# Coefficient in Sparse Signal Recovery

To understand the idea of sparse signal recovery described in Sparsity and Incoherence in Compressive Sampling, I decided to build a toy problem. Suppose the sparse signal that we want to reconstruct $$x^0$$ is $$n=1000$$ dimensional and only $$S=9$$ of its entries are nonzero. For simplicity, I pick the transform matrix $$U$$ to be a DFT matrix and I randomly sample $$m$$ values from $$Ux^0$$ (transform coefficients) to use them as our measurements $$y$$. In other words, $$y = U_{\Omega} x^0.$$

Based on the theorem given in the paper, signal recovery is possible with constrained $$\ell_1$$ norm minimization if the number of measurements $$m$$ satisfies the following condition: $$m \geq C \mu^2(U) S \log n$$ for a positive constant $$C$$. In my toy problem, I know that $$\mu(U)=1$$, $$S=9$$ and $$n=1000$$, but I do not know the constant $$C$$! I looked at the paper over and over again and I think I am missing something. Could you please explain to me how to determine the value of $$C$$?