Sparse system identification using orthogonal rational functions

We consider the problem of reconstructing a real sparse coefficient θ of a transfer function under the orthogonal rational functions from a limit number of linear measurements. Given m randomly selected samples of Φθ, where Φ is a sample matrix constructed by orthogonal rational functions at samples in the unit circle, we show that ℓ₁ minimization can recover the coefficient θ from the real part model or the imaginary part model for the real and imaginary part of Φ have the similar structure of the orthogonal matrix.