**5.1 Capacity of multiple antenna channel for ICIR and ICSIT**

It is assumed that channel is deterministic and noise is zero-mean complex Gaussian and circular symmetric random variable denoted as *nk* � *<sup>N</sup>* 0, *<sup>σ</sup>*<sup>2</sup> *nINR* � �, instantaneous channel state information is available at the receiver (ICSIR) and transmitter knows instantaneous channel knowledge (ICSIT) [3]. So for this scenario, the MIMO channel capacities are build.


$$\begin{aligned} I(\mathbf{s}; \mathbf{r}) &= \mathbf{h}(\mathbf{r}) - h(\mathbf{r}\zeta) \\ &= \mathbf{h}(\mathbf{r}) - h(n) \end{aligned}$$

where h rð Þ is the differential entropy of received signal vector r and since noise, received symbols are independent *h <sup>r</sup> =<sup>s</sup>* ð Þ¼ h nð Þ.

MIMO channel capacity *CMIMO* is expressed as.

$$\mathbf{C\_{MIMO}} = \max\_{Tr(R\_n) = N\_T} \log\_2 \left( \det \left( I\_{N\_k} + \frac{E\_\mathcal{S}}{N\_T \sigma\_n^2} H \mathbf{R}\_n H^H \right) \right) \tag{16}$$

Where, *Rss* is the covariance matrix of transmitted signal and is represented as,

$$\mathcal{R}\_{\mathfrak{s}} = E\left[\mathfrak{s}^{H}\right]$$

after substituting 's' matrix, the expression is

$$R\_{\mathfrak{s}} = E\left( \begin{bmatrix} s\_1 \\ s\_2 \\ \vdots \\ s\_{N\_T} \end{bmatrix} \begin{bmatrix} s\_1^\* & s\_2^\* & \cdots & s\_{N\_T}^\* \end{bmatrix} \right).$$

*Performance Analysis of Multiple Antenna Systems with New Capacity Improvement Algorithm… DOI: http://dx.doi.org/10.5772/intechopen.98883*

the covariance matrix *Rss* now becomes

$$R\_{\mathcal{S}} = \begin{bmatrix} \frac{E\_{\mathcal{S}}}{N\_T} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \frac{E\_{\mathcal{S}}}{N\_T} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \frac{E\_{\mathcal{S}}}{N\_T} \end{bmatrix} = \frac{E\_{\mathcal{S}}}{N\_T} I\_{N\_T} \tag{17}$$

So the total signal power constraint is the trace of the covariance matrix *Rss* and is equal to

$$E\_S \operatorname{Tr}(R\_{\mathfrak{s}}) = \sum\_{i=1}^{N\_T} E|s\_i|^2 = E\_S.$$
