lecture 6: signal processing iii
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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) - PowerPoint PPT PresentationTRANSCRIPT
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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 SystemsHexidecimal Binary Decimal
0 0000 01 0001 12 0010 2
3 0011 34 0100 45 0101 56 0110 67 0111 78 1000 8
9 1001 9A 1010 10B 1011 11C 1100 12D 1101 13E 1110 14F 1111 15
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3-bits worth of Pigeonholes
Decimal number (k) Binary number
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
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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