The square wave Fourier series provides fundamental insight into the nature of Fourier series expansions. You will learn the relationship between the width or duty cycle of a square wave and the concentration of the Fourier series coefficients. You will also be introduced to the sinc function, a function that has widespread significance in all forms of Fourier analysis. You will encounter both the square wave and sinc function repeatedly in your signal-processing studies. Developing your insight and understanding of the square wave Fourier series and the sinc function will pay dividends as you work in this field.

## Prerequisites

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Lily Ngo says

in "Concepts and Screenshots for Square Wave Fourier Series and the Sinc Function", page 3/12;

how can I get sin(k2πTo/T)/kπ from 2 sin(kΩoTo)/

kΩoT?

Thank you very much for your help!

Lily Ngo says

because when I replaced Ωo = 2π/To, I could not get the same result. In the video, you said to replace Ωo = 2π/T. I'm confused here: I thought I should use To to calculate Ωo and T to calculate Ω.

Barry Van Veen says

If you look at the line above, T is the period of the square wave, so . I apologize for too many T's subscript "o" and so on - this is a case where and do not correspond. So if you use the equations match up.

Lily Ngo says

in "Concepts and Screenshots for Square Wave Fourier Series and the Sinc Function", page 3/12:

square wave x(t) ->(FS: omega=2pi/T) -> X[k]=sin(k.pi.2To/T)/(k.pi)

Therefore, what I solved for the quiz question 1:

To=5

T = 20

then, X[k] = sin(k*pi*2*5/20)/(k.pi)

= sin(k.pi/2)/(k.pi)

My answer did not match any option.

Could you please explain how it is sin(k.pi/4)/(k.pi)

In addition, this question is similar to the example in your video "Screenshot - page 8/12" 50% duty cycle: To/T= 5/20 = 1/4.

Thank you!

Barry Van Veen says

Note that in this problem one period is defined based on so here . That is, one period of is defined for an interval ranging from to .

Lily Ngo says

As I compared the formula #1 on page 3/12 and the graph on page 7/12, I thought if the definition for x(t) in the formula for page 3/12 were modified to be x(t) = either 1 for |t|<To or 0 for To<|t|<T/2,

then I solved for the quiz question #1, I got the result as same as option (b) because T=40, not 20.

Barry Van Veen says

Sorry about that - there is a mistake on p 3/12. The text says the period is T, but then, as you point out, the equation defines x(t) for a period of 2T. Thank you for catching that error. I've fixed the document and uploaded a new one.