# Inter Channels interference in MIMO system

If we have MIMO system with $$2$$ transmit antennas, ($$T_1$$ and $$T_2$$) and $$2$$ receive antennas ($$R_1$$ and $$R_2$$). The received signal of the first received antennas can be explained:

$$r_1$$ = $$h_{11}x_1$$ + $$h_{21}x_2$$ + $$v_1$$

where $$h_{11}$$ denotes the channel between $$T_1$$ and $$R_1$$ , $$h_{21}$$, denotes the channel between $$T_2$$ and $$R_1$$ , $$x_1$$ is the signal transmitted from $$T_1$$ and $$x_2$$ is the signal transmitted from $$T_2$$, and finally $$v_1$$ the AWGN on the $$R_1$$.

The signal received on the second received antennas can also be explained using same way of the above equation.

as I know that $$h_{21}x_1$$ is considered as interference on the first received antenna. Is that right ?

and my question, That interference should be varied in function of distance between the receive antennas, but how can we model that interference? I mean it should be big when the distance between the receive antennas is short and small when the distance is far. So, how can I model that interference in function of distance too ?

• It's not accurate to say that one term is interference and the other isn't; rather, both $h_{11}x_1$ and $h_{21}x_2$ interfere on each other. The receiver will combine $r_1$ and $r_2$ to obtain good estimates of $x_1$ and $x_2$. – MBaz Sep 24 '20 at 14:37
• Can you clarify why do you think that the distance between transmit antennas will affect the interference? In the model you're using, the channels are modeled by $h_{ij}$, and Tx antenna distance does not play a role. – MBaz Sep 24 '20 at 14:38
• @MBaz First, I mean that $h_{21}$$x_2$ will be considered as an interference in the first receive antenna and vise versa. Second, I think that distance between transmit antennas affect the interference since the interference will be higher when the antennas are near from each other. It means a correlation will happen between them. Is that right? Moreover, if the antennas are far way from each other, it means that no interference will happen. – New_student Sep 24 '20 at 15:42
• @New_student but that's exactly the wrong way of looking at it: there's no signal "meant" for either antenna with the other interfering. That's really not the case! – Marcus Müller Sep 24 '20 at 17:56
• Can't put it better than @MBaz did: the two separate receive signals only emerge by combining the observations of both receive antennas. Individually, there's nothing to decode: The information simply isn't there if you're just observing one antenna. – Marcus Müller Sep 24 '20 at 18:32

The point of increasing the distance between receive antennas is not to reduce the interference; it is to uncorrelate the channels.

In the system you're studying, the signal $$x_1$$ travels by two paths. One has gain $$h_{11}$$ and the other has gain $$h_{12}$$. Both of these gains are random.

A channel is considered bad when the magnitude of its gain, $$|h|$$, is small. The reason is that, in this case, the signal is likely to be overwhelmed by the noise.

If you had a single receiver antenna, then the probability of having a bad channel is very large. With two receiver antennas, there are two channels; as long as one if them is not bad, the transmitted signal can be recovered. Now, the probability of having two bad channels is much less than the probability of having one bad channel. Think of it like this: the probability of rolling two 1s is much less than the probability of rolling one 1.

The above is true as long as the two channels are uncorrelated. If the channels are correlated, then they tend to be either both good, or both bad. In this case there is little gain to be obtained by having two antennas.

Imagine that the two receiver antennas are in the same exact place. Obviously, in this case the channels are perfectly correlated. If the antennas are widely separated, then the channels will be completely uncorrelated.

A general rule of thumb is that, in a rich-scattering environment (that is, with lots of signal reflections and scattering), the receiver antennas should be spaced at least $$\lambda/2\,\text{m}$$, where $$\lambda$$ is the wavelength of the transmitted signal.

• Please, I have a question regarding to that, .. Is that interference I mentioned above called also cochannel interference?? I mean in rich multi-path environment. – New_student Sep 28 '20 at 14:12
• Not in the most common usage: normally, co-channel interference results when an unrelated transmitter interferes with another, often because of malfunction. – MBaz Sep 28 '20 at 18:52
• Is there a mathematical expression for the co-channel interference or clear explanation I can understand it? thanks in advance – New_student Sep 29 '20 at 1:40
• Id suggest asking a new question. – MBaz Sep 29 '20 at 3:12
• I asked new question here dsp.stackexchange.com/questions/70583/… – New_student Sep 29 '20 at 6:47