Work individually.
Beijing Dublin International College
Assignment 3
Digital filters and Gibbs ringing
Spring 2020
John Healy and Wang Yue
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.
EEEN3004J Digital Signal Processing
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 14207109 15206092 15206120 15206134 15206137 15206141 15206154 15206160 15206164 15206168 15206304 16206535 16206539 16206553 16206560 16206564 16206565 16206570 16206573 16206574 16206709
Name
Wang Xiaozhi Deng Zida
Liu Yunhe
Sun Tierui Tian Xiaoyang Wang Jiyu Yang Weiqin ZHAO ZHAO Zhang Yupeng Zheng Lingruo Jiang Canhui Bai Wenyuan Cui Jinkai
Liu Ziyang Wang Xiaoxin Wu Siyuan Wu Wenqi Zhang Runmin Zhang Zhelin Zhu Lei
Yao Xiyao
Filter type
Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I Chebyshev filter II Elliptic filter
Digital Biquad filter Kaiser filter Blackman window Bessel filter
Parzen window Gaussian Filter Exponential filter Erfc-Log Filter Savitzky-Golay filtering
16206716
16206749
16206798
16206802
16206807
16206810
16206812
16206814
16206820
16206823
16206829
16206832
16206835
16206868
16206955
17205857
17205858
17205859
17205860
17205861
17205862
17205865
17205866
17205867
17205868
17205869
17205870
17205871
17205872
17205873
17205874
17205877
17205878
17205879
17205880
17205881
17205882
17205883
17205884
17205885
17205886
17205888
17205889
17205890
17205892
Sun Yuqing Ma Chi
Sun Yiran
Lv Jiaming Wang Tong Chen Qipei Xiao Xiangyu Zhang Mingyu Lu Tianyang Zhu Chensi Ren Zeyu Chen Yuqiao Zhao Yuxin Zhang Jinming Feng Haoze Gu Chenran
Li Xinyu
Han Jinfang Wang Shuyi Zhang Xiaofei Zhu Ziming
Li Tianhao Zhang Cenyue Wang Zichen Zhang Manlin Wang Jianan Shi Bo
Zhang Zichen Zou Xueping Qi Wanpeng Sun Yifeng Zhang Guangzhen Hao Tingting Li Nan
Wang Pinhua Cao Yuan
Xu Jiaming Yuan Xiling
Li Zichen
Zou Yang
Li Jiashu
Hu Jiayi
Bai Wanfeng Li Xinghao Fang Xiang
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I Chebyshev filter II Elliptic filter
Digital Biquad filter Kaiser filter Blackman window Bessel filter
Parzen window
Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I Chebyshev filter II Elliptic filter
Digital Biquad filter Kaiser filter Blackman window Bessel filter
Parzen window Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I Chebyshev filter II Elliptic filter
Digital Biquad filter Kaiser filter Blackman window
17205893
17205894
17205897
17205898
17205900
17205901
17205904
17205905
17205906
17205907
17205908
17205909
17205910
17205911
17205912
17205913
17205914
17205915
17205916
17205918
17205919
17205920
17205921
17205922
17205924
17205925
17205926
17205927
17205928
17205930
17205931
17205932
17205933
17205935
17205950
17205952
17205953
17205954
17205955
17205956
17205957
17206005
17206012
17206013
17206014
17206015
Guo Haoran Chen Yixiao Xu Zhikun Han Sanyue Zhu Yanxing Yang Ruicui Qiu Sitao
Li Yuan
Zhao Zijie
Zhang Youwu Zhang Zhengyan Wu Bochen Zhang Xinyan Yuan Xiaoran Zhang Yuhui Wang Zhengpu Gong Chen Wang Siqi
Wang Zhining Bian Yuhan
Gao Yuzhe Zhang Qiyue Ma Siteng
Xu Yiruo
Lu Jiacheng Zhao Yuting
Jia Zixuan
Xiao Shibang Wang Weixing Fang shicheng Wang Ziyi
Cao Yunfeng Wang Kaize Zhang Aoran Yang Feifan
Fu Ziyi
Zhang Ran
Qi Tianzhuo
Li Jinglin
Wu Ming Yang Zhu Yucheng Luo Yuzhao Chen Hanming Zhou Puqi
Jian Dingding
Li Jiahua
Bessel filter
Parzen window
Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I
Chebyshev filter II
Elliptic filter
Digital Biquad filter
Kaiser filter
Blackman window
Bessel filter
Parzen window
Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I
Chebyshev filter II
Elliptic filter
Digital Biquad filter
Kaiser filter
Blackman window
Bessel filter
Parzen window
Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I
17206016
17206018
17206019
17206020
17206021
17206022
17206023
17206024
17206040
17206041
17206151
17206185
17206205
17206206
17206208
17206209
17206210
17206211
17206221
17206238
Tian Feng Cheng Litao Guo Xu
Que Chencan Wang Peizhao Li Xiang
Chen Haixin Wei Lian Chen Xiang Lu Jiahe
Gu Zhenlei Wen Yannuo Wang Xuliang Zhang Yuxiang Chen Dingrui Li Chengjin Sun Buwei
Li Ruijie
Tang Song Wu Haochang
Chebyshev filter II Elliptic filter
Digital Biquad filter Kaiser filter Blackman window Bessel filter
Parzen window
Gaussian Filter
Exponential filter
Erfc-Log Filter
Savitzky-Golay filtering
Digital Total Variation Filtering Hann and Hamming windows The Vandeven filter Parks-McClellan optimal filter Butterworth filter
Chebyshev filter I Chebyshev filter II Elliptic filter
Digital Biquad filter