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 Shibang 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