Description:
The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analysed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining ‘extremal’ statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.
Description:
The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analysed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining ‘extremal’ statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.
Description:
For a given collection H of subsets of a set X we examine the convexity on X generated by H. We use a special type of connectedness of H and X for investigation of separation of convex sets by elements of H. In particular, we give a description of convex sets, which can be represented as the intersection of a subfamily of H. As an application, we give a description of abstract convex functions and sets. We also describe the abstract convex hull of a finite union of abstract convex sets.
Description:
Blind Source Separation techniques are widely used in the field of wireless communication for a very long time to extract signals of interest from a set of multiple signals without training data. In this paper, we investigate the problem of separation of the human voice from a mixture of human voice and sounds from different musical instruments. The human voice may be a singing voice in a song or may be a part of some news, broadcast by a channel with background music. This paper proposes a generalized Short Time Fourier Transform (STFT)-based technique, combined with filter bank to extract vocals from background music. The main purpose is to design a filter bank and to eliminate background aliasing errors with best reconstruction conditions, having approximated scaling factors. Stereo signals in time-frequency domain are used in experiments. The input stereo signals are processed in the form of frames and passed through the proposed STFT-based technique. The output of the STFT-based technique is passed through the filter bank to minimize the background aliasing errors. For reconstruction, first an inverse STFT is applied and then the signals are reconstructed by the OverLap-Add method to get the final output, containing vocals only. The experiments show that the proposed approach performs better than the other state-of-the-art approaches, in terms of Signal-to-Interference Ratio (SIR) and Signal-to-Distortion Ratio (SDR), respectively.