Fourier Varieties

The questions below are due on Thursday March 13, 2025; 02:00:00 PM.
 
You are not logged in.

Please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.mit.edu) to authenticate, and then you will be redirected back to this page.

Enter all answers as Python expressions. sin, cos, e, j, and pi are available, as are functions Re (which computes the real part of its argument) and Im (which computes the imaginary part of its argument).

Fourier Series

Determine the Fourier series coefficients of the following signal, which is periodic in T=10.

X[0]=~

For k \neq 0, X[k]=~

Determine the Fourier transform of the following signal, which is zero outside the indicated range. Enter your answer as a Python expression in terms of omega.

X_2(\omega)=~

What is the relation between the answers to parts a and b? In particular, derive an expression for X[k] (the solution to part 1) in terms of X_2(\omega) (the solution to part 2). Enter your answer as a Python expression in terms of X_2 (capitalized), k, and any other necessary constants.

X[k]=~

Determine the time waveform that corresponds to the following Fourier transform, which is zero outside the indicated range.

x_3(t)=~

What is the relation between the answers to parts b and d? (not graded)