In this series:

In my last blog post, I provided a short introduction to multiple-input multiple-output (MIMO) technology. In this blog post, I discuss the fundamental limit of the maximum error-free data rate that can be supported by the MIMO channels. The maximum error-free data rate that a channel can support is called the channel capacity. The channel capacity for additive white Gaussian noise (AWGN) channels was first derived by Claude Shannon in 1948

In this blog series, I discuss the following aspects of MIMO channel capacity:

- MIMO channel capacity – Information theoretic derivation
- MIMO channel capacity – Importance of channel knowledge
- MIMO channel capacity – Ergodic capacity

## MIMO channel capacity – Information theoretic derivation

Consider a MIMO channel with *M _{t}* transmit antennas and

*M*receive antennas. For simplicity, the channel is considered to be frequency flat and the channel is assumed to have a bandwidth of 1 Hz. The channel transfer matrix is denoted by

_{r}*H*with dimension

*M*×

_{r }*M*. The input-output relation for the MIMO channel is given as

_{t}where *y* is the *M _{r }*×1 received signal vector,

*s*is the transmit signal vector of dimension

*M*×1, and

_{t}*n*is the

*M*×1 spatial-temporal white zero mean circularly symmetric complex Gaussian (ZMCSCG) noise vector with variance

_{r}*N*in each dimension.

_{o}*E*is the total average energy available at the transmitter over a symbol period (this is equal to the total average transmit power since the symbol period is 1 second). The covariance matrix of

_{s}*s, R*=

_{ss }*E*{

*ss*}, (

^{H}*s*is assumed to have zero mean) must satisfy

*Tr*(

*R*) =

_{ss}*M*in order to constrain the total average energy transmitted over a symbol period.

_{t}Assuming a deterministic channel, the capacity of the MIMO channel is defined as [2] [3]

where *f(s)* is the probability distribution of the vector *s*, and *I (s;y)* is the mutual information between vector *s* and *y*. Note that

where *H(y)* is the differential entropy of the vector y, while *H*(*y*│*s*) is the conditional differential entropy of vector *y* given knowledge of vector *s*. Since vector *s* and *n* are independent, *H*(*y*│*s*)=*H*(*n*). Eq. (3) simplifies to

As we have no control over the noise, maximizing *I(s;y)* reduces to maximizing *H(y)*. The covariance matrix of y, *R _{yy }= E{yy^{H} }*, satisfies

Among all vector *y* with covariance matrix *R _{yy}*, the differential entropy

*H(y)*is maximized when

*y*is ZMCSCG [4]. This implies that

*s*must be a ZMCSCG vector, the distribution of which is completely characterized by

*R*. The differential entropies of

_{ss}*y*and

*n*are given by

Therefore, *I (s;y)* reduces to

Thus, the capacity of the MIMO channel is given by [3]

The capacity *C* in (9) is also referred to as the error-free spectral efficiency, or the data rate per unit bandwidth that can be sustained reliably over the MIMO link. Thus, given a bandwidth of *W* Hz, the maximum achievable data rate over this bandwidth using the MIMO channel is simply *WC* bps.

## References

[1] C. Shannon. A mathematical theory of communication.*Bell Labs Technical Journal*, 1948.[2] G. Foschini. Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas.

*Bell Labs Technical Journal*, 1996.[3] E. Telatar. Capacity of multi-antenna gaussian channels.

*European Transaction on Telecommunications*, 10, 1999.[4] F. Neeser and J. Massey. Proper complex random process with applications to information theory.

*IEEE Transaction of Information Theory*, vol. 39 p:1293-1302, 1993.