I would like to see more of these articles cover the phase portion.
What do you mean?
Understanding The Fourier Transform
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Re: Understanding The Fourier Transform
#12Re: Understanding The Fourier Transform
#13That's so conceptually simple I hate myself for not getting it before.
Re: Understanding The Fourier Transform
#14This is a nice way to see how the DFT is computed, however I find the view of the FT as a change of basis as even more important - generalizes easily to other bases and and one can understand easily wavelets and their advantages. Basically, the sinusoids form a basis of the vector space of functions (every 'non-pathological' function can be written as a possibly infinite sum of them) and the numbers computed by the FT are coefficients for the respective basis vectors - the magnitude of these coefficients is interpreted as the strength of the corresponding wave in the original signal.
Another way to see the FT is as the basis where the convolution operators are diagonal - this is used in image processing, where computing the FFT of a filter + entry-wise multiplication can be much faster than running the convolution at each pixel of the input image.
Re: Understanding The Fourier Transform
#15The Fourier Transform can also be thought of as part of Linear Algebra, because it's actually funding a representation of a given function in the basis consisting of sin and cos functions (or complex exponentials).
See, the collection of non-pathological functions is a vector space. We add elements by adding the functions pointwise, we multiply by a constant in the obvious way, and the other requirements can be checked. Functions form a vector space.
And vector spaces have bases. One basis for the vector space of functions is the collection of sin and cos functions. Thus we can see that finding the Fourier Transform is just finding how much of each basis vector we need to make the function.
And as we know, the amount of basis vector u needed in the representation of a vector v is v.u, the dot product.
Thinking of it this way starts to make connections between all sorts of ideas.
Added in edit: I see the same sort of point made by dropdownmenu in http://news.ycombinator.com/item?id=4862228
Re: Understanding The Fourier Transform
#16You can also think of the Fourier Transform as a projection (dot product) of a signal onto the space of all sinusoids. That's the explanation that made everything click for me.
I like this way of thinking about it, but I think it is not quite accurate for discrete Fourier Transforms. In this case, we're not projecting onto the space of all sinusoids, only the space of sinusoids whose period is a multiple of (1/N). We could probably prove (if we wanted to try) that those form a basis for the vector space of N-long complex vectors, so using any more sinusoids would be redundant.
However, I believe the continuous Fourier Transform works exactly like that.
Re: Understanding The Fourier Transform
#17Re: Understanding The Fourier Transform
#18You can also think of the Fourier Transform as a projection (dot product) of a signal onto the space of all sinusoids. That's the explanation that made everything click for me.
As is explained, you don't even need to project. What you do is a change of basis, which is to say no information is lost.
Re: Understanding The Fourier Transform
#19If someone (like me) is having trouble visualizing, http://dave.whipp.name/tutorial/fourier.html
Re: Understanding The Fourier Transform
#20This is a pretty good intro to DFTs. I was excited to see that others were also motivated by the connection to music and math! I looked into this quite in linear algebra in college and found Benson[1] and Smith[2] to be really thorough and interesting resources on the topic. Hope they help others!
1. http://homepages.abdn.ac.uk/mth192/pages/html/maths-music.ht... (free pdf)
2. https://ccrma.stanford.edu/~jos/mdft/ (skip down to applications and the digital audio number systems for a preview)