Abstract
Compressive sensing has attracted significant interest of researchers providing an alternative way to sample and reconstruct the signals. This approach allows us to recover the entire signal from just a small set of random samples, whenever the signal is sparse in certain transform domain. Therefore, exploring the possibilities of using different transform basis is an important task, needed to extend the field of compressive sensing applications. In this paper, a compressive sensing approach based on the Hermite transform is proposed. The Hermite transform by itself provides compressed signal representation based on a smaller number of Hermite coefficients compared to the signal length. Here, it is shown that, for a wide class of signals characterized by sparsity in the Hermite domain, accurate signal reconstruction can be achieved even if incomplete set of measurements is used. Advantages of the proposed method are demonstrated on numerical examples. The presented concept is generalized for the short-time Hermite transform and combined transform.
Highlights
The Hermite polynomials and Hermite functions have attracted the attention of researchers in various fields of engineering and signal processing [1,2,3,4,5,6,7], such as in quantum mechanics, ultra-high band telecommunication channels, and ECG data compression using Hermite functions representation of the QRS complexes
We are especially interested in a class of signals that are sparse in the Hermite transform domain
In the light of compressive sensing (CS) theory [8,9,10,11,12], we propose the method for efficient reconstruction of signals from its incomplete set of samples using the Hermite transform
Summary
The Hermite polynomials and Hermite functions have attracted the attention of researchers in various fields of engineering and signal processing [1,2,3,4,5,6,7], such as in quantum mechanics (harmonic oscillators), ultra-high band telecommunication channels, and ECG data compression using Hermite functions representation of the QRS complexes. We are especially interested in a class of signals that are sparse in the Hermite transform domain Note that, generally, such signals are not sparse in the Fourier transform domain. In the light of compressive sensing (CS) theory [8,9,10,11,12], we propose the method for efficient reconstruction of signals from its incomplete set of samples using the Hermite transform. The theory is illustrated through examples showing that the Hermite transform based CS for certain types of signals can outperform the Fourier transform related reconstructions.
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