I really appreciate the way you explained technically. Hopefully, concepts of equal gain combining (EGC), maximum gain combining (MGC) in digital communications are coming soon. Much respect!
Eigenvalue decomposition can help you to invert the channel, but the SVD enables much more. I've got this topic on my "to do" list, so keep an eye out for it.
Wow. Your explanations are so clear, that you are fooling most of us into thinking, "Yep, now I know this stuff..:") So do you have any references you can recommend, when you aren't here to explain to us, what we think we know? Thanks! And these vids really are great stuff.
Thanks for your comment. I hear what you're saying. I always try to encourage people to test themselves by trying to give their own explanation (even if it's just to themselves), after they think they understand a topic. In terms of references, I'm reluctant to provide reference lists for these videos, since in many cases I have made the video precisely because I have not found existing texts to be particularly clear on a specific topic. Of course, there are some excellent text books out there.
Thank you so much for this video. I have a question about the feedback path to the transmitter that you are mentioning at 9:19 and also in the description of the video. How does such a feedback path work? If you want to feed the information about the channel matrix H to the transmitter, the feedback transmission itself is also corrupted by the channel matrix of the MIMO channel from the Rx to the Tx. Consequently, it seems like a chicken-and-egg situation.
Thank you very much for the insightful content you’re sharing, Professor Collings. I have a question regarding ZF precoding: I understand that in this technique, the vector 𝑥 is multiplied by the inverse of the channel matrix before transmission. While this approach avoids noise amplification (which is the case of ZF filter at the receiver) when the channel is ill-conditioned, wouldn’t a similar issue arise with the transmitted symbols? Specifically, if the channel matrix is ill-conditioned, wouldn’t the PAPR of the transmitted vector (after ZF precoding) become high, potentially leading to amplifier non-linearity problems? I would greatly appreciate it if you could confirm whether my understanding is correct or if there’s something I might be overlooking. Thank you :)
I have some doubts on the concept of Periodogram, digital filters,holt forecasting,leakage effect,MA(q) stochastic process,Markov process and dyadic grid .I hope you add this concepts in future.It really makes me very happy to understand the concept in detail.
@6:05 the equation for MMSE, [(HH^H+r2I)^-1.H]^H.. In the references I have seen, the last Hermitian applies only to the last H matrix, ie (HH^H+r2I)^-1.H^H. PS: Your videos are excellent, thank you. PS2: I am 666th viewer :)
You may be confusing the hermitian with an inverse. Note that the "last H matrix" moves to the left hand side when the "last Hermitian" gets applied. There is an identity that can then be applied if you want to move that H matrix back to the right hand side, but then it comes with an inverse power. Details can be seen on the following Wikipedia page, under the heading "Linear MMSE estimator for linear observation process": en.wikipedia.org/wiki/Minimum_mean_square_error
The equations in this video are for the baseband representation. There are many more details that would need to be added, if I were to include the full transmission chain, including pulse shaping filters, D-to-A converters, amplifiers, analog filters, ... etc. One of those things would also be a matched filter at the receiver - matched to the pulse shape. For more details on the baseband model, see: "How are Complex Baseband Digital Signals Transmitted?" ruclips.net/video/0lkRJgnywkg/видео.html and "What is a Baseband Equivalent Signal in Communications?" ruclips.net/video/etZARaMNN2s/видео.html
Dear Iain, very nice and simple video at the same time. I have a quick question. For the Z.F. precoding you said the disadvantage is that "you need to tell the receiver what the value of the channel is". First of all, did you mean the transmitter? I suppose the receive needs to estimate the channel H and then this needs to be fed back to the transmitter to precode x with H^-1. Am I right? The second question is that even with SVD, either H or V needs to be fed back to the transmitter to be able to precode x. So why is this a disadvantage for Z.F. precoding but not for SVD? Thanks!
Sorry, can you be a bit more precise? What exact time in the video are you referring to. (sorry, I don't have time to re-listen to my videos to try to work out what you're asking about.)
Ah, OK, that's a "voice typo". Yes, you're correct, I meant to say that you need to tell the _transmitter_ what the channel is. And yes, the SVD also needs information to be fed back, so has the same requirement as the ZF precoder. In that comment I made, I was only talking about the relative pros and cons of the ZF precoder compared to the ZF receiver, but I probably should have also pointed that out for the SVD as well. Thanks for noticing that.
@@iain_explains Thank you very much. Got it. Last question: what is the advantage of the SVD compared to the ZF precoder? Has this something to with the computation of the inverse of H?
Yes, exactly! With the SVD, the precoder is a unitary matrix, so the elements are finite (unlike the case for the ZF precoder when the channel is not invertible, or close to not being invertible). Also, you can choose to not send any data on the "spatial subchannels" that have a zero (or very small) gain (ie. singular values), but for the ZF precoder you don't have the concept of spatial subchannels (ie. all "subchannels" have equal gain, since when you invert the channel you are left with the identity matrix ... assuming you are able to do the inversion).
It is somewhat historical, from Control Systems. In control theory, "zero forcing" filters (or feedback controllers) "force" a "zero" to be located at the same place as the system has a "pole" - in order to cancel out the pole, and make the overall system stable. In communications, the ZF receiver "forces" the interference to "zero", in a similar way. More details on ZF receivers can be found here: "How are Matched Filter (MF), Zero Forcing (ZF), and MMSE Related?" ruclips.net/video/U3qjVgX2poM/видео.html
I have benefited a lot from your DSP & Digital Communications playlists Prof. Iain Any plans to couple your work with Matlab examples! This is in some much demand You are are doing a wonderful service
Thanks for the suggestion. Yes, I've got a plan to put some more Matlab and Python code together to go with my videos. I've already done it for a few videos, but I need to find time to do more.
Great explanation. If I got this right, via SVD we convert MIMO communication to M SISO channels provided that Tx=Rx=M. I have some questions though. In real propagation conditions how possible is to transmit only to 1 receiver antenna (due to scattering)? Also does SVD make sense when Rx>Tx (massive MIMO)? I think that x_est = D*x + U'*w makes sense only for Tx=Rx systems but in the end of video when beamforming analyzed (x_est1 = u1'*H*v1*x1) does it make sense for both Tx=Rx and Rx>Tx systems? A series of videos about MIMO would be perfect.
Thanks, during the S.V.D. you say we have beamforming when we transmit just one symbol. But that isn't the criterion for beamforming I assume. We can also transmit two beams with two symbols at a time. How does that translate into the equation? We have now x1 and x2? We could get to vectors v1 and v2 -> so it is a matrix v then? Then it would fit again. The more symbols, the more the energy spreads perhaps to more beams.
Hello, First of all I would like to thank you for this fantastic video. Secondly, I have a question: When you started talking about beam forming and you said it would be like sending only one value of x so your matrix would look like [x1 0 0 0 0 0 0]. Isn't beam forming sending the same data using different antennas so that the transmitted signals constructively add up at the receiver? wouldn't the matrix look something like this: [x1 x1 x1 x1 x1 x1 x1]?
If you are using a precoding matrix (V in the video), then [x1 0 0 0 0 0 0] sends the same data (for user 1) on all antennas (first column of V times the scalar x1). If you are using a precoding vector instead (v1 in the video), then you only need to multiply it by the scalar x1.
There is MMSE receiver writter, we can also precode via MMSE. Is that correct? So that these 3 options (MMSE, ZF and SVD) can be applied at the transceiver and the receiver, yes?
Yes that's right. Here are some related videos: "How are Beamforming and Precoding Related?" ruclips.net/video/iMIqEpzxN9Y/видео.html and "How are Matched Filter (MF), Zero Forcing (ZF), and MMSE Related?" ruclips.net/video/U3qjVgX2poM/видео.html
Thank you for nice video sir. Question. You explain some receivers. In real situation such as 5G, how to select a receiver of them?. I think it depends on whether BS knows channel matrix or not. So, BS tells the receiver method to the UE. Right?
When communications are done according to a Standard, the receiving UE knows what to expect, in terms of all the signal parameters. So it can choose to implement whatever type of receiver it thinks will do the best job of receiving that signal.
This maybe a stupid question but when they said linear processing (under the context of MIMO) does that imply the kind of processing that was done by multiplying the signal with some matrix ?
Thank you very much. In the line for the Z.F. precoder, we have a term H*H^(-1)*x H*H^(-1) is not 1? (Because it is a matrix, and not a scalar?) Thanks.
Yes, H*H^(-1) = 1. That's the whole point! But maybe you're getting confused by not realising that the "H" is the mathematical representation of what happens to the signal during transmission over the channel (ie. this is not an operation that is done in the transmitter or the receiver), and "H^(-1)" is an operation that the transmitter does to the signal before it is transmitted (so that when it goes through the channel, the result will be that the channel cancels out the H^(-1) and the receiver just receives the signal "x" plus a noise term). If the transmitter didn't implement any "precoding", then the receiver would receive Hx plus a noise term.
With ZF does the interference always vanish completely? It sounds a bit difficult to get it completely away. Or is that not this interference that results too wide beams that overlap then? Thanks
Yes, the interference is completely cancelled in ZF. H^(-1)H=I ... assuming H can be inverted (some channels can be, some can't). This video gives more details: "How are Matched Filter (MF), Zero Forcing (ZF), and MMSE Related?" ruclips.net/video/U3qjVgX2poM/видео.html
It depends on what you mean by "perfectly correlated". Do you mean that all the channels between all the transmit and receive antennas are the same? This video might help: "Channel Estimation for Mobile Communications" ruclips.net/video/ZsLh01nlRzY/видео.html
In MIMO communications, assuming that each Tx antenna transmits a different signal, do the Tx antennas transmit their independent symbols simultaneously in the same frequency? or there is a frequency spacing?
Hello, Can you make videos from a industry product perspective? Or just extend the video on basic concept to include what is used in the present industry. Thankyou For example currently modern 5g antenna systems use 64TRx with a maximum of 16 layer mimo in downlink, and a simple explanation about its working.
These techniques are used in present industry (eg. 802.11n WiFi and 4/5G mobile), although of course there are many detailed technical issues that also come into the chip implementations that I haven't covered here. I'll try to mention this aspect more in future videos. Thanks for the suggestion.
Yes. It really depends on how much detail you want to add into your program. Matlab has functional blocks for each of the main elements of a communication system.
Dear Sir, Thanks for this very informative video. I have few questions, could you please answer 1) As singular value sigma1>sigma2...in D Matrix, always x1 will be travelling through the channel with more power(weight) when compared to x2, x3.... Am I right? 2) Also you explained about including Power matrix along with Precoding matrix D, does this power matrix together with D is also termed as Precoding (which results in flexibility in transmit power to each antenna of BS)? 3) What is the difference or advantage of SVD Precoding over ZF Precoding? - Is it that SVD decomposes into multiple parallel channels, multiple data streams transmitted without Interference, resulting in spatial multiplexing, whereas in ZF precoding we simply get our transmit vector back without Interference. - i.e in ZF always HH^(-1) = I, => Power Matrix should be used in addition in order to send symbols with different powers. - Whereas in case of SVD, we have Diagonal matrix (sigma in diagonal) multiplying with symbols and also in addition Power Matrix can be used to have different power on symbols. Is my understanding correct? Also could you please make video on i) H matrix of the model and H_est matrix obtained after acquiring CSI. ii) SVD Precoding in MU-MIMO. Thanks for your time.
Thanks, the entries of the Precoding/ Beamforming Matrix are complex and they are multiplied with the complex entries of the channel matrix H - is that right? Many thanks.
I really appreciate the way you explained technically. Hopefully, concepts of equal gain combining (EGC), maximum gain combining (MGC) in digital communications are coming soon. Much respect!
Thanks for the suggestion. I've added it to my "to do" list. I'm glad you like the videos.
A must-have channel for communication students. Thank you so much for sharing.
I'm so glad you hear that you like the channel, and recommend it to others. Thanks.
@@iain_explains Will do that, definitely. Thanks, professor.
You mention the SVD for precoding, can we also use the eigenvalue decomp. for these purposes?
Eigenvalue decomposition can help you to invert the channel, but the SVD enables much more. I've got this topic on my "to do" list, so keep an eye out for it.
This is gold....
This is so amazing video! Very clear explanation
Glad it was helpful!
Wow. Your explanations are so clear, that you are fooling most of us into thinking, "Yep, now I know this stuff..:") So do you have any references you can recommend, when you aren't here to explain to us, what we think we know? Thanks! And these vids really are great stuff.
Thanks for your comment. I hear what you're saying. I always try to encourage people to test themselves by trying to give their own explanation (even if it's just to themselves), after they think they understand a topic. In terms of references, I'm reluctant to provide reference lists for these videos, since in many cases I have made the video precisely because I have not found existing texts to be particularly clear on a specific topic. Of course, there are some excellent text books out there.
Hello,
Your videos are very helpful and well explained.
Glad you like them!
Thank you so much for this video. I have a question about the feedback path to the transmitter that you are mentioning at 9:19 and also in the description of the video. How does such a feedback path work? If you want to feed the information about the channel matrix H to the transmitter, the feedback transmission itself is also corrupted by the channel matrix of the MIMO channel from the Rx to the Tx. Consequently, it seems like a chicken-and-egg situation.
thanks
Thank you very much for the insightful content you’re sharing, Professor Collings. I have a question regarding ZF precoding: I understand that in this technique, the vector 𝑥 is multiplied by the inverse of the channel matrix before transmission. While this approach avoids noise amplification (which is the case of ZF filter at the receiver) when the channel is ill-conditioned, wouldn’t a similar issue arise with the transmitted symbols? Specifically, if the channel matrix is ill-conditioned, wouldn’t the PAPR of the transmitted vector (after ZF precoding) become high, potentially leading to amplifier non-linearity problems?
I would greatly appreciate it if you could confirm whether my understanding is correct or if there’s something I might be overlooking.
Thank you :)
Yes, you're correct. Nothing comes for free.
I have some doubts on the concept of Periodogram, digital filters,holt forecasting,leakage effect,MA(q) stochastic process,Markov process and dyadic grid .I hope you add this concepts in future.It really makes me very happy to understand the concept in detail.
Thanks for the suggestions. I'll see what I can do.
@6:05 the equation for MMSE, [(HH^H+r2I)^-1.H]^H.. In the references I have seen, the last Hermitian applies only to the last H matrix, ie (HH^H+r2I)^-1.H^H. PS: Your videos are excellent, thank you. PS2: I am 666th viewer :)
You may be confusing the hermitian with an inverse. Note that the "last H matrix" moves to the left hand side when the "last Hermitian" gets applied. There is an identity that can then be applied if you want to move that H matrix back to the right hand side, but then it comes with an inverse power. Details can be seen on the following Wikipedia page, under the heading "Linear MMSE estimator for linear observation process": en.wikipedia.org/wiki/Minimum_mean_square_error
Hello and thanks a lot for your amazing videos. I have a question, wasn't the first one at the receiver side a Matched Filter (instead of ZF)?
The equations in this video are for the baseband representation. There are many more details that would need to be added, if I were to include the full transmission chain, including pulse shaping filters, D-to-A converters, amplifiers, analog filters, ... etc. One of those things would also be a matched filter at the receiver - matched to the pulse shape. For more details on the baseband model, see: "How are Complex Baseband Digital Signals Transmitted?" ruclips.net/video/0lkRJgnywkg/видео.html and "What is a Baseband Equivalent Signal in Communications?" ruclips.net/video/etZARaMNN2s/видео.html
THanks a lot Sir!
Most welcome!
Dear Iain, very nice and simple video at the same time. I have a quick question. For the Z.F. precoding you said the disadvantage is that "you need to tell the receiver what the value of the channel is". First of all, did you mean the transmitter? I suppose the receive needs to estimate the channel H and then this needs to be fed back to the transmitter to precode x with H^-1. Am I right? The second question is that even with SVD, either H or V needs to be fed back to the transmitter to be able to precode x. So why is this a disadvantage for Z.F. precoding but not for SVD? Thanks!
Sorry, can you be a bit more precise? What exact time in the video are you referring to. (sorry, I don't have time to re-listen to my videos to try to work out what you're asking about.)
@@iain_explains Sure, I totally understand :) Here: ruclips.net/video/TC19gMQ6azE/видео.html (9:19)
Ah, OK, that's a "voice typo". Yes, you're correct, I meant to say that you need to tell the _transmitter_ what the channel is. And yes, the SVD also needs information to be fed back, so has the same requirement as the ZF precoder. In that comment I made, I was only talking about the relative pros and cons of the ZF precoder compared to the ZF receiver, but I probably should have also pointed that out for the SVD as well. Thanks for noticing that.
@@iain_explains Thank you very much. Got it. Last question: what is the advantage of the SVD compared to the ZF precoder? Has this something to with the computation of the inverse of H?
Yes, exactly! With the SVD, the precoder is a unitary matrix, so the elements are finite (unlike the case for the ZF precoder when the channel is not invertible, or close to not being invertible). Also, you can choose to not send any data on the "spatial subchannels" that have a zero (or very small) gain (ie. singular values), but for the ZF precoder you don't have the concept of spatial subchannels (ie. all "subchannels" have equal gain, since when you invert the channel you are left with the identity matrix ... assuming you are able to do the inversion).
Hello sir, are you considering making a video on QAM and QPSK? These may seem like really simple concepts but I have yet to find a good video on them
Great suggestion, thanks. I've added them to my "to do" list (up near the top).
Thanks for these videos! Why is it called a 'zero forcing' receive?
It is somewhat historical, from Control Systems. In control theory, "zero forcing" filters (or feedback controllers) "force" a "zero" to be located at the same place as the system has a "pole" - in order to cancel out the pole, and make the overall system stable. In communications, the ZF receiver "forces" the interference to "zero", in a similar way. More details on ZF receivers can be found here: "How are Matched Filter (MF), Zero Forcing (ZF), and MMSE Related?" ruclips.net/video/U3qjVgX2poM/видео.html
Thanks for the response! I'll check out the other videos.
I have benefited a lot from your DSP & Digital Communications playlists Prof. Iain
Any plans to couple your work with Matlab examples! This is in some much demand
You are are doing a wonderful service
Thanks for the suggestion. Yes, I've got a plan to put some more Matlab and Python code together to go with my videos. I've already done it for a few videos, but I need to find time to do more.
Great explanation.
If I got this right, via SVD we convert MIMO communication to M SISO channels provided that Tx=Rx=M. I have some questions though.
In real propagation conditions how possible is to transmit only to 1 receiver antenna (due to scattering)? Also does SVD make sense when Rx>Tx (massive MIMO)?
I think that x_est = D*x + U'*w makes sense only for Tx=Rx systems but in the end of video when beamforming analyzed (x_est1 = u1'*H*v1*x1) does it make sense for both Tx=Rx and Rx>Tx systems?
A series of videos about MIMO would be perfect.
Thanks for your comment and questions. Yes, modified things need to be done when Tx does not equal Rx. I'll add this topic to my "to do" list.
Thanks, during the S.V.D. you say we have beamforming when we transmit just one symbol. But that isn't the criterion for beamforming I assume. We can also transmit two beams with two symbols at a time. How does that translate into the equation? We have now x1 and x2? We could get to vectors v1 and v2 -> so it is a matrix v then? Then it would fit again.
The more symbols, the more the energy spreads perhaps to more beams.
Beamforming sends a single data stream on the largest eigenvector.
Thanks for sharing
Would there be videos for
1-signal whitening for detection and estimation
2-multiuser MIMO
Thanks for the suggestions. I've added these topics to my "to do" list.
Here's my new video on MU-MIMO ruclips.net/video/0ncIWlhsu1A/видео.html
Hello,
First of all I would like to thank you for this fantastic video. Secondly, I have a question:
When you started talking about beam forming and you said it would be like sending only one value of x so your matrix would look like [x1 0 0 0 0 0 0]. Isn't beam forming sending the same data using different antennas so that the transmitted signals constructively add up at the receiver? wouldn't the matrix look something like this: [x1 x1 x1 x1 x1 x1 x1]?
If you are using a precoding matrix (V in the video), then [x1 0 0 0 0 0 0] sends the same data (for user 1) on all antennas (first column of V times the scalar x1). If you are using a precoding vector instead (v1 in the video), then you only need to multiply it by the scalar x1.
There is MMSE receiver writter, we can also precode via MMSE. Is that correct?
So that these 3 options (MMSE, ZF and SVD) can be applied at the transceiver and the receiver, yes?
Yes that's right. Here are some related videos: "How are Beamforming and Precoding Related?" ruclips.net/video/iMIqEpzxN9Y/видео.html and "How are Matched Filter (MF), Zero Forcing (ZF), and MMSE Related?" ruclips.net/video/U3qjVgX2poM/видео.html
Thank you for nice video sir. Question. You explain some receivers. In real situation such as 5G, how to select a receiver of them?. I think it depends on whether BS knows channel matrix or not. So, BS tells the receiver method to the UE. Right?
When communications are done according to a Standard, the receiving UE knows what to expect, in terms of all the signal parameters. So it can choose to implement whatever type of receiver it thinks will do the best job of receiving that signal.
This maybe a stupid question but when they said linear processing (under the context of MIMO) does that imply the kind of processing that was done by multiplying the signal with some matrix ?
Yes, that's right.
Hi Professor...can you please upload a video on IRS communication?
Thanks for the suggestion. I'll add it to my "to do" list.
Thank you very much. In the line for the Z.F. precoder, we have a term H*H^(-1)*x
H*H^(-1) is not 1?
(Because it is a matrix, and not a scalar?)
Thanks.
Yes, H*H^(-1) = 1. That's the whole point! But maybe you're getting confused by not realising that the "H" is the mathematical representation of what happens to the signal during transmission over the channel (ie. this is not an operation that is done in the transmitter or the receiver), and "H^(-1)" is an operation that the transmitter does to the signal before it is transmitted (so that when it goes through the channel, the result will be that the channel cancels out the H^(-1) and the receiver just receives the signal "x" plus a noise term). If the transmitter didn't implement any "precoding", then the receiver would receive Hx plus a noise term.
With ZF does the interference always vanish completely? It sounds a bit difficult to get it completely away.
Or is that not this interference that results too wide beams that overlap then? Thanks
Yes, the interference is completely cancelled in ZF. H^(-1)H=I ... assuming H can be inverted (some channels can be, some can't). This video gives more details: "How are Matched Filter (MF), Zero Forcing (ZF), and MMSE Related?" ruclips.net/video/U3qjVgX2poM/видео.html
sir, how to estimate channel in case of mimo systems with perfectly correlated channel coefficient h.
It depends on what you mean by "perfectly correlated". Do you mean that all the channels between all the transmit and receive antennas are the same? This video might help: "Channel Estimation for Mobile Communications" ruclips.net/video/ZsLh01nlRzY/видео.html
In MIMO communications, assuming that each Tx antenna transmits a different signal, do the Tx antennas transmit their independent symbols simultaneously in the same frequency? or there is a frequency spacing?
Yes, all antennas use the same frequency band.
Hello,
Can you make videos from a industry product perspective? Or just extend the video on basic concept to include what is used in the present industry.
Thankyou
For example currently modern 5g antenna systems use 64TRx with a maximum of 16 layer mimo in downlink, and a simple explanation about its working.
These techniques are used in present industry (eg. 802.11n WiFi and 4/5G mobile), although of course there are many detailed technical issues that also come into the chip implementations that I haven't covered here. I'll try to mention this aspect more in future videos. Thanks for the suggestion.
Hi mr.
I have a question about spatially correlated channel in massive MIMO. What does it means?
All my respect
Have you tried watching my video: "Statistical Modelling of MIMO Communication Channels" ruclips.net/video/Q38bHrygPZg/видео.html
could you use matlab to do a simulation which include the whole process (modulation ,OFDM,RF and so on)about MIMO?
Yes. It really depends on how much detail you want to add into your program. Matlab has functional blocks for each of the main elements of a communication system.
Sir can you make videos about LTI properties and differential equations
Thanks for the suggestions. I've added them to the "to do" list.
Dear Sir, Thanks for this very informative video. I have few questions, could you please answer
1) As singular value sigma1>sigma2...in D Matrix, always x1 will be travelling through the channel with more power(weight) when compared to x2, x3.... Am I right?
2) Also you explained about including Power matrix along with Precoding matrix D, does this power matrix together with D is also termed as Precoding (which results in flexibility in transmit power to each antenna of BS)?
3) What is the difference or advantage of SVD Precoding over ZF Precoding?
- Is it that SVD decomposes into multiple parallel channels, multiple data streams transmitted without Interference, resulting in spatial multiplexing, whereas in ZF precoding we simply get our transmit vector back without Interference.
- i.e in ZF always HH^(-1) = I, => Power Matrix should be used in addition in order to send symbols with different powers.
- Whereas in case of SVD, we have Diagonal matrix (sigma in diagonal) multiplying with symbols and also in addition Power Matrix can be used to have different power on symbols.
Is my understanding correct?
Also could you please make video on i) H matrix of the model and H_est matrix obtained after acquiring CSI.
ii) SVD Precoding in MU-MIMO.
Thanks for your time.
Thanks, the entries of the Precoding/ Beamforming Matrix are complex and they are multiplied with the complex entries of the channel matrix H - is that right?
Many thanks.
Yes, that's right.
done