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Definition of Orthogonal Set
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Good morning class! Today, we're discussing orthogonal sets of functions. Can anyone tell me what they think might define an orthogonal set?
I think it has something to do with functions being perpendicular in some way.
Exactly! An orthogonal set is a collection of functions where every distinct pair of functions is orthogonal over a specific interval. This means that their inner product is zero. Can someone recall what we use as the inner product for functions?
Isn't it the integral of the product of the two functions over a specified interval?
That's right! For example, if we have two functions f1(t) and f2(t), we define their inner product over an interval [a, b] as the integral from a to b of f1(t) times f2(t) dt. If that equals zero, they are orthogonal. This property is fundamental in Fourier series because it allows us to uniquely determine the coefficients.
So, if we use sine and cosine functions, they wonβt affect each other when we find their coefficients?
Correct! They can exist together without interfering because they are orthogonal over their interval. Remember the analogy: think of them like vectors in space that don't overlap. Let's summarize: An orthogonal set helps us in isolating the contributions from different functions with respect to one another.
Examples of Orthogonal Sets
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Now that we understand what an orthogonal set is, let's look at some examples. Who can name an orthogonal set used in Fourier series?
The set of sine and cosine functions!
Exactly! The set {1, cos(Οβt), sin(Οβt)} forms an orthogonal set over one period of the signal. Can anyone explain why these functions are orthogonal?
Because when you integrate the product of a sine function with a cosine function over a complete period, it results in zero!
Exactly! And this is crucial for calculating the Fourier coefficients easily. What about the complex exponential basis?
The set {e^(jkΟβt)} is also orthogonal but it looks more elegant mathematically!
Thatβs correct! Each member's inner product over a period yields zero when k is not equal to m. Remember, these orthogonal sets provide a basis for the space of periodic functions, allowing us to craft Fourier series effectively.
Complete Sets of Functions
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Letβs move on to the concept of complete sets of functions. Can someone indicate what a complete set does?
Isn't it the one that can represent any well-behaved function?
Yes! For a set to be complete, any square-integrable function can be expressed as a linear combination of the functions within that set. For instance, using our prior orthogonal sets, any function that meets the criteria can be crafted from them. Why do you think completeness is vital?
It ensures that we can work with real-world signals effectively and have accurate representations!
Exactly! Completeness ensures the Fourier series can represent a wide variety of periodic signals encountered in engineering. This foundational concept is what allows us to utilize Fourier methods in practice.
So, without completeness, we wouldnβt be able to fully represent some signals?
Correct! Completeness guarantees that even complex or unconventional signals can still be managed through our basis functions. Let's recap: orthogonality defines the independence of functions, while completeness ensures we can form any necessary representation from them.
Introduction & Overview
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Quick Overview
Standard
Orthogonal sets consist of functions where each pair is orthogonal over a specified interval. Complete sets, or basis functions, can represent any square-integrable function through linear combinations, forming the foundation for Fourier series.
Detailed
Orthogonal Sets and Complete Sets of Functions
In this section, we delve into the significance of orthogonal sets and complete sets of functions in the context of Fourier series analysis. An orthogonal set is a collection of functions where every distinct pair of functions is orthogonal over a defined interval; mathematically, this is expressed when their inner product equals zero. Examples include the trigonometric basis set {1, cos(Οβt), sin(Οβt)} over an interval equal to the period, Tβ, and the complex exponential basis set {e^(jkΟβt)}.
Furthermore, a set of functions is considered complete if any square-integrable function can be represented as a linear combination of the functions within that set. Completeness is essential as it ensures that Fourier series can accurately represent a broad spectrum of periodic signals encountered in various engineering fields. The relationship established allows the Fourier series to be viewed as a type of generalized Fourier series, wherein any well-behaved function x(t) can be expressed through its basis functions. This foundational knowledge is crucial for effectively leveraging the Fourier series in practical applications.
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Definition of Orthogonal Set
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An orthogonal set is a collection or family of functions, {phi_k(t)}, where every distinct pair of functions within the set is orthogonal over a specific interval [a, b]. That is, the inner product of phi_i(t) and phi_j(t) is zero for all i not equal to j.
Detailed Explanation
An orthogonal set of functions means that for any two different functions within the set, when you calculate their inner product over a given interval, the result is zero. This indicates that the functions do not overlap or influence one another over that interval. Essentially, they are 'independent' in the function space.
Examples & Analogies
Think of two people dancing in a room. If they are dancing in separate corners without interfering with each other's space, they are like orthogonal functions. Their movements do not affect one another, similar to how the product of these functions results in zero.
Examples Relevant to Fourier Series
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Trigonometric Basis: The set {1, cos(omega_0 t), sin(omega_0 t), cos(2 * omega_0 t), sin(2 * omega_0 t), ...} is an orthogonal set over any interval whose length is equal to the fundamental period T_0 = 2pi / omega_0. Intuitively, these functions oscillate in such a way that their positive and negative products perfectly cancel out over a full period when multiplied by any other distinct member of the set.
Complex Exponential Basis: The set {e^(j * k * omega_0 * t)} for integer values of k (from negative infinity to positive infinity) forms an orthogonal set over any interval of length T_0. This set is particularly elegant mathematically. For example, if you take the inner product of e^(j * i * omega_0 * t) and e^(j * j * omega_0 * t) for i not equal to j, the integral over T_0 will yield zero.
Detailed Explanation
The trigonometric basis consists of functions like sine and cosine that oscillate and cancel each other out, leading to orthogonality. The complex exponential basis uses the math of complex numbers and shows a similar independence, making it useful for Fourier analysis. In both cases, the functions maintain orthogonality over specific intervals of periodicity.
Examples & Analogies
Imagine a pendulum swinging back and forth (sine) and a wheel rotating (cosine). They move in different ways and don't affect one anotherβs trajectoriesβlike orthogonal functions. When you combine their motions, they add up neatly, without interference, similar to the cancellation seen in these orthogonal sets.
Complete Set (Basis Functions)
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The concept of a "complete set" is crucial. An orthogonal set is considered complete if any arbitrary "well-behaved" function (specifically, any square-integrable function, meaning its energy is finite, over the interval) can be accurately represented as a linear combination (a weighted sum) of functions from that set. This completeness guarantees that the Fourier series can indeed represent a wide and practical range of periodic signals encountered in engineering. The functions in a complete orthogonal set are often called "basis functions" because they form a fundamental set from which other functions in the space can be constructed, similar to how 'i', 'j', 'k' form a basis for 3D vectors.
Detailed Explanation
A complete set of functions means you can express any reasonable function in that space by summing the basis functions multiplied by appropriate coefficients. This ensures that the Fourier series can represent a broad variety of periodic functions, hence making the analysis and application of these tools in engineering and other fields very powerful.
Examples & Analogies
Think of a set of musical notes on a piano as a complete basis. With just a few notes, you can play a wide range of songs (any arbitrary well-behaved melody). Each note corresponds to a basis function, and by adjusting how long you hold each note (the coefficients), you can create any music you want.
Generalized Fourier Series
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Briefly, the Fourier series is a specific example of a broader concept known as a generalized Fourier series. In general, if you have a complete orthogonal set of functions {phi_k(t)}, any function x(t) can be represented as:
x(t) = Sum from k of [C_k * phi_k(t)]
The coefficients C_k are found by projecting x(t) onto each basis function:
C_k = [Inner product of x(t) and phi_k(t)] / [Inner product of phi_k(t) and phi_k(t)]
This provides a deeper mathematical context for the specific formulas used for Fourier series coefficients.
Detailed Explanation
The Generalized Fourier Series expands the idea of representing any function in terms of basis functions. The coefficients that multiply these basis functions tell us how much of each function we need to recreate x(t). We find these coefficients by determining how much x(t) correlates with each basis function, a process involving inner products.
Examples & Analogies
Imagine inviting a chef to recreate your favorite dish. The chef has various ingredients (basis functions) and needs to know how much of each to use (coefficients). By tasting your dish and adjusting the ingredients accordingly (akin to finding inner products), the chef will masterfully replicate it. Similarly, the coefficients help build the original function from the complete orthogonal set.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Orthogonality: Functions are orthogonal if their inner product equals zero over a specific interval.
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Inner Product: An integral operation that measures correlation between two functions.
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Complete Set: A set of functions where every square-integrable function can be expressed as a linear combination of them.
Examples & Real-Life Applications
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Examples
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The functions sin(t) and cos(t) are orthogonal over the interval [0, 2Ο].
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The trigonometric functions {1, cos(Οβt), sin(Οβt)} form an orthogonal set in Fourier series.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Orthogonal set, where zero's the bet, functions align, with no overlap to fret.
π Fascinating Stories
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Imagine a dance party where different dance styles donβt interfere with each other's space, just like orthogonal functions.
π§ Other Memory Gems
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C.O.D.E. - Complete, Orthogonal, Determined, & Expressed; core principles in understanding function sets.
π― Super Acronyms
O.C.E.A.N. - Orthogonal Complete Expressive Aggregate of Functions.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Orthogonal Set
Definition:
A collection of functions where each distinct pair is orthogonal over a specified interval, meaning their inner product is zero.
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Term: Complete Set
Definition:
A set of functions is complete if any well-behaved function can be represented as a linear combination of the functions within that set.
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Term: Inner Product
Definition:
An operation that combines two functions to yield a scalar, often defined as an integral of their product over a specific interval.
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Term: SquareIntegrable Function
Definition:
A function whose integral of the square over an interval is finite, ensuring that it has a well-defined energy.