**4. Equivalent decomposition model of MIMO system**

In the previous section, the concepts of SVD, EVD and the significance of eigenvalues have been discussed. As aforementioned, the receiver and transmitter symbols relies on precoding and combining matrices obtained from channel

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

decomposition [1]. In this section, the basic structure of the channel and importance of channel state information are explained.

In the **Figure 2**, *<sup>H</sup>* <sup>¼</sup> *<sup>U</sup>*Σ*V<sup>H</sup>* is the channel matrix, the columns of U and V are the unitary matrices and diagonal matrix Σ is a *NR* � *NT* dimension with singular values σk, where *σ<sup>k</sup>* is the kth singular value of the Σ ordered decreasingly (*σ*<sup>1</sup> >*σ*<sup>2</sup> > … >*σr*) and rank 'r' of the matrix. For a input *s* ¼ *V*~*s* using precoder at transmitter is sent in the channel and received at receiver as

$$r = \sqrt{\frac{E\_s}{N\_T}} \\ \text{Hs} + n = \sqrt{\frac{E\_s}{N\_T}} \\ U\Sigma V^H V\tilde{s} + \tilde{n}$$

also expressed as,

$$r = \sqrt{\frac{E\_s}{N\_T}} U \Sigma \bar{s} + \bar{n}$$

The output of channel is multiplied by U<sup>H</sup> at receiver, resulting in

$$
\tilde{r} = \sqrt{\frac{E\_t}{N\_T}} U^H U \Sigma \tilde{s} + U^H \tilde{n} = \sqrt{\frac{E\_s}{N\_T}} \Sigma \tilde{s} + \tilde{n} \tag{14}
$$

further expressed as,

$$r\_i = \sqrt{\frac{E\_s}{N\_T}} \sigma\_i s\_i + \tilde{n}\_i = \sqrt{\frac{E\_s}{N\_T}} \sqrt{\lambda\_i} s\_i + \tilde{n}\_i \tag{15}$$

where *<sup>σ</sup><sup>i</sup>* <sup>¼</sup> ffiffiffiffi *λi* <sup>p</sup> , *<sup>λ</sup><sup>i</sup>* is the eigenvalue.

$$\text{trace } \text{of } R\_{\mathfrak{s}} = Tr(R\_{\mathfrak{s}}) = N\_T \text{ and } E(\tilde{n}\tilde{n}^H) = \sigma\_n^2 I\_r$$

From the above expression the number of non-zero elements in the diagonal matrix Σ corresponds to the number of independent channels. The processing of U matrix to the noise elements is not going to change the variance in the noise components *<sup>U</sup>Hn*<sup>~</sup> <sup>¼</sup> *<sup>n</sup>*~. In order to find U and V matrices, the channel matrix H is needed i.e. to do transmit precoding CSI at transmitter (CSIT) and CSIR for receiver shaping is needed [8].

#### **4.1 CSI at receiver (CSIR)**

It has all the channel coefficients i.e. transmitted from transmitter to receiver. It is estimated by using pilot symbols insertion in the signal sent from the transmitter. It is assumed that the receiver is having perfect CSI to do receiver shaping. In open loop MIMO system the CSI is available only at the receiver but not at the transmitter.

**Figure 2.** *MIMO system decomposition model for perfect CSIT.*
