10 頻率會隨著時間而變化的例子： Frequency Modulation (FM) Signal Speech Music Others (Animal voice, Doppler effect, seismic waves, radar system, optics, rectangular function) In fact, in addition to sinusoid-like functions, the instantaneous frequencies of other functions will inevitably vary with time.
12 時頻分析理論發展年表 AD 1785 The Laplace transform was invented AD 1812 The Fourier transform was invented AD 1822 The work of the Fourier transform was published AD 1910 The Haar Transform was proposed AD 1927 Heisenberg discovered the uncertainty principle AD 1929 The fractional Fourier transform was invented by Wiener AD 1932 The Wigner distribution function was proposed AD 1946 The short-time Fourier transform and the Gabor transform was proposed. (In the same year, the computer was invented) AD 1961 Slepian and Pollak found the prolate spheroidal wave function AD 1966 Cohen’s class distribution was invented
13 AD 1971 Moshinsky and Quesne proposed the linear canonical transform AD 1980 The fractional Fourier transform was re-invented by Namias AD 1981 Morlet proposed the wavelet transform AD 1982 The relations between the random process and the Wigner distribution function was found by Martin and Flandrin AD 1988 Mallat and Meyer proposed the multiresolution structure of the wavelet transform; In the same year, Daubechies proposed the compact support orthogonal wavelet AD 1989 The Choi-Williams distribution was proposed; In the same year, Mallat proposed the fast wavelet transform AD 1990 The cone-Shape distribution was proposed by Zhao, Atlas, and Marks AD 1993 Mallat and Zhang proposed the matching pursuit; In the same year, the rotation relation between the WDF and the fractional Fourier transform was found by Lohmann
14 AD 1994 The applications of the fractional Fourier transform in signal processing were found by Almeida, Ozaktas, Wolf, Lohmann, and Pei AD 1995 L. J. Stankovic, S. Stankovic, and Fakultet proposed the pseudo Wigner distribution AD 1996 Stockwell, Mansinha, and Lowe proposed the S transform AD 1998 N. E. Huang proposed the Hilbert-Huang transform AD 1999 Candes, Donoho, Antoine, Murenzi, and Vandergheynst proposed the directional wavelet transform AD 2000 The standard of JPEG 2000 was published by ISO AD 2002 Stankovic proposed the time frequency distribution with complex arguments AD 2003 Pinnegar and Mansinha proposed the general form of the S transform AD 2007 The Gabor-Wigner transform was proposed by Pei and Ding
16 時頻分析的大家族 (1) Short-time Fourier transform (STFT) (rec-STFT, Gabor, …) square spectrogram improve S transform (2) Wigner distribution function (WDF) combine Gabor-Wigner Transform improve windowed WDF improve Cohen’s Class Distribution (Choi-Williams, Cone-Shape, Page, Levin, Kirkwood, Born-Jordan, …) improve Pseudo L-Wigner Distribution (4) Time-Variant Basis Expansion Matching Pursuit Prolate Spheroidal Wave Function (5) Hilbert-Huang Transform (3) Wavelet transform Haar and Daubechies Coiflet, Morlet Directional Wavelet Transform ( 唯一跳脫 Fourier transform 的架構 ) Asymmetric STFT
17 (1) Short-Time Fourier Transform (2) Wigner Distribution Function
18 Simulations x(t) = cos(2 t) by WDF by short-time Fourier transform f-axis t-axis
19 (3) Wavelet Transform x 1,L [n] x 1,H [n] g[n] x[n]x[n] h[n] 2 x[n] 的低頻成份 x[n] 的高頻成份 lowpass filter highpass filter down sampling xL[n]xL[n] xH[n]xH[n] N-points L-points
20 例子： 2-point Haar wavelet g[n] = 1/2 for n = −1, 0 g[n] = 0 otherwise h = 1/2, h[−1] = −1/2, h[n] = 0 otherwise n g[n] -3 -2 -1 0 1 2 3 ½ n h[n] -3 -2 -1 0 1 2 3 ½ -½ then ( 兩點平均 )( 兩點之差 )
21 x[m, n] g[n]g[n] h[n]h[n] 2 along n v 1,L [m, n] v 1,H [m, n] g[m]g[m] h[m]h[m] along m 2 x 1,L [m, n] 2 along m x 1,H1 [m, n] g[m]g[m] h[m]h[m] along m 2 x 1,H2 [m, n] x 1,H3 [m, n] 2-D 的情形 m 低頻, n 低頻 m 高頻, n 低頻 m 低頻, n 高頻 m 高頻, n 高頻 L-points M ×N n m
22 原影像 2-D DWT 的結果 x 1,L [m, n] x 1,H1 [m, n] x 1,H2 [m, n] x 1,H3 [m, n]
30 3-3 S Transform [Ref] R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform,” IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, Apr. 1996. 比較：原本的 short-time Fourier transform 當 w(t) = exp( t 2 ) 時 f , window width f , window width
31 x(t) = cos( t) when t < 10, x(t) = cos(3 t) when 10 t < 20, x(t) = cos(2 t) when t 20 Using the short-time Fourier transform
32 x(t) = cos( t) when t < 10, x(t) = cos(3 t) when 10 t < 20, x(t) = cos(2 t) when t 20 Using the S transform
33 3-4 Gabor-Wigner Transform 如何同時達成 (1) high clarity (2) no cross-term 的目標？ by WDF by short-time Fourier transform f-axis t-axis cos(2 t)
34 Short-time Fourier transform Wigner [Ref] S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans. Signal Processing, vol. 55, no. 10, pp. 4839-4850, Oct. 2007.
36 Bandlet 根據物體的紋理或邊界，來調整 wavelet transforms 的方向 Stephane Mallet and Gabriel Peyre, "A review of Bandlet methods for geometrical image representation," Numerical Algorithms, Apr. 2002.
37 3-6 Hilbert-Huang Transform ( 國產 ) 時頻分析，何必要用到那麼複雜的數學？ (Step 2) Find the local peaks 0 1 2 y(t) (Step 1) Initial: y(t) = x(t), (x(t) is the input) n = 1, k = 1 為中研院黃鍔院士於 1998 年提出
38 (Step 3) Connect local peaks 0 1 2 IMF 1; iteration 0 通常使用 B-spline ，尤其是 cubic B-spline 來連接
39 (Step 4) Find the local dips (Step 5) Connect the local dips
40 (Step 6-1) Compute the mean (pink line) (Step 6-2) Compute the residue -1.5 -0.5 0 0.5 1 1.5
41 Step 7 Repeat Steps 1-6 to determine the intrinsic mode function (IMF) Step 8 Repeat Steps 1-7 to further determine x(t) Step 9 Determine the instantaneous frequency for each IMF (just calculating the number of zero-crossing during [t-1/2, t+1/2])