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The idea of OFDM is to spread data across multiple overlapping subcarriers. As they are orthogonal it is still possible to demodulate them. That means, all other subcarriers are 0 at the evaluation point of a specific subcarrier.

Orthogonality is achieved by generating a rectangular subcarrier pulse. By applying a IFFT a $\sin(x)/x$ emerges which allows orthogonality.

Another requirement for orthogonality concerns the subcarrier spacing. It must be equal to the reciprocal of the symbol period. Howver, I do not understand why. I thought it is most important to create these $\sin(x)/x$ that are shifted across the x axis on an unique position for each subcarrier.

Question: Why would orthogonality break if the subcarrier spacing was not reciprocal of the symbol period?

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There are many intepretation of OFDM to answer your question. One way I find intuitive is looking at the resolution of (Discrete) Fourier Transform.

By capturing a signal during time $T$, and by assuming periodicity outside $T$, DTFT is able to provide a frequency resolution $1/T$. Thus if you choose a subcarrier spacing $\Delta f$, you will want a frequency resolution at least $\Delta f$. As a consequence, signal must be captured at least $T = 1/\Delta f$.

One may argue that I capture then $T > 1/\Delta f$. It is fine, but the extra capturing time may catch the next symbol and cause ISI. Assuming that in a good system design, e.g. with CP or zero guard time, so that ISI is avoided, the resolution of DTFT (and DFT/IFFT) $1/T < \Delta f$. The issue here is that if $\Delta f$ is not a multiple of $1/T$, the output of IFFT will be at frequencies $k/T, k \in \mathbb{Z}$ outside the subcarrier positions and this output will be sinc-interpolation of data.

So, at the end the requirement is $\Delta f=k/T, k \in \mathbb{N_+}$ or $T=k/\Delta f$. For the sake of economy, we choose the smallest $T$ possible at $k=1$ yeilding the famous thumb rule "subcarrier spacing is reciprocal of the symbol period".

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