# Relationships among OFDM Bandwidth, Sample rate and Symbol period

I am studying OFDM and have some questions to ask.

Assuming my WiFi network operating on 5GHz with OFDM channel BW of 20MHz and N = 10 subcarriers:

1. Does the RF spectrum start at 5GHz (frequency shifting by fc=5GHz, single sideband) or centered at 5GHz +- 10MHz (fc = 5.01GHz double sideband)?
2. Why is BW equal to the sampling frequency fsa? Shouldn’t BW be smaller than fNyquist = fsa/2?
3. Tsym (symbol duration or sample time) = NTsa = N/fsa= 10/20MHz = 0.5 µsec
4. fspacing = fsa/N = 1/Tsym = 2 MHz. Does this mean that, within the sample acquisition time Tsym, the first subcarrier fspacing will have fspacing x Tsym= 1 complete cycle, the 2nd subcarrier 2fspacing have 2 cycles and the last subcarrier Nfspacing have N cycles?
5. What is the BW of a subcarrier? Is it the fspacing = 2 MHz here?
6. Is the total bit rate equal to fsa (channel BW = 20 MHz) * bits/symbol?
7. The main lobe of FT (rect(Tsym)) is sinc(f)=[-1/Tsym: 1/Tsym]. So, its BWo=2/Tsym. But OFDM’s orthogonality between two subcarriers’ frequency bins would make fspacing=BW= 0.5 BWo=1/Tsym. Regular FDM would get at most N=20MHz/4MHz=5 subcarriers compared to OFDM’s N=10 here. Is this 2x band saving an OFDM advantage? But I guess it won’t make the theoretical speed faster if fsa or BW doesn’t increase.
• Hi Leon! Welcome here. Your first question reads as if you haven't had a proper basic introduction to OFDM; could you tell us based on what you're learning OFDM? I think pointing you to a good resource is worth more than answering 7 questions that mostly amount to "yes, this is a basic concept, you need to read up on that". Commented Apr 27, 2023 at 8:53
• I am interested in learning DFT, and sampling theory in relation to BW and speed of wireless/WiFi/fiber optics media. I have done some studies on OFDM. But it would be appreciated if you could point me to a few systematic introductions to OFDM basics with mathematical proofs. Commented May 7, 2023 at 3:15

I'm going to quickly answer your questions. These answers will not be as helpful as I think you hoped! The problem is that you're really asking stuff that someone who's read an introduction to OFDM should know. So, my guess is you're coming from a different field and just have not been given a basic introduction. Please reply to my comment on what your background is, and we can recommend literature that works for you :)

1. DSB, this sounds like you need to refresh even the very basics of complex baseband!
2. no; complex sampling.
3. $$T_{\text{sym}}$$ is not the sample time; this feels like you don't understand the frequency-domain nature of OFDM. The formula, as easy as it looks, seems correct, but neglects the necessary guard interval. So, correct, if according to the definitions of what you learn.
4. yes. You need to realize this is just the DFT -> really really central concept of OFDM
5. Define "bandwidth"; different people have different definitions of that. I tend to say that the bandwidth is $$f_{sa}$$ for all practical purposes in such a small-$$N$$ OFDM system, because the sinc shape of the subcarriers has significant sidelobes all over the Nyquist band. Other people mean "bandwidth between spectral zeros", in which case the subcarrier spacing is right. But it really just depends on your definition of bandwidth, which isn't uniquely defined.
6. Yes, but only if you neglect the guard interval. Note that this is the central takeaway from the basic principle of OFDM: A change of orthogonal base, as done in OFDM with the DFT, does not change the amount of mutual information of a channel.
7. This question is levels of understanding above 1. to 6.!
You're assuming a definition of "bandwidth" here that you can apply to answer 5.
What you're saying about FDM is wrong. You can pack the carriers in any FDM system as tightly as in OFDM, but you need intense filtering / channelization and might still need to do inter-carrier interference (ICI) cancellation, depending on your acceptable level of interference. Note that having a good channelized multicarrier system means you don't need the guard interval that OFDM usually requires to not have inter-symbol interference (ISI), which can play out as a datarate advantage. It really depends on the channel and your desired data rates! Note that your example ($$N=10$$, $$f_{\text{sa}}=20\,\text{MHz}$$ is a bit unusual arounbd $$f_{carrier} = 5.8\,\text{GHz}$$ ("5-GHz Wifi" occupies 20, 40, or 80 MHz wide channels in the 5.725 to 5.875 GHz band), as coherence bandwidths are usually smaller than 2 MHz in an indoor channel in these bands. WiFi uses subcarrier spacings of 312.5 kHz.
• You don't need me to tell you why answering that question is a problem, and I don't want to downvote this much detail. But if you do answer, you should at least make it clear to the user that this is not normal and OP should do better next time. Other networks are suspending such answerers for a year or more - not that I agree, just to put in perspective. Commented Apr 27, 2023 at 10:55
• Can you elaborate on your answers to 1. and 2. please? What is "complex sampling"? Do you mean the symbols are complex? Why is complex sampling fsa = BW? Commented May 7, 2023 at 3:15
• @LeonChang You'll need to look that up. Complex sampling means you sample the complex baseband synchronously in I and Q. I don't know how you can have read something about OFDM without having seen complex samples, so I think you'll probably already know this, but don't realize. Commented May 7, 2023 at 7:33