Consider a MIMO (multiple input multiple output) system of transmitters and receivers and assume

where , and are column vectors of real random variables representing received signals, transmitted signals and noise.

The capacity of the MIMO channel is given by [1]

where is the p.d.f. of the transmitted signal vector . It turns out [1, 2] that where is the differential entropy of the noise and satisfies

Here is the covariance matrix of the noise. Thus Eq. (2) becomes

since the entropy of does not depend on the probability density function of the input variable .

In the SISO (single input single output) problem the mutual information was maximized while holding the variance of the output p.d.f. constant [3]. In the MIMO problem, the mutual information is maximized while holding the covariance matrix of the output signal vector constant and setting the mean of to zero. The method for finding the maximum is outlined in the references [2, 3, 4].

As has been shown [1, 2, 4] is maximized when is the p.d.f. of jointly-normal random variables. Using the jointly-normal p.d.f for the maximum indicated in Eq. (4) produces

Next, the covariance matrix of the received signal is evaluated in terms of the covariance matrix of the input signal and the covariance matrix of the noise using the relationship of Eq. (1). Thus

So

where two terms have been dropped from Eq. (5) because the noise and the transmitted signal are uncorrelated and the noise is assumed to have zero mean. Using Eq. (6) in Eq. (4) shows

the capacity per channel use or capacity in bits per sample. The capacity per unit time (or bits/second) is therefore [3]

where is the signal bandwidth and is the sampling rate.

In the SISO case, this reduces to

where is the variance of the input signal and is the ratio of the input signal power to noise power. Thus the result for in the MIMO case reduces to for the SISO case [3] as expected.

A more detailed derivation is given in [2] and is available by clicking below.

**References**

[1] J. R. Hampton, *Introduction to MIMO Communications\/*, Cambridge University Press, N.Y. (2014).

[2] H. L. Rappaport, “Derivation of MIMO Log-Det Formula for Channel Capacity with Real Input / Output Signals,” 7G Communications, 7GCTN05, November 5, 2014. real

[3] H. L. Rappaport, “Notes on Information Theory II and the Geometric Interpretation of the Shannon Channel Capacity,” 7G Communications, 7GCTN02, October 2014. infoII

[4] I. E. Telatar, “Capacity of Multi-Antenna Gaussian Channels,” Euro. Trans. Telecommun., 10 (6), (1999), pp. 585 – 595.