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Anonymous Wed 07 Oct 10:28:50 2020 No.12203532 View Reply Original Report

Quoted By: >>12203785 >>12204026

Is there something special about the sine function where the frequency domain representation of a function would only be related to sine waves. Why do the points on a frequency domain plot only associate with sines and cosines. Why can't they represent other periodic functions like square waves or triangle waves.

Anonymous

Anonymous Wed 07 Oct 2020 11:38:00 No.12203656 Report

Quoted By:

They can. See Lebesgue Integration.

Anonymous

Anonymous Wed 07 Oct 2020 12:55:00 No.12203785 Report

Quoted By: >>12203815

>>12203532
The concept of Fourier Series is just a special case of an orthonormal basis expansion of a function in a bounded domain. In other words, absolutely nothing special with sines and cosines, it's just a convention. As long as you can define another proper orthonormal basis, you can potentially reconstruct all Fourier theory using your functions.

Anonymous

Anonymous Wed 07 Oct 2020 13:07:44 No.12203815 Report

Quoted By: >>12203831 >>12205385

>>12203785
>orthonormal basis
what does this mean in terms an engineerlet can understand

Anonymous

Anonymous Wed 07 Oct 2020 13:14:19 No.12203831 Report

Quoted By: >>12203835

>>12203815
Under some (very realistic) circumstances you can express any function in a continuous domain $[a,b]$ using a linear combination of basis functions, like when you express vectors with basis vectors.
In the case of Fourier Series, it's a countably infinite set of sines and cosines at different frequencies. Such sines and cosines are orthonormal exactly like the basis vectors of like $R^3$ .

Anonymous

Anonymous Wed 07 Oct 2020 13:16:07 No.12203835 Report

Quoted By: >>12203846 >>12204020

>>12203831
Thank you, I think I understand now. Would taylor/maclaurin series be another example of this?

Anonymous

Anonymous Wed 07 Oct 2020 13:21:10 No.12203846 Report

Quoted By: >>12205174

>>12203835
Afaik No, Taylor/McLaurin is another thing as they do not deal with bases.

Anonymous

Anonymous Wed 07 Oct 2020 14:48:39 No.12204020 Report

Quoted By:

>>12203835
No but google orthogonal polynomials
They are many.

Anonymous

Anonymous Wed 07 Oct 2020 14:51:26 No.12204026 Report

Quoted By:

>>12203532
What I always wondered how programs isolate the different frequencies in all the noise.

Anonymous

Anonymous Wed 07 Oct 2020 20:51:55 No.12205174 Report

Quoted By:

>>12203846
Actually, they do. Partial derivatives are the basis vectors of tangential spaces of manifolds. This directly explains why Taylor series only exactly represent analytical functions.

Anonymous

Anonymous Wed 07 Oct 2020 22:06:40 No.12205385 Report

Quoted By: >>12205678

>>12203815
Remember when you did LinAlgebra and you worked on constructing an orthonormal base? A set of, in that case, vectors whose norm was 1 and whose product was 1 if they were the same vector, 0 if else. Any vector could be expressed as a sum of these.

Basically the same with functions. A set of functions whose integrals (loosely said) from -inf to +inf are 1 or 0, depending on whether they are the same function or not, and that can build any other function when acting as a linear combination.

Anonymous

Anonymous Wed 07 Oct 2020 23:46:49 No.12205678 Report

Quoted By:

>>12205385
amazingly enough my EE degree doesn't require linear algebra, so I actually don't
Though I do get the gist of the idea now, orthonormal bases and all

Anonymous

Anonymous Wed 07 Oct 2020 23:59:19 No.12205713 Report

Quoted By:

Fourier transform is defined that way. Make your own transform for square waves etc.

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