CHAPTER 1: Number CHAPTER 1: Number Systems and CodesSystems and CodesIntroductionIntroductionDigital circuits, processing discrete information, Digital circuits, processing discrete information, are found in an astonishingly wide rang of electronic are found in an astonishingly wide rang of electronic systems. They have provided system designers with systems. They have provided system designers with more functions contained in less space, thereby more functions contained in less space, thereby improving system reliabilityimproving system reliability and and increasing speedincreasing speed. . Digital systems operate on numbers that represent some Digital systems operate on numbers that represent some real logical or arithmetic function. It is the designerreal logical or arithmetic function. It is the designer s s task to establish the relationship between the numbers task to establish the relationship between the numbers and the task being performed. This chapter develops and the task being performed. This chapter develops the use of numbers and how they represent real world the use of numbers and how they represent real world conditions in digital system.conditions in digital system.P1P11.1 Digital and Analog: Basic 1.1 Digital and Analog: Basic ConceptsConceptsP1P1 ContinuousContinuous: Without breaks, smooth, no interruptions; a range of values forming a line or curve without gaps or discontinuities DiscreteDiscrete: Broken down into pieces; opposite of continuous; a single part or point that can unambiguously be defined AnalogAnalog: Refer to something that is analogous, has a likeness, or is similar to something else. Analog electronic signals are continuous, whereas voltage(电压), current(电流), or some other electrical parameter is used to represent a physical parameter. For instance, voltage changes may represent changes in temperature. DigitalDigital: Refer to digits or numbers in any radix; digital electronics usually involves binary or base-2 number systems. Digital signals are discrete, not continuousP1P11.1 Digital and Analog: Basic 1.1 Digital and Analog: Basic ConceptsConceptsP2P21.1 Digital and Analog: Basic 1.1 Digital and Analog: Basic ConceptsConceptsVc(0)=0V, Initial voltage at t0 is 0VVc=Va(1-e-t/RC)P2P21.1 Digital and Analog: Basic 1.1 Digital and Analog: Basic ConceptsConcepts(a) Analog representation(b) Discrete representationDiscrete voltage values for time intervalP3P31.1 Digital and Analog: Basic 1.1 Digital and Analog: Basic ConceptsConceptsAnalog volt meterDigital volt meterP9P91.2 Digital Systems Overview1.2 Digital Systems OverviewSSI (small-scale integration) contains 0-9 gatesMSI (medium-scale integration) contains 10-99gatesLSI (large-scale integration) contains 100 or more gatesVLSI (very large-scale integration ) contains more than 1000 gatesP9P91.3 Introduction to Number System1.3 Introduction to Number System Digital systems represented information in a discrete form, and Number systems provide the means of quantifying information for processing by digital system.Positional Number Systems Number system conversion Binary Codes ArithmeticNumber Systems Number Systems - - DecimalDecimal “base” 10 (radix is 10) 10 digits: 0.9536.15910=(5102)+(3101)+(6100)+(110-1)+(510-2)+(910-3)WeightWeightRadix or BaseRadix or BaseNumber Systems Number Systems Decimal (cont.)Decimal (cont.) In general, a decimal number with n digits to the left of the decimal point, and m digits to the right of the decimal point is written as:An-1An-2 A1A0. A-1A-2 A-m+1A-mwhere Aiis a coefficient between 0.9, and i denotes the weight (=10i) of Ai.Number Systems Number Systems GeneralGeneral “base” r (radix r) r digits N = An-1 rn-1 + An-2 rn-2+ + A1 r +A0+ A-1 r-1+ A-2 r-2+ + A-m r-mMost Significant Bit (MSB)Least Significant Bit (LSB)Let r=radix or base of a number system; Let c=character from the character set of the radix; Let N=number to be represented in radix; Let n=the number of digits in the integer portion of N Let m=the number of digits in the fractional portion of N( (N)N)r r=(c=(cn n- -1 1r rn n- -1 1+c+cn n- -2 2r rn n- -2 2+ +c+c1 1r r1 1+c+c0 0r r0 0+c+c- -1 1r r- -1 1+c+c- -2 2r r- -2 2+ + +c c- -mmr r- -mm) )( (N)rN)r= = c ci ir ri iPositional Number SystemsPositional Number SystemsP11P11Number Systems Number Systems General (cont.)General (cont.) e.g. let r = 6(312.4)6= 3 62 + 1 61 + 1 60 + 4 6-1 = (116.66)10 Conversion from n-radix (any system with radix n) to decimal follows similar process as aboveNumber Systems (cont.)Number Systems (cont.) Most common number systems for computers:Binary (r = 2) Octal (r = 8) Hexadecimal (r = 16)Binary NumbersBinary Numbers- -Base 2Base 2 Computers represent all data as “strings