Frequency estimation with missing measurements under impulsive noise

The frequency recovery problem from impulsive noise corrupted signal with missing data is considered. The main motive of this work is to explore the signal sparse property that is proven to be advantageous if it is properly utilized. To that end, first, a transformation domain, namely frequency domain, is constructed in which multiple sinusoids have a sparse representation. Second, the data missing problem is reformulated in a way that is represented by a sparse vector containing only zeros and ones. Third, thanks to the exciting finding of the nearly-sparse property of the impulsive noise in the time domain, the noise reduction can be designed as well to explore its sparsity to cancel the noise. By utilizing the sparsity from the frequency, missing pattern and impulsive noise, a joint estimation approach is designed that allows simultaneously to perform frequency and missing pattern estimation under the impulsive noise. Numerical studies demonstrate that joint estimate offers robust and consistent results compared to non-joint estimate (without noise reduction).