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Lecture 6: Signal Processing III

Lecture 6: Signal Processing III. EEN 112: Introduction to Electrical and Computer Engineering. Professor Eric Rozier, 2/ 25/ 13. PIGEONS AND HOLES. Pigeonholes. The Pigeonhole Principle. First formalized by Johann Dirichlet in 1834 Schubfachprinzip (drawer principle)

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Lecture 6: Signal Processing III

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  1. Lecture 6: Signal Processing III EEN 112: Introduction to Electrical and Computer Engineering Professor Eric Rozier, 2/25/13

  2. PIGEONS AND HOLES

  3. Pigeonholes

  4. The Pigeonhole Principle • First formalized by Johann Dirichlet in 1834 • Schubfachprinzip (drawer principle) • Given n items, which must be put into m pigeonholes, with n > m, at least one pigeon hole must contain more than one item.

  5. The Pigeonhole Principle • Seems simple, right? But has some non-obvious consequences. • A typical person has aroung 150,000 hairs. • A reasonable assumption is that no one has more than 1,000,000 hairs. • All people have between 0 and 1,000,000 hairs. • There are 5,564,635 people in Miami • Consequences?

  6. The Pigeonhole Principle • The Birthday Paradox • How likely is it that two people in our class share the same birthday? • How would we know?

  7. The Pigeonhole Principle • How many “holes” do we have that can be filled? • Each person is equally likely to inhabit any one hole.

  8. Birthday Probabilities

  9. Birthday Probability • Imagine everyone has a deck of cards with 365 possible values. We each draw independently. • Let’s think about the likelyhood…

  10. Pigeons and Holes • We have “pigeons” in signal processing, and “holes” we want to put them into.

  11. Pigeons and Holes • In a N-bit system, how many holes do we have?

  12. Pigeons and Holes • Think of the bits as labels we put on the holes, and k as the decimal number equivalent. Our classification rule gives us a way to know what hole to put each pigeon into… and we have a LOT of pigeons…

  13. Labeling our Pigeonholes • We can label our pigeon holes with decimal integers • This is what k is in our equation • But why use decimals? What are decimals?

  14. Numeral Systems • In mathematics, we talk about the base of a numeral system. Decimals are a base-10 numeral system. • Why?

  15. Numeral Systems • Decimal uses 10 numerals • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Once we exhaust the numerals, we add a more significant digit • 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 • 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

  16. Numeral Systems • What base is binary? Why?

  17. Numeral Systems • Binary enumeration • 0, 1 • 10, 11 • 100, 101 • 110, 111

  18. There are 10 types of people in this world.Those who can count in binary and those who can’t!

  19. Numeral Systems • We can pick any base we want, even large than base-10! • Hexadecimal, base-16 • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • (Actually a very useful system in ECE…)

  20. Numeral Systems

  21. 3-bits worth of Pigeonholes

  22. Classification Rule • Let’s say we have one pigeon for every real number between 0 and 1. • How many pigeons? • Actually we have more than simply an infinite number of pigeons… • We have uncountably infinite pigeons

  23. Thinking about infinity • Let’s say we had a number of pigeonholes equal to the cardinality of the set of natural numbers (0, 1, 2, …). How many do we have? • Let’s say we have a number of pigeons equal to the cardinality of the set of integers (…, -2, -1, 0, 1, 2, …) • Do we have a hole for each pigeon?

  24. Thinking about infinity • Let’s say we had a number of pigeonholes equal to the cardinality of the set of natural numbers (0, 1, 2, …). How many do we have? • Let’s say we have a number of pigeons equal to the cardinality of the set of real numbers (…, -1, …, -0.333333, …, 0, …, 1, …, 2.9756, …) • Do we have a hole for each pigeon?

  25. Ordinal Numbers

  26. Thinking about Infinity • Countably infinite • Uncountably infinite - c

  27. Quantization • Classification and Reconstruction

  28. Types of Functions • Functions can be classified by how the elements of the domain and codomain relate • F: X -> Y

  29. Types of functions • Injective (one-to-one) • Preserves distinctiveness

  30. Types of functions • Surjective (onto) • Every element

  31. Types of functions • Bijection (both) • Injective and surjective

  32. Quantization • Quantization is surjective

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