tailieunhanh - Phân tích tín hiệu P7

Short-Time Fourier Analysis A fundamental problem in signal analysis is to find the spectral components containedin a measuredsignal z ( t ) and/or to provide information about the time intervals when certain frequencies occur. An example of what we are looking for is a sheet of music, which clearly assigns time to frequency, see Figure . The classical Fourier analysis only partly solves the problem, because it does not allow an assignment of spectralcomponents to time. Therefore one seeks other transforms which give insight into signal properties in a different way. . | Signal Analysis Wavelets Filter Banks Time-Frequency Transforms and Applications. Alfred Mertins Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4 Chapter 7 Short-Time Fourier Analysis A fundamental problem in signal analysis is to find the spectral components contained in a measured signal x t and or to provide information about the time intervals when certain frequencies occur. An example of what we are looking for is a sheet of music which clearly assigns time to frequency see Figure . The classical Fourier analysis only partly solves the problem because it does not allow an assignment of spectral components to time. Therefore one seeks other transforms which give insight into signal properties in a different way. The short-time Fourier transform is such a transform. It involves both time and frequency and allows a time-frequency analysis or in other words a signal representation in the time-frequency plane. Continuous-Time Signals Definition The short-time Fourier transform STFT is the classical method of timefrequency analysis. The concept is very simple. We multiply x t which is to be analyzed with an analysis window 7 t r and then compute the Fourier 196 . Continuous-Time Signals 197 Figure . Time-frequency representation. transform of the windowed signal co x t 7 t - r dt. 00 7-1 The analysis window 7 t r suppresses ar t outside a certain region and the Fourier transform yields a local spectrum. Figure illustrates the application of the window. Typically one will choose a real-valued window which may be regarded as the impulse response of a lowpass. Nevertheless the following derivations will be given for the general complex-valued case. If we choose the Gaussian function to be the window we speak of the Gabor transform because Gabor introduced the short-time Fourier transform with this particular window 61 . Shift Properties. As we see from the analysis equation a time shift x t x t to .