Fractional Fourier Transform and Time-Frequency Analysis and Apply to Acoustic Signals
Date Issued
2008
Date
2008
Author(s)
Kuo, Kuo-Cyuan
Abstract
Fourier analysis takes an important role in signal processing, and we often use it to decompose frequencies for further applications. However, in the time-frequency analy-sis, the Fourier transform is not good enough. Since the Fourier transform can only deal with the stationary signals but can not deal with non-stationary or time-varying signals. Although we can develop time-frequency analysis to help us deal with time-varying signals, we also can extend the math of Fourier transform itself – fractional Fourier transform.This thesis mainly has two parts: the first part is to extend the math of the Fourier transform. This includes the theorem of the fractional Fourier transform and the linear canonical transform. The fractional Fourier transform can combine time-frequency analysis, and extend to applications of optics and Radar systems. We also derive the uncertainty principles of the fractional Fourier transform and the linear canonical trans-form, and discrete the continuous fractional Fourier transform with ei-gen-decomposition. One dimensional fractional Fourier transform can produce filters which traditional Fourier transform cannot and also reduce the sampling rate and en-cryption…. Two dimensional fractional Fourier transform can deal with image and op-tics.The second part will introduce the time-frequency distribution systematically. We list pros and cons of each time-frequency distribution and propose three methods to re-duce the computation, including adaptive time-frequency distribution. We will apply these tools to music and acoustics signals. We are going to realize some parts of the auto-transcription and discuss the problems we face and the solutions, including exceed the harmonics and computation. Finally, there are our results and simulations.
Subjects
fractional Fourier transform
linear canonical transform
uncertainty principle
time-frequency distribution
adaptive time-frequency distribution
Type
thesis
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