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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.

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

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

This leads to a fourier coefficient equation of the following:

[itex]a_{k} = \frac{1}{T_{0}} [∫ (-1)*e^{jk\omega_{0}t} + ∫ (1)*e^{jk\omega_{0}t}][/itex]

To compute the general equation for [itex] a_k [/itex] 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: [itex] a_k = \frac{1}{jkπ} (1 - (-1^{k}))[/itex]

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

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# Fourier Series - Asymmetric Square Wave

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