Beijing Dublin International College EEEN3004J Digital Signal Processing Spring 2020 Assignment 3 Digital filters and Gibbs ringing John Healy and Wang Yue Work individually. You are to solve the problems given below, and to submit your report on Brightspace. See Brightspace for the submission deadline. Late reports will be penalized according to UCD policy. Gibb’s Phenomenon should be familiar to you. It’s the ringing or oscillation that happens when we try to reconstruct a signal with a discontinuity from its Fourier coefficients. Fig. 1. The signal in red has a discontinuity at t = π and, because it is periodic, also at t = 0 = 2π. The four subfigures show a reconstruction (the blue curve) which uses the lowest 16, 32, 64, or 128 Fourier coefficients to reconstruct the signal. Notice the oscillations near the boundary, which are inevitable because of the reconstruction from Fourier coefficients. A popular method to suppress Gibbs’ phenomenon is called filtered Fourier reconstruction. In this approach, we multiply the truncated Fourier coefficients by the transfer function of a low pass filter with a more graduate transition into the stopband. This approach is illustrated in Fig. 2. Fig. 2. The filtered Fourier reconstruction (1st column, 4th row) shows none of the ringing of the Fourier reconstruction (1st column, 2nd row). However, this filter, a Gaussian (2nd column, 3rd row), has also noticeably altered the shape of the signal. Your task in this assignment is to find the best filter you can to meet two conflicting goals: 1. Reduce the Gibbs’ ringing. 2. Change the signal as little as possible. Problems 1. Each student has been assigned a particular type of filter. See Appendix 1 for the assignments. You will find some helpful information about the filters in a document my phd student prepared for you that is also included on Brightspace. For your type of filter, design and implement the filter. For any parameters that the filter has, find the best parameters you can. (E.g. in the example above, the width of the Gaussian window is a parameter.) 2. You may then freely explore any other filter types you wish to. Some filter types you are assigned may be difficult to implement. Those students will receive a lot of marks for part 1 and don’t need to do as much work on part 2. Other filter types are really easy because they are implemented in MATLAB. Those students will receive more marks for part 2. The MATLAB file included in the assignment will call a MATLAB function called myfilter, use it on four test examples, and calculate some metrics for how well those examples were reconstructed. You should write your own myfilter. You should use those test examples and those metrics to evaluate any filters you design. There are four metrics in the test: • The Mean Squared Error (MSE) provides a measure of the distortion introduced in an image. A good reconstruction will have low MSE. • PSNR is a ratio of the maximum sample power to the power of the reconstruction error. A good reconstruction will have high PSNR. • Entropy is a measure of the information content in an image, and the reduction in entropy introduced by a filter is therefore a proxy measure of the loss of fidelity. For example, the entropy of a signal will decrease with blurring. A good reconstruction will have high entropy. • The variance of a signal is a measure of the difference between the samples and the signal mean; the variance of the reconstruction error will increase if it is corrupted with noise. A good reconstruction will have low variance. You are welcome to search websites and research journal papers for advice about the best filter design. You should reference any information you find in your report. You will submit two files: • a report detailing your investigation, and • a copy of your best myfilter file to support your claims. You don’t need to zip them together, but name them myfilter1234 and report1234, where 1234 is the last four digits of your UCD student number. Include your name and student number at the beginning of the report and the code. Appendix 1 Assignments Student ID Name Filter type 14207109 Wang Xiaozhi Gaussian Filter 15206092 Deng Zida Exponential filter 15206120 Liu Yunhe Erfc-Log Filter 15206134 Sun Tierui Savitzky-Golay filtering 15206137 Tian Xiaoyang Digital Total Variation Filtering 15206141 Wang Jiyu Hann and Hamming windows 15206154 Yang Weiqin The Vandeven filter 15206160 ZHAO ZHAO Parks-McClellan optimal filter 15206164 Zhang Yupeng Butterworth filter 15206168 Zheng Lingruo Chebyshev filter I 15206304 Jiang Canhui Chebyshev filter II 16206535 Bai Wenyuan Elliptic filter 16206539 Cui Jinkai Digital Biquad filter 16206553 Liu Ziyang Kaiser filter 16206560 Wang Xiaoxin Blackman window 16206564 Wu Siyuan Bessel filter 16206565 Wu Wenqi Parzen window 16206570 Zhang Runmin Gaussian Filter 16206573 Zhang Zhelin Exponential filter 16206574 Zhu Lei Erfc-Log Filter 16206709 Yao Xiyao Savitzky-Golay filtering 16206716 Sun Yuqing Digital Total Variation Filtering 16206749 Ma Chi Hann and Hamming windows 16206798 Sun Yiran The Vandeven filter 16206802 Lv Jiaming Parks-McClellan optimal filter 16206807 Wang Tong Butterworth filter 16206810 Chen Qipei Chebyshev filter I 16206812 Xiao Xiangyu Chebyshev filter II 16206814 Zhang Mingyu Elliptic filter 16206820 Lu Tianyang Digital Biquad filter 16206823 Zhu Chensi Kaiser filter 16206829 Ren Zeyu Blackman window 16206832 Chen Yuqiao Bessel filter 16206835 Zhao Yuxin Parzen window 16206868 Zhang Jinming Gaussian Filter 16206955 Feng Haoze Exponential filter 17205857 Gu Chenran Erfc-Log Filter 17205858 Li Xinyu Savitzky-Golay filtering 17205859 Han Jinfang Digital Total Variation Filtering 17205860 Wang Shuyi Hann and Hamming windows 17205861 Zhang Xiaofei The Vandeven filter 17205862 Zhu Ziming Parks-McClellan optimal filter 17205865 Li Tianhao Butterworth filter 17205866 Zhang Cenyue Chebyshev filter I 17205867 Wang Zichen Chebyshev filter II 17205868 Zhang Manlin Elliptic filter 17205869 Wang Jianan Digital Biquad filter 17205870 Shi Bo Kaiser filter 17205871 Zhang Zichen Blackman window 17205872 Zou Xueping Bessel filter 17205873 Qi Wanpeng Parzen window 17205874 Sun Yifeng Gaussian Filter 17205877 Zhang Guangzhen Exponential filter 17205878 Hao Tingting Erfc-Log Filter 17205879 Li Nan Savitzky-Golay filtering 17205880 Wang Pinhua Digital Total Variation Filtering 17205881 Cao Yuan Hann and Hamming windows 17205882 Xu Jiaming The Vandeven filter 17205883 Yuan Xiling Parks-McClellan optimal filter 17205884 Li Zichen Butterworth filter 17205885 Zou Yang Chebyshev filter I 17205886 Li Jiashu Chebyshev filter II 17205888 Hu Jiayi Elliptic filter 17205889 Bai Wanfeng Digital Biquad filter 17205890 Li Xinghao Kaiser filter 17205892 Fang Xiang Blackman window 17205893 Guo Haoran Bessel filter 17205894 Chen Yixiao Parzen window 17205897 Xu Zhikun Gaussian Filter 17205898 Han Sanyue Exponential filter 17205900 Zhu Yanxing Erfc-Log Filter 17205901 Yang Ruicui Savitzky-Golay filtering 17205904 Qiu Sitao Digital Total Variation Filtering 17205905 Li Yuan Hann and Hamming windows 17205906 Zhao Zijie The Vandeven filter 17205907 Zhang Youwu Parks-McClellan optimal filter 17205908 Zhang Zhengyan Butterworth filter 17205909 Wu Bochen Chebyshev filter I 17205910 Zhang Xinyan Chebyshev filter II 17205911 Yuan Xiaoran Elliptic filter 17205912 Zhang Yuhui Digital Biquad filter 17205913 Wang Zhengpu Kaiser filter 17205914 Gong Chen Blackman window 17205915 Wang Siqi Bessel filter 17205916 Wang Zhining Parzen window 17205918 Bian Yuhan Gaussian Filter 17205919 Gao Yuzhe Exponential filter 17205920 Zhang Qiyue Erfc-Log Filter 17205921 Ma Siteng Savitzky-Golay filtering 17205922 Xu Yiruo Digital Total Variation Filtering 17205924 Lu Jiacheng Hann and Hamming windows 17205925 Zhao Yuting The Vandeven filter 17205926 Jia Zixuan Parks-McClellan optimal filter 17205927 Xiao Shib
ang Butterworth filter 17205928 Wang Weixing Chebyshev filter I 17205930 Fang shicheng Chebyshev filter II 17205931 Wang Ziyi Elliptic filter 17205932 Cao Yunfeng Digital Biquad filter 17205933 Wang Kaize Kaiser filter 17205935 Zhang Aoran Blackman window 17205950 Yang Feifan Bessel filter 17205952 Fu Ziyi Parzen window 17205953 Zhang Ran Gaussian Filter 17205954 Qi Tianzhuo Exponential filter 17205955 Li Jinglin Erfc-Log Filter 17205956 Wu Ming Yang Savitzky-Golay filtering 17205957 Zhu Yucheng Digital Total Variation Filtering 17206005 Luo Yuzhao Hann and Hamming windows 17206012 Chen Hanming The Vandeven filter 17206013 Zhou Puqi Parks-McClellan optimal filter 17206014 Jian Dingding Butterworth filter 17206015 Li Jiahua Chebyshev filter I 17206016 Tian Feng Chebyshev filter II 17206018 Cheng Litao Elliptic filter 17206019 Guo Xu Digital Biquad filter 17206020 Que Chencan Kaiser filter 17206021 Wang Peizhao Blackman window 17206022 Li Xiang Bessel filter 17206023 Chen Haixin Parzen window 17206024 Wei Lian Gaussian Filter 17206040 Chen Xiang Exponential filter 17206041 Lu Jiahe Erfc-Log Filter 17206151 Gu Zhenlei Savitzky-Golay filtering 17206185 Wen Yannuo Digital Total Variation Filtering 17206205 Wang Xuliang Hann and Hamming windows 17206206 Zhang Yuxiang The Vandeven filter 17206208 Chen Dingrui Parks-McClellan optimal filter 17206209 Li Chengjin Butterworth filter 17206210 Sun Buwei Chebyshev filter I 17206211 Li Ruijie Chebyshev filter II 17206221 Tang Song Elliptic filter 17206238 Wu Haochang Digital Biquad filter
