**5.3 New capacity improvement algorithm for perfect CSIT**

The system model after equivalent decomposition as described in Section 4 allow for the characterization of Perfect CSIT MIMO channel capacity. The MIMO capacity is

$$\operatorname{Tr}(\mathsf{R}\_{\mathsf{SS}}) = \sum\_{k=1}^{r} E|s\_k|^2 \le \overline{P} \tag{27}$$

from Eq. (15), for the ith received signal, *ri* the Perfect CSIT MIMO capacity is.

$$\mathbf{C}\_{\text{MIMO}}^{\text{Perfect CST}} = \sum\_{i=1}^{r} \log\_2 \left( \mathbf{1} + \frac{E\_{\text{S}}}{N\_T} \frac{\lambda\_i}{\sigma\_n^2} \chi\_i \right) (\text{bit/s/Hz}) \tag{28}$$

where, *<sup>γ</sup><sup>i</sup>* is the transmit power at the ith transmit antenna, *<sup>γ</sup><sup>i</sup>* <sup>¼</sup> *E si* j j<sup>2</sup> h i, *<sup>i</sup>* <sup>¼</sup>

1, 2, … *r*. Where in No CSIT case the transmit covariance is *Rss* ¼ *INT* i.e. each of them are having equal power allocated, but in case of Perfect CSIT system with transmit power *γ<sup>i</sup>* indicates that there is certain amount of power to be given to each of transmitting signal with transmit constraint P*<sup>r</sup> <sup>i</sup>*¼<sup>1</sup>*γ<sup>i</sup>* <sup>¼</sup> *NT*. In the case of Perfect CSIT, the singular values of the sub channel matrix is calculated for the MIMO system and transmitter can allocate variable power levels to them (SVD based IWFA algorithm) for maximization of capacity or mutual information. As the transmitter has full information about the channel matrix H with given set of eigenvalues *λi*, it can allocate different energy across the sub channels to maximize the capacity. So the capacity maximization problem is

$$\mathcal{C} = \sum\_{\gamma\_i=N\_T}^{\text{max}} \sum\_{i=1}^r \log\_2 \left( 1 + \frac{E\_S}{N\_T} \frac{\lambda\_i}{\sigma\_n^2} \boldsymbol{\gamma}\_i \right), \\ \text{subject to constraint} \\ \sum\_{i=1}^r \boldsymbol{\gamma}\_i = N\_T \tag{29}$$

Since the optimization function is concave and constraints is also defined. Using method of Lagrange multipliers [11], define a new variable, *ζ*, with objective function as

$$L(\boldsymbol{\chi}\_i) = \sum\_{i=1}^r \log\_2 \left( \mathbf{1} + \frac{E\_S}{N\_T} \frac{\lambda\_i}{\sigma\_n^2} \boldsymbol{\chi}\_i \right) + \zeta \left( N\_T - \sum\_{i=1}^r \boldsymbol{\chi}\_i \right) \tag{30}$$

where, this function L is known as Lagrangian and the new variable, '*ζ*' is the Lagrange multiplier. The unknown transmit power is obtained by taking derivative of *L γ<sup>i</sup>* ð Þ with respect to *γ<sup>i</sup>* and equating it to zero.

$$\frac{\partial \mathcal{L}(\chi\_i)}{\partial \chi\_i} = \mathbf{0} \tag{31}$$

by substituting Eq. (30) in Eq. (31) results,

$$\frac{\partial}{\partial \boldsymbol{\gamma}\_i} \left[ \log\_2^\epsilon \sum\_{i=1}^r \log \left( \mathbf{1} + \frac{E\_\mathcal{S}}{N\_T} \frac{\lambda\_i}{\sigma\_n^2} \boldsymbol{\chi}\_i \right) + \boldsymbol{\zeta} \left( N\_T - \sum\_{i=1}^r \boldsymbol{\chi}\_i \right) \right] = \mathbf{0}$$

by simplifying the above expression,

$$\frac{\log\_2^{\epsilon}}{1 + \frac{E\_S}{N\_T} \frac{\lambda\_i}{\sigma\_n^2} \boldsymbol{\gamma}\_i} \frac{E\_S}{N\_T} \frac{\lambda\_i}{\sigma\_n^2} - \boldsymbol{\zeta} = \mathbf{0},$$

Solving further the expression becomes

$$\log\_2^\epsilon \frac{E\_S \lambda\_i}{N\_T \sigma\_n^2 + E\_S \lambda\_i \gamma\_i} = \zeta$$

$$\left(\log\_\epsilon^2\right) \frac{N\_T \sigma\_n^2 + E\_S \lambda\_i \gamma\_i}{E\_S \lambda\_i} = \frac{1}{\zeta}$$

and the expression for *γ<sup>i</sup>* is

$$\gamma\_i = \frac{1}{\zeta \log\_e^2} - \frac{N\_T \sigma\_n^2}{E\_S \lambda\_i}.$$

The power allocation expression is

$$\gamma\_i = \mu - \frac{N\_T \sigma\_n^2}{E\_S \lambda\_i} \tag{32}$$

where, *<sup>μ</sup>* <sup>¼</sup> <sup>1</sup> *ζ* log <sup>2</sup> *e* .

So an IWFA algorithm is used for finding optimal power *γ opt <sup>i</sup>* for achieving maximum capacity, as *γ opt <sup>i</sup>* corresponds to the power allocated to the ith branch and it should be always greater than or equal to zero its value is

$$\gamma\_i^{\text{opt}} = \left(\mu - \frac{N\_T \sigma\_n^2}{E\_s \lambda\_i}\right)^+ \tag{33}$$

where, ð Þ*<sup>k</sup>* <sup>þ</sup> <sup>¼</sup> *k for k*><sup>0</sup> 0 *for k*≤0 �

and to find the value of constant *μ* referring to the constraint P*<sup>r</sup> <sup>i</sup>*¼<sup>1</sup>*<sup>γ</sup> opt <sup>i</sup>* ¼ *NT*

$$\sum\_{i=1}^{r} \left( \mu - \frac{N\_T \sigma\_n^2}{E\_i \lambda\_i} \right) = N\_T$$

Further solving the above expression

$$\left(r\mu - \frac{N\_T \sigma\_n^2}{E\_s} \sum\_{i=1}^r \frac{\mathbf{1}}{\lambda\_i}\right) = N\_T$$

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

and can be simplified as

$$r\mu = N\_T \left( \mathbf{1} + \frac{\sigma\_n^2}{E\_s} \sum\_{i=1}^r \frac{\mathbf{1}}{\lambda\_i} \right)^2$$

the threshold value now for optimal power allocation *γ opt <sup>i</sup>* is

$$\mu = \frac{N\_T}{r} \left( \mathbf{1} + \frac{\sigma\_n^2}{E\_s} \sum\_{i=1}^r \frac{\mathbf{1}}{\lambda\_i} \right) \tag{34}$$

So the Perfect CSIT IWFA MIMO capacity is

$$\mathbf{C}\_{\text{IVFA\\_MIMO}}^{\text{Perfect CSIT}} = \sum\_{i=1}^{r} \log\_2 \left( \mathbf{1} + \chi\_i \left( \mu - \frac{N\_T \sigma\_n^2}{E\_i \lambda\_i} \right)^+ \right) \tag{35}$$

The IWFA strategy therefore allocates power to those spatial channels with positive non-zero singular values i.e. good quality channels and discards the lower eigenmodes channels resulting in maximum capacity of MIMO systems when CSI is known at the transmitter. Usually the statistical fading channels are random in nature, so the expected value of capacity should be computed to obtain the average capacity or ergodic capacity. In the capacity analysis for MIMO channels, the ergodic capacity and outage capacity are obtained from the instantaneous capacity.
