# 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
26,398
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
26,398
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
26,398
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?