  This paper considers the reconstruction of a sparse coefficient vector
{\theta} for a rational transfer function, under a pair of FIR and
Takenaka-Malmquist (TM) bases and from a limited number of linear
frequency-domain measurements. We propose to concatenate a limited number of
FIR and TM basis functions in the representation of the transfer function, and
prove the uniqueness of the sparse representation defined in the infinite
dimensional function space with pairs of FIR and TM bases. The sufficient
condition is given for replacing the l_0 optimal solution by the l_1 optimal
solution using FIR and TM bases with random samples on the upper unit circle,
as the foundation of reconstruction. The simulations verify that l_1
minimization can reconstruct the coefficient vector {\theta} with high
probability. It is shown that the concatenated FIR and TM bases give a much
sparser representation, with much lower reconstruction order than using only
FIR basis functions and less dependency on the knowledge of the true system
poles than using only TM basis functions.
