**5.2 MIMO Channel capacity with UPA algorithm for No CSIT with perfect CSIR**

As discussed in Section 3, the decomposition of MIMO channel using SVD into r parallel channels helps in increasing the capacity of MIMO systems. Optimal Power allocation plays a significant function in the computation of enhanced MIMO capacity [1]. In this section the amount of power allocated to each transmitting antennas based on Perfect CSIR and No CSIT availability is presented.

The signals, *<sup>s</sup>* <sup>¼</sup> *<sup>s</sup>*1, *<sup>s</sup>*2, … *sNT* ½ �*<sup>T</sup>*is transmitted from the source and have no knowledge about what lying out there in the channel for the transmitter i.e. No CSIT. So vector 's' is statistically independent i.e.

$$R\_{\mathfrak{X}} = I\_{N\_T} \tag{18}$$

So substituting Eq. (18) in Eq. (16) the capacity expression is.

$$\mathbf{C}\_{\text{MIMO}}^{\text{No CSIIT}} = \log\_2 \left( \det \left( I\_{N\_\text{R}} + \frac{E\_\text{S}}{N\_T \sigma\_\text{n}^2} H H^H \right) \right) \tag{19}$$

From Section 3.2, it is observed that decomposition of Hermitian matrix HH<sup>H</sup> is *<sup>Q</sup>* � <sup>Λ</sup> � *<sup>Q</sup><sup>H</sup>*, where Q is an Orthonormal matrix and <sup>Λ</sup> is a non-zero diagonal matrix. So the capacity expression becomes.

$$\mathbf{C}\_{\text{MIMO}}^{\text{No CSIT}} = \log\_2 \left( \det \left( I\_{N\_k} + \frac{E\_\text{S}}{N\_T \sigma\_n^2} Q \Lambda Q^H \right) \right) \tag{20}$$

Now using matrix identity detð Þ¼ *IN* þ *AB* detð Þ *IN* þ *BA* , the above equation can be written as

$$\mathbf{C}\_{\mathrm{MIMO}}^{\mathrm{No\,CST}} = \log\_2 \left( \det \left( I\_{N\_\mathrm{R}} + \frac{E\_\mathrm{S}}{N\_T \sigma\_n^2} \Lambda \mathbf{Q} \mathbf{Q}^H \right) \right) \mathrm{(bit/s/Hz)}$$

also from Eq. (11) *QQ<sup>H</sup>* <sup>¼</sup> *<sup>Q</sup>HQ* <sup>¼</sup> *INR* ,

$$\mathbf{C}\_{\text{MIMO}}^{\text{No CSIIT}} = \log\_2 \left( \det \left( I\_{N\_R} + \frac{E\_S}{N\_T \sigma\_n^2} \Lambda \right) \right) \left( \text{bit/s/Hz} \right) \tag{21}$$

where Λ ¼ *diag*f g *λ*1, *λ*2, … *λNR* and *INR* are the diagonal matrix. In Eq. (21) the sum of two diagonal matrices is again a diagonal matrix with the elements as <sup>1</sup> <sup>þ</sup> *ES NTσ*<sup>2</sup> *n λi* � �diagonal entries [1]. The determinant of matrix is the product of diagonal values or eigenvalues i.e. det <sup>¼</sup> <sup>Q</sup>*<sup>r</sup> <sup>i</sup>*¼1*λ<sup>i</sup>* and log of product is sum of log of individuals. So the capacity expression of Eq. (21) is given as.

$$\mathbf{C}\_{\text{MIMO}}^{\text{No CST}} = \sum\_{i=1}^{r} \log\_2 \left( \mathbf{1} + \frac{E\_{\text{S}}}{N\_T \sigma\_n^2} \boldsymbol{\lambda}\_i \right) \text{ (bit/s/Hz)} \tag{22}$$

Therefore, the capacity of a MIMO channel can be interpreted in terms of sum of 'r' number of SISO channels each having a signal strength of *λi*, *i* ¼ 1, 2, … *r* corresponding to eigenvalue of that particular link and each of channel is excited by *ES NT* i.e. certain fraction of total transmitted power.

It is also observed that the maximum capacity is a function of channel coefficients and channel properties. The capacity of MIMO with No CSIT Eq. (22) and fixed total channel power Eq. (23) includes eigenvalues *λ<sup>i</sup>* which are obtained from parallel decomposition of HH<sup>H</sup> [7, 9, 10]. So here, MIMO systems with Perfect CSIT and No CSIT were considered for the development of a novel SVD-based IWFA algorithm (IWFAA) [1].
