Finding the orthonormal basis for cosine function

In summary, the given function can be represented as a set of signals with dimensionality N = 4. To find the orthonormal basis functions, the Gram-Schmidt orthogonalization procedure can be used. To find the basis functions for sine waves, the values of cosine at different angles (α + π/4, α + π/2, α + 3π/4, and α + π) can be evaluated.
  • #1
aiq25
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Homework Statement


si(t) = √(((2*E)/T)*cos(2*π*fc*t + i*(π/4))) for 0≤t≤T and 0 otherwise. Where i = 1, 2, 3, 4 and fc = nc/T, for some fixed integer nc.
What is the dimensionality, N, of the space spanned by this set of signal? Find a set of orthonormal basis functions to represent this set of signals. Plot the locations of si(t) (i = 1, 2, 3, 4) in the signal space.

2. The attempt at a solution
Dimensionality N = 4.
I know how to find the orthonormal basis functions for square waves, because so far we were just given square waves. We used the Gram-Schmidt orthogonalization procedure. My question is how do I find it when the function is a sine wave?

Just looking for some directions. :)

Thank You.
 
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  • #2
What is ##\cos(\alpha + {\pi\over 4})##, what is ##\cos(\alpha + {\pi\over 2})##, ##\cos(\alpha + {3\pi\over 4})## and ##\cos(\alpha + \pi)## when evaluated ?
 

Related to Finding the orthonormal basis for cosine function

1. What is an orthonormal basis for a cosine function?

An orthonormal basis for a cosine function is a set of vectors that are both orthogonal (perpendicular) to each other and have a magnitude of 1. These vectors can be used to represent any point or direction in a space, and they form the basis for the cosine function.

2. How do you find the orthonormal basis for a cosine function?

To find the orthonormal basis for a cosine function, you can start by finding the standard basis vectors for the space in which the cosine function is being used. Then, you can use the Gram-Schmidt process to orthogonalize these vectors and normalize them to have a magnitude of 1. This will result in an orthonormal basis for the cosine function.

3. Why is it important to find the orthonormal basis for a cosine function?

Finding the orthonormal basis for a cosine function is important because it allows us to represent any point or direction in a space using a set of orthogonal vectors. This can simplify calculations and make it easier to understand and analyze the behavior of the cosine function.

4. Can the orthonormal basis for a cosine function be used for other functions as well?

While the orthonormal basis for a cosine function is specifically designed for the cosine function, the Gram-Schmidt process can be used to find an orthonormal basis for any set of vectors. These vectors can then be used to represent other functions, as long as they have the same number of dimensions as the original set of vectors.

5. Are there any real-world applications of finding the orthonormal basis for a cosine function?

Yes, there are many real-world applications of finding the orthonormal basis for a cosine function. For example, in computer graphics, an orthonormal basis can be used to represent the orientation of an object in 3D space. In signal processing, an orthonormal basis can be used to decompose a signal into its frequency components. It also has applications in physics, engineering, and many other fields.

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