  Gridless direction-of-arrival (DOA) estimation with multiple frequencies can
be applied in acoustics source localization problems. We formulate this as an
atomic norm minimization (ANM) problem and derive an equivalent
regularization-free semi-definite program (SDP) thereby avoiding regularization
bias. The DOA is retrieved using a Vandermonde decomposition on the Toeplitz
matrix obtained from the solution of the SDP.
  We also propose a fast SDP program to deal with non-uniform array and
frequency spacing. For non-uniform spacings, the Toeplitz structure will not
exist, but the DOA is retrieved via irregular Vandermonde decomposition (IVD),
and we theoretically guarantee the existence of the IVD. We extend ANM to the
multiple measurement vector (MMV) cases and derive its equivalent
regularization-free SDP. Using multiple frequencies and the MMV model, we can
resolve more sources than the number of physical sensors for a uniform linear
array. Numerical results demonstrate that the regularization-free framework is
robust to noise and aliasing, and it overcomes the regularization bias.
