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

2. 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

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

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

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

6. 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) For example, Binary Alphabet: 0, 1. 10101

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

8. Relationship between Analogue and Digital systems 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 5 Volt 0 Volt 0.8 0.4 2.4 2.8 Input Range for 1 Input Range for 0 Output Range for 0 Output Range for 1

9. 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

10. Binary Inputs and Outputs Coding:  A single binary input can only have two values: True or False (Yes or No) (1 or 0)

11. 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 2 n combinations

12. Combinations Example 1:  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?”

13. 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: 403 10 = 4x10 2 + 0x10 1 + 3x10 0 = 400 + 0 + 3

14. Binary Inputs and Outputs  Both Decimal and Binary numbers use a positional weighting system, eg: 1010 2 = 1x2 3 +0x2 2 +1x2 1 +0x2 0 = 1x8 + 0x4 + 1x2 + 0x1 = 10 10 decimal100 (10 2 )10 (10 1 )1 (10 0 ) 403400 + 0 + 3 binary8 (2 3 )4 (2 2 )2 (2 1 )1 (2 0 ) 10018 + 0 + 0 + 1

15. Binary to decimal Multiply each 1 bit by the appropriate power of 2 and add them together. ??1286432168421 10000011 101001100 10000011 2 = ………………. 10 ? 101001100 2 = …………………… 10 ?

16. Binary Inputs and Outputs Number Representation - Binary to decimal A decimal number can be converted to binary by repeated division by 2 number/2remainder 155771Least Significant Bit 77381 190 91 941 420 210 101Most Significant bit 155 10 = 10011011 2

17. 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… 1286432168421 1 1001

18. 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

19. Binary Inputs and Outputs  6 of the possible 16 values unused  example 453 10 = 0100 0101 0011 BCD  Note that BCD code is longer than a direct representation in natural binary code:  453 = 111000101

20. 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. octal512 (8 3 )64 (8 2 )8 (8 1 )1 (8 0 ) 1207512 + 128 + 7 hexadecimal256 (16 2 )16 (16 1 )1 (16 0 ) 1AF256 + 160 + 15 Notice that hexadecimal requires 15 symbols (each number system needs 0 – base-1 symbols) and therefore A – F are used after 9.

21. Octal as shorthand for Binary Each octal digit corresponds to 3 binary bits binaryoctal 0000 0011 0102 0113 1004 1015 1106 1117 To convert a binary string: 10011101010011 Split into groups of 3: 010 011 101 010 011 2 3 5 2 3 Thus 10011101010011 2 = 23523 8

22. Similarly with Hexadecimal Each hex digit corresponds to 4 binary bits binaryhex 00000 00011 00102 00113 01004 01015 01106 01117 To convert a binary string: 10011101010011 Split into groups of 4: 0010 0111 0101 0011 Thus 10011101010011 2 = …………… 16 ? binaryhex 10008 10019 1010A 1011B 1100C 1101D 1110E 1111F

23. 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?

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

25. 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. DecGray 0000 1001 2011 3010 4110 5111 6101 7100

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

27. 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.