# On the use of OMP for sparse channel estimation based on pilots

Given an OFDM frame whose length is 256 sub-carriers, the length of guard interval CP is 64, and the transmitted known pilots 'in red' are 64 too as below:

where the received signal is suppose to be :

$$Y = Filter(h,1,x)$$

where $$Y$$ is the received signal of length $$1$$x$$(N+CP)$$, $$h$$ is the sparse channel of length $$1$$x$$64$$, and $$x$$ is the transmitted signal of length $$1$$x$$(N+CP)$$. my question is related to the sparse channel reconstruction using the known algorithm OMP (Orthogonal Matching pursuit).

As known, the OMP algorithm is used to reconstruct the sparse channel, but we need as inputs the following, the dictionary (or measurement matrix), the pilots data which is of size $$1$$x$$64$$ and the number of non-zeros taps in the sparse channel.

I ask about building the measurement matrix , how can we build it based on the received data? what its size is suppose to be?

NP: In basic OMP algorithm, measurement matrix is supposed to be same size of $$x$$, but in our case we have only pilots of size $$(x/4)$$ but not $$x$$ ! so how it can be built?

Thank you