1. Lecture 1 Binary Representation Topics Terminology Base 10, Hex, binary Fractions Base-r to decimal Unsigned Integers Signed magnitude Two’s complement August 24, 2015 CSCE 211H Digital Design

2. – 2 – CSCE 211H Fall 2015 Overview Readings Chapter 1 Overview of Course Analog vs Digital Conversion Base-r to decimal Conversion decimal to Base-r Conversion of Fractions base-r  decimal Unsigned Arithmetic Signed Magnitude Two’s Complement Excess-1023

3. Click to edit Master subtitle style Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 1 Introduction

4. – 4 – CSCE 211H Fall 2015 Course Outcomes 1.Represent numbers and perform arithmetic in bases 2, 8, and 16 2.Encode symbols and numbers in binary codes 3.Add and subtract using 2’s complement code 4.Evaluate and simplify logical functions using Boolean algebra 5.Represent logical functions in Canonical form and with Gates 6.Analyze and design combinatorial 7.Simplify combinatorial circuits using Karnaugh 8.Implement functions with NAND-NAND and NOR-NOR logic 9.Analyze and design modular combinatorial logic circuits containing decoders, multiplexers, demultiplexers, 7-segments display decoders and adders 10.Use the concepts of state and state transition for analysis and design of sequential circuits 11.Use the functionality of flip-flops for analysis and design of sequential circuits 12.Software-hardware co-design (Arduino)

5. – 5 – CSCE 211H Fall 2015 Comp. Arch. Quantitative Approach - H&P Chapter 1: Figure1.1 Performance Growth since 1978

6. – 6 – CSCE 211H Fall 2015 Analog vs digital: Why Binary

7. – 7 – CSCE 211H Fall 2015 Other bases Binary = Base 2 Binary = Base 2 Hexadecimal = Base 16 Hexadecimal = Base 16 Octal = Base 8 Octal = Base 8 Notations Notations Subscript In C Conversions between bases Conversions between bases

8. – 8 – CSCE 211H Fall 2015 Note that the number one less than 2 n consists of n 1’s (for example, 2 4 – 1 = 1111 = 15 and 2 5 – 1 = 11111 = 31).

9. – 9 – CSCE 211H Fall 2015 Base-r to Decimal Conversions Converting base-r to decimal by definition d n d n-1 d n-2 …d 2 d 1 d 0 (base r) = d n r n + d n-1 r n-1 … d 2 r 2 +d 1 r 1 + d 0 r 0 Example 4F0C 16 = 4*16 3 + F*16 2 + 0*16 1 + C*16 0 4F0C 16 = 4*16 3 + F*16 2 + 0*16 1 + C*16 0 = 4*4096 + 15*256 + 0 + 12*1 = 16384 + 3840 + 12 =20236

10. – 10 – CSCE 211H Fall 2015 Decimal to Base-r Conversion Repeated division algorithm Justification: d n d n-1 d n-2 …d 2 d 1 d 0 = d n r n + d n-1 r n-1 … d 2 r 2 +d 1 r 1 + d 0 r 0 Dividing each side by r yields (d n d n-1 d n-2 …d 2 d 1 d 0 ) / r = d n r n-1 + d n-1 r n-2 … d 2 r 1 +d 1 r 0 + d 0 r -1 So d 0 is the remainder of the first division ((q 1 ) / r = d n r n-2 + d n-1 r n-3 … d 3 r 1 +d 2 r 0 + d 1 r -1 So d 1 is the remainder of the next division and d 2 is the remainder of the next division …

11. – 11 – CSCE 211H Fall 2015 Decimal to Base-r Conversion Example Repeated division algorithm Example Convert 4343 to hex 4343/16 = 271 remainder = 7 271/16 = 16 remainder = 15 271/16 = 16 remainder = 15 16/16 = 1 remainder = 0 16/16 = 1 remainder = 0 1/16 = 0 remainder = 1 1/16 = 0 remainder = 1 So 4343 10 = 10F7 16 To check the answer convert back to decimal 10F7 = 1*16 3 + 15*16 + 7*1 = 4096 + 240 + 7 = 4343

12. – 12 – CSCE 211H Fall 2015 Hex to Binary One hex digit = four binary digits Example 3FAC = 0011 1111 1010 1100 (spaces just for readability) 3FAC = 0011 1111 1010 1100 (spaces just for readability) Binary to hex four binary digits  one hex digit (group from right!!!)Example 101101001111010 = 0010 1101 0011 1010 101101001111010 = 0010 1101 0011 1010 = 2 D 3A

13. – 13 – CSCE 211H Fall 2015 Hex to Decimal Fractions.d -1 d -2 d -3 …d –(n-2) d –(n-1) d -n = d -1 r -1 + d -2 r -2 …d -(n-1) r -(n-1) +d 1 r -n Example.1EF 16 = 1*16 -1 + E*16 -2 + F*16 -3.1EF 16 = 1*16 -1 + E*16 -2 + F*16 -3 = 1*.0625 + 14*.003906025 + 15*2.4414e-4 =.117201… (probably close but not right) =.117201… (probably close but not right)

14. – 14 – CSCE 211H Fall 2015 Example: Hex Fractions to decimal Convert.3FA to decimal.3FA 16 = 3*16 -1 + F*16 -2 + A*16 -3.3FA 16 = 3*16 -1 + F*16 -2 + A*16 -3 = 3*.0625 + 15*.00390625 +10* (1/4096) =.191162109

15. – 15 – CSCE 211H Fall 2015 Decimal Fractions to hex.d -1 d -2 d -3 …d –(n-2) d –(n-1) d -n = d -1 r -1 + d -2 r -2 …d -(n-1) r -(n-1) +d 1 r -n Multiplication by r yields r *(.d -1 r -1 + d -2 r -2 …d -(n-1) r -(n-1) +d 1 r -n ) = d -1 r 0 + d -2 r -1 …d -(n-1) r -(n-2) +d 1 r -(n-1) Whole number part = d -1 r 0 Multiplying again by r yields d -2 r 0 as the whole number part … till fraction = 0

16. – 16 – CSCE 211H Fall 2015 Example Decimal Fraction to hex By repeated multiplication

17. – 17 – CSCE 211H Fall 2015 Unsigned integers What is the binary representation of the biggest integer representable using n-bits(n binary digits)? What is its value in decimal? Special cases 8 bits 16 bits 16 bits 32 bits 32 bits

18. – 18 – CSCE 211H Fall 2015 Arithmetic with Binary Numbers 10010110 100101101101 +00110111 - 00110111 x 101 +00110111 - 00110111 x 101 Problems with 8 bit operations 10010110 +10010110 +10010110

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20. – 20 – CSCE 211H Fall 2015 Signed integers How do we represent? Signed-magnitude Excess representations w bits  0 <= unsigned_value < 2 w In excess-B we subtract the bias (B) to get the value. In excess-B we subtract the bias (B) to get the value. example example

21. – 21 – CSCE 211H Fall 2015 Complement Representations of Signed integers One’s complement Two’s complement

22. – 22 – CSCE 211H Fall 2015 Two’s Complement Operation One’s complement + 1 or Find rightmost 1, complement all bits to the left of it. Examples 01001110000000010000000000000010 01001110000000010000000000000010

23. – 23 – CSCE 211H Fall 2015 Two’s Complement Representation Consider a two’s complement binary number d n d n-1 d n-2 …d 2 d 1 d 0 If d n, the sign bit = 0 the number is positive and its magnitude is given by the other bits. If d n, the sign bit = 1 the number is negative and take its two’s complement to get the magnitude. Weighted Sum Interpretation 0 1 1 2 … n-1 2 n-2 n -2 n-1

24. – 24 – CSCE 211H Fall 2015 Two’s Complement Representation Consider a two’s complement binary number d n d n-1 d n-2 …d 2 d 1 d 0 If d n, the sign bit = 0 the number is positive and its magnitude is given by the other bits. If d n, the sign bit = 1 the number is negative and take its two’s complement to get the magnitude. Weighted Sum 0 1 Example 10010011 = 1 2 … n-1 2 n-2 n -2 n-1

25. – 25 – CSCE 211H Fall 2015 Two’s Complement Representation What is the 2’s complement representation in 16 bits of –5? +7?-1?0-2

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27. – 27 – CSCE 211H Fall 2015 Arithmetic with Signed Integers Signed Magnitude Addition if the signs are the same add the magnitude if the signs are different subtract the smaller from the larger and use the sign of the larger if the signs are different subtract the smaller from the larger and use the sign of the largerSubtraction? Two’s complement Just add signs take care of themselves

28. – 28 – CSCE 211H Fall 2015 Overflow in Two’s Complement

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30. – 30 – CSCE 211H Fall 2015 Binary Code Decimal Floating point – IEEE 754

31. – 31 – CSCE 211H Fall 2015 Representations of Characters

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33. – 33 – CSCE 211H Fall 2015 ASCII Message on the Wall Princeton CS Building West Wall Bricks This brick pattern is located on the west wall of the Computer Science building, and dates back to 1989, when the building was constructed. The pattern is read top to bottom and consists of five 7-bit ASCII values. The vertical lines to the left and right are "framing" bits. (Table 1.8 page 18 previous slide) x 1 0 1 0 0 0 0 x x 0 1 1 1 1 0 1 x x 1 0 0 1 1 1 0 x x 1 0 1 0 0 0 0 x x 0 1 1 1 1 1 1 x This pattern asks _________? http://www.cs.princeton.edu/general/bricks

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35. – 35 – CSCE 211H Fall 2015 Basic Gates ANDORNOT

36. – 36 – CSCE 211H Fall 2015 Basic Gates NANDNORXOR

37. – 37 – CSCE 211H Fall 2015 Half Adder Circuit; Full Adder

38. – 38 – CSCE 211H Fall 2015 Homework Due at the start of class Wednesday Page 25 1.1.*a, b,f 2.2h 3.3b 4.4b 5.5a, 5c 6.7a,b 7.Convert 137.125 to hex and then to binary 8.Convert 137.1 to hex, rounding to 6 hex digits after the “decimal” point.