*5.2.1. Uniform power allocation (UPA) algorithm for No CSIT MIMO Channel capacity*

Therefore, in MIMO system with No CSIT, the most optimal choice of power allocation is to use UPA algorithm i.e. to allocate power equally to all eigenvalues of HH<sup>H</sup> channel. The fixed total channel power (i.e. the eigenvalues of channel are equal) is expressed as,

$$\|\|H\|\|\_{F}^{2} = \sum\_{i=1}^{r} \lambda\_{i} = \xi \tag{23}$$

If the full rank channel matrix H is *NT* ¼ *NR* ¼ *N*, then the capacity can be maximized when the eigenvalues, *<sup>λ</sup><sup>i</sup>* <sup>¼</sup> *<sup>λ</sup> <sup>j</sup>* <sup>¼</sup> <sup>⋯</sup> <sup>¼</sup> *<sup>ξ</sup> <sup>N</sup>* , *i*, *j* ¼ 1, 2, … *N*. To achieve this, channel H should be orthogonal matrix

$$H H^H = H^H H = \frac{\xi}{N} I\_N \tag{24}$$

then the maximized capacity for UPA No CSIT is.

$$\begin{split} \mathbf{C}\_{\text{UPAMMOO}}^{\text{No GST}} &= \sum\_{i=1}^{N} \log\_2 \left( 1 + \frac{E\_S}{N \sigma\_n^2} \frac{\xi}{N} \right) \\ &= N \log\_2 \left( 1 + \frac{E\_S}{\sigma\_n^2} \frac{\xi}{N^2} \right) \end{split} \quad (\text{bit/s/Hz})$$

now if the diagonal elements of H is j j *<sup>H</sup>* <sup>2</sup> <sup>¼</sup> 1, then k k *<sup>H</sup>* <sup>2</sup> *<sup>F</sup>* <sup>¼</sup> *<sup>ξ</sup>* <sup>¼</sup> *<sup>N</sup>*<sup>2</sup>

$$\mathbf{C}\_{UPA\ MMO}^{\text{No CST}} = N \log\_2 \left( 1 + \frac{E\_S}{\sigma\_n^2} \frac{N^2}{N^2} \right).$$

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

similarly the UPA capacity for No CSIT is

$$\mathbf{C}\_{\text{UPA MIMO}}^{\text{No CSI}} = N \log\_2 \left( \mathbf{1} + \frac{E\_S}{\sigma\_n^2} \right) \tag{25}$$

Therefore, the capacity of orthogonal MIMO channel is N times the capacity of SISO channel. If at transmitter CSI is not available, then the UPA is used and is given as,

$$\mathbf{C}\_{\text{UPA MIMO}}^{\text{No CSIIT}} = \max\_{\mathrm{Tr}(\mathbb{R}\_n) = \overline{\mathbb{P}}} \log\_2 \left( \det \left( I\_{N\_\mathbb{R}} + \frac{E\_t}{N\_T} H H^H \right) \right) \tag{26}$$
