Home / Problem Set 2 / Trigonometric Series for Square Wave

Trigonometric Series for Square Wave

The questions below are due on Monday March 01, 2021; 10:00:00 PM.
 
You are not logged in.

If you are a current student, please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.csail.mit.edu) to authenticate, and then you will be redirected back to this page.
Note that this exercise is intended to be solved by hand, without the use of computation.

1) Square Wave

Consider the following signal, which is periodic with a fundamental period of T=1 second:

Let c_k and d_k be the Fourier series coefficients (trigonometric form) associated with this signal, such that

x(t)= c_0 + \sum_{k=1}^\infty \big(c_k\cos (k\omega_ot) + d_k\sin (k\omega_ot)\big)

What is the DC term, c_0?

c_0 =~

For k \geq 1, what is the value of c_k? Your answer may include the variable k, and you can use pi to refer to \pi.

c_k =~

For k \geq 1, what is the value of d_k? Your answer may include the variable k, and you can use pi to refer to \pi.

d_k =~

Which of the following are true?
c_k = 0 for all k
d_k = 0 if k is even
|d_k| decreases like k^2
there are an infinite number of non-zero d_k