Subcarrier spacing in OFDM

Question

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?

Answer 1

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".