# Explanation a flat fading channel in case MIMO OFDM

I am trying to understand flat fading and frequency selective channels.

As I understood, in the case of flat fading channels MIMO channel elements, $$h_{i,j}$$, are random variables.

Which assumption can I make, if the channel flat fading of frequency selectiv for channel matrix?

Actually, I am reading about MIMO OFDM. In all paper which I have read, frequency selective channel model is given for MIMO OFDM. But in my task I need to assume a flat fading. Can i do such assumption?

Frequency-flat fading essentially means that the channel's transfer function is (approximately) constant in your frequency band of interest, which means you can treat it as if it would not depend on frequency at all. This means two things:

• Your MIMO transmission model is simply $$\mathbf y(t) = \mathbf H \mathbf x(t) + \mathbf w(t)$$ with a transmit signal $$\mathbf x(t)$$, a MIMO channel matrix $$\mathbf H$$, a received signal $$\mathbf y(t)$$ and your additive noise $$\mathbf w(t)$$. Note that $$\mathbf x(t)$$ needs to be narrow-band for this to make sense.
• Your wouldn't use OFDM in this case. It just doesn't make sense: Every subcarrier would see exactly the same channel and you would gain nothing from the OFDM pre/post processing.

As for modelling the elements of your channel matrix, you can use statistical models that assume some random distribution that the elements are drawn from but be wary of the assumptions those are derived under. For instance, if you assume dense multipath from all directions, you end up with a Rayleigh distribution (resembling a case where you have rich scattering and no line of sight component). For cases with line of sight you might want to consider a Rician distribution. This is fine for not too high frequencies. For mmWave bands, this would be a bad model, since there the multipath is typically quite sparse and your channel matrix tends to be low rank.

• you have mentioned mmWave bands and low-rank channel matrix. Could you please suggest to me what I can read to find a detailed explanation about the low-rank channel matrix? – user36610 Jun 12 '19 at 6:24
• Have you tried google scholar? The first few hits look quite promising, e.g., this one, eqn. (9), or this one eqn (1). – Florian Jun 12 '19 at 8:09
• I have looked " low rank channel matrix" and didn't find good explanation. ok, will try with your suggestion – user36610 Jun 12 '19 at 8:24

In frequency-selecting model, the received samples for SISO channel are given by

$$y_n = \sum_{l=0}^Lh_lx_{n-l}+z_n$$

$$y_n = h_n x_n +z_n$$

where $$\{h_l\}$$ are the channel coefficients, $$\{x_n\}$$ are the data symbols, and $$\{z_n\}$$ are AWGN samples.

In the first case you need equalization or OFDM to combat the intersymbol interference, in the second case you don't need OFDM, but if you are using spatial multiplexing MIMO, then you would need equalization, but the interference in this case is coming from other streams, not from within the stream. If you are using OSTBC MIMO, then no need for equalization.

By the way, since you are reading MIMO-OFDM papers, you will of course see the assumption that the channel is frequency-selective. But this doesn't mean all papers assume frequency-selective channels, but those which talk about MIMO-OFDM will most certainly make this assumption, that is why they use OFDM to combat the frequency selectivity. So, you can find papers which don't make frequency-selective assumption.

As for what to assume, frequency-selective channels probably are more realistic in high data rate systems, but for analysis purposes flat-fading is usually used to get some insights. If you are talking about channel coefficients statistical distribution, both frequency-selective and flat fading channel coefficients are random variables. The exact distribution depend on the environment you are studying. In many cases for cellular mobile communication usually Rayleigh fading is assumed because of the absence of a strong line-of-sight (LOS).

In a multipath channel, the channel response is varying in frequency. However, in a OFDM system, we get a flat response for each subcarrier bandwidth. Even if it is not stricly the case, thanks to a cyclic prefix, we get a model as if it is strictly the case.

Then, in MIMO OFDM, we consider a MIMO scheme for each subcarrier, independently from one subcarrier to the others. Therefore, we get a collection a flat fading MIMO transmissions, one for each subcarrier.