  This paper studies the problem of reconstructing spectrally sparse signals
from a small random subset of time domain samples via low-rank Hankel matrix
completion with the aid of prior information. By leveraging the low-rank
structure of spectrally sparse signals in the lifting domain and the similarity
between the signals and their prior information, we propose a convex method to
recover the undersampled spectrally sparse signals. The proposed approach
integrates the inner product of the desired signal and its prior information in
the lift domain into vanilla Hankel matrix completion, which maximizes the
correlation between the signals and their prior information. Theoretical
analysis indicates that when the prior information is reliable, the proposed
method has a better performance than vanilla Hankel matrix completion, which
reduces the number of measurements by a logarithmic factor. We also develop an
ADMM algorithm to solve the corresponding optimization problem. Numerical
results are provided to verify the performance of proposed method and
corresponding algorithm.
