The Theory of Compressive Sensing Matching Pursuit Considering Time-domain Noise with Application to Speech Enhancement

Compressive sampling matching pursuit (CoSaMP) is an efficient compressive sensing algorithm holding rigorous estimation error bounds and low computational complexity, when it deals with an additive noise signal model in the observation domain. However, in some applications, e.g., speech enhancement (SE), noise is added to a signal in the time domain, where the conventional CoSaMP cannot be directly applied. In this paper, we establish the theory of CoSaMP to address the time-domain noise, referred to as Tdn-CoSaMP, which extends the canonical theory of CoSaMP. In particular, we prove the existence of a new upper bound of Tdn-CoSaMP, which is found to be larger than that of the conventional CoSaMP by appending two additional terms: a multiplier
$1+\sqrt{{N\over s}}$
, where
$N$
is the dimension of the signal, and an
${\ell_1}$
norm of the noise
${1\over\sqrt{s}}\Vert {\mbi{e}}\Vert_1$
scaled by the sparse level
$s$
of the signal. We also apply Tdn-CoSaMP to the SE task based on the sequential denoising of overlapped frames in the discrete cosine transform (DCT) domain. The proposed system, CoSaMP-based speech enhancement (CoSaMPSE), has been evaluated in terms of both objective and subjective criteria on various types of noise. Positive results have been achieved for denoising stationary and nonstationary white Gaussian noise (WGN) and are comparable to other SE methods. Moreover, due to its low computational complexity, CoSaMPSE is possible to be combined with optimally modified log-spectrum amplitude estimation (OMLSA) and able to achieve complementary denoising effects in various noisy conditions.