ENEL 111 Digital Electronics Richard Nelson G.1.29 richardn@cs.waikato.ac.nz

Second Half of ENEL 111 Digital Electronics Number Systems and Logic Electronic Gates Combinational Logic Sequential Circuits ADC – DAC circuits Memory and Microprocessors Hardware Description Languages

Weekly Structure Lectures Monday, Tuesday, Wednesday Slides in ppt and pdf format on support website: http://wand.cs.waikato.ac.nz/~111/2005/ (follow link from course website) Friday Tutorials - Sample Questions on website.

The lecture today Digital vs Analog data Binary inputs and outputs Binary, octal, decimal and hexadecimal number systems Other uses of binary coding.

Analog/Analogue Systems V(t) can have any value between its minimum and maximum value V(t)

Digital Systems Digital Systems V(t) must take a value selected from a set of values called an alphabet Binary digital systems form the basis of almost all hardware systems currently V(t) 1 1 1 For example, Binary Alphabet: 0, 1.

Slide example Consider a child’s slide in a playground: a set of discrete steps continuous movement levels

Relationship between Analogue and Digital systems 5 Volt 0 Volt 0.8 0.4 2.4 2.8 Input Range for 1 for 0 Output Advantages of Digital Systems Analogue systems: slight error in input yields large error in output Digital systems more accurate and reliable Computers use digital circuits internally Interface circuits (for instance, sensors and actuators) are often analogue

Exercise Explain whether the following are analog or digital: A photograph or painting A scanned image Sound from a computer’s loud speaker Sound file stored on disc

Binary Inputs and Outputs Coding: A single binary input can only have two values: True or False (Yes or No) (1 or 0) A volume control on a stereo requires more than two positions (off and full volume) 2 inputs can represent 4 values 3 inputs can represent 8 values 4 inputs can represent 16 values 5 inputs can represent 32 values ... Example 1 There would be 210 = 1024 = 1K combinations Example 2 Need to represent 10 digits Using three bits only allows us to represent 8 and 4 bits allows us to represent 16 - have to use 4 inputs.

Binary More bits = more combinations 0 0 0 1 1 0 1 1 Each additional input doubles the number of combinations we can represent i.e. with n inputs it is possible to represent 2n combinations

Combinations Example 1: Example 2: How many combinations are possible with 10 binary inputs? Example 2: What is the minimum number of bits needed to represent the digits ‘0’ to ‘9’ as a binary code?”

Decimal systems Number Representation Difficult to represent Decimal numbers directly in a digital system Easier to convert them to binary There is a weighting system: eg 403 = 4 x 100 + 0 x 10 + 3 x 1 or in, powers of 10: 40310= 4x102 + 0x101 + 3x100 = 400 + 0 + 3

Binary Inputs and Outputs Both Decimal and Binary numbers use a positional weighting system, eg: 10102 = 1x23+0x22+1x21+0x20 = 1x8 + 0x4 + 1x2 + 0x1 = 1010 decimal 100 (102) 10 (101) 1 (100) 4 3 400 + 0 + 3 binary 8 (23) 4 (22) 2 (21) 1 (20) 1 8 + 0 + 0 + 1

Binary to decimal Multiply each 1 bit by the appropriate power of 2 and add them together. ? 128 64 32 16 8 4 2 1 100000112 = ……………….10 ? 1010011002 = ……………………10 ?

Binary Inputs and Outputs Number Representation - Binary to decimal A decimal number can be converted to binary by repeated division by 2 number /2 remainder 155 77 1 Least Significant Bit 38 19 9 4 2 Most Significant bit 15510 = 100110112

Decimal to Binary An alternative way is to use the “placement” method 128 goes into 155 once leaving 27 to be placed So 64 and 32 are too big (make them zero) 16 goes in once leaving 11 and so on… 128 64 32 16 8 4 2 1 1 1

Representations There are different ways of representing decimal numbers in a binary coding BCD or Binary Coded Decimal is one example. Each decimal digit is replaced by 4 binary digits

Binary Inputs and Outputs 6 of the possible 16 values unused example 45310 = 0100 0101 0011BCD Note that BCD code is longer than a direct representation in natural binary code: 453 = 111000101

Binary Inputs and Outputs Hexadecimal and Octal Writing binary numbers as strings of 1s and 0s can be very tedious Octal (base 8) and Hexadecimal (base 16) notations can be used to reduce a long string of binary digits. octal 512 (83) 64 (82) 8 (81) 1 (80) 1 2 7 512 + 128 + 7 hexadecimal 256 (162) 16 (161) 1 (160) A F 256 + 160 + 15 Octal group into lots of 3 binary digits starting from the LSB Hex group into lots of 4 binary digits starting from the LSB Notice that hexadecimal requires 15 symbols (each number system needs 0 – base-1 symbols) and therefore A – F are used after 9.

Octal as shorthand for Binary Each octal digit corresponds to 3 binary bits binary octal 000 001 1 010 2 011 3 100 4 101 5 110 6 111 7 To convert a binary string: 10011101010011 Split into groups of 3: 010 011 101 010 011 2 3 5 2 3 Thus 100111010100112 = 235238

Similarly with Hexadecimal Each hex digit corresponds to 4 binary bits To convert a binary string: 10011101010011 Split into groups of 4: 0010 0111 0101 0011 Thus 100111010100112 = ……………16 ? binary hex 0000 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 binary hex 1000 8 1001 9 1010 A 1011 B 1100 C 1101 D 1110 E 1111 F

Binary inputs and outputs Colour codes You often see hex used in graphic design programs for the red, blue and green components of a colour: FF0000 represents red, for example. How many bits are used to represent each colour? How many different colours can be represented?

Binary Inputs and Outputs Characters Three main coding schemes used: ASCII (widespread use), EBCDIC (not used often) and UNICODE (new) ASCII table (in hex) :

Gray Codes Other codes exist for specific purposes Gray codes provide a sequence where only one bit changes for each increment Allows increments without ambiguity due to bits changing at different times. E.g. changing from 3 to 4, normal binary has all three bits changing 011 -> 100. Depending on the order in which the bits change any intermediate value may be created. Dec Gray 000 1 001 2 011 3 010 4 110 5 111 6 101 7 100

Summary Support website Analogue and Digital Binary Number Systems Coding schemes considered were: Natural Binary BCD Octal representation Hexadecimal representation ASCII

Exercises You should practice conversions between binary, octal, decimal and hexadecimal. You should be able to code decimal to BCD (and BCD to decimal). You should be able to explain and give examples of digital and analogue data.