# How do we find the capacity of MIMO wideband point to point system?

This question is in the context of digital communication .

In many text books (wireless communication), when people derive the capacity of point to point communication system (one transmit one receive end) with multiple transmit and multiple receive antennas, people consider the narrow band flat fading channel. Below I write down the output equation of system with $N$ transmit antennas and $M$ receive antennas

$${\bf Y_{M\times 1} = H_{M\times N}X_{ N\times 1}+ Z_{M\times 1}}$$

The solution to find the maximum capacity is to do Singular Value Decomposition (SVD) of the channel as follows

$${\bf H = U D V^H}$$

The transmitter uses matrix ${\bf V}$ that is the transmit signal is $${\bf X'= VX}$$ while the receiver uses matrix ${\bf U}$ as follows

$${\bf Y'= U^HY}$$

My question is what about wideband channels, time varying frequency selective channels that is systems with channel such as

$${\bf H_{M\times N}(t)} \,\,\,\,\,\, t =0... T_{Delay}$$

How do we find the MIMO capacity of such channels. In one case, I think once can assume multicarrier (OFDM) system and divide the wideband channel into smaller narrow band and apply the algorithm I mentioned above. But what if its not multicarrier system? What is the optimal scheme, what should the transmitter and receiver do?

Thank you in advance

• I think "what is the optimal scheme" is an open question. I suggest to start your research with Tulino, Lozano, Verdu, "Capacity of multi-antenna channels in the low-power regime", and Verdu, "Spectral efficiency in the wideband regime". – MBaz Nov 25 '15 at 2:46
• so capacity of wide band MIMO systems is an open problem? I am not sure this is correct.. the book by David Tse Wirless Communication Chapter 7 says that capacity of wideband MIMO frequency selective channels are direct extension of results presented for narrowband channels...in that book however it was only stated as a remark... no results were presented – Henry Nov 25 '15 at 2:53
• @MBaz Please check, this link eecs.berkeley.edu/~dtse/Chapters_PDF/… In particular please check the second page of this pdf file where the following remark has been made: "Our focus throughout is on flat fading MIMO channels. The extensions to frequency-selective MIMO channels are straightforward and are developed in the exercises" – Henry Nov 25 '15 at 3:03
• I have done little work on wideband channels, so I'm not confident enough to give a certain answer. However, when I said that it's an open problem, I didn't mean that the theoretical capacity was unknown, but rather what's the best modulation scheme to make efficient use of it (i.e. OFDM vs Multicarrier vs. Massive MIMO, etc -- each have drawbacks). – MBaz Nov 25 '15 at 17:38