Impulsive noise suppression in the case of frequency estimation by exploring signal sparsity

The frequency estimation problem is addressed in this work in the presence of impulsive noise. Two typical scenarios are considered; that is, the received data are assumed to be uniformly sampled, i.e., without data missing for the first case and data are randomly missed for the second case. The main objective of this work is to explore the signal sparsity in the frequency domain to perform frequency estimation under the impulsive noise. Therefore, to that end, a DFT-like matrix is created in which the frequency sparsity is provided. The missing measurements are modeled by a sparse representation as well, where missing samples are set to be zeros. Based on this model, the missing pattern represented by a vector is indeed sparse since it only contains zeros and ones. The impulsive noise is remodeled as a superposition of a unknown sparse vector and a Gaussian vector because of the impulsive nature of noise. By utilizing the sparse property of the vector, the impulsive noise can be treated as a unknown parameter and hence it can be canceled efficiently. By exploring the sparsity obtained, therefore, a joint estimation method is devised under optimization framework. It renders one to simultaneously estimate the frequency, noise, and the missing pattern. Numerical studies and an application to speech denoising indicate that the joint estimation method always offers precise and consistent performance when compared to the non-joint estimation approach.