Abstract
We survey the definitions and use of rank-revealing matrix decompositions in single-channel noise reduction algorithms for speech signals. Our algorithms are based on the rank-reduction paradigm and, in particular, signal subspace techniques. The focus is on practical working algorithms, using both diagonal (eigenvalue and singular value) decompositions and rank-revealing triangular decompositions (ULV, URV, VSV, ULLV, and ULLIV). In addition, we show how the subspace-based algorithms can be analyzed and compared by means of simple FIR filter interpretations. The algorithms are illustrated with working Matlab code and applications in speech processing.
Highlights
The signal subspace approach has proved itself useful for signal enhancement in speech processing and many other applications—see, for example, the recent survey [1]
The algorithms are illustrated with working Matlab code and applications in speech processing
Throughout the literature of signal processing and applied mathematics, these methods are formulated in terms of different notations, such as eigenvalue decompositions, Karhunen-Loeve transformations, and singular value decompositions
Summary
The signal subspace approach has proved itself useful for signal enhancement in speech processing and many other applications—see, for example, the recent survey [1]. The central idea is to approximate a matrix, derived from the noisy data, with another matrix of lower rank from which the reconstructed signal is derived. As stated in [5]: “Rank reduction is a general principle for finding the right trade-off between model bias and model variance when reconstructing signals from noisy data.”. Throughout the literature of signal processing and applied mathematics, these methods are formulated in terms of different notations, such as eigenvalue decompositions, Karhunen-Loeve transformations, and singular value decompositions. All these formulations are mathematically equivalent, but the differences in notation can be an obstacle to understanding and using the different methods in practice
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