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Feel free to work on the assignment with other students in the class, however please write up your own work and code

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Feel free to work on the assignment with other students in the class, however please write up your own work and code. All code must be turned in as a .IPYNB file AND a link to your Google Collab notebook. Code will not be graded without both of these. When you are asked to create a function to solve some problem, do not use some built-in function. Non-coding solutions may either be turned in as a PDF of handwritten work or a PDF of a INIBdocument. No file types other than PDF will be accepted. Please show all of your work. All problems or parts of problems are worth 10 points.

1. Give the discrete Fourier Transform of the following vectors:

(a)

(b)

2. (Coding) Check your answer to Question 1 using Python (scipy.fft is a good option).
3. Apply the inverse discrete Fourier transform to each of your answers from question 1 to show that you get the original vector back.
4. Using your answers from question 1, determine the trigonometric function of best fit through the following points. Plot your answer. (a) (1, O), (2, 1/4), (3, 1/2), (4, 3/4)

(b) (1, o), (1, 1/4), (1, 1/2, (1, 3/4)

5. (Coding) Use Python (scipy.fft is a good option) to find the order n 4 and order n 8 trigonometric interpolating function for f (t) — e^t at n evenly spaced points in [0, 1]. Plot the graphs and comment on their fit of the function.