# Power spectrum of periodic signal

#### xxxyyyba

Joined Aug 7, 2012
289
This is not really homework question, but this category looks most appropriate to me
We know that periodic function can be writen in terms of complex Fourier coefficients:
$$f(t)=F_n_=_0+\sum_{n=-\infty,n\neq 0}^{n=\infty}F_ne^{jnw_0t}$$, where $$F_n=\frac{1}{T}\int_{\tau}^{\tau+T}f(t)e^{-jnw_0t}dt$$ and $$F_n_=_0$$ is DC component. Power spectrum of signal is defined as $$S_1_1(nw_0)=\left | F_n \right |^{2}$$, where $$\left | F_n \right |$$ is modulus of complex Fourier coefficient $$F_n$$.
We can draw power spectrum as function of both negative and positive discrete frequencies nw0. My question is, if I want to draw it only for positive discrete frequencies, I should multiply $$\left | F_n \right |^{2}$$ by 2, right? Should $$\left | F_n_=0 \right |^{2}$$ be also multiplied by 2?

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#### WBahn

Joined Mar 31, 2012
26,398
Yes and no, respectively.

Having said that, I don't understand your first expression (not that it isn't even an equation).

#### xxxyyyba

Joined Aug 7, 2012
289
I edited my post, I hope it is clear now

#### WBahn

Joined Mar 31, 2012
26,398
Why exclude n=0 from the sum? Why not just get rid of the first term and let it appear naturally as part of the sum?