introduction digital systems · 2012-12-23 · 1 introduction to digital systems by dr. ahmet...

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IntroductionIntroduction toto DigDigital ital SystemsSystems

byDr. Ahmet ÖZKURT

for IZMIR University of ECONOMICS2006

Based on the notes ofJaeyoung Choi, [email protected] fall 2000

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Digital Computers and Digital Computers and NumberNumber SystemsSystems

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3/43Digital ComputersDigital Computers

Digital Computers‘information age’

a prominent and growing role in modern society'generality'

follow a sequence of instruction, called a program,that operates on given dataperform a variety of information-processing tasks

Digital computer the best-known example of a digital systemmanipulate discrete elements of information

(ex) 10 decimal digits, 26 letters of the alphabets, ....


Signalselectrical signals such as voltages and currentstwo discrete values

High (output) 4.5~5.5 (input) 3.0~5.5Low (output) -0.5~1.0 (input) -0.5~2.0

High & Low (H & L), True & False, 1 & 0

Binary Number Systema binary digit is called a bitinformation is represented in group of bitsuse various coding techniques

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Computer Structure

Figure 1-2: Block Diagram of a Digital Computer


Basic Structurememory unit: stores programs, input, output, dataALU Unit (Arithmetic Logic Unit): performs arithmetic and other dataprocessing operations, as specified by the programcontrol unit: supervises the flow of information between units(CPU = control unit + data path)input device: key boardoutput device: CRT, LCD

MoreFPU (floating-point unit)MMU (memory management unit)(Memory: MMU + internal cache + external cache + RAM)(See Text p.2)

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7/43Number SystemsNumber Systems

decimal number (base 10 or radix 10)724.5 = 7 x 102 + 2 x 101 + 4 x 100 + 5 x 10-1

In general, AnAn-1....A1A0.A-1A-2....A-m+1A-m

Each Ai coefficient is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

base r or radix rexpressed with a power series in rAn rn + An-1 rn-1 + .... + A1 r1 + A0 r0 + A-1 r-1 + A-2 r-2 + .… + A-m+1 r-m+1 + A-m r-m

expresses in positional notationAnAn-1....A1A0.A-1A-2....A-m+1A-m

. is called radix point


An is the most significant digit (msd)A-m is the least significant digit (lsd)enclose coefficients in parentheses and place a subscript

(312.4)5 = 3 x 52 + 1 x 51 + 2 x 50 + 4 x 5-1

= 75 + 5 + 2 + 0.8 = (82.8)10

in computer work, binary, octal, and hexadecimal is popular

Binary Numbersbase 2 with two digits: 0 & 1(11010)2 = 1x24 + 1x23 + 0x22 + 1x21 + 1x20 = (26)10

digits in a binary numbers are called bitspowers of two are listed in Table 1-1.

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conversion of decimal to binarysuccessively subtracts powers of two from the decimal number

(625)10 = ( ? )2

625 - 512 = 113 512 = 29

113 - 64 = 49 64 = 26

49 - 32 = 17 32 = 25

17 - 16 = 1 16 = 24

1 - 1 = 0 1 = 20

(625)10 = 29 + 26 + 25 + 24 + 20 = (1001110001) 2


Octal and Hexadecimal Numberoctal number - base 8 (0, 1, ..., 6, 7)(127.4)8 = 1x82 + 2x81 + 7x80 + 4x8-1 = (87.5 )10

hexadecimal number - base 16 (0,1,....,9,A,B,C,D,E,F)(B65F)16 = 11x163 + 6x162 + 5x161 + 15x160 = (46687)10

1 octal digit = 3 binary digits1 hexa digit = 4 binary digits

conversion(0010 1100 0110 1011. 1111 0000 0110)2 = (2C6B.F06)16

(3A6.C)16 = 0011 1010 0110. 1100 = (1110100110.11)2

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Table 1-2 Numbers with Different Bases

12/43Arithmetic OperationsArithmetic OperationsConversion from Decimal to Other Base(Ex1.3) (153)10 = ( ? )8

153/8 = 19 + 1/8 .... 119/8 = 2 + 3/8 .... 32/8 = 0 + 2/8 .... 2

(153)10 = (231)8

(Ex1.5) (0.6875)10 = ( ? )2

0.6875 x 2 = 1.375 .... 10.375 x 2 = 0.75 .... 00.75 x 2 = 1.5 .... 10.5 x 2 = 1.0 .... 1

(0.6875)10 = (0.1011)2

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13/43Decimal CodesDecimal Codesdecimal number system (people are accustomed to)(vs) binary number system (natural for computer)

2 waysconvert decimal numbers to binary

perform all arithmetic calculation in binaryand then convert the binary results back to decimal

perform the arithmetic operations with decimal numberswhen they are stored in coded form

n-bit binary codea group of n bits up to 2n distinct combinations of 1's & 0's2-bit binary code: 00, 01, 10, 11

BooleanBoolean AlgebraAlgebra

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15/43Binary Logic and GatesBinary Logic and Gates

Digital circuitshardware components that manipulate binary informationimplemented using transistors and interconnections in ICeach basic circuit is called logic gate

performs a specific logical operation

Boolean Algebramathematical notation to specify the operation of each gateused to analyze and design circuits


Binary Logictake on two discrete values &

with the operations of mathematical logic

three logical operations

1) AND Z = X • Y (Z is equal to X and Y)

2) OR Z = X + Y (Z is equal to X or Y)

3) NOT Z = X' (Z is equal to NOT X)

-- complement operation (0 => 1 & 1 => 0)

AND/OR is similar to multiplication/addition

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logical OR logical AND logical NOT0 + 0 = 0 0 • 0 = 0 0' = 10 + 1 = 1 0 • 1 = 0 1' = 01 + 0 = 1 0 • 1 = 01 + 1 = 1 1 • 1 = 1

18/43Binary Logic and GatesBinary Logic and GatesLogic Gates

electronic circuits that operate on one or more input signals to produce an output signalvoltage operated circuits: logic 0 & logic 1 intermediate region is crossed during state transition

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19/43Boolean AlgebraBoolean Algebra

deal with binary variables and logic operations

with three basic logic operations AND, OR, NOTexpress logical relationship between binary variables

Consider F = X + Y' Z represented in a truth tabletransformed from an algebraic expression into a circuit diagram composed of logic gates

(Fig 2.3)

F = X + Y' Z

20/43Boolean AlgebraBoolean Algebra

Basic Identities of Boolean Algebramost basic identities of Boolean algebradual - obtained by interchanging OR and AND,

and replacing 1's by 0's and 0's by 1's

Table 2-3. Basic Identities of Boolean Algebra

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21/43Standard FormsStandard Forms

Product of Sumsanother standard algebraic expressionobtained by forming a logical product of sum terms

(Ex) F = X(Y'+Z)(X+Y+Z')needs 2 OR gates and one AND gates

22/43Map SimplificationMap Simplification

Karnough map (K-map)to simplify Boolean functions of up to 4 variables(5 or 6 variables can be drawn, but cumbersome to use)a diagram of squares, each representing one mintermsimplified expressions are in sum-of-products or product-of-sums

two-level implementations

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Two-Variable Mapfour minterms for a Boolean function with 2 variables

(Ex)

m1+m2+m3 = X'Y+XY'+XY = X+Y

(by algebra) => X'Y+X(Y'+Y) = X'Y+Y = X+Y


Three-variable Map8 minterms for 3 variables

Ex2.3 F(X,Y,Z) = Σ m(2,3,4,5)

(from K-map) F = X'Y + XY'

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Ex2.4 F1(X,Y,Z) = Σ m(3,4,6,7) F2(X,Y,Z) = Σ m(0,2,4,5,6)

(Ex)

F2(X,Y,Z) = Σ m(1,3,4,5,6)= X'Z + XZ' + XY'

or = X'Z + XZ' + Y'Z

26/43LogicLogic GatesGates

Boolean functions are expressed in terms of AND, OR, NOTstraight forward to implement the function with these gatesOther useful logic gates

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27/43Integrated CircuitsIntegrated Circuits

Integrated Circuits (IC)small silicon semiconductor crystal, called a chipcontains electronic components for the digital gates

Levels of IntegrationSSI (small scale integration), < 10 gatesMSI, 10 ~ 100 gatesLSI, 100 ~ 1000sVLSI, > 1000s

28/43Integrated CircuitsIntegrated Circuits

Digital Logic FamiliesRTL, DTL - earliest logic familiesTTL - widespread, considered as standardECL - high speed operationMOS - high component densityCMOS - low power consumptionBiCMOS - CMOS + TTL, used selectively

Positive and Negative LogicNormal Convention:

Positive Logic/Active HighLow Voltage = 0; High Voltage = 1

Alternative Convention sometimes used: Negative Logic/Active Low

Low Voltage = 1; High Voltage = 0

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Combinational Logic Design

30/43Combinational CircuitsCombinational Circuits

logic circuits for digital systems: combinational vs sequential

Combinational Circuit (Chap 3)outputs are determined by the present applied inputsperforms an operation, which can be specified logically

by a set of Boolean expressions

Sequential Circuit (Chap 4)logic gates + storage elements (called flip-flops)outputs are a function of the inputs &

bit values in the storage elements (state of storage elements is a function of previous inputs)

output depends on the present values of inputs & past inputs

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31/43Combinational CircuitsCombinational Circuits

Combinational Circuit (Fig 3.1)consist of input/output variables, logic gates, & interconnectionsn inputs: 2n possible combinationsm outputs: circuit can be described by m Boolean expressionslogic gate transforms binary information from input to outputs

CombinationalCircuit

32/43ExamplesExamples

3 inputs, X, Y, Z; 2 outputs C, S (00 ~ 11)

C is 1 if XY, XZ, or YZ = 11;C is 0 otherwise

Ex) binary adder

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33/43ExamplesExamples

Ex3.3) BCD to Seven-Segment DecoderLED (light emitting diodes), or LCD (liquid crystal display)accept a decimal digit in BCD & generate the appropriate outputs

outputs (a, b, c, d, e, f, g)7 four-variable maps

34/43Decoders

Decodersa binary code of n bits represents 2n distinct elementsconvert binary info of n coded inputs to 2n (max) unique outputs

n-to-m-line decoders (m < 2n)

generate 2n (or fewer) minterms of n input variables3-to-8-line decoder (3 inputs & 8 outputs)

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35/43Decoders

36/43Multiplexersselects binary information from one of many input lines,

and directs it to a single output lineNormally, 2n input lines and n selection variables4-to-1-line multiplexer

called data selector or MUX2n-to-1-line multiplexer is constructed from n-to-2n decoder

by adding 2n input lines

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37/43Binary MultipliersBinary Multipliers

a 2-Bit by 2-Bit Binary Multiplier

Sequential Sequential CircuiCircuit t DesignDesign

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39/43Sequential Circuit DefinitionsSequential Circuit Definitions

sequential circuitcombinational circuit + storage elementsstorage elements

store binary information state of the sequential circuit at given stateoutputs are a function of the inputs & present state of the storage elementsnext state of storage elements is also a function of the inputs & the present state

40/43Sequential Circuit DefinitionsSequential Circuit Definitions

two typessynchronous sequential circuit

behavior is defined from the knowledge of its signals at discrete instants of time

asynchronous sequential circuitbehavior depends on the inputs at any instance of time & the order in continuous time in which the inputs change,

clock generatorsynchronous sequential circuit has a timing deviceproduce a periodic train of clock pulsesstorage elements are affected only upon the arrival of each pulseclock pulses are applied with other signalsthe outputs can change their value only in the presence of clock pulsesclocked sequential circuits

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41/43Sequential Circuit DefinitionsSequential Circuit Definitionsflip-flop

storage elements employed in clocked sequential circuitsa binary storage device capable of storing one bit of infoNormally, a sequential circuit uses many flip-flops the transition from one state to the other occurs only at predetermined time intervals dictated by the clock pulsestwo outputs: normal & complemented values

42/43LatchesLatches

A storage element can maintain a binary state indefinitely until directed by an input signal to switch states

Latch most basic types of flip-flops simple & most often used within flip-flopsused with more complex clocking methods

to implement sequential circuits

SR Latch a circuit with 2 cross-coupled NOR (or NAND) gates2 inputs: S (set) & R (reset)

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43/43FlipFlip--FlopsFlopsCharacteristic Tables

logical properties of a Flip-Flop in tabular formdefine the next state as a function of the inputs and present state

T (toggle) flip-flopwhen inputs J & K are tied togetherwhen T=0 (J=K=0), no changewhen T=1 (J=K=1), toggle the state of F-F

introduction digital systems · 2012-12-23 · 1 introduction to digital systems by dr. ahmet...

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