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.
In order to apply dictionary learning on audio data, it is helpful to regard a time-frequency representation which subdivides the problem into smaller time frames and highlights the frequency characteristics of the signal. Classically, such a representation is computed via application of the short-time Fourier transform (STFT) (cf. @cite_8 ) on the audio signal, and the resulting spectrogram (i.e., the magnitude of the STFT) is then decomposed via non-negative matrix factorization (NMF) @cite_22 in order to obtain a dictionary. This approach was initially studied by Smaragdis and Brown @cite_19 for the purpose of polyphonic music transcription and then applied to audio source separation by Wang and Plumbley @cite_27 .
@cite_19 @cite_27 @cite_22 @cite_8
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We consider inference in a general data-driven object-based model of multichannel audio data, assumed generated as a possibly underdetermined convolutive mixture of source signals. We work in the short-time Fourier transform (STFT) domain, where convolution is routinely approximated as linear instantaneous mixing in each frequency band. Each source STFT is given a model inspired from nonnegative matrix factorization (NMF) with the Itakura-Saito divergence, which underlies a statistical model of superimposed Gaussian components. We address estimation of the mixing and source parameters using two methods. The first one consists of maximizing the exact joint likelihood of the multichannel data using an expectation-maximization (EM) algorithm. The second method consists of maximizing the sum of individual likelihoods of all channels using a multiplicative update algorithm inspired from NMF methodology. Our decomposition algorithms are applied to stereo audio source separation in various settings, covering blind and supervised separation, music and speech sources, synthetic instantaneous and convolutive mixtures, as well as professionally produced music recordings. Our EM method produces competitive results with respect to state-of-the-art as illustrated on two tasks from the international Signal Separation Evaluation Campaign (SiSEC 2008). Non-negative Matrix Factorisation (NMF) based algorithms have found application in monaural audio source separation due to their ability to factorize audio spectrogram into additive part-based basis functions, which typically corresponds to individual notes or chords in music. These separated basis functions are usually greater in number than the active sources, hence clustering is needed for individual source signal synthesis. Although, many attempts have been made to improve the clustering of the basis functions to sources, much research is still required in this area. Recently, Shifted NMF based methods have been proposed as a means to avoid clustering these pitched basis functions to sources. However, the Shifted NMF algorithm uses a log-frequency spectrogram with a xed number of frequency bins per octave which compromises the quality of separated sources.We show that by replacing the method used to calculate the log-frequency spectrogram with a recently proposed invertible Constant Q Transform (CQT), we can considerably improve the separation quality of the individual sound signals. In this paper, a signal is shown to be uniquely represented by the magnitude of its short-time Fourier transform (STFT) under mild restrictions on the signal and the analysis window of the STFT. Furthermore, various algorithms are developed which reconstruct signal from appropriate samples of the STFT magnitude. Several of the algorithms can also be used to obtain signal estimates from the processed STFT magnitude, which generally does not have a valid short-time structure. These algorithms are successfully applied to the time-scale modification and noise reduction problems in speech processing. Finally, the results presented here have similar potential for other application areas, including those with multidimensional signals. Non-negative Matrix Factorization (NMF) has found use in single channel separation of audio signals, as it gives a parts-based decomposition of audio spectrograms where the parts typically correspond to individual notes or chords. However, a notable shortcoming of NMF is the need to cluster the basis functions to their sources after decomposition. Despite recent improvements in algorithms for clustering the basis functions to sources, much work still remains to further improve these algorithms. To this end we present a novel clustering algorithm which overcomes some of the limitations of previous clustering methods. This involves the use of Shifted Nonnegative Matrix Factorization (SNMF) as a means of clustering the frequency basis functions obtained from NMF. Results show that this gives improved clustering of pitched basis functions over previous methods.