A wired MIMO channel

In order to introduce the MIMO channel, as a preliminary step let us consider the SISO (Single Input, Single Output) channel, represented by a simple transmission line.

The received data are affected by Additive White Gaussian Noise (AWGN) n(t) having sigma^2 variance .

We suppose a passband communication system operating with narrowband signals. In the following the complex equivalent baseband representation of the system is used. 

The signal at the output of the RX, y(t), is related to the input of the TX, x(t), by a linear relationship:

wherein h is the channel response (including the TX and RX antennas response). For the sake of simplicity, the channel is considered deterministic. . 

The dependence of x, y and n on t will be dropped in the following.

If P_T is the power at the input of the TX element, supposed to be perfectly matched, the Signal to Noise Ratio (SNR) at the receiving sensor is

Consequently, the channel capacity in bits/s/Hz, e.g. the maximum data rate that can be reached with an arbitrarily small detection error, is

and logarithmically increases with P_T in case of SNR>>1.

Let us consider a cable consisting of M transmission lines. We suppose that no mutual coupling exists between the transmission lines, obtaining the following model:

and apex T denotes transpose.

In this case we can divide the input data stream into M different data streams, associating each stream with a different transmission line. Let us suppose that the available transmitted power is constrained to be P_T. Furthermore, we suppose that the transmitter has no knowledge about the channel. In this case the best strategy is to equidistribute the available power P_T among the M channels, so that the input power to each channel is P_T/M and the SNR at the output of the m-th receiver is

The channel capacity (bits/s/Hz) is the sum of the channel capacity of each SISO channel, i.e.

bits/s/Hz

The above capacity is the maximum data rate that can be sent with arbitrarily small detection error when the transmitter has no knowledge about the channel (note that if the transmitter has knowledge about the channel, it is possible to obtain a higher data rate with arbitrarily small detection error with a proper distribution of P_T given by the so called “waterpouring algorithm”). 

If we suppose that

the channel capacity is 

and almost linearly increases with M.