What Konstantinos Dovelos is saying is correct. But one thing to bear in mind is that the array response vectors describe the propagation in the far-field of an array. As you increase the number of antennas, the far-field will also move further and further away. So the limit cannot be approached in practice. You should rather view as follows: when you have a large array, you get better and better at separating signals from different angles. A perfect separation is not achievable, but if you apply zero-forcing, you can separate the signals with a negligible SNR loss.

Hi Neeraj,

In a limited scattering environment, e.g., mmWave propagation, you have a small number of paths. Thus, the channel can effectively be modelled as the weighted sum of array response vectors; the weights are the path gains. In contrast, a rich scattering environment would yield a channel vector modelled as a complex Gaussian vector, i.e., Rayleigh fading. This answers your question about poor scattering.

Now, by taking the normalised product of any two array response vectors, you can show that it goes to zero as the number of antennas N goes to infinity as long as the angles-of-arrival of the paths are not equal (or their sine difference does not scale as 1/N). The inequality (b) in the Proof of Lemma 1 follows from basic algebra:

|1 - exp(jx)| = sqrt(1 - exp(jx))(1 - exp(jx)^*)

= sqrt(1 - exp(-jx) - exp(jx) + 1)

= sqrt(2 - 2Re{ exp(jx)})

= sqrt(2)sqrt(1 - cos(x))

< sqrt(2)sqrt(2) = 2.

The last inequality holds because cos(x) is in [-1,1], and hence sqrt(1 - cos(x)) takes its maximum value for cos(x) = -1.

What Konstantinos Dovelos is saying is correct. But one thing to bear in mind is that the array response vectors describe the propagation in the far-field of an array. As you increase the number of antennas, the far-field will also move further and further away. So the limit cannot be approached in practice. You should rather view as follows: when you have a large array, you get better and better at separating signals from different angles. A perfect separation is not achievable, but if you apply zero-forcing, you can separate the signals with a negligible SNR loss.

If you need to have orthogonal signals you should use orthogonal frequencies or orthogonal coding of signals in all transmitter channels. In this case, you can have little quantity of channels in MIMO system.