How are OFDM Sub Carrier Spacing and Time Samples Related?

Explains the relationship between subcarrier spacing and time domain sample rate for OFDM digital transmission.
Related videos: (see iaincollings.com)
• OFDM and the DFT • OFDM and the DFT
• Why is Subcarrier Spacing Bigger in 5G Mobile Communications? • Why is Subcarrier Spac...
• How are Complex Baseband Digital Signals Transmitted? • How are Complex Baseba...
• Pulse Shaping and Square Root Raised Cosine • Pulse Shaping and Squa...
• How does the Discrete Fourier Transform DFT relate to Real Frequencies? • How does the Discrete ...
• What is a Cyclic Prefix in OFDM? • What is a Cyclic Prefi...
• How does OFDM Overcome ISI? • How does OFDM Overcome...
• What is SC-FDMA? And why is it used for the Uplink of 4G/5G Mobile? • What is SC-FDMA? And w...
• Why is SC-FDMA called "Single Carrier"? • Why is SC-FDMA called ...
• What is a Constellation Diagram? • What is a Constellatio...
For a full list of Videos and Summary Sheets, goto: www.iaincollings.com
.

[@jemussi7842]

Excellent. I wish everyone on YT doing educational STEM videos were as clear and understandable as this. No fancy widgets, no garbled audio or annoying soundtracks just pure useful content. Thank you.

[@iain_explains]

Thanks for your nice comment. I'm so glad you like the approach I take to my videos.

[@mahmoudshata1105]

you are such an amazing person sharing these priceless information for free ,Thank you

[@iain_explains]

Thanks for your very nice comment. I'm glad you like the videos.

[@datenraffzahn6094]

This channel is a treasure trove of ODFM information, thank you very much for your brilliant videos!

[@iain_explains]

Thanks for your nice comment. I'm so glad to hear that you like the videos.

[@tomwalsh7547]

you are so good at explaining this!, I will be doing some work on SDR's in space and on the ground, while a competent EE and Embedded FSW and GSW engineer working for various programs, your videos rock to tie it all together!

[@iain_explains]

I'm so glad my videos are helpful! Sounds like you've got a pretty interesting job!

[@lxinrui3]

Dear Professor Iain, For the vector at 0:10 you are saying that you put the constellation point into each sub-carrier, so if we are using 16-QAM, just like you said in the "OFDM and DFT" video, there's diginal bits 0100, 1111 somthing like this in each block? Or there should be in I&Q component form like Acos + Bsin in frequency domain?

[@iain_explains]

It might help to watch this: "OFDM and the DFT" kzbin.info/www/bejne/kGWvepqEnLN0oqs

[@wesamamiri2323]

Thank you for this vedio! I have one general question, please. In case we generated any baseband OFDM signal and we need to know the duration of the ofdm symbol. This can be done by number of fft points (M) by the duration of pulse shaping filter (T). Is that correct?

[@iain_explains]

Yes.

[@roanage]

Starting around 6:30 when you are describing how the subchannels are fitting into the 1/2T bandwidth (on the lower left diagram) and are then drawing in the sinc functions that correspond to the constellation points of the data, are those overlapping sinc functions that you draw supposed to be thought of as time domain waveforms even though youre drawing on a frequency axis? I'm confused because you even refer back to the box-shaped frequency domain figure on the right but then continue to draw sinc functions on the left.

[@iain_explains]

No, those sinc functions are in the frequency domain. But I understand the confusion you're having (which is why I made the video - but perhaps I didn't explain it well enough?) The "samples" that go into the channel (in the time domain) are at a rate of 1/T. If they are sent with a sinc pulse shape (ie. the bottom middle figure), then the overall bandwidth will have a flat spectrum (the bottom right hand figure). Now, within that bandwidth there will be sub-carriers that are generated as a result of the IFFT. Since the OFDM symbol lasts for MT time, and since the constellation point in each subcarrier changes abruptly from one symbol to the next (ie. they have a square shape in the time domain, of width MT - and remember that they are all in parallel, due to the IFFT), therefore the subcarriers have a spacing of 1/MT and they have a sinc shape (in the frequency domain). I hope that's clearer. Perhaps these videos will help: "OFDM Waveforms" kzbin.info/www/bejne/fGelZX6vn5elgtk and "OFDM and the DFT" kzbin.info/www/bejne/kGWvepqEnLN0oqs

[@rishabhkumar1050]

sir what is the duration of T of pulse shaping filter in microseconds for which it's multiplied with carrier for upconversion. I am referring to the "practical value " used in OFDM systems

[@iain_explains]

I'm not sure what you mean. I gave a practical value in the video. IEEE802.11a WiFi has an option for a 20MHz bandwidth for the OFDM symbol, which means T = 1/(20x10^6) = 50 ns. The 4G LTE mobile communications standard also has a 20MHz BW option. The 5G standard has BW options up to 400MHz. And IEEE802.11ac and ax WiFi both has BW options up to 160MHz.

[@user-hh2vh9sw5y]

Hi Ian! I am trying to understand how OFDM works. But I don´t have the mathematical background so I can´t always follow your explanations all the way. So I try to understand in, let’s say, "a more practical way". I think I understand the general concepts now, except for one thing: Let say we have 64 subcarriers of 312,5kHz bandwidth. Doesn't this means that there are subcarriers in the range from 312,5kHz to 20MHz? If so, why are the I- and Q-signals that are generated by the IFFT only 10MHz wide? Don't they both contains the "contribution" from all the subcarrier between 10 to 20 MHz too? ( Please, explain without too much mathematics if possible. )

[@user-ex2iw4hi3f]

Should frequency vector apply ifftshift Before doing IFFT?

[@iain_explains]

No. The frequency vector that I showed is in the standard (non-ifftshifted) format. This video gives more details: "How does the Discrete Fourier Transform DFT relate to Real Frequencies?" kzbin.info/www/bejne/pnqpq2tqpM9smaM

[@iceinmylean3947]

Thanks again for the awesome videos! I am confused about pulse shaping. In all of the other videos (in fact all material I can find), OFDM is not combined with pulse shaping. If the pulse is non-constant over one period of the carrier(s), wouldnt that mean our orthogonality relations are now likely violated? The orthogonality relations only hold because integrals over products of sines/cosines vanish, but not in general when those are multiplied with other time-dependent functions, i.e. the pulse shape in this case. Also, wikipedia states that pulse shaping is usually applied after modulation. I guess that is mathematically equivalent here, but do you know whether there is any practical reason for this? Thanks in advance!

[@iain_explains]

Great question. Think about the M length real-valued vector that comes out of the IDFT in the transmitter, that needs to be "sent" on the cos wave carrier (top right hand picture in my summary sheet drive.google.com/file/d/1ASUhoAXJXpGc8Vp44cvPLgzPv8lDdatm/view ). It's a vector of discrete values. Think of it as a discrete-time signal. Imagine it as a discrete time sampled version of a continuous-time signal. In order to "reconstruct" the continuous-time signal exactly, you would need a sinc pulse shape filter. If you are happy to have a low-resolution reconstruction, you could use a zero-order-hold filter. People don't talk about this when they talk about OFDM because either they don't understand it (often the case) or they consider it to be "just implementation details". Here are a couple of my videos that might help: "How are Signals Reconstructed from Digital Samples?" kzbin.info/www/bejne/mnWceXZnfLmbkLs and "Pulse Shaping and Square Root Raised Cosine" kzbin.info/www/bejne/h5abf4Suac6Ve5o And to your point about the orthogonality, it's also a very good point. But think about breaking down the pulse shape into small time segments each of duration 1/f_c Each of these time segments is only as long as the period of the carrier. This is massively shorter than T. Over each of these time segments, the "segment pulse shape" is essentially constant (almost no matter what the "overall" pulse shape is), and so the sin and cos components are orthogonal. The overall integral from 0 to T is just the addition of each of these components, which is zero (... I think that's right, although I haven't checked it mathematically).

[@iceinmylean3947]

@@iain_explains Thanks for the quick and detailed response! Your point on orthogonality seems reasonable. If the carrier frequency is very high compared to the pulse shape variations, the pulse would be approximately constant over one period of the carrier. I hadn't thought of that! Regarding the first part: This sounds to me as if pulse shaping should be done before modulation onto the carrier though, or am I missing something?

[@iain_explains]

I'm not an "implementation" person, but I've always thought of it as being done before multiplying by the carrier (surely it has to be - otherwise it would be necessary to implement shaping filters at the carrier frequency, which would be a more complicated filter - at least that's what I'd expect - but again, I don't really know about hardware implementation aspects).

[@2011HPS]

@@iain_explains Hi Prof., This explanation is very interesting(and convincing) but now I am confused because in many of the SDR implementations such GNU Radio 802.11g, and Openairinterface-LTE/5G, I never saw pulse shaping of the IFFT output before they are modulated with carriers.

[@stringstoparadise2392]

hii professor first of all thanks for video , my question is why we are considering the -ve frequency subcarriers why we not taking +ve subcarriers only, and my 2nd question is the why subcarrier spacing is 1/2T , if we have the symbol duration of T then it should be 1/T

[@iain_explains]

If you only use the +ve frequency subcarriers (at the baseband) then you'll only be using half the bandwidth (in the passband). And for your second question, the subcarrier spacing is not 1/2T or 1/T. As I wrote half way down the page, the (sub)carrier spacing is 1/(TM).

[@Hotrod768]

Just so it's clear, g(t) is not the pulse shaping filter, but the reconstruction filter used for interpolation when doing the analog-to-digital conversion?

[@iain_explains]

g(t) is the pulse shaping filter. Another description might be that it is the filter that turns the digital (discrete time) value into a continuous-time waveform that can be transmitted (... I'm not sure if this is what you mean by "reconstruction filter" or not?). Have you seen my video on pulse shaping?: "Pulse Shaping and Square Root Raised Cosine" kzbin.info/www/bejne/h5abf4Suac6Ve5o

[@Hotrod768]

@@iain_explains There are these old lectures from Alan Oppenheim's Digital Signal Processing series (or from his textbooks) where he shows that you can treat reconstruction of a sampled signal through essentially the same idea of pulse shaping (convolution of a pulse train with an "interpolation" filter). The idea of interpolation and pulse shaping are identical when I think about it more. OFDM view point just makes is very clear. Thanks for the lectures, they are very helpful and intuitive.

[@Hotrod768]

I guess my understanding is, when you typically think of turning an I/Q symbol into an analog signal to transmit, each time instance has a pulse for a different symbol that you convolve with the pulse shaping filter to get get the analog output, which can then be mixed with the carrier. For OFDM (or any other orthogonal transmission), instead of having a pulse train (that is orthogonal) you have each symbol occupying the full sampled time, but with orthogonal signals. In the case of OFDM we have the sampled sinusoids with integer multiple frequencies for each symbol. which so happens to by the output of the IDFT. These orthogonal samples are filtered by the pulse shaping filter in the same way a pulse train is. This process is the same as reconstructing a sampled signal (sampled into a pulse train). Here is the lecture from Alan Oppenheim if you're interested: kzbin.info/www/bejne/o57OgHSDhNxqabc

[@iain_explains]

Yes, that's right. Glad you liked the video.

[@ibrahf5911]

Hi, I have a question . Towards the end of the video you mention that you are using just 48 of the 64 subcarriers for data. Why do you still need to multiply by 3/4? Does that 3/4 represent something else?

[@iain_explains]

3/4 relates to the error correction coding rate. The elements are labeled on this pdf summary sheet on my webpage: drive.google.com/file/d/1ASUhoAXJXpGc8Vp44cvPLgzPv8lDdatm/view

[@ibrahf5911]

@@iain_explains Thank you very much

[@nanjiang4158]

@@iain_explains What about the other 16 subcarriers. Are they for control signalling?

[@DrAiden121]

Based on my understanding, there are M OFDM symbols in time domain and each symbol is transmitted over T time by g(t). Is it right?

[@iain_explains]

Normally the term "OFDM symbol" refers to the whole "M-sample length time sequence". Each time _sample_ is sent in T seconds, using the pulse shaping filter g(t).

[@cubbyhoo]

You mention the different pulse shaping filters which can be used and their different bandwidths in the Fq domain, 2/T for the square and 1/T for the sink. In your explanation of this concept you are pointing at the square time domain filter (top right) and then saying the the fq domain bw is 1/T but this isn't correct? Or have I misunderstood? Perhaps I don't fully understand the pulse shaping filters part. Additionally could you explain how the sink functions in the channels would look in the example of the square pulse shaping filter (the sink function in the Fq domain?) would the amplitudes be kind of curved to match that fq domain response?

[@iain_explains]

This video might help: "Pulse Shaping and Square Root Raised Cosine" kzbin.info/www/bejne/h5abf4Suac6Ve5o

[@cubbyhoo]

@@iain_explains thanks a lot, just watched that, excellent explanation as per. I hadn't heard of the Square Root Cosine but that was a great explanation

[@arjunsnair4986]

Thank you sir

[@iain_explains]

You're welcome

[@amahbubul85]

@Iain Explains Signals, Systems, and Digital Comms lain Hi lain, Hope you are doing fine. I want to ask you some questions not related to this video: 1) We see bit error probability vs SNR curves in digital communications. What I know is that a threshold of this error probability exists which is application specific, and below which the communication fails. By a failed communication, here I mean dedoding a packet incorrectly. Am I right? If the SNR is below the SNR corresponding to the threshold of bit error probability, then the packet is considered to be decoded incorrectly? Is this understanding correct? 2) I am developing a simulator for NR V2X mode 2 (sidelink) where the desired user senses the resources to detect transmission on these resources by other users. To do this, the desired user needs to decode the SCI received from other users. If the SCI is undecoded or if it is decoded but the sensed received power is less than a power threshold, then that resource is considered to be available. My question is, what if more than one users use a particular resource? How to deal with that? How to decode them? What to consider? Can you please give me some light on this matter. Thanks

[@ravindratomar9916]

Dear professor, If we increase the number of subcarriers but keep the bandwidth constant , cyclic prefix constant then will the data rate increase?? Also if we double the bandwidth , while keeping the number of subcarriers constant then will data rate changes?? I am very confused..for the first case I have calculated and find that data rate remains almost unchanged...this is due to constant bandwidth ?? Please clarify. Thanks.

[@iain_explains]

It's not a simple relationship, since the overall data rate depends on many factors, including the constellation size you decide to use in each subcarrier, the SNR in each subcarrier, and the characteristics of the channel.

[@ravindratomar9916]

@@iain_explains Yes Sir, I understood that point. Thank you.

[@tihihoang]

Dear Professor, at 4:30, could you explain why f=1/2T please? I tried my assumption that 1/T = B, then it leads to B = N_fft * subc_spacing, which is not correct in some 5G settings, like 20MHz, subc_spacing = 30kHz, 1024 fft, and then the T_sample = 1/ (N_fft * subc_space) = 3.2552e-8, which is different to 1/B

[@tihihoang]

Dear Professor, I just calculated that, the sample rate of 1/T = N_fft * subc_space is greater than the sample rate of just 1/B, which is fine, but my question now is that, why don't why set it to 1/2B ? thank you so much

[@tihihoang]

I just realized that, when we map the frequencies from - N_fft/2 * sub_space to +N_fft/2 * sub_space, the maximum frequency now is N_fft/2 * sub_space, so the sample rate is 2* (N_fft/2 *sub_space) = N_fft* sub_space. The point now is how we can differentiate the signals in negative frequencies and the signals in positive frequencies, I will look for it

[@iain_explains]

This video should help: "What is Negative Frequency?" kzbin.info/www/bejne/nauZcn6NYrdpb9U

[@tihihoang]

@@iain_explains thank you so much Professor

[@muhammadahmedtariq2357]

One thing is not clear. You are fitting M frequency domain since functions of null-to-null bandwidth 2/T into system bandwidth of 1/T.

[@iain_explains]

No, that's not right. I should have made it clearer in the video, but the null-to-null bandwidth of each sub-channel is 1/(2MT). This is because the duration of each sub-channel symbol (in the time domain), equals the entire OFDM symbol time (MT seconds). Don't forget, the QAM constellation point in each sub-channel is fixed/constant within an OFDM symbol. In other words, the QAM constellation points in each sub-channel only change at the boundaries of the OFDM symbol times (ie. every MT seconds), and they change abruptly/instantly, which means that in the frequency domain the sub-channels have a sinc shape around the subcarrier frequency.

[@muhammadahmedtariq2357]

Thanks Prof Iain. If you could please make a video on mimo ofdm when each antenna has multiple subcarriers, then it would be great. Particularly how data is distributed among different subcarriers across different antennas ? Subcarriers' mapping ? Reception and detection back to frequency domain ! Carrier Aggregation !

[@iain_explains]

Thanks for the suggestion. It's on my "to do" list (but it's a long list!)

[@benjaminsamaniegoerazo3993]

My dummie question is, every subcarrier modulate at a rate of 4us and then all components are +?

[@benjaminsamaniegoerazo3993]

Finally they are sent in the GHz band?

[@iain_explains]

The OFDM symbol length is 4us (including the cyclic prefix), which means that each of the narrow band subcarriers changes its constellation point (digital information) every 4us (since they are all in parallel). Sorry, I don't know what you mean when you ask if "all components are +". And yes, the carrier frequency is generally in the GHz range (frequencies of the sin and cos waveforms indicated at the top right hand side of the screen). drive.google.com/file/d/1ASUhoAXJXpGc8Vp44cvPLgzPv8lDdatm/view

[@benjaminsamaniegoerazo3993]

@@iain_explains very clear now, I watch all the videos related to OFDM, thanks professor :)

[@amahbubul85]

@Iain Explains Signals, Systems, and Digital Comms lain Hi lain, Hope you are doing fine. I want to ask you some questions not related to this video: 1) We see bit error probability vs SNR curves in digital communications. What I know is that a threshold of this error probability exists which is application specific, and below which the communication fails. By a failed communication, here I mean dedoding a packet incorrectly. Am I right? If the SNR is below the SNR corresponding to the threshold of bit error probability, then the packet is considered to be decoded incorrectly? Is this understanding correct? 2) I am developing a simulator for NR V2X mode 2 (sidelink) where the desired user senses the resources to detect transmission on these resources by other users. To do this, the desired user needs to decode the SCI received from other users. If the SCI is undecoded or if it is decoded but the sensed received power is less than a power threshold, then that resource is considered to be available. My question is, what if more than one users use a particular resource? How to deal with that? How to decode them? What to consider? Can you please give me some light on this matter. Thanks

[@iain_explains]

Answers: 1) Yes. 2) Decoding information from other users depends on the particular signalling protocol. So I can't really answer your question, since the protocol is not specified.

[@amahbubul85]

@@iain_explains thanks for your reply. Regarding protocol, can you please name some so that I can search what is used for the one I am talking about?