Research Institute for Future Media Computing 未来媒体技术与计算研究所 Research Institute for Future Media Computing http://futuremedia.szu.edu.cn 4. Fourier Transform 江健民，国家千人计划特聘教授 深圳大学未来媒体技术与计算研究所所长 Office Room: 409 Email: jianmin.jiang@szu.edu.cn http://futuremedia.szu.edu.cn

4.1 Fourier Transform Theorem-1: Any function f(t) can be decomposed into harmonic terms, which contain one constant and infinite number of sinusoidal and cosinusoidal terms; Where ω=2πf is known as the fundamental angular frequency, f=1/T is the fundamental frequency, and T is the period of the corresponding harmonic element inside f(t).

Theorem-2(Fourier transform definition): Any function of time t can be transformed into frequency domain and vice versa, which is referred to as Fourier transform; Theorem-3 (discrete Fourier transform): Any discrete function, sampled at a regular time interval T, can be transformed into frequency domain via discrete Fourier transform, which is defined as:

Example: Given a sequence x(nT)={x(0), x(T), x(2T), x(3T)}={1, 0, 0, 1}, calculate its DFT and plot the signal’s strength with respect to the frequency kΩ. Solution:

3Ω 4Ω 2Ω Ω kΩ +45 -45 Plots: 3Ω 4Ω 2Ω Ω kΩ Inverse transform:

Fast Fourier Transform(FFT)

Fourier transform applications (1) System analysis: (2) Signal analysis: (3) Filtering: (4) Research: [1] Wei D. and Li Y. ‘Generalized sampling expansions with multiple sampling rates for lowass and bandpass signals in the fractional Fourier transform domain’, IEEE Transactions on Signal Processing, Vol 64, pp4861-4874, Sept. 2016; [2] Zhao Y., Yu H. and Wei G. ‘Parameter estimation of wideband underwater acoustic multipath channels based on fractional Fourier transform’, IEEE Transactions on Signal Processing, Vol 64, pp5396-5408, Oct 15, 2016; [3] Tian N., Zhang X. et al ‘Two-dimensional discrete fractional Fourier transform-based content removal algorithm’, Signal Image & Video Processing, Vol 10, pp1311-1318, 2016; [4] Akin B. et al ‘FFTs with near-optimal memory access through block data layouts algorithm architecture and design automation’, Journal of Signal Processing Systems for Signal Image & Video Technology, Vol 85, pp67-82, 2016;