Coefficient of Complex Exponential Fourier Series ( Cn)

Started by Basim 2 weeks ago4 replieslatest reply 2 weeks ago136 views

The formula for Coefficient Cn in Complex Exponential Fourier Series for a signal x(t) can be found through a well known formula, that is : 

Cn=1/T ∫f(t)exp(-jw0t) dt

However, for periodic signals I noticed that Cn is given in a different expression like :


Any thoughts about the background behind the second Cn formula, and why it is different from the first formula?


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Reply by shafie7February 11, 2024

Hello Basim,

Hope all is well.

If x(t) is periodic then X(w) [Fourier Series of x(t)] is discrete.

Would you point to the sources of the first and second equation?




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Reply by BasimFebruary 12, 2024

Thanks Shahram for your informative answer, the sources for both are from You-tube organized and submitted online by Neso Academy.

Still, I need to make sure that Cn=X(nw0)/T is valid and applicable and if so what is the background of such a formula and why it is different from the main one 

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Reply by LmseleFebruary 12, 2024

This may only be tangent to the discussion, but may lend some insight to the question.


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Reply by BasimFebruary 12, 2024

Thank you Lmsele for your interaction, this question is simple if you have background about Exponential Complex Fourier Series.

In the formula used to find such series, there is an important coefficient denoted by Cn, this has also formula which is :

Cn=1/T ∫f(t)exp(-jw0t) dt

Now I found other sources using the following formula:


The first one is the standard and available in most textbooks, but regarding the second one, this is the first time I have seen it, and the related sources are using it in specific digital signal processing scenarios.

Any thoughts about the second formula, is it equivalent for the first one? how much reliable is the second formula,?