I am learning about OFDM and QAM.
Every explanation I have read illustrates how the separate carriers are sinc functions that are arranged to overlap in a nice orthogonal way so as to cancel out interference from neighbouring carriers.
The part I don't understand is how is it possible to generate these neat perfectly overlapping sinc functions, when they contain constantly changing symbols. Each transition from 1 symbol to the next, in the time domain is a random change in phase and amplitude. This must correspond to a random change in the sideband pattern of the carrier in the frequency domain, and therefore disrupt the orthogonal arrangement.
Best Answer
These sinc functions, as you've noticed, have zeros in a distance of the subcarrier spacing \$\Delta f\$.
Remember how these sincs come to be (the texts you've been reading most definitely mention that!): The sinc function is the Fourier transform of the rectangle function. Scaled to yield zeros in frequency domain every \$\Delta f\$, the width \$T\$ of that rectancgle must be \$T=\frac1{\Delta f}\$.
So, that answers your question: all your sincs are just the result of having a rectangle in time domain, and multiplying it by \$e^{j2\frac{n\cdot\Delta f}{f_\text{sample}}t}\$, so to shift it in frequency to yield the \$n\$th subcarrier. The QAM symbol is just a complex factor you multiply the result with – that is just a constant factor and doesn't change the shape, neither in time nor frequency domain.
Now, what's \$\Delta f\$, when you think about it? In OFDM, you use the \$N\$-point DFT to divide your Nyquist bandwidth (complex!) \$f_\text{sample}\$ into \$N\$ equally large subcarriers, so \$\Delta f = \frac{f_\text{sample}}{N}\$. Therefore, the width of the rectangle \$T=\frac1{\Delta f}=N\cdot\frac{1}{f_\text{sample}} = N\cdot T_\text{sample}\$. That very simply means that the sinc shapes are just the effect of turning on a (complex) oscillation of frequency \$n\cdot\frac{f_\text{sample}}N\$ for exactly \$N\$ samples.
Such a transition simply doesn't happen within one OFDM symbol: For the duration of one of these rectangles, the symbol for each subcarrier is constant. So, you use \$N\$ samples to send a single symbol, but you gain the ability to send \$N\$ symbols at once. So, nothing lost, nothing gained here!