# Compressive sensing based sparse vector estimation

I am newbie in compressive sensing (CS), I read about compressive sensing and its use for sparse vector estimation. As I understood CS can be used either in time or frequency domain. For me, The part I couldn't understand in my case is how to build the measurement matrix for my vector required to be estimated. (In my case the sparse vector $$h$$ needs to be estimated)

I have a vector $$X$$ where some values of it are known (let's say the values located at $$1:4:end$$), called pilots, that vector was convoluted with a sparse vector $$h$$, to have:

$$y = X*h$$

$$*$$ denotes the convolution operation.

My goal is to estimate the sparse vector $$h$$ based on the known values (pilots) in $$X$$. How to build the measurement matrix in that case?

• In order to be able to use CS, it's better to change the convolution into multiplication, such that $y= X*h = XH$ where $H$ is the toeplitz matrix of h, but I don't know how to deal with your measurement matrix exactly. Jun 27 '20 at 2:43
• Ok. I think we can write one of them since they are the same Jun 27 '20 at 11:01
• I don't know if that means it's not possible to use compressive sensing in that case or my question was n't read by someones who know about the compressive sensing. Jun 29 '20 at 2:53
• I think it's difficult to use CS in your case, but I'm not sure! Jul 3 '20 at 11:52