphic symbols of gates shown below Gates are blocks of hardware that produce proper 0 and 1 signals given proper input signals Each gate is made up by a series of transistors Some gates require more transistors and therefore more space Current technology only allows for transistors to be constructed on a single plane Roberto Muscedere Images 2013 Pearson Education Inc. 67

68 Logic Gates The input signals x and y in the AND and OR gates may exist in one of four possible states: 00, 10, 11, or 01 These input signals are shown together with the corresponding output signal for each gate Timing diagrams illustrate the idealized response of each gate to the four input signal combinations Horizontal axis represents the time, and the vertical axis shows the signal as it changes between the two possible voltage levels In reality the transitions between logic values occur quickly, but not instantaneously nor are they perfectly square Low level represents logic 0 High level represents logic Roberto Muscedere Images 2013 Pearson Education Inc. 68

69 Logic Gates AND and OR gates may have more than two inputs The three input AND gate responds with logic 1 output if all three inputs are logic 1; otherwise the output produces logic 0 The four input OR gate responds with logic 1 if any input is logic 1; otherwise the output produces a 0 In practice no more than 3 inputs are used since it slows down the gate Multiple gates are cascaded to generate the same result Roberto Muscedere Images 2013 Pearson Education Inc. 69