convolution - Get an input signal from LTI system output, find impulse response

I just study signal processing and I have some questions.

1. How to find impulse response $h[n]$ sequence of the system if I have $x[n]$ and $y[n]$ (input and output) sequences? Hope there is a formula to do that.


2. How to find $x[n]$ input sequence if I have $h[n]$ and $y[n]$?


3. Is it possible to find $x[n]$ if I just have set of $y[n]$ values, but I don't know impulse response $h[n]$? How to do that if so?

Maybe you can also suggest something for further reading about this topics.

Answer

Ok. You are touching on the topic of deconvolution. You can do this most easily by transforming the data into the frequency domain, doing some basic math and then converting back to the time domain.

For example for case 1, if you transform the signals you have $ H(f) = Y(f)/X(f) $ so $h[n] = IDFT[Y(f)/X(f)]$

You can do similar manipulations to handle case 2.

I'm not sure case 3 is possible. I think you need more information to produce x[n]

Here is a site that discusses deconvolution: http://terpconnect.umd.edu/~toh/spectrum/Deconvolution.html

This is a VERY simplified treatment of this question. I'm not sure what level of detail is being requested in the posted question.