Correlated Random variable for shadow fading

On Matlab I would like to simulate a wireless mobile phone cell, in particular I want to reproduce the effect of shadow fading on the link between two users. So if I consider a certain 2D map, with some random users distributed along the map, I need to obtain from each user perspective a vector of random variable each on reffered to the path to the other users. All this variables are correlated due to the fact that users that are very close should experience approximately the same attenuation. Think of one transmitter A and two receiver which are very close to each other: fadin from A to RX1 and to RX2 should be very close. This relation can be expressed as an exponential: exp(- d/d0) with d the relative distance between A and RX and d0 a reference distance. So the longer the distance the larger will be the random number.

Another example: Consider a 30m x 30m area with let’s say 10 person in 10 random positions. For each person I want to generate a vector of 10 random numbers. Each entry of the vector is a number that is influenced by the relative distance of that user with the other. For example for user number 1 the vector will have 0 in the first position because the distance of user 1 from user 1 is zero. Second entry for example is 0.5 for a distance of 10m and third entry is 0.57 for a distance of 11m.

How can I reproduce this? I realized a correlation matrix whose elements are exp(-d/d0) with d a matrix of all the distances. Then I used the Cholensky decomposition.But the random realization does not follow the criteria I wanted.

Answer 1

[Thiago Henrique Gomes Lobato]

exp(-0/d0) equals 1, not 0, so I don't understand why you think the raw exp(-d/d0) would produce those results, since values will decrease with distance. If you want them to increase you can do somehting like 1-exp(-d/d0). Which, for it to be 0.5 at a distance of 10, should have d0 = 14.4270.

Answer 2

[Riccardo Giaffreda]

Let me clarify: if you think of a value of the shadow fading in one particular point and then you move from that point of a short distance d, the new value is correlated to the previous according to the exp(-d/d0) law. So if you move a short distance it is pretty similar (highly correlated), if you move for d>>d0 it changes completely.

[Thiago Henrique Gomes Lobato]

If I understood properly shadow fading is a fading effect in wirelless transmissions which originates from unknown obstacles along the way. If you have close cellphones, the chances of an obstacle between them is very low, and thus there's a very low amount of noise, while this is the opposite for larger distances since many obstacles can happen. exp(-d/d0) is a power decay law which would indicate the power between cellphones without any obstacle between them, it is not direct connected with actually correlation (maybe you misuse the term to mean similar?).

If you want to model the obstacles as a random process that increases with distances, an alternative is to use a log-normal distribution with a given standard deviation that depends of the log-distance between the cellphones, so closer phones will follow with higher probability the real exp(-d/d0) law and phones very far from each other would be more suitable to have noisy results arived from random objects in the path. In code this may look like something like this (you can try different values of sigma):

M(phone1,phone5) = exp( sigma*log10(d)*randn(1) ) + exp(-d/d0)

[Riccardo Giaffreda]

Shadow fading is referred to shadowing from obstacles affecting the wave propagation. The thing that I want to highlight and reproduce is that if you consider three users A, B,C where B and C are close. The transmission from A to B and from A to C must have a very similar shadowing value since the path of the wave propagating is pretty much the same and goes through the same obstacles. This should work for a large number of users.

[Thiago Henrique Gomes Lobato]

Yes, I understood that. Depending of the considerations you want to have in your model this problem can be extremely hard to code. An easy aproximation just considering the exp(-d/d0) relation as you originally asked can be done as described in my original answer for deterministic shadowing, the problem is that, for example, if you have a square place and A is in the botton-left corner, B upper-left and C botton-right, they will have, in your model, the same shadowing, which actually doesn't make much sense since C and B are actually very far apart, so this could be used for a very simplified model.

A better approach in my opinion is to calculate stochastic values for every microphone where the degree of randomness increases with distance, as my comment above. If you want to go a step further you can pre-calculate some grid points with the random approach and them interpolate between your calculated values to get the cellphone values, this is basically the same as creating fix obstacles in your field and should produce more realistic results.

[Riccardo Giaffreda]

Thank you for your answer. I really like the last approach you were suggesting, could you please walk me through it and help me? Thanks