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.
The Heisenberg uncertainty principle (cf. @cite_8 @cite_6 ) sets a fundamental bound to the time-frequency resolution achievable. If we use Gaussian windows where @math is the standard deviation of the window in the time domain and @math is the standard deviation of its Fourier transform, then we have @math . If, for instance, @math , then @math . For speech processing, this may be sufficient, but considering music that contains frequencies as low as @math , this deviation equals almost half an octave (or a tritone).
@cite_6 @cite_8
2963959545 2035340268
Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design, lead to algorithms for reconstructing missing information via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of graph signals in the vertex and graph spectral domains, our approach generalizes the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform. One challenge we highlight is that the local structure in a small region of an inhomogeneous graph can drastically affect the uncertainty bounds, limiting the information provided by global uncertainty principles. Accordingly, we suggest new notions of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the atom's center vertex. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals. The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck’s constant 2 as demonstrated by Heisenberg’s thought experiment using a g-ray microscope. Here it is shown that this common assumption is not universally true: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below Planck’s constant when the intervention is dependent. A model of measuring interaction with dependent intervention shows that Heisenberg’s lower bound for the noise-disturbance product is violated even by a nearly nondisturbing precise position measurement. An experimental implementation is also proposed to realize the above model in the context of optical quadrature measurement with currently available linear optical devices.