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Lecture 6: Signal Processing III EEN 112: Introduction to Electrical and Computer Engineering Professor Eric Rozier, 2/25/13
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PIGEONS AND HOLES
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Pigeonholes
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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.
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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?
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The Pigeonhole Principle The Birthday Paradox How likely is it that two people in our class share the same birthday? How would we know?
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The Pigeonhole Principle How many “holes” do we have that can be filled? Each person is equally likely to inhabit any one hole.
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Birthday Probabilities
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Birthday Probability Imagine everyone has a deck of cards with 365 possible values. We each draw independently. Let’s think about the likelyhood…
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Pigeons and Holes We have “pigeons” in signal processing, and “holes” we want to put them into.
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Pigeons and Holes In a N-bit system, how many holes do we have?
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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…
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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?
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Numeral Systems In mathematics, we talk about the base of a numeral system. Decimals are a base-10 numeral system. – Why?
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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
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Numeral Systems What base is binary? Why?
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Numeral Systems Binary enumeration – 0, 1 – 10, 11 – 100, 101 – 110, 111
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There are 10 types of people in this world. Those who can count in binary and those who can’t!
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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…)
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Numeral Systems HexidecimalBinaryDecimal 000000 100011 200102 300113 401004 501015 601106 701117 810008 910019 A101010 B101111 C110012 D110113 E111014 F111115
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3-bits worth of Pigeonholes Decimal number (k)Binary number 00 11 210 311 4100 5101 6110 7111
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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
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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?
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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?
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Ordinal Numbers
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Thinking about Infinity Countably infinite Uncountably infinite - c
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Quantization Classification and Reconstruction
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Types of Functions Functions can be classified by how the elements of the domain and codomain relate F: X -> Y
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Types of functions Injective (one-to-one) – Preserves distinctiveness
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Types of functions Surjective (onto) – Every element
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Types of functions Bijection (both) – Injective and surjective
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Quantization Quantization is surjective
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