Signal Compaction Using Polynomial EVD for Spherical Array Processing With Applications

Abstract

Multi-channel signals captured by spatially separated sensors often contain a high level of data redundancy. A compact signal representation enables more efficient storage and processing, which has been exploited for data compression, noise reduction, and speech and image coding. This paper focuses on the compact representation of speech signals acquired by spherical microphone arrays. A polynomial matrix eigenvalue decomposition (PEVD) can spatially decorrelate signals over a range of time lags and is known to achieve optimum multichannel data compaction. However, the complexity of PEVD algorithms scales at best cubically with the number of channel signals, e.g., the number of microphones comprised in a spherical array used for processing. In contrast, the spherical harmonic transform (SHT) provides a compact spatial representation of the 3-dimensional sound field measured by spherical microphone arrays, referred to as eigenbeam signals, at a cost that rises only quadratically with the number of microphones. Yet, the SHT's spatially orthogonal basis functions cannot completely decorrelate sound field components over a range of time lags. In this work, we propose to exploit the compact representation offered by the SHT to reduce the number of channels used for subsequent PEVD processing. In the proposed framework for signal representation, we show that the diagonality factor improves by up to 7 dB over the microphone signal representation with a significantly lower computation cost. Moreover, when applying this framework to speech enhancement and source separation, the proposed method improves metrics known as short-time objective intelligibility (STOI) and source-to-distortion ratio (SDR) by up to 0.2 and 20 dB, respectively.

Extension of Power Method to Para-Hermitian Matrices: Polynomial Power Method

“…The proposed method finds potential applications in areas such as data compaction [18], single broadband source to sensors transfer and source spectral density estimation [29], broadband transient signal detection [30]- [32], and the calculations of an approximate PEVD through deflation via subsequent extractions and eliminations of the principal eigenpairs. The approach can also be generalised to calculating a polynomial singular value decomposition [33].…”

“…Exploiting this, we now define a metric as α(Ω) = ∠{v (k) (e jΩ ), v (k−1) (e jΩ )}; note that we retain the normalisation in (18) in case of errors due to truncation. If two successive estimates are aligned, we have α(Ω) = 0 ∀Ω.…”

“…Over recent years, PEVD algorithms have seen use in MIMO communications design [11], [12], subband coding [3], blind signal separation [2], speech enhancement [13], beamforming [14]- [16], broadband angle of arrival estimation [17], and many more. In cases such as coding or compaction, which work by extracting the dominant signal component from multichannel data [18], a complete PEVD may be unnecessary, and it often suffices to extract the largest eigenvalue and its corresponding eigenvector, here termed the principal eigenpair. For the standard EVD, this can be accomplished e.g.…”