OFDM and the DFT

Shows how Orthogonal Frequency Division Multiplexing (OFDM) is implemented with a Discrete Fourier Transform (DFT), and how it relates to single carrier digital communications.
Related videos: (see: www.iaincollings.com)
• How are OFDM Sub Carrier Spacing and Time Samples Related? • How are OFDM Sub Carri...
• Why is the OFDM Symbol Prefix Shorter in 5G Mobile and 802.11ac WiFi? • Why is the OFDM Symbol...
• OFDM Waveforms: • OFDM Waveforms
• Why is Subcarrier Spacing Bigger in 5G Mobile Communications? • Why is Subcarrier Spac...
• What is a Cyclic Prefix in OFDM? • What is a Cyclic Prefi...
• How does OFDM Overcome ISI? • How does OFDM Overcome...
• Orthogonal Basis Functions in the Fourier Transform: • Orthogonal Basis Funct...
For a full list of Videos and Summary Sheets, goto: www.iaincollings.com
** Note: I gave the continuous-time version of the Inverse Fourier Transform equation because it's more intuitive to show how the waveforms (at the different frequencies) add up. But if you substitute t=(n/N)T, then you get the standard IDFT equation (in terms of the discrete-time samples, indexed by the variable n). This is because in the IDFT, there are N time-domain samples, which is because there are N frequency-domain subcarriers. I also didn't show the usual scaling by a factor of 1/N (which I probably should have mentioned. ... but it's just a scaling, so it doesn't change any of the intuition, which is what I am trying to show in the video).

[@luuhinz]

It all makes sense now! Thank you so much! I've been wondering for too long why an IFFT shows up in the block diagram of an OFDM transmitter (I thought, why take a time domain signal, our data stream, and apply IFFT which usually takes frequency domain as an input!!) but you explained the idea perfectly and now I can finally go to bed in peace after a long day of studying... :)

[@iain_explains]

I'm so glad it helped you understand how OFDM works. It's great to hear when people find the videos useful.

[@rahulshiv2945]

Your videos are the reason I passed my digital signal processing class. Thank you!

[@iain_explains]

I'm so glad to hear that the videos were helpful.

[@hanklong6426]

This video is soooo helpful! Thank you so much!

[@sanimukhtar5973]

Being new to DSP and communications, this is one of the most finest channel in this regards. Your succinct and didactic explanations are nonesuch!...You have a new subscriber , indeed! :).....please don't stop making those amazing videos, sir.

[@iain_explains]

Thanks, and welcome to the channel.

[@BrotherLuke2008]

Iain, thankyou for these videos. With your help I have successfully blagged my way through a month in my new job.

[@iain_explains]

Glad to help. Best of luck with it.

[@mohamedtalha9790]

I have watched a couple or more of your videos, and glad to subscribe. Keep going , you’re doing great 👍

[@iain_explains]

I'm glad you've found the videos useful.

[@sansaskater]

thanks for the great explanation. helps a lot with my last examen in my electronics bachelor :)

[@iain_explains]

Glad it helped!

[@deepfedi4883]

best vid i've ssen on this

[@sagarraj4521]

Why are you so amazing!!!! Just a pen and paper with great explanation!!!!

[@iain_explains]

Thank you so much 😀

[@andreasmiller5448]

You have a very good series of videos on the subject of OFDM. Thank you very much for your great effort!

[@iain_explains]

Glad you like them!

[@heitorsousa1830]

Amazing video! Thank you professor!!

[@iain_explains]

Glad you liked it!

[@houyao2147]

so impressive!

[@m.furkanisk8858]

Very good explanation

[@animeshsrivastava5067]

Just to add more info- The output of IDFT is x[n] so you'll be obtaining discrete data at output of IDFT (you can compute this IDFT fast using IFFT process) now when you combine all data using parallel to serial you need to convert it to analog and then send. So add a DAC there and mix it with RF to send.

[@iain_explains]

Thanks, yes, I probably should have mentioned that.

[@Mindandsoul1996]

just great

[@linn8007]

Well-explained video! thanks! and please make more videos...

[@iain_explains]

Thank you, I will

[@jacobprice3079]

What an excellent video. Bravo!

[@iain_explains]

Glad you liked it.

[@nicholaselliott2484]

Amazing as usual!

[@iain_explains]

Thanks. It's great to hear when people find the videos helpful.

[@shaghayeghsamadzadeh5793]

Thanks for this video.

[@iain_explains]

You're welcome

[@davidstelte931]

Excellent videos on this and other subjects. Could I suggest some future videos on some related subjects? 1. Carrier clock recovery for OFDMA wireless systems; 2. Symbol rate clock recovery for OFDMA wireless systems; 3. Receiver gain/level control for accurate decoding of QAM constellation points. All of these have obviously been solved quite satisfactorily for 4G/5G and 802.11ac/ax systems, but there isn't too much info out there on how it is done.

[@iain_explains]

Thanks for the suggestions. These are actually already on my "to do" list, but it's a long list.

[@janiyatin]

Great explanation! You have a new subscriber :) Had one question , Why does the signal need to be transformed into a time domain signal before transmitting on to the channel?

[@iain_explains]

Because you need to send the signal out onto the channel "in time" - ie. one sample after another.

[@falguni.bonaparte]

Thank you Sir!

[@iain_explains]

You are welcome!

[@kyungtaekim217]

Well-explained video. You have a new subscriber :)

[@iain_explains]

Welcome aboard!

[@thiswillwork18]

Wonderful videos! Thank you. Can you make a video on the DFT-s OFDMA algorithm? I don't understand how it can reduce PAPR over CP-OFDM.

[@iain_explains]

Thanks for the suggestion. I've put it on my "to do" list. Hopefully I'll have something on this soon, because I know a few people have asked about it.

[@hs4hf]

Thank you every much for such a beautiful explanation. I have a quick question. Instead of doing IFFT, Since we know the frequency and sampled waveform of the i_th subcarrier, can we just store the waveform in ROM. Then recall the waveform and multiply with the i_th symbol instead. Or the IFFT is already more efficient to do the OFDM.

[@iain_explains]

Yes, you could do that, but then you've got to factor in the time it takes to load each waveform (vector) from memory, and you'd still need to add that waveform (vector) to all the others for the other subcarriers. Overall it's quicker to do the IFFT.

[@williamtang6394]

Thank you for such a great video. If possible, I would like to ask a question. At 5:10, I see that the two sub channels (S1(t) & S2(t)) are orthogonal and they don’t influence each other after we take the integration in time-domain. However, in frequency domain, the sinc functions have side-lobes that spread into neighboring channels. For example, the side lobe of S1(F) would overlap with the main lobe of S2(F). In that case, would the S1(F) still have some effects on S2(F)? If so, it seems to contradict our conclusion in the time domain.

[@iain_explains]

You're right, they do overlap in the frequency domain, however they are exactly aligned (in the frequency domain), such that the contributions from all neighbouring sinc functions equals zero at each of the subcarrier frequency values. Hopefully this video will help: "OFDM Waveforms" kzbin.info/www/bejne/fGelZX6vn5elgtk

[@yasserothman4023]

does the phase noise problem affect the carrier frequency Fc in ofdm or does it affect the subcarrier frequencies as well ? i mean the phase noise makes the power spectrum density looks like skirt shape unlike impulse in case of ideal oscillator

[@malini50]

Hello! great video again. One simple query, can we also say cos and sine are orthogonal to each other?

[@iain_explains]

Yes, people generally say this. For more details, see: "Orthogonal Basis Functions in the Fourier Transform" kzbin.info/www/bejne/pGPOlqaCmLWMbdE

[@yuzhechen1324]

Thanks for this great video. I am just confused by the P/S function after IDFT for a long time. It seems that we are already transferring X0 to X_N-1 in parallel by adding up multiple signal together in time domain. Could you please give a explanation about this?

[@iain_explains]

Well the P/S is really only shown there to indicate that the frequency domain symbols are processed in parallel in the DSP/Processor, whereas the time domain samples that result from having performed the IDFT, need to be clocked out into the transmitter in series (one after the other).

[@wesamamiri2323]

Thank you so much for this helpful vedio. I have one question please. The signal x(t) is a baseband signal, is there any need to multiply/modulate it by/with fc for transmission? I have seen some block diagrams include multiplication by fc after the P/S block.

[@iain_explains]

Yes, it needs to be shifted up the the carrier frequency. See: "How are Complex Baseband Digital Signals Transmitted?" kzbin.info/www/bejne/Zp3Og32do96qock

[@mordehaym3239]

I appreciate your great videos, Professor! Could you please explain why this operation is referred to as DFT? DFT that I am familiar with has N in the exponential, not T.

[@iain_explains]

Yes, you're right. I gave the continuous-time version of the equation because it's more intuitive to show how the waveforms (at the different frequencies) add up. But if you substitute t=(n/N)T, then you get the equation you are talking about (in terms of the discrete-time samples, indexed by the variable n). This is because in the IDFT, there are N time-domain samples, because there are N frequency-domain subcarriers. I also didn't show the usual scaling by a factor of 1/N (which I probably should have mentioned. ... but it's just a scaling, so it doesn't change any of the intuition, which is what I am trying to show in the video).

[@ahmetserdr2920]

Thank you sir. Can you make a video that is exolained IQ Signal and sampling rate formula?

[@iain_explains]

I'm not sure exactly what you mean by IQ signal and sampling rate formula. Can you please be a bit more specific?

[@hieutrung2883]

Great explanation ! Subcriber from Viet Nam

[@iain_explains]

Thanks and welcome

[@menoone2042]

Can we do the same to PSK modulation instead of ASK ?

[@kishorks5664]

IDFT or DFT should be applied to the discrete signals. But the basis used in the IDFT formula in the video is continuous. Is there any block digital to analog conversion that happens?

[@iain_explains]

Yes, you're right, I skipped over a technical aspect here. The equation I showed is not _exactly_ the IDFT. For the IDFT, the continuous time variable _t_ should be replaced by the discrete time values _t=(T/N)n_ . In practice, the actual transmitted signal x(t) is formed by putting those discrete time values/samples (ie. x(Tn/N) ) into a pulse shaping filter. See this video for more details: "How are OFDM Sub Carrier Spacing and Time Samples Related?" kzbin.info/www/bejne/oZ_NloulaLuNrMU

[@gamingandmusic9217]

in OFDM we use FFT and IFFT. how does the frequency resolution, sampling rate and FFT length N are related to the subcarriers ? please make a video explaining how sampling rate and choosing N for FFT and frequency resolution of FFT related to the original analog frequency of a signal. i hope i asked my doubt in an understanding way, thanks.

[@iain_explains]

Hopefully this video will help to explain it: "How does the Discrete Fourier Transform DFT relate to Real Frequencies?" kzbin.info/www/bejne/pnqpq2tqpM9smaM

[@dimitrisv.1729]

Is it valid to say that the role of IFFT at the transmitter is to multiplex the data streams onto orthogonal subcarriers?

[@iain_explains]

Yes, exactly.

[@cubbyhoo]

Hi Iain, great explanation! I am just checking for my own understanding, what exactly is the X_n data that is being fed into the IDFT? In the higher up section it is 1/-1 data which is being encoded on a given frequency wave by either being +ve or -ve to represent binary data, but that is not in the frequency domain? So is it a 1/-1 that is being IDFT'd or is it the +ve/-ve sinusoid that is then being IDFT'd? Just not 100% on that bit. EDIT: Just watched the next video where you explain this exact thing. Cheers for the excellent videos!

[@iain_explains]

Great, I'm glad you found the answer - and that you have found the videos helpful.

[@thrillscience]

How does the P/S and S/P step work? You need to transmit the signal over the channel to preserve the orthogonality.

[@iain_explains]

Sorry, I'm not really sure what you're asking. Sometimes, when a vector needs to be transmitted, people draw a Parallel-to-Serial box, to indicate that the vector is stored in memory elements in the digital device (computer/phone/...) which can be thought of as being "in parallel", and the output signal needs to be sequentially clocked out of the transmit amplifier in time order, which can be thought of as being "serial". But you don't really need to show those boxes since they are just an implementation issue.

[@louisli9480]

Great explanation, thx, I am wondering the sub carrier frequencies are f0, f0+15khz, f0+30khz for example, how are they regarded as 2pi*k/T in idft , seems hard for me to understand with a beginning frequency for sub carrier as f0, shall we just do like 15khz, 30khz, 45khz and then use a mixer to add f0 which is GHz

[@iain_explains]

Yes, exactly. The baseband subcarriers are multiples of the fundamental (first) frequency, and then the whole waveform needs to be up-converted to the passband. This video gives more details: "How are OFDM Sub Carrier Spacing and Time Samples Related?" kzbin.info/www/bejne/oZ_NloulaLuNrMU

[@hs4hf]

May I ask you more question. How are the number of Xk the number of x[n], and the number of subcarriers related? I think there are identical, but sometimes I see the number of subcarriers is less then the others. Can it be.

[@iain_explains]

This video should help: "How does the Discrete Fourier Transform DFT relate to Real Frequencies?" kzbin.info/www/bejne/pnqpq2tqpM9smaM

[@alikadhimal-janabi9484]

Firstly, thank you for these useful lectures. The o/p of the IDFT is x(t). So, why we use the P/S after it

[@iain_explains]

The IDFT is a matrix operation. It takes in a vector, and puts out a vector. The IDFT output only becomes x(t) when it is "played out in a serial fashion" and convolved with a transmit filter. This video hopefully provides more insights: "How are OFDM Sub Carrier Spacing and Time Samples Related?" kzbin.info/www/bejne/oZ_NloulaLuNrMU

[@DavidePecile]

Very good explanation, but i have a doubt, in BPSK modulation the equivalent complex moduated signal is given by y(t)= Real{b*(j2*pi*f0*t)} where b is the binary data baseband waveform (e.g. a rect of duration T_s), so shouldn't be the same here? also if you consider a DSP implementation of the modulation you have to feed the DAC with a real value, but the IDFT may give you complex numbers at the output. Is this right? Also, if i consider the numeric implementation of the OFDM, shouldn't be in the following form? y(nT)=real{ sum of k from 0 to N -1 of b_k(nT)*exp((j2*pi*k*n*T)/T_s) } and 1/T_s = 1/N*Tch where Tch is channel bandwith , T_s is the symbol duration, T is the interpolation frequency (T

[@iain_explains]

I probably should have also explained that everything in the video is baseband. The transmitted signal x(t) still needs to be multiplied by a carrier waveform in order to be transmitted at the allocated frequency (eg. 2.4GHz for WiFi). This is why there are complex numbers in the IDFT. The carrier waveform has a cos component and a sin component.

[@je8969]

Hello Sir. Thank you for the video! I don't understand why x(t) is the DFT. If the signal length of X would be infinite then at every frequency component there would be a Dirac with the hight of the value of each symbol. Then I would understand why x(t) is the DFT. But in this case the frequency components are sinc Funktions. So why ist the Fourier Transformation discrete? Sorry for the bad English.

[@iain_explains]

Hi Sorry, I'm not exactly sure what you're asking. As I mention at the 9:11 min point of the video, x(t) is the _inverse_ DFT of the frequency components X(f). (note that I originally said it was the DFT, but I corrected myself at the 9:11 min mark). The components X are in the frequency domain, and there are only a finite number of them, since we are only wanting to send our data over a finite bandwidth. I'm planning another video to explain how the subcarrier spacing is related to the digital sample rate, so keep an eye out for that one.

[@nikhilprasad717]

Sir, can you please tell that the signals X1 and X2 are carriers or mesage signal?

[@iain_explains]

They are complex numbers from a modulation constellation, that carry the digital data/message, that are going to be sent on the first and the second subcarriers.

[@lucidasser7153]

Thanks. At the beginning of the block, we give in digitale data (here 1and -1). These are the points on the IQ plane, right? With 4-PSK, there are 4 possible points I can give to the frequencies. With 16 QAM, I have 16 possible points I can give to the frequencies. Is that right?

[@iain_explains]

Yes, that's right.

[@dimitrisv.1729]

What the S/P block achieves in transmitter? I thought that we had N parallel data streams (N: number of subcarriers). Is this time between each transmission so small that we consider parallel transmission?

[@iain_explains]

Yes, the S/P can be confusing. Often people (including me) leave it out of block diagrams. Really it's just indicating that the OFDM modulator takes in a data stream (serial data) and distributes it into frequency sub channels (which we think of as being in parallel, because they are orthogonal to each other), and then converts that into a time domain signal using an inverse Fourier transform (which is serial again).

[@srk3567]

❤❤❤❤❤❤

[@natanijelvasic]

For QAM-16, the constellation points before the IFFT are in a regular grid. But after the IFFT, the constellation points would look very scattered and random, right?

[@iain_explains]

After the IFFT there are no "constellation points". Each constellation point (in the frequency domain) corresponds to a sinusoidal waveform over the digital symbol period, T. See this video for more details: "What is a Constellation Diagram?" kzbin.info/www/bejne/oZetln9qgbZnadU

[@natanijelvasic]

​@@iain_explains Oh yes - sorry. I got confused for a second thinking that the IFFT output was a parallel vector, and not a stream of values over time. Thanks.

[@anjurs7777]

Sir Will it function the same if I swap the idft and dft blocks together ?

[@iain_explains]

Great question. The answer is "yes" it will still work, but the phases of the subcarriers will all be rotated, in the time domain signal that is actually sent. For more insights into the difference between the FT and the IFT, see: "Fourier Trfm and Inv FT: What's the difference?" kzbin.info/www/bejne/hGm1h2mNiZmmirs

[@jenniferandrea7996]

Professor, Here X1 and X2 are bits in time domain. How can we have IDFT there ? Input to IDFT is Frequency domain symbols right ? I have this query for a long time. Didn't find proper convincing explanation anywhere.. can you help me out with this Thank you.

[@iain_explains]

X1 is a data value (complex constellation point) that is being sent on a sinusoidal carrier waveform with a frequency f1. X2 is a data value (complex constellation point) that is being sent on a sinusoidal carrier waveform with a frequency f2. In this sense, X1 and X2 are "in the frequency domain", since they are being sent (at the same time) at different frequencies. More details about constellation points can be found here: "What is a Constellation Diagram?" kzbin.info/www/bejne/oZetln9qgbZnadU

[@DavidG2P]

Wow, the beauty AND power of applied math! I wonder when OFDM was invented?

[@iain_explains]

It was invented in 1966. Glad you found the video useful. You might like to check out some of my other videos listed on iaincollings.com

[@kingsman428]

@@iain_explains Dude, that is a nice collection of videos you have there which is a bit of a bummer because it means I've now got to start watching them and having to learn the maths again. 😁

[@luckyboy-ih5hd]

Sir this is what confuses me, because we know that IFFT converts signal from frequency domain to time domain, so why frequency on transmitter side. Hope to see your explanation and thank you for all you have done for this community.

[@iain_explains]

In OFDM, each "data stream" is sent in a separate orthogonal frequency sub-channel, so the data starts in the frequency domain and then gets converted it into the time domain so it can be sent over an OFDM-symbol time period. Perhaps this video will help to explain it more: "OFDM Waveforms" kzbin.info/www/bejne/fGelZX6vn5elgtk

[@luckyboy-ih5hd]

@@iain_explains thank you, sir

[@nithinbabu4962]

Great video, thanks, Professor! I have a doubt though: from an implementation point of view, after IFFT, we get a discrete time-domain signal with N-samples whose values are complex numbers. How are we transmitting these then?

[@iain_explains]

Great question. Here are two videos that I'm pretty sure will explain it (if you watch them both): "How are OFDM Sub Carrier Spacing and Time Samples Related?" kzbin.info/www/bejne/oZ_NloulaLuNrMU and "How are Complex Baseband Digital Signals Transmitted?" kzbin.info/www/bejne/Zp3Og32do96qock

[@felipehugobragabittar2806]

oh yeah. same doubt over here. WHY people don't use a SUMMATION of X_k * e^ (i thetha) + e^ -(i theta)/ 2 (thats a cossine in EULER formula). that seens prety more clear and direct !

[@tihihoang]

Dear Professor, I have some questions, could you have some time to explain them? 1. In the OFDM settings, there is a parameter which is the subcarrier spacing, which is 30kHz. As I understand, the signal at subcarrier 2 would be X[k=2]*exp(j*2pi* 30e3 *t), and signal at the subcarrier 3 would be X[k=3]*exp(j*2pi* 60e3 *t),... so if we take the sum of them, is that different to the IDFT, which is the sum of X[k=2]*exp(j * 2pi /1024 *t), ... (with 1024 IDFT) ( I mean, the frequencies are 30e3*1024 difference). So do we need any technique between them? 2. Assume that we can map X[1] from subcarrier 1 to exp(j * 2pi *0/1024 *t), X[2] from subcarrier 30kHz to exp(j * 2pi /1024 *t),..., X[1024] from subcarrier 1023*30kHz to exp(j * 2pi *1023 /1024 *t), so the highest frequency here is 1023*30kHz, so the sampling rate should be 2 times of it? then it is different from the video kzbin.info/www/bejne/oZ_NloulaLuNrMU (How are OFDM Sub Carrier Spacing and Time Samples Related?) Is there something I am misunderstanding here? Could you explain it, or have one video to explain that? Thank you so much

[@iain_explains]

I'm not sure you are understanding the DFT/IDFT properly. The DFT/IDFT doesn't know the frequency spacing. It only takes a sequence of complex numbers and produces another sequence of complex numbers. It's up to us to interpret what those numbers relate to (ie. which frequencies they relate to). Hopefully this video will help: "Discrete / Fast Fourier Transform DFT / FFT of a Sinusoid Signal" kzbin.info/www/bejne/oqi0hYGZjNmBbdk

[@tihihoang]

@@iain_explains thank you so much, Professor. I found this video also: kzbin.info/www/bejne/sIDdmpyljdOhY80 , and my questions are answered at 20:00 of that video. your videos and his video are all great ❤ thank you so much and wish you a lot of health