Fourier Series

The questions below are due on Thursday February 20, 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.

Let X_i[k] represent the k^{th} coefficient in the Fourier Series expansion of a periodic signal x_i(t)=x_i(t+T), so that

X_i[k]={1\over T}\int_T x_i(t)e^{-j2\pi kt/T}\,dt

Part 1

Determine the Fourier series coefficients X_1[k] for x_1(t) shown below.

Enter your answers in the boxes below. Use Python notation for multiplication, exponentiation, etc. Some mathematical constants and functions are available (pi, j, e, sin, cos, etc), as well as the independent variable k.

X_1[0] =

For k \neq 0, X_1[k] =

Part 2

Determine the Fourier series coefficients X_2[k] for x_2(t) shown below.

X_2[0] =

For k \neq 0, X_2[k] =

Part 3

Determine the Fourier series coefficients X_3[k] for x_3(t) shown below (hint: how does x_3 relate to the signals above?):

X_3[0] =

For k \neq 0, X_3[k] =

Part 4

Determine the Fourier series coefficients X_4[k] for x_4(t) shown below (hint: how does this signal relate to x_3(\cdot)?):

X_4[0] =

For k \neq 0, X_4[k] =