We propose a novel method for the blind separation of audio signals produced by musical instruments. While the approach of applying non-negative matrix factorization (NMF) has been studied in many papers, it does not make use of the pitch-invariance that the sounds of instruments exhibit. This limitation can be overcome by using tensor factorization, in which context the use of log-frequency spectrograms was initiated, but this still requires the specific tuning of the instruments to be hard-coded into the algorithm. We develop a time-frequency representation that is both shift-invariant and frequency-aligned, with a variant that can also be used for wideband signals. Our separation algorithm exploits this shift-invariance in order to find patterns of peaks related to specific instruments, while non-linear optimization enables it to represent arbitrary frequencies and incorporate inharmonicity, and the reasonability of the representation is ensured by a sparsity condition. The relative amplitudes of the harmonics are saved in a dictionary, which is trained via a modified version of ADAM. For a realistic monaural piece with acoustic recorder and violin, we achieve qualitatively good separation with a signal-to-distortion ratio (SDR) of 12.5 dB, a signal-to-interference ratio (SIR) of 25.7 dB, and a signal-to-artifacts ratio (SAR) of 12.7 dB, averaged.
Another approach is the computation of the Mel spectrogram @cite_4 , which applies logarithmically spaced windows in the frequency direction. This, however, unfavorably amplifies transients in the higher frequencies.
@cite_4
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Mel-frequency cepstral coefficients (MFCCs) are efficient audio descriptors providing spectral energy measurements over short time windows of length 23 ms. These measurements, however, lose non-stationary spectral information such as transients or time-varying structures. It is shown that this information can be recovered as spectral co-occurrence coefficients. Scattering operators compute these coefficients with a cascade of wavelet filter banks and modulus rectifiers. The signal can be reconstructed from scattering coefficients by inverting these wavelet modulus operators. An application to genre classification shows that second-order cooccurrence coefficients improve results obtained by MFCC and Delta-MFCC descriptors. 1