# What is the physical meaning of coherence bandwidth in mmWave massive MIMO channel?

I am doing some research about millimeter wave massive MIMO channel now. We first take account about static MIMO channel, cause we only have SISO measurement system now. We use scanner to move our antenna to simulate it as array antenna to measure the static channel. Afterwards, using signal processing method to simulate static MIMO channel. There are some paper that doing the similar research, they analysis the coherence bandwidth at different antenna element. But I can't realize what the purpose of studying that issue. Is that related to some other MIMO techniques?

As far as I know, coherence bandwidth is corresponding to multi-path delay spread, and is an indicator to distinguish whether the channel is flat-fading or not.

In the other point side of view, I also study multi-path MIMO channel model. The model is represent as below It shows that multi-path would cause difference phase shift owing to the steering array. However, I can't realize what the model is associated with coherence bandwidth at each element.

reference ： R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh and A. M. Sayeed, "An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems," in IEEE Journal of Selected Topics in Signal Processing, vol. 10, no. 3, pp. 436-453, April 2016.

You are right, coherence bandwidth is the frequency domain counterpart of delay spread.

However, to achieve diversity across antennas the spacing between the antennas is important "relative to the environment".Let me explain that a bit more. In a user mobile phone you would typically find the order of wavelengths seperation between antennas sufficient to get independent fading accross them. However, the same is not true at the base station. Typically at the base station the seperation between the antennas has to be a 10s of order of wavelength. Why so? Because a user equipment near to ground sees a much richer scattering environment, which are enough to cause phase changes within a wavelength. However at the base station (usually placed high up in the air) the scattering environmental is not that rich and hence more separation is required.

So, to study the effect of independent fading with antenna separations one can use the methodology to see if other antennas are seeing the same coherence bandwidths or not. It is all part of channel profiling whenever new frequencies are deployed.

Looking at the equation you have posted, it seems $$l$$ is the variable for $$N_p$$ multipaths and for each path, there will be a constant delay $$\tau_l$$, an attenuation $$\alpha_l$$, a doppler shift in the carrier $$\nu_l$$ , angle of arrival at the receiver $$\mathbb a_{\mathbf R}$$ and angle of departure at transmission $$\mathbb a_{\mathbf T}$$. And both of these angles will depend on the antenna array configuration.

In 3-dimensional channel models, we require a pair of angles to describe the direction of coming in(to receiver) or going out(of transmitter). That is why you see $$\theta, \phi$$ pair associated with both angles $$\mathbb a_{\mathbf T}$$ and $$\mathbb a_{\mathbf R}$$

And one time-constant attenuation per path $$\alpha_l$$ and one time constant delay per path $$\tau_l$$, kind of associates the model with Coherence Bandwidth, which would be reciprocal of the delay spread of the channel. And, delay spread depends on these $$\tau_l$$.

In contrast to your previous question(Why multi-path channel has linear phase within the coherence bandwidth?), the coherence BW in mmWave may not be relevant considering delay spread alone. If you see eq (7) of the reference, the $$H$$ has contribution from all the $$N_p$$ paths, even though channel is mentioned as narrow band (flat fading). This is true till eq (15) of the reference. So to summarize, the coherence bandwidth is not dependent on the channel delay spread alone but also the array steering vectors. This is because, even if delay spread is same, a simple direction change of antenna will change the phase of your signal. But this model is too complex for practical purpose. In real world, after beamforming, you can train the direction such that you get a single strong tap channel. Hence in eq (16) of the reference, you do not see the mention of $$N_p$$ paths. They simplify the model and consider single path due to beamforming. This still does not fully answer your question though (influence of steering array vector on coherence bandwidth) but for practical purpose you can use model of eq (16). Reference 36 of this paper is also a good read.