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 many cases, a single musical instrument can generate different sounds which are perceptually similar but only vary in the pitch of the tones. In the STFT spectrogram, different pitch manifests in linear scaling of the distances between the peaks in the frequency axis, which is computationally hard to handle. Therefore, @cite_2 applied the constant-Q transform (CQT) @cite_28 , which turns scaling into shifts, and developed the shifted non-negative matrix factorization (SNMF), that is actually a tensor factorization, in order to train a shift-invariant dictionary. This approach was later refined by @cite_24 @cite_13 @cite_15 .
@cite_28 @cite_24 @cite_2 @cite_15 @cite_13
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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. 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. A shifted non-negative matrix factorisation algorithm is derived, which offers advantages over previous matrix factorisation techniques for the purposes of single channel source separation. It represents a sound source as translations of a single frequency basis function. These translations approximately correspond to notes played by an instrument. Results are presented for a set of synthetic data, and on a single channel recording of piano and clarinet. Though the system is aimed at musical recordings, the technique can be applied to any data which contains shifted versions of an underlying factor, and so the algorithm could possibly be used in other applications such as image processing An unsupervised learning algorithm for the separation of sound sources in one-channel music signals is presented. The algorithm is based on factorizing the magnitude spectrogram of an input signal into a sum of components, each of which has a fixed magnitude spectrum and a time-varying gain. Each sound source, in turn, is modeled as a sum of one or more components. The parameters of the components are estimated by minimizing the reconstruction error between the input spectrogram and the model, while restricting the component spectrograms to be nonnegative and favoring components whose gains are slowly varying and sparse. Temporal continuity is favored by using a cost term which is the sum of squared differences between the gains in adjacent frames, and sparseness is favored by penalizing nonzero gains. The proposed iterative estimation algorithm is initialized with random values, and the gains and the spectra are then alternatively updated using multiplicative update rules until the values converge. Simulation experiments were carried out using generated mixtures of pitched musical instrument samples and drum sounds. The performance of the proposed method was compared with independent subspace analysis and basic nonnegative matrix factorization, which are based on the same linear model. According to these simulations, the proposed method enables a better separation quality than the previous algorithms. Especially, the temporal continuity criterion improved the detection of pitched musical sounds. The sparseness criterion did not produce significant improvements Historically, papers have been physically bound to the journal in which they were published; but in the digital age papers are available individually, no longer tied to their respective journals. Hence, papers now can be read and cited based on their own merits, independently of the journal's physical availability, reputation, or impact factor (IF). We compare the strength of the relationship between journals' IFs and the actual citations received by their respective papers from 1902 to 2009. Throughout most of the 20th century, papers' citation rates were increasingly linked to their respective journals' IFs. However, since 1990, the advent of the digital age, the relation between IFs and paper citations has been weakening. This began first in physics, a field that was quick to make the transition into the electronic domain. Furthermore, since 1990 the overall proportion of highly cited papers coming from highly cited journals has been decreasing and, of these highly cited papers, the proportion not coming from highly cited journals has been increasing. Should this pattern continue, it might bring an end to the use of the IF as a way to evaluate the quality of journals, papers, and researchers.  2012 Wiley Periodicals, Inc.