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For an OFDM system, it is necessary to know the number of subcarriers. For example for the subcarrier $k^e$, we must have: $B_k <B_c$ with $B_c$ the channel coherence bandwith. And $B_k=\frac{1} {NT+I}$ with $I$ is the guard interval, $T$ is the symbol time and $N$ the number of subcarriers.

Here is my questions:

  1. For example, for $N=3$, we have $B_k<B_c$. Is the number of subcarriers small? Because I always see, N = 64, 1024, ...
  2. Is there a minimum number of subcarriers?
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  • $\begingroup$ 0. what is $B_k$ and how do you come up with this derivation? 1. yes, N=3 is smaller than 64 and than 1024, but maybe it is already enough ... 2. the minimum number is 1 and this is what called monotone OFDM is recent industrial specs. $\endgroup$ – AlexTP Jul 13 '18 at 9:16
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You've got that right – when designing an OFDM system, you pick the number of subcarriers usually just large enough to make the subchannels look flat.

You avoid overly large subcarrier numbers, because more subcarriers means longer symbols, and you need to restrict the symbol length to stay below coherence time of the channels.

Now, 3 is indeed a choice that is very small. I wouldn't build an OFDM system with 3 subcarrieres, simply because the smallest cyclic prefix that you can build is then $\frac 13$ of the symbol length – and chances are, you need more than one $\frac1{B_{subcarrier}}$ as guard interval to "capture" the whole channel impulse response.

Then again, your channel seems to be very benevolent, since it has such a high coherence bandwidth, so maybe that's really the case.

However, in a situation where you'd do 3- or 4-subcarrier OFDM, you'd simply not do OFDM alltogether and would probably either simply use a single-carrier system with a proper equalizer (might be easier, even), which will not waste as much spectrum as OFDM, or you'd go for filterbank multicarrier systems, because you will have a very bad time¹ trying to achieve synchronicity in such an OFDM system. Schmidl&Cox will be very ambiguous prior to first lock, and the SNR of any timing symbols will be less-than-good with just three values to work with per OFDM symbol.


¹ yes, that's a pun.

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