tailieunhanh - Báo cáo hóa học: " Research Article On the Characterization of Slowly Varying Sinusoids"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Characterization of Slowly Varying Sinusoids | Hindawi Publishing Corporation EURASIP Journal on Audio Speech and Music Processing Volume 2010 Article ID 941732 7 pages doi 2010 941732 Research Article On the Characterization of Slowly Varying Sinusoids Xue Wen EURASIP Member and Mark Sandler Centre for Digital Music Queen Mary University of London Mile End Road London E1 4NS UK Correspondence should be addressed to Xue Wen Received 4 March 2010 Revised 29 April 2010 Accepted 6 July 2010 Academic Editor Roger Dannenberg Copyright 2010 X. Wen and M. Sandler. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We give a brief discussion on the amplitude and frequency variation rates of the sinusoid representation of signals. In particular we derive three inequalities that show that these rates are upper bounded by the 2nd and 4th spectral moments which in a loose sense indicates that every complex signal with narrow short-time bandwidths is a slowly varying sinusoid. Further discussions are given to show how this result helps providing extra insights into relevant signal processing techniques. 1. Introduction Sinusoid representations of signals have been widely used in various signal processing areas including speech 1 music 2 3 telecommunications 4 . Given any complex variable x x t its sinusoid representation is x t a t ejv t a t x t p t Arg x t 1 where the real variables a t and q t are the amplitude and phase angle or phase for short of x. By this definition every nonzero complex variable has a unique sinusoid representation up to the polarization of a and 2kn shift of p k e Z. In practice these ambiguities are relieved by assuming that a and p be continuous and smooth 5 . The parameter variations considered in this paper are the first and second derivatives of a and the second derivative of p which respectively .