A new discrete Tchebichef-like transform and its application in speech denooising

A new discrete orthogonal transform for input data sequences of composite length is proposed in this paper. The introduced transform is called the discrete Tchebichef-like transform (DTLT). This transform generalizes the discrete Tchebichef transform (DTT). The DTLT matrix is constructed as a Kronecker product of small-sized DTT matrices. A fast algorithm for computing the proposed transform was developed. An example of constructing a DTLT algorithm for a source data sequence whose length is the product of positive integers is presented, and the corresponding data-flow graph is shown. The proposed transform was applied to speech signal denoising. A comparison of the DTLT with the discrete cosine transform (DCT) showed that the DTLT is more robust to white Gaussian noise. In addition, the DTLT has lower computational complexity than the DCT. For signals corrupted by correlated noise, the performance of the transform-domain approach depends on the noise power spectral density. It is more effective when the noise exhibits dominant spectral peaks. Among the considered transforms, the DTT provides the best overall performance, while the DTLT closely follows and demonstrates behavior comparable to that of the DTT. Experimental results demonstrate that the proposed DTLT and DTT achieve better speech enhancement than wavelet denoising on speech signals corrupted by low-level additive white Gaussian noise. However, wavelet denoising shows an advantage at higher noise levels (0-5 dB). For speech signals corrupted by additive colored noise, the proposed DTLT and DTT achieve speech intelligibility and perceptual quality comparable to wavelet denoising while providing more effective noise suppression, since wavelet denoising yields ISNR values close to zero. In addition, the DTLT requires significantly fewer arithmetic operations than the DCT and the discrete wavelet transform, particularly in terms of the number of multiplications, making it an efficient solution for speech denoising.