Background 3: Complex Baseband Model

[@Musiclover5258]

Hello Professor Emil Björnson, this lecture series is superb!! Thank you so much for sharing; I am trying to digest the material slowly. I have a quick question. At 25:43, in a general MCS (e.g., 32 QAM constellation), the input symbols x[k] would be complex valued, i.e., you would be modulating both the amplitude and phase of p(t-k/B). Even if we choose the sync function for p(t) then I was thinking that p((l-k)/B) may not necessarily be zero when l!=k, under arbitrary phase shifts. Am I right, and if so how is the general complex valued modulation handled? Thank you for any advice you can provide !

[@Musiclover5258]

On thinking further, the answer to my question might probably be that for a given modulating symbol considered in isolation, both the amplitude and phase angle of the carrier are continually ramped from (0,0) to the respective position in the constellation (A, theta), and then back to (0,0), in accordance with the smooth curvature of a sync pulse?

[@WirelessFuture]

​@@Musiclover5258 If the Nyquist criterion (i.e., p(k/B)=0 for integer k≠0) is satisfied, then the pulse makes sure that the symbols transmitted at the discrete time instances k and l are not affecting each other. You are then free to choose the complex symbols x[k] and x[l] independently of each other. The term x[k] p((l-k)/B) will be zero for any value of k since p((l-k)/B)=0. The signal's phase and amplitude will change continuously in between the sample instances, and be affected by multiple adjacent symbols. It is only at the sample instances where only one x[l] affect the signal. For that reason, it is important to synchronize the transmitter and receiver so that can agree on when to sample the signal.

[@Musiclover5258]

@@WirelessFuture Ok, thank you very much for the more detailed explanation. I am trying to visualize it this way. For a complex symbol sequence, there is a distinct baseband signal corresponding to the real axis and another distinct baseband signal corresponding to the imaginary axis. Each of these is formed exactly as described on page 8 at 15:43 (my apologies, I accidently referred to the wrong time earlier), and hence they do meet the Nyquist criterion. Then, when you up-convert to the passband signal, the instantaneous amplitude and phase of a carrier (say 3GHz) are modulated in accordance with the polar equivalent of these two baseband waveforms. Hope I am right.

[@WirelessFuture]

@@Musiclover5258 yes, this sounds correct

[@Musiclover5258]

@@WirelessFuture Thank you !!

[@alirezaqazavi]

very thanks, if is possible share presentations files so

[@WirelessFuture]

Yes, we have uploaded them here: github.com/emilbjornson/multiple_antenna_communications

[@PoonamDevi-qy4un821]

Thank you very much ... you are really true master ..... love from india😀😀😀

[@mohanadahmed2819]

Interesting lecture. - Is there any criteria for when we should we consider a system as a narrowband vs wideband? - And what happens if the narrowband assumption is broken? will the y_l = g x_l + n_l formula still hold?

[@WirelessFuture]

The general model is provided on Slide 9, where the received signal at time l depends on the transmitted signal at time l and at different times k ≠ l. As said on Slide 11, the latter dependence is negligible when all the propagation delays 𝜏_i are almost the same. Why do we call this narrowband versus wideband? Because the propagation delays are multiplied with B on Slide 11. When the narrowband property isn't satisfied, we need to use the general formulation on Slide 9 in what is known as "single-carrier transmission" or we can transform the system into multiple narrowband channels using OFDM.

[@mohamedabdirizakisse5018]

Thank you for the lecture. I have one basic question. Why is the noise effect additive, and not multiplicative like the channel effects?

[@WirelessFuture]

The channel describes how the signal is attenuated and phase-shifted on the way to the receiver, so it is directly connected to the signal. The noise is caused by random motion of electrons in the receiver hardware. It happens even in the lack of a signal.

[@mohamedabdirizakisse5018]

@@WirelessFuture Thank you so much for the response. And I cannot thank you enough for the lectures.

[@dennisting5209]

Hi, Is there a typo regarding the upper and lower bounds of the convolution integral on page 3? I think the negative infinity and plus infinity should be switched ? Thanks.

[@WirelessFuture]

Yes, you are right about that. Thank you for noticing! I have now corrected this in the slides on Github.

[@dennisting5209]

Thanks for the reply, I think there is also the same typo on page 7. The part explaining the equivalent complex-baseband systems. By the way, these are the best lectures I’ve seen on multiple antenna communications, thank you so much for sharing these lectures!

[@getsamik1019]

If we consider sampled signal as time series then is it possible to estimate frequency (s) from the sampled signal without using Fourier Transform

[@WirelessFuture]

Yes, you can probably correlate the time series with pre-generated sampled sinusoidal signals with different frequencies, to determine which one provides the best match.

[@juanminango7064]

Excellent explaniation

[@elsayed_mostafa]

Thanks a lot for such great lectures! Will the whole course (Multiple Antenna Communications) be shared also?

[@WirelessFuture]

Yes, they will be shared in the coming weeks.

[@AbdulUofG]

Nice...

[@ooppl-k8t]

thanks~