X. Other Applications of Time-Frequency Analysis (1) Finding Instantaneous Frequency (2) Signal Decomposition (3) Filter Design (4) Sampling Theory (5) Modulation and Multiplexing (6) Electromagnetic Wave Propagation (7) Optics (8) Radar System Analysis (9) Random Process Analysis (10) Music Signal Analysis (11) Biomedical Engineering (12) Accelerometer Signal Analysis (13) Acoustics (14) Data Compression (15) Spread Spectrum Analysis (16) System Modeling (17) Image Processing (18) Economic Data Analysis (19) Signal Representation (20) Seismology (21) Geology (22) Astronomy (23) Oceanography

10-1 Sampling Theory Number of sampling points == Area of time frequency distribution + The number of extra parameters How to make the area of time-frequency smaller? (1) Divide into several components. (2) Use chirp multiplications, chirp convolutions, fractional Fourier transforms, or linear canonical transforms to reduce the area. [Ref] X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Processing Letters, vol. 3, no. 3, pp. 72-74, March 1996. [Ref] J. J. Ding, S. C. Pei, and T. Y. Ko, “Higher order modulation and the efficient sampling algorithm for time variant signal,” European Signal Processing Conference, pp. 2143-2147, Bucharest, Romania, Aug. 2012.

shearing Area

+ Step 1 Separate the components (a) (b) Step 2 Use shearing or rotation to minimize the “area” to each component Step 3 Use the conventional sampling theory to sample each components

傳統的取樣方式 重建： 新的取樣方式 k = 1, 2, …, K k = 1, 2, …, K

重建：

嚴格來說，沒有一個信號的 時頻分佈的「面積」是有限的。 Theorem: If x(t) is time limited (x(t) = 0 for t < t1 and t > t2) then it is impossible to be frequency limited If x(t) is frequency limited (X(f) = 0 for f < f1 and f > f2) then it is impossible to be time limited 但是我們可以選一個 “threshold”  時頻分析 |X (t, f)| >  或 的區域的面積是有限的 實際上，以「面積」來討論取樣點數，是犧牲了一些精確度。

只取 t  [t1, t2] and f  [f1, f2] 犧牲的能量所佔的比例 X1(f) = FT[x1(t)], x1(t) = x(t) for t  [t1, t2] , x1(t) = 0 otherwise  For the Wigner distribution function (WDF) = energy of x(t).

A B C D f-axis D f2 B A t-axis t1 t2 f1 C

10-2 Modulation and Multiplexing With the aid of the Gabor transform (or the Gabor-Wigner transform) horizontal shifting and vertical shifting, dilation, tilting, and rotation. [Ref] C. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A, vol. 14, pp. 558-562, Mar. 1997. [Ref] S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” vol. 55, issue 10, pp. 4839-4850, IEEE Trans. Signal Processing, 2007.

Example FT We want to add f(t) into G(u) (no empty band)

◎ Conventional Modulation Theory The signals x1(t), x2(t), x3(t), ……., xK(t) can be transmitted successfully if Allowed Bandwidth  Bk: the bandwidth (including the negative frequency part) of xk(t) ◎ Modulation Theory Based on Time-Frequency Analysis The signals x1(t), x2(t), x3(t), ……., xK(t) can be transmitted successfully if Allowed Time duration  Allowed Bandwidth  The interference is inevitable. How to estimate the interference? Ak: the area of the time-frequency distribution of xk(t)

10-3 Electromagnetic Wave Propagation Time-Frequency analysis can be used for Wireless Communication Optical system analysis Laser Radar system analysis Propagation through the free space (Fresnel transform): chirp convolution Propagation through the lens or the radar disk: chirp multiplication

Fresnel Transform ：描述電磁波在空氣中的傳播 (See page 220) 電磁波包括光波、雷達波、紅外線、紫外線……… Fresnel transform == LCT with parameters 思考： (1) STFT 或 WDF 哪一個比較適合用在電磁波傳播的分析？ (2) 為何波長越短的電磁波，在空氣中散射的情形越少？

(4) Spherical Disk y-axis x-axis direction of wave propagation R radius of the disk = R plane Disk 相當於 LCT 的情形

RA RB disk A disk B D 相當於 LCT 的情形

10-4 Music and Acoustic Signal Analysis Music Signal Analysis Acoustic Voiceprint (Speaker) Recognition Speech Signal ： (1) 不同的人說話聲音頻譜不同 (2) 同一個人但不同的字音，頻譜不一樣 (3) 語調 (第一、二、三、四聲和輕聲) 不同，則頻譜 變化的情形也不同 (4) 即使同一個字音，子音和母音的頻譜亦不相同 (5) 雙母音本身就會有頻譜的變化  王小川， “語音訊號處理”，第二章，全華出版，台北，民國94年。

large energy middle energy small energy large energy Typical relations between time and the instantaneous frequencies for (a) the 1st tone, (b) the 2nd tone, (c) the 3rd tone, and (d) the 4th tone in Chinese. X. X. Chen, C. N. Cai, P. Guo, and Y. Sun, “A hidden Markov model applied to Chinese four-tone recognition,” ICASSP, vol. 12, pp. 797-800, 1987.

ㄚ1, ㄚ2, ㄚ3, ㄚ4

10-5 Accelerometer Signal Analysis The 3-D Accelerometer (三軸加速規) can be used for identifying the activity of a person. z-axis y: 0 z: -9.8 z-axis y-axis y-axis tilted by θ x-axis z-axis y-axis y: -9.8sinθ z: -9.8cosθ

Using the 3D accelerometer + time-frequency analysis, one can analyze the activity of a person. Walk, Run (Pedometer 計步器) Healthcare for the person suffered from Parkinson’s disease

The result of the short-time Fourier transform 3D accelerometer signal for a person suffering from Parkinson’s disease The result of the short-time Fourier transform Y. F. Chang, J. J. Ding, H. Hu, Wen-Chieh Yang, and K. H. Lin, “A real-time detection algorithm for freezing of gait in Parkinson’s disease,” IEEE International Symposium on Circuits and Systems, Melbourne, Australia, pp. 1312-1315, May 2014

10-6 Other Applications 時頻分析適用於頻譜會隨著時間而改變的信號 Biomedical Engineering (ECG, EMG, ……) Communication and Spread Spectrum Analysis Economic Data Analysis Seismology Geology Astronomy Oceanography Satellite Signal

Short-time Fourier transform of the power signal from a satellite 福爾摩沙衛星三號 C. J. Fong, S. K. Yang, N. L. Yen, T. P. Lee, C. Y. Huang, H. F. Tsai, S. Wang, Y. Wang, and J. J. Ding, “Preliminary studies of the applications of HHT (Hilbert-Huang transform) on FORMOSAT-3/COSMIC GOX payload trending data,” 6th FORMOSAT-3/COSMIC Data Users' Workshop, Boulder, Colorado, USA, Oct. 2012

時頻分析的應用範圍 astronomy satellite over 700 km communication human life vocal signal, ECG vocal signal oceanography over 1000m geology ocean crust

附錄十：幾個常見的資料蒐尋方法 (1) Google 學術搜尋 http://scholar.google.com.tw/ (太重要了，不可以不知道) 只要任何的書籍或論文，在網路上有電子版，都可以用這個功能查得到 再按「搜尋」，就可找到想要的資料 輸入關鍵字，或期刊名，或作者

(2) 尋找 IEEE 的論文 http://ieeexplore.ieee.org/Xplore/guesthome.jsp (3) Google (4) Wikipedia (5) 數學的百科網站 http://eqworld.ipmnet.ru/index.htm 有多個 tables，以及對數學定理的介紹 (6) 傳統方法：去圖書館找資料 台大圖書館首頁 http://www.lib.ntu.edu.tw/ 或者去 http://www.lib.ntu.edu.tw/tulips

(7) 查詢其他圖書館有沒有我要找的期刊 台大圖書館首頁 其他聯合目錄 全國期刊聯合目錄資料庫 如果發現其他圖書館有想要找的期刊，可以申請「館際合作」，請台大圖書館幫忙獲取所需要的論文的影印版 台大圖書館首頁 館際合作 (8) 查詢其他圖書館有沒有我要找的書 「台大圖書館首頁」 「其他圖書館」 (9) 找尋電子書 「台大圖書館首頁」 「電子書」 或「免費電子書」

(10) 中文電子學位論文服務 http://www.cetd.com.tw/ec/index.aspx 可以查到多個碩博士論文 (尤其是 2006年以後的碩博士論文) 的電子版 (11) 查詢一個期刊是否為 SCI Step 1: 先去 http://scientific.thomson.com/mjl/ Step 2: 在 Search Terms 輸入期刊全名 Search Type 選擇 “Full Journal Title”，再按 “Search” Step 3: 如果有找到這期刊，那就代表這個期刊的確被收錄在 SCI

(12) 想要對一個東西作入門但較深入的了解: 看 journal papers 或 Wikipedia 會比看 conference papers 適宜 看書會比看 journal papers 或 Wikipedia 適宜 如果實在沒有適合的書籍，可以看 “review”， “survey”，或 “tutorial” 性質的論文 (13) 有了相當基礎之後，再閱讀 journal papers (以 Paper Title， Abstract， 以及其他 Papers 對這篇文章的描述， 來判斷這篇 journal papers 應該詳讀或大略了解即可)