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 HexidecimalBinaryDecimal 000000 100011 200102 300113 401004 501015 601106 701117 810008 910019 A101010 B101111 C110012 D110113 E111014 F111115

21. 3-bits worth of Pigeonholes Decimal number (k)Binary number 00 11 210 311 4100 5101 6110 7111

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