When we compute the Discrete time Fourier series of a discrete time periodic signal , why don't we get the same number of negative complex exponentials and positive complex exponentials ? Even though the magnitude spectrum is periodic throughout , if you take just the fundamental period , why the magnitude graph is not symmetric with respect to y axis for some discrete time periodic signals ?
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$\begingroup$ Did Hilmar answer your question? If so, please check it off as complete to close this out. Thanks and welcome to DSP.SE! $\endgroup$– Dan BoschenCommented Jul 15 at 13:02
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why don't we get the same number of negative complex exponentials and positive complex exponentials
But we do. We get an infinite number of positive and negative complex exponentials, i.e. $X[k]$ is well defined for all $k \in \mathbb{Z}$
the magnitude spectrum is periodic throughout
Exactly. Specifically
$$X[k] = X[k + mN] \forall m \in \mathbb{Z}$$
where $N$ is the DFT length.
why the magnitude graph is not symmetric with respect to y
But it is. It really depends on how you draw it. Since the DFT is periodic in $N$, any $N$ consecutive values will completely describe the entire DFT. The choice of which $N$ points to use is somewhat arbitrary and you can choose whatever suits your specific application best. The "fundamental period" simply chooses $[0,N-1]$ but you can also do $[-N/2,N/2-1]$, $[-3,N-4]$, etc. There is nothing "fundamental" about the so called fundamental period. It's just a convention that's convenient in some cases and less so in others. You don't have to stick with it, if it's not a good choice for your task.
