Capturing n basis vectors by single one

[valjok]

Normally, you need how system transforms n basis vectors to say how it transforms arbitrary vector. For instance, when your signal is presented in Fourier basis, you need to know how system responds to every sine. But, I have noted that it is not true for the simplest standard basis. You just measure or compute a single Impulse Response (green's function?), for input <1000...> and you already know the responses for all other basis vectors: <0100...>, <0010...> and etc. You use it in convolution. How it is possible?

Indeed, the delta-impulse, if you take its Fourier transform, is a combination of sines of constant amplitude. It is <111111...> in the Fourier basis. But, all sines are entangled here. So, instead of measuring response for every separate sine, we can measure response for all sines at once. Then, we can project the response to every separate sine and, thus, figure out the response per every separate sine. Is it right?

 

[chiro]

Hey valjok.

It might help if you state what you want to do mathematically (either get comments on or show or something else).

In terms of the Fourier basis for a periodic-signal decomposition, the basis elements for sin and cosine basis are all orthogonal to each other which means that all components corresponding to all basis are independent from each other due to orthogonality.

This means that when you project your function to a particular basis element of your full basis (sine and cosine functions in L^2 inner product), then it means that you won't get any entangled information when you say project your continuous signal (or function mapping) to the relevant sine or cosine basis.

Just for clarification, are you talking about the Fourier transform over the whole real line or a periodic-type transform over some finite interval for finite-interval signal data?

 

[valjok]

Revising the Linear Algebra lecture 30, where G. Strang explains the need to transform basis vectors separately, I suddenly realized that this is not needed for delta-pulse basis, where n components of a single vector can capture all nxn components of the transform matrix. And, yes, I decided to listen from more advanced mathematicians, how such compression is possible?

 

[Stephen Tashi]

valjok,

You are asking a question about linear algebra using the jargon of engineering. If you want a mathematical answer, you'll have to give a mathematical definition of a "delta pulse" etc.

You didn't quote what Strang's book said. Was he talking about infinite dimensional vector spaces?

 

[valjok]

You didn't quote what Strang's book said. Was he talking about infinite dimensional vector spaces?

I refer a video lecture. He explains transforms using nxn matrices as an example. Indeed, my computer science education has more engineering bias, so to me, delta-function is a good name for the elements of standard basis.

 

[valjok]

To me, in N-dimentional space, we can represent a linear transformation using only n numbers when basis is eigenspace. Combining eigenvectors, transforming the combination, and projecting the response back to every eigenbasis vector will produce n eigenvalues. But, this cannot work for any orthogonal matrix because, I'm sure that generally projecting response to one basis vector will not kill all contributions from other basis vectors. Since I was taught that impulse response method works for any linear system, I must conclude that Fourier space is eigenspace for any linear system. Earlier, I also felt that people define linear system as one that has a constant amplification ("eighenvalue") for every frequency component. But, is it right? Why Fourier basis (i heard, it is an eighenspace only for Topplitz operators)?

 

[Hawkeye18]

Hi Valjok,

indeed, a CAUSAL TIME-INVARIANT system is completely defined by its impulse response function, because if you know the response on (1, 0, 0,0,0,...) you know the response to (0,1,0,0,0,...), (0,0,1,0,0,...) etc.
But here you need the fact that the system is time invariant, so if your signal is 1 at t=k and 0 otherwise, you know the response for t\ge k. And since the system is causal, this signal does not influence the past, so the response before time k is 0.

If one does not require causality, a time invariant system is still completely defined by its impulse response to the signal (1,0,0,0,...), but you need to measure the response for all times k from -\infty to +\infty. So formally the signal should be written as (...0,0,0,1,0,0,0,...).

If your system is not time invariant, you need to measure the response to all vectors in the basis.

 

[valjok]

Thank you Hawkeye18 for clarification. Now, we'll get the answer to my question much sooner.

 

[valjok]

Yes, they report that LTI corresponds to a Toeplitz matrix, which has Fourier eighenbasis!