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.