I have been reading about OFDM modulation, it's not clear to me how you can have band-limited OFDM but given its prevalence in wireless communication there must be some designation of approximately band-limited that OFDM satisfies.
So my question is how is the bandwidth of OFDM systems calculated and if it's an approximation based upon the largest spectrum lobe, why is that good enough?
It doesn't even look like OFDM systems are pulse shaped to reduce the side-lobe strengths so what's going on?
This isn't a question specific to OFDM:
All transmissions need to be bandlimited by law. Hence, law defines band-limited here.
Typically, this happens in the shape of spectrum masks (often expressed in textual form in regulations); these say something like (this is just an example, actual legislation varies across regions and is definitely different):
"not more than -40 dBm/Hz within 10 MHz of the carrier frequency, not more than -90 dBm/Hz between 10 and 20 MHz of the carrier frequency, and -120 dBm/Hz more than 20 MHz away"
Regarding the specialties of OFDM:
You indeed can't pulse shape OFDM without losing its benefits. (You can do a bit of a tradeoff, and trade a bit of orthogonality for out-of-band radiation suppression, but you'd typically avoid that)
That's why you typically have "empty" carriers at the edges. You simply leave room for the sinc to decrease.
A simple way to compute the "occupied bandwidth" of an OFDM signal is $$BW_{\rm occupied} = N_{\rm active-subcarriers} \times \Delta f \ ,$$ where $ N_{\rm active-subcarriers}$ denotes the number of active subcarriers in the considered IFFT size and $\Delta f$ corresponds to the subcarrier width or spacing.
Moreover, you also need to do some sort of spectrum shaping, e.g., windowing or low-pass filtering in order to meet the desired adjacent channel leakage ratios (specified by the standards normally). Not considering, the non-linearity of the dirty RF herein.
Theoreticaly of course you are right; no practical system can be bandlimited. Then the OFDM is not perfectly orthogonal. Yet we are dealing with practical definition which is indeed extremely satisfying up to accepted error bounds.
Hence bandlimited in the practice of communication systems means that most of the energy of the transmission resides in a specified frequency range. When this holds, what you achieve is reliable communication (instead of a perfect-ideal communication).
For digital communication systems, reliability is measured (quantified) by the metric of bit error rate which indicates how clean your transmission medium is. Most typical sources of bit errors are the internal noise, external interference,..., and cross-over from adjacent channels.
This last source of noise is highly related with associated channel bandwidths for the given transmission system. The tradeoff is that instead of an infinite (and ideal) bandwidth per channel, one should use a finite but acceptible bandwidth which would unfortunately yield a non-ideal, errenous, distorting tranmission that is however transparent to the receivers as long as they are ok with the associated bit-error rates.
You may be confusing to O in OFDM as orthogonality in the mathematical sense with orthogonality in the engineering sense. The subcarriers only need to be close enough to orthogonal to each other that their cross correlations are below and within the link noise budget for whatever S/N (etc.) produces the appropriately low enough statistical error rate. That allows bashing the outer side bands of the OFDM channel and thus each subcarrier to meet legal/engineering requirements.
Imperfect/Non-linear amplification/hetrodyning (etc.) will also damage orthogonality in the real world.
Thus, legal OFDM is not orthogonal (mathematically), but is (enough to work).
