  This paper investigates recovery of an undamped spectrally sparse signal and
its spectral components from a set of regularly spaced samples within the
framework of spectral compressed sensing and super-resolution. We show that the
existing Hankel-based optimization methods suffer from the fundamental
limitation that the prior of undampedness cannot be exploited. We propose a new
low rank optimization model partially inspired by forward-backward processing
for line spectral estimation and show its capability in restricting the
spectral poles on the unit circle. We present convex relaxation approaches with
the model and show their provable accuracy and robustness to bounded and sparse
noise. All our results are generalized from the 1-D to arbitrary-dimensional
spectral compressed sensing. Numerical simulations are provided that
corroborate our analysis and show efficiency of our model and advantageous
performance of our approach in improved accuracy and resolution as compared to
the state-of-the-art Hankel and atomic norm methods.
