31–40 of 45 posts
I would like to see more of these articles cover the phase portion.
http://www.youtube.com/watch?v=M0Sa8fLOajA
(Gilbert Strang - my first time watching him teach. Such a great paced lecturer)
The 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 check…
The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions.
So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by some combination of vectors in the basis u?
That is, suppose I describe some (infinite) set of a functions. How do I determine the set of functions that it spans? Is there an intuitive way to picture why the set of sin and cos functions forms a suitable basis?
The 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 check…
This is a great explanation. The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions. So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by…
The 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 check…
This is a great explanation. The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions. So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by…
So start with a periodic wave and look at how much sin(x) is in it. Do this by integrating:
Integral from 0 to 2.pi of f(x)*sin(x) dx
Interesting note:
integral sin(k.x)*sin(m.x)
So the question is: what functions can be reached by adding and subtracting multiples of sine (and cosine) functions of different frequencies? In Linear Algebra terms, what is the space of functions spanned by these basis functions?
That's harder, but it turns out to be "everything non-pathological".
http://en.wikipedia.org/wiki/Fourier_series#Hilbert_space_in...
http://en.wikipedia.org/wiki/Hilbert_space#Fourier_analysis
Edited for typos
You can also think about the Fourier Transform in terms of its physical properties. For example, the Fourier transform is behind quantum uncertainty (dp.dx>h). Think of it this way: the inverse Fourier transform of a frequency impulse (zero extent) is a sine wave of infinite duration. Truncate the infinite sine wave and its spectrum ceases being an impulse, broadening into the shape of the windowing function used to…
That is also a really well phrased one line point of commonality to talk to a telecom / RF / EE type person about communications bandwidth theory. If you just wedge in signal to noise ratio / bit error rate, and look at what you phrase "time definition" and "frequency broadening" in the right way, then you pretty much have Shannons famous paper.
The FT shows up all over the place in science and anywhere you find it, you can analogize it into a totally different field of study. One "design pattern" hundreds of "implementations".
The 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 check…
This is a great explanation. The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions. So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by…
There are infinite expansions of trig functions to solve the finite vs infinite apparent mismatch, so don't worry about that.
An intuitive way to look at it, is on a 2-d graph there's no spot on the graph that can't be hit by a point of a triangle aka trig function collection, so it doesn't particularly matter how you pick any one spot, a triangle can always hit that one spot because it can hit all spots.
Comp sci analogy would be something like x=x+1 starting at x=0 will hit all the positive integers eventually...
Earlier quoted context omitted.
This is a great explanation. The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions. So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by…
"That is, suppose I describe some (infinite) set of a functions.... Is there an intuitive way to picture why the set of sin and cos functions forms a suitable basis?" There are infinite expansions of trig functions to solve the finite vs infinite apparent mismatch, so don't worry about that. An intuitive way to look at it, is on a 2-d graph there's no spot on the graph that can't be hit by a point of a triangle aka t…
The problem with your comment is that yes, for any given point you can get as close as you like, but the hard part is getting close nearly everywhere all at once. More, you then have to show that there's some sort of convergence, and that where you are close now is a superset of where you are close when you demand to be closer.
It's not straight forward.
Earlier quoted context omitted.
This is a great explanation. The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions. So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by…
This was Fourier's big theorem, that every periodic, non-pathological function is "simply" the sum of phase-shifted sine waves of all necessary frequencies. You can remove the "periodic" if you allow infinitely many frequencies, and so on. The phase-shifting is taken care of by having a sine and a cosine at the same frequency, but (possibly) different amplitudes. So start with a periodic wave and look at how much sin…
I think I need to go home and have a play with this now!