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 2049686551 2018778093
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. Abstract A set of simple new procedures has been developed to enable the real-time manipulation of speech parameters. The proposed method uses pitch-adaptive spectral analysis combined with a surface reconstruction method in the time–frequency region. The method also consists of a fundamental frequency (F0) extraction using instantaneous frequency calculation based on a new concept called fundamentalness'. The proposed procedures preserve the details of time–frequency surfaces while almost perfectly removing fine structures due to signal periodicity. This close-to-perfect elimination of interferences and smooth F0 trajectory allow for over 600 manipulation of such speech parameters as pitch, vocal tract length, and speaking rate, while maintaining high reproductive quality. Consider the problem of recovering a measure @math supported on a lattice of span @math , when measurements are only available concerning the Fourier Transform @math at frequencies @math . If @math is much smaller than the Nyquist frequency @math and the measurements are noisy, then, in general, stable recovery of @math is impossible. In this paper it is shown that if, in addition, we know that the measure @math satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length @math there are at most R elements. Indeed, if the (unknown) support of @math is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like @math as @math . This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astrono...