I evaluated the Fourier series of $\delta(x-nd)$ for integer n between infinity and minus infinity I think this is an expression for an infinite array of delta functions separated by d. When I evaluated the Fourier transform into q space (reciprocal partner to x) I got $\sum_{n=-\infty}^\infty exp(-iq(nd))$ which is also $1+2\sum_{n=1}^\infty cos(ndq)$. I've been told that the Fourier transform should be an array of delta functions separated 1/d but I can't see how either of these expressions can geometrically represent that array.
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Dirac comb for some references. – Raymond Manzoni Nov 07 '15 at 09:23
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1duplicate of http://math.stackexchange.com/questions/1514258/intuition-behind-alternate-expression-of-impulse-train/1514413#1514413 – Fabrice NEYRET Nov 07 '15 at 09:47
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As I said in the similar post:
just play with this interactive Desmos Graph and see. The point is constructive and destructive interferences : every cos contributes around the Dirac comb support or progressively cancel each other at other places, resulting into the Dirac comb.
Fabrice NEYRET
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One of the definitions of dirac delta functions is $\frac 1{2\pi} \int_{-∞}^∞ {e^{iw(t-t_0)}}dw)$
Then going back to your questions what you derived is some function f(q) = $\sum_{n=-∞}^∞ exp^{-iqnd}$
which is equivalent
is equivalent to $\sum_{n=-∞}^∞ \Bigl( \int_{-∞}^∞ \bigl( {e^{(\frac{x_0}{nd}-x)iq}}\bigr)^{nd} dx \Bigr)$, in which we have multiple Dirac functions separated by distance nd and of amplitude nd.
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