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Error Control Techniques: Hamming Code and Floating-Point Representation

This lecture covers Hamming error correction code and floating-point representation for real numbers, providing examples and explanations. It also discusses the concept of Hamming distance and its use in error detection and correction.

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Error Control Techniques: Hamming Code and Floating-Point Representation

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  1. CS 3501 - Chapter 2 Dr. Clincy Professor of CS Lecture 6

  2. Error Control Techniques Approaches Detection • Simple Parity (Double Parity) – already covered • Cyclic Redundancy Checksum– already covered Error correction • Hamming Approach Lecture 6

  3. Understanding Hamming Distance Concept Lecture 6

  4. D-001 C-011 A-000 B-010 H-101 G-111 F-110 E-100 Error Correction – Hamming Code Concept • Computers make errors occasionally due to voltage spikes and etc. • Recall Encoding Concept – codes representing characters • Hamming Distance of 1: change in 1 bit creates a new code What happens with 1 bit in error ? Lecture 6

  5. 001 B-011 A-000 010 D-101 111 C-110 100 Hamming Distance of 2 What happens with 1 bit in error ? What happens with 2 bits in error ? Lecture 6

  6. 001 011 A-000 010 101 B-111 110 100 Hamming Distance of 3 What happens with 1 bit in error ? What happens with 2 bits in error ? What happens with 3 bits in error ? Lecture 6

  7. Hamming Error Correcting Approach • 1st: Determine the number of parity bits to add to the code word for checking • 2nd: determine bit positions of each added parity bits • 3rd: Determine what each parity bit checks Lecture 6

  8. Hamming Code Example(Understand the “how” vs “why”) • Example: Given a 4-bit code and even parity request – recall parity • Determine number of parity bits to add: 20=1, 21=2, 22=4: also determine bit positions of parity bits – 1, 2 and 4 (with labels C1, C2 and C4) • Let the 4-bit code have labels I3, I5, I6 and I7 due to the parity bit positions • Therefore the seven bits would be transmitted in the following order: C1 C2 I3 C4 I5 I6 I7 • Determine what each parity bit checks: add the parity positions to determine this (must be less than or equal to 7 in this case): C1 case: 1+2=3, 1+4=5, 1+2+4=7;C2 case: 2+1=3, 2+1+4=7, 2+4=6;C4 case: 4+1=5, 4+2=6, 4+2+1=7 • For example, transmitting the 4-bit code of 0101 (I3I5I6I7) would be 0100101 (red bits are parity bits) • If I3 was corrupted during transmission, C1 and C2 would detect it (1+2=3) • If I5 was corrupted during transmission, C1 and C4 would detect it (1+4=5) • If I6 was corrupted during transmission, C2 and C4 would detect it (2+4=6) • If I7 was corrupted during transmission, C1 , C2 and C4 would detect it (1+2+4=7) Lecture 6

  9. Floating Point Numbers Lecture 6

  10. Floating-Point Representation • The signed magnitude, one’s complement, and two’s complement representation that we have just presented deal with signed integer values only. • Without modification, these formats are not useful in scientific or business applications that deal with real number values. • Floating-point representation solves this problem. • For example, how would 123.44 be stored compared to 12344 ? • How can we use the existing infrastructure of the computer to store real numbers ? Lecture 6

  11. Scientific-Notation Vs Floating-Point Representation • Computers use a form of scientific notation for floating-point representation • Numbers written in scientific notation have three components: • Computer representation of a floating-point number consists of three fixed-size fields: Size determines range Size determines precision Lecture 6

  12. Floating-Point Representation • Floating-point numbers allow an arbitrary number of decimal places to the right of the decimal point. • For example: 0.5  0.25 = 0.125 • They are often expressed in scientific notation. • For example: 0.125 = 1.25  10-1 5,000,000 = 5.0  106 Lecture 6

  13. The Concept Lecture 6

  14. The Simple Model • We introduce a hypothetical “Simple Model” to explain the concepts • In this model: • A floating-point number is 14 bits in length • The exponent field is 5 bits • The significand field is 8 bits • The significand is always preceded by an implied binary point. • Thus, the significand always contains a fractional binary value. • The exponent indicates the power of 2 by which the significand is multiplied. Lecture 6

  15. Simple Model Illustrated • Example: • Express 3210 in the simplified 14-bit floating-point model. • We know that 32 is 25. So in (binary) scientific notation 32 = 1.0 x 25 = 0.1 x 26. • In a moment, we’ll explain why we prefer the second notation versus the first. • Using this information, we put 110 (= 610) in the exponent field and 1 in the significand as shown. Lecture 6

  16. Problems with the Simple Model • The illustrations shown at the right are all equivalent representations for 32 using our simplified model. • Not only do these synonymous representations waste space, but they can also cause confusion. • Another problem with our system is that we have made no allowances for negative exponents. We have no way to express 0.5 (=2 -1)! (Notice that there is no sign in the exponent field.) All of these problems can be fixed with no changes to our basic model. Lecture 6

  17. To resolve the problem of synonymous forms, we establish a rule that the first digit of the significand must be 1, with no ones to the left of the radix point. This process, called normalization, results in a unique pattern for each floating-point number. In our simple model, all significands must have the form 0.1xxxxxxxx For example, 4.5 = 100.1 x 20 = 1.001 x 22 = 0.1001 x 23. The last expression is correctly normalized. Normalization In our simple instructional model, we use no implied bits. Lecture 6

  18. To provide for negative exponents, we will use a biased exponent. A bias is a number that is approximately midway in the range of values expressible by the exponent. We subtract the bias from the value in the exponent to determine its true value. In our case, we have a 5-bit exponent. We will use 16 for our bias. This is called excess-16 representation. In our model, exponent values greater than 16 are positive. an exponent of 4, would be 16+4=20 an exponent of 10, would be 16+10=26 In our model, exponent values less than 16 are negative, representing fractional numbers. an exponent of -4, would be 16-4=12 an exponent of -10, would be 16-10=6 Negative Exponents Lecture 6

  19. Examples • Example 1: • Express 3210 in the revised 14-bit floating-point model. • We know that 32 = 1.0 x 25 = 0.1 x 26. • To use our excess 16 biased exponent, we add 16 to 6, giving 2210 (=101102). • So we have: • Example 2: • Express 0.062510 in the revised 14-bit floating-point model. • We know that 0.0625 is 2-4. So in (binary) scientific notation 0.0625 = 1.0 x 2-4 = 0.1 x 2 -3. • To use our excess 16 biased exponent, we add 16 to -3, giving 1310 (=011012). Lecture 6

  20. Another Example • Example: • Express -26.62510 in the revised 14-bit floating-point model. • We find 26.62510 = 11010.1012. Normalizing, we have: 26.62510 = 0.11010101 x 2 5. • To use our excess 16 biased exponent, we add 16 to 5, giving 2110 (=101012). We also need a 1 in the sign bit. Lecture 6

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