Exploring Fourier Series and Trigonometric Interpolating Polynomials in MATLAB
Summary
This Assignment on MATLAB Grader is related to "Trigonometric Interpolating Polynomials" in Discrete Fourier Transform.
Students work individually on the exercise after receiving a lecture on the related theory. They will create sampled data from a given continuous function and complete a pre-structured DFT MATLAB script in the Learner platform to find the coefficients in interpolating trig polynomials, ak and bk and reconstruct the original data.
The output is the reconstructed data and it should be in good agreement with the original continuous function.
The student will also learn how changing the sampling frequency affects the accuracy of the reconstruction. They will put the Nyquist criteria into test for themselves to understand its logic better.
CollapseLearning Goals
1- Understand the concept of Trigonometric Polynomial Approximation.
2- Create a discrete sampled set from a given continuous function with the specified sampling rate.
3- Use proper MATLAB functions/commands to apply related theory.
4- Create proper plots to compare the reconstructed data to the original data.
5- Analyze the effect of sampling frequency on the accuracy of the reconstructed data.
6- Find the minimum required sample points to effectively reconstruct the highest frequency component included in the approximation.
Context for Use
This exercise is one of the four exercises for a lab activity for the course: "Scientific Computing 2" for third year BTech students in Electrical Engineering Program.
The course consists of 7 labs for 7 weeks on numerical techniques and computational challenges.This exercise covers Fourier Series, trigonometric Interpolating polynomials and Nyquist frequency.
The students need basic MATLAB knowledge to complete the previously provided MATLAB script in the learner platform. The students also need intermediate level knowledge of the Fourier Series which they receive as a pre-lab lecture.
This exercise can be easily integrated to other courses/activities.
Description and Teaching Materials
This lab is based on the theory presented in the course text book (given in References and Resources section).
In the pre-lab lecture, I talk about many real life situations where we need to use sampled data (audio/visual processing, noise cancelation,etc) and what we can extract from the set of sampled data. Could we separate the noise? Can we find the frequencies present in the original signal?
What is the relation between the number of sampled data points we had and the maximum frequency we can detect and include in our Trig polynomial? This activity provides the opportunity for the students to answer such questions.
Teaching Notes and Tips
Teaching Discrete Fourier Transform requires a background on calculation of Fourier Series and this is what I usually cover before this activity.
I will also visit students one by one while they are working on the exercise to share their joy of finding answers :) and also give them feedback.
And since teaching is better when combined with learning, I discuss things with the students and use their thoughts and creativity to better my assessments for later.
Assessment
This activity is an exercise in a Lab for the course. The activity has 4 assessments which test if the student was able to calculate the coefficients for Fourier Interolating Polynomials correctly and reconstruct the data. A summary of the assessment is copied here Assessment Description.pdf (Acrobat (PDF) 355kB Oct29 24).
References and Resources
Text Book for the course: Numerical Analysis, Richard L. Burden, J. Douglas Faires
Lecture Notes Based on Chapter 8, Approximation Theory, 8.5 : Trigonometric Polynomial Approximation.