I will not repeat the answer of my colleague Navid. I would like to add some conceptual comments which may be useful for the issue.

Which channel model is dominant depends on the transmission media. If the media is homogeneous the line of sight model is dominant. If the medium contains large obstacles such as building then lognormal channel model will prevail. If the medium is scattering rich it will be multipath dominant with Rayleigh fading is the most suitable model. In case of making a study one can study the effect of such propagation models on the system performance.

As far the variation of the channel model with time represented by resource element, depends on the time variation of the channel. Generally the channel can be considered consatnt during the correlation time which is dictated by the Doppler shift.

Best wishes

Hi, Viktor Mishin

Question1.

Yes, what prof. Bjornson mentioned is absolutely correct. The ((variance)) of the channel gains are obtained from the following channel model known as "combined path loss and shadowing" [see, Wireless Communication, Gold Smith, Eq.(2.51)]:

[variance]dB = 10.log(Go) - 10.gamma.log(d/d0) + [PHI]dB

variance = 10^([variance]dB/10)

and substituted in the following MatLab code in order for generating the random values for the channel gains:

channel = sqrt(variance/2)*(randn(M,1)+1i*randn(M,1))

Question2.

In the combined path loss and shadowing model mentioned above, the random variable [PHI]dB in decibels is a log-normal random variable hence, in the combined path loss and shadowing model you must consider [PHI]dB as a log-normal random variable or equivalently you could consider PHI as a Normal random variable. It is more convenient to directly work with the log-normal random variable [PHI]dB.

Question3.

The article assumes that the channel gains are not independent through different resource blocks. I guess this dependence has nothing to do with the model used for determining the variance of these random variables. I guess the channel gains could be dependent or independent random variables while the variance of these random variables are obtained from the combined path loss and shadowing channel model.

I will not repeat the answer of my colleague Navid. I would like to add some conceptual comments which may be useful for the issue.

Which channel model is dominant depends on the transmission media. If the media is homogeneous the line of sight model is dominant. If the medium contains large obstacles such as building then lognormal channel model will prevail. If the medium is scattering rich it will be multipath dominant with Rayleigh fading is the most suitable model. In case of making a study one can study the effect of such propagation models on the system performance.

As far the variation of the channel model with time represented by resource element, depends on the time variation of the channel. Generally the channel can be considered consatnt during the correlation time which is dictated by the Doppler shift.

Best wishes

Dear Navid Pourjafari ,

Thank you very much for your detailed answer.

Thanks for the book, it is excellent.

I think that now I have no misunderstanding regarding the first and second questions.

In my third question, perhaps I did not express myself exactly, I apologize. I’ll clarify my question. Do I understand correctly that the resource block that was sent on the channel at time t will affect the next resource block going on this channel at time t + 1 (for example, the fading of the first block will overlap the signal of the second block). Or is this not taken into account?

I tried to find such dependencies. In particular, in article [3], formula (2), found it is explicitly indicated. H (t + 1, f) = alpha * H (t, f) + scrt (1 - alpha ^ 2) * W (as I understand, W is a fading)

and it seems that in equation (3) they denote how to find alpha. alpha = expectation (h*_{i,j}(t + 1, f) * h_{i,j}(t, f)). where h_{i,j} is an element of H channel matrix.

And you said correctly ("I guess this dependence has nothing to do with the model used for determining the variance of these random variables."), But I had a misunderstanding how to find this alpha, what is h*? Therefore, in my question, I doubted the reproducibility of this approach, because I decided that it was true only for their algorithm (Figure 1) and their channel approach. Now I have the assumption that h* is an element of the channel matrix at time t + 1 (for example, as you said channel = sqrt (variance / 2) * (randn (M, 1) + 1i * randn (M, 1)) ). Then we show how the RB, which sent at the time t, will influence this matrix (we find alpha through the correlation of the "raw" matrix of the channel t + 1 and the matrix of the channel t, and then substitute alpha into formula (2). seems like the truth?

Please allow me to clarify your answer to the third question.

"I guess the channel gains could be dependent or independent random variables while the variance of these random variables are obtained from the combined path loss and shadowing channel model."

I also think so, that's why I want to understand what is the relationship between H (t + 1, f + 1) and H (t, f) , where H is the “channel” in our notation above, (which already takes into account pass loss and fading). So I understand that at time t a group of resource blocks can be sent, so that a dependence is also possible in frequency.

Or do you mean ("the variance of these random variables are obtained from the combined path loss and shadowing channel model") that to determine this relationship, we just need to change the variance for the random variable [PHI] dB in "channel"t+1?

Can you say, please,

- Have you seen such dependences?

- Can you mathematically suggest how variance will change for the random variable [PHI] dB in "channel"t+1?

- Are such dependencies used to simulate signal transmission in the channel? Or is this not necessary?

Dear Abdelhalim Zekry

Thank you for your recommendations regarding the types of media and related model options. I will use them in modeling different scenarios.

I understand what you mean by saying that a channel is constant if the transmission time is less than the coherence time.

But I wanted to know how we can consider the effect of the transmitted signal at time t, on the signal transmitted at time t + 1. For example, due to the multipath propagation of the first signal, which can overlap the second. Is such a correlation between resource blocks considered?

Thank you very much in advance.

Dear Viktor,

My understanding for fading correlation for the consequent symbols is that the channel is correlated at the time t and at the time t+1. The symbols them selves are assumed uncorrelated as they represent the signal to be transmitted. This what i tried to point out in my first comment.

Best wishes

Viktor Mishin

YOU ASKED: "Do I understand correctly that the resource block that was sent on the channel at time t will affect the next resource block going on this channel at time t + 1 (for example, the fading of the first block will overlap the signal of the second block) "

ANSWER: A channel is represented with a function of time and frequency that characterizes the relation between the input and output signals of a channel which varies over time and frequency domain. A frequency range along with time duration in which this function returns an almost constant value is denoted as a RB. In other words channel gain is almost constant over certain frequency/time ranges. If fading exists then this constant value would be random in each RB otherwise it would be deterministic and could be calculated. It is common that only one RB is taken in to consideration and the results are averaged over different RBs. If this is the case, then it is implicitly assumed that there is no dependence among channel gains over different RBs. Also this assumption is fine for most scenarios but it would be much realistic to consider dependence between channel gains over different RBs since channel does not change instantly over frequency and time domains. The alpha that is obtained by determining the expectation in [3, eq.3] is in fact the covariance of two channel gains over two RBs (t+1,f) and (t,f) where h* denotes the conjugate of random variable "h" and E[h*(t+1,f).h(t,f)] = cov( h(t+1,f) , h(t,f) ) = dependence of channel gain "h" over two RBs (t+1,f) and (t,f).

YOU SAID: Therefore, in my question, I doubted the reproducibility of this approach, because I decided that it was true only for their algorithm (Figure 1)

COMMENT: I guess [3, Eq.3] is a fine representation for dependence of channel gains over different RBs.

YOU SAID: Now I have the assumption that h* is an element of the channel matrix at time t + 1 (for example, as you said channel = sqrt (variance / 2) * (randn (M, 1) + 1i * randn (M, 1)) ).

COMMENT: Yes, h*(t+1,f) is the conjugate of chaanel gain in RB(t+1,f) and is randomly generated by sqrt (variance / 2) * (randn (M, 1) + 1i * randn (M, 1)) in any RB.

YOU SAID: Then we show how the RB, which sent at the time t, will influence this matrix (we find alpha through the correlation of the "raw" matrix of the channel t + 1 and the matrix of the channel t, and then substitute alpha into formula (2). seems like the truth?

COMMENT: Yes, I guess it is correct to start with one H in one RB and then obtain the other H matrices of other RBs by iteratively obtaining alpha from [3, Eq.3] and substitute it in to [3, Eq.2]. In this context sqrt (variance / 2) * (randn (M, 1) + 1i * randn (M, 1)) is not applicable since MATLAB function randn() does not take the dependence in to account while generating random values unless you provide randn() function with proper seed which I personally haven't done before.

YOU SAID: Please allow me to clarify your answer to the third question...

COMMENT: No matter the channel gains are dependent or independent random variables, the variance is determined from the path loss and shadowing model and then substituted in h(t,f) = sqrt(variance/2)*( randn (M, 1) + 1i * randn (M, 1) ). In case of independent gains, randn() function generates random values which are independent anytime they are generated but in case of dependence, the randn() function must satisfy the proper dependence among the random values it generates every time.

YOU ASKED:

Can you say, please,

- Have you seen such dependences?

- Can you mathematically suggest how variance will change for the random variable [PHI] dB in "channel"t+1?

- Are such dependencies used to simulate signal transmission in the channel? Or is this not necessary?

ANSWERS:

- No, I haven't but as I mentioned it is much realistic to consider them dependent.

- The variance is conventionally obtained empirically. I am not sure if the dependence of channel gains introduces any constraints on the distribution of [PHI]dB because [PHI]dB belongs to the combined path loss and shadowing model and as I guess it has nothing to do with the dependence of the channel gains themselves. The dependence should be maintained by the randn() function.

- It is not necessary in most cases but if taken in to consideration then the simulation would be much realistic

Dear Navid Pourjafari

You cannot even imagine how much I would like to shake your hand. Thank you for your time and detailed answers.

In your first answer (ANSWER: A channel is represented ...) you clearly described a process where, indeed, "it would be much realistic to consider dependence between channel gains over different RBs since channel does not change instantly over frequency and time domains ".. I also apologize to you for inaccurate use of the terms that some resource blocks affect others. I meant the process that you described above.

 I am very glad that in the next two comments you confirm that alpha is a fine representation for dependence. And that h* is the conjugate element of the generated channel matrix before we're taking into account the dependence.

In our simulation, the channel matrix is ​​obtained “honestly”, without generation (randn (M, 1) + 1i * randn (M, 1)). We model reference signal, consider precoding, etc. I did not mention this before, in order not to complicate the discussion and speak in the same terms of the model mentioned by the professor Bjornson. Sorry if this was misleading. I understood your caution that the matlab function is not suitable. But we will act without it, i.e.

1) h(t + 1, f) we get honestly from reference signal, precoding, etc.

2) Multiply it by sqrt (variance / 2), where variance = 10 ^ (- pathloss / 10);

3) The pathloss will be, as you pointed out from the book Wireless Communication, Gold Smith, Eq. (2.51) or from the quadriga b1 scenario documentation.

4) Then we find its conjugated h*(t + 1, f)

5) Find the alpha [3, Eq.3] - correlation with h(t, f).

6) Finally, we get the correlated H(t + 1, f) [3, Eq.2].

Everything became clear.

My question "Can you mathematically suggest how variance will change .." was caused because I misunderstood your answer

"Question3.

The article assumes that the channel gains are not independent through different resource blocks. I guess this dependence has nothing to do with the model used for determining the variance of these random variables. I guess the channel gains could be dependent or independent random variables while the variance of these random variables are obtained from the combined path loss and shadowing channel model. "

I got the impression that the channel correlation does not occur through [3, Eq.3], but through some factor in the variance of the lognormal random variable when generating the next h(t + 1). Therefore, I asked you to indicate what a factor it might be. I am sorry.

Now I will use the sequence 1-6 above. So, in the end, did I understand you correctly?

Dear Abdelhalim Zekry

Sir, I completely apologize for operating the wrong terms. I absolutely agree with You and had in mind the same thing.

"The channel is correlated at the time t and at the time t+1", and how exactly this correlation can be determined mathematically by a formula is what I would like to know.

I am totally grateful to Your expertise and Your answers.

Viktor Mishin, you are quite welcome.

And that h* is the conjugate element of the generated channel matrix before we're taking into account the dependence.

- If you are referring to h*(t+1,f) then it is the conjugate of the channel gain at the time (t+1) ,i.e., the other RB and is definitely correlated with h(t,f). Any two channel gains in two RBs are correlated with each other in essence and I'm not sure if there is any before or after correlations.

in our simulation, the channel matrix is ​​obtained “honestly”, without generation (randn (M, 1) + 1i * randn (M, 1)). We model reference signal, consider precoding, etc.

- I guess you are referring to channel estimation here. Probably you mean preamble or pilot signals by the reference signal. Or you might be referring to channel matrices taken from real scenarios or measurements.

1) h(t + 1, f) we get honestly from reference signal, precoding, etc.

- I guess you mean you pick the channel gain from a real scenario or you mean channel estimation

2) Multiply it by sqrt (variance / 2), where variance = 10 ^ (- pathloss / 10);

3) The pathloss will be, as you pointed out from the book Wireless Communication, Gold Smith, Eq. (2.51) or from the quadriga b1 scenario documentation.

- Not sure why you intend to multiply with the variance, maybe the picked value from the real scenario was normalized with variance

4) Then we find its conjugated h*(t + 1, f)

5) Find the alpha [3, Eq.3] - correlation with h(t, f).

6) Finally, we get the correlated H(t + 1, f) [3, Eq.2].

- Maybe I'm missing a point here but I'm not sure why channel matrix is reconstructed from the alpha values while the alpha values themselves have been obtained from the channel gains.

Now I will use the sequence 1-6 above. So, in the end, did I understand you correctly?

- I'm afraid I think the 1 - 6 sequence is not clear enough to me. Maybe I'm missing some important points here.