Fourier Series - Asymmetric Square Wave

[DmytriE]

Good morning everyone,

I am taking a signals and systems course where we are now studying the Fourier series. I understand that this is for signals that are periodic. But I get hung up when determining the Fourier coefficients. In the video by Alan Oppenheim, he derives the equation for the Fourier series. Below is the analysis equation.

a_{k} = \frac{1}{T_{0}} ∫ x(t)*e^{jk\omega_{0}t}

He goes through an example using an asymmetric square wave with an amplitude of 1. I understand the bounds that he chooses (-T_0/2, 0) and (0, T_0/2).

This leads to a Fourier coefficient equation of the following:
a_{k} = \frac{1}{T_{0}} [∫ (-1)*e^{jk\omega_{0}t} + ∫ (1)*e^{jk\omega_{0}t}]

To compute the general equation for a_k should I treat the ±1 as a function and use u-substitution / integration by parts? If so, can someone at least show the first step or 2? I haven't done I.B.P. or U-substitution in some time. If I can treat the x(t) as a constant then how can I integrate to get the answer below?

General equation: a_k = \frac{1}{jkπ} (1 - (-1^{k}))

This example is from M.I.T. Open Courseware Alan V. Oppenheim Signals and Systems course Lecture 7 approximately 20 minutes into the video.

 

[AlephZero]

This leads to a Fourier coefficient equation of the following:
a_{k} = \frac{1}{T_{0}} [∫ (-1)*e^{jk\omega_{0}t} + ∫ (1)*e^{jk\omega_{0}t}]

To compute the general equation for a_k should I treat the ±1 as a function and use u-substitution / integration by parts?[/itex]

You don't need to do all that. "1" and "-1" are just constants. If you are studying this, you should know $\int e^{at}dt$, and that formula works when $a$ is a complex number as well as when $a$ is real.

You should also know $e^{j\omega t} = \cos \omega t + j \sin \omega t$.

You also have an equation connecting $T_0$ and $\omega_0$, which is why they both disappeared in the final equation for $a_k$ (and that's also where the $\pi$ came from).

If you are confused by the $(-1)^k$ part, just work out the first few values of $a_1$, $a_2$, etc and see what happens.

 

[DmytriE]

If you are confused by the $(-1)^k$ part, just work out the first few values of $a_1$, $a_2$, etc and see what happens.

This is precisely what I am confused about. Is there a rule that applies to the (-1)^k or do I have to input a couple numbers for a_k?