Relating discrete fourier series coefficient to the energy of the orthogonal signal set

confuseddesigner

Joined Mar 14, 2017
9
The last page of the PDF shows how the DFS coefficient is related to the energy of a signal in a orthogonal signal set. What I don't understand is how is
Σgk[n]gm*[n] equal to Ek.δ [k-m], and how is the whole thing equal to CmEm?
Notation: Ek is the energy of gk[n], Em is the energy of gm[n]. cm and ck are the mth and kth discrete fourier series coefficient.

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WBahn

Joined Mar 31, 2012
27,392
What is the integral of two sinusoidal signals of different frequencies over an integral number of periods?

confuseddesigner

Joined Mar 14, 2017
9
If they share a common period then the integral would be 0 if integrated over an integer number of that period. How does that relate to my question?

WBahn

Joined Mar 31, 2012
27,392
Well, which terms survive and which terms don't?

confuseddesigner

Joined Mar 14, 2017
9
I see now that the sum would be non-zero only if k=m and δ [k-m] is there to ensure that.
In the case that k=m, we will have Ek=Em, Ck=Cm, but how was the summation Σk=0-->N-1 removed?

WBahn

Joined Mar 31, 2012
27,392
I see now that the sum would be non-zero only if k=m and δ [k-m] is there to ensure that.
In the case that k=m, we will have Ek=Em, Ck=Cm, but how was the summation Σk=0-->N-1 removed?
What does a summation reduce to if only one of the terms is non-zero?