**2. Multiple antenna system model**

In this section the system model and channel matrix, 'h' of multiple antenna system are introduced representing the complex channel gain which incorporates the channel fading effect. In wireless mobile communications, multi antenna systems are four types: single input single output (SISO), SIMO, MISO, and MIMO systems. With one antenna on either side, SISO provides no diversity protection against fading. When compared to SISO, the use of multiple antenna configurations will improve the reliability and capacity of the system.

A Multiple antenna system with NT transmitting and NR receiving antennas is shown in **Figure 1**. It has multi element antenna arrays at both the transmitter and the receiver side of a radio link to drastically improve the capacity over more traditional SIMO system. SIMO channels can offer diversity gain, array gain, and interference reduction [1]. In addition to these advantages, MIMO links can offer a multiplexing gain by parallel spatial data channels within the same frequency band at no additional power expenditure [2–4]. It creates receiver and transmitter diversity, with beam forming on both sides of antennas, improves SINR, and provides greater spectral efficiency [5].

The MIMO channel can be expressed as

**Figure 1.** *Schematic of NT* � *NR MIMO system.*

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

can also be represented as,

$$r^a = \sqrt{\frac{E\_s}{N\_T}} H s^a + n^a \tag{2}$$

where the received signal at instant <sup>α</sup> is *<sup>r</sup><sup>α</sup>* <sup>¼</sup> *<sup>r</sup><sup>α</sup>* <sup>1</sup> *r<sup>α</sup>* <sup>2</sup> ⋯ *r<sup>α</sup> NR* h i*<sup>T</sup>* , the transmitted signal is *s <sup>α</sup>* <sup>¼</sup> *<sup>s</sup> α* <sup>1</sup> *s α* <sup>2</sup> ⋯ *s α NT* � �*<sup>T</sup>* , the superscript 'T'stands for the matrix transpose, Es is the signal power across the transmitting antennas NT and *n<sup>α</sup>* is Additive white Gaussian noise with zero mean and variance σ<sup>n</sup> 2 . For a message from transmitter i, the jth receiving antenna is weighted by AWGN and the channel coefficient H. The above Eq. (2) represents a MIMO model with or without perfect CSI. The NR � NT channel response is as follows

$$H = \begin{bmatrix} h\_{1,1} & \dots & h\_{1,N\_T} \\ \dots & \dots & \dots \\ h\_{N\_R,1} & \dots & h\_{N\_R,N\_T} \end{bmatrix}\_{N\_R \times N\_T} \tag{3}$$

where hj,i is the complex channel coefficient between the jth receiver and the ith transmit antenna with zero mean complex Gaussian and circular symmetric channel at time instant α. With pilot symbols the receiver can estimate H matrix. The received signal at time α is given by

$$
\begin{bmatrix} r\_1^a \\ r\_2^a \\ \vdots \\ r\_{N\_k}^a \end{bmatrix}\_{N\_{\mathbb{R}} \times \mathbb{1}} = \begin{bmatrix} h\_{1,1} & \dots & h\_{1,N\_T} \\ \dots & \dots & \dots \\ \vdots \\ h\_{N\_k,1} & \dots & h\_{N\_R,N\_T} \end{bmatrix}\_{N\_{\mathbb{R}} \times N\_T} \begin{bmatrix} s\_1^a \\ s\_2^a \\ \vdots \\ s\_{N\_T}^a \end{bmatrix}\_{N\_{\mathbb{T}} \times \mathbb{1}} + \begin{bmatrix} n\_1^a \\ n\_2^a \\ \vdots \\ n\_{N\_k}^a \end{bmatrix}\_{N\_{\mathbb{R}} \times \mathbb{1}} \tag{4}
$$

and the vector form of received signal *r t*ð Þ in MIMO systems is

$$[r(t)]\_{N\_R \times \mathbf{1}} = [h(t)]\_{N\_R \times N\_T} [s(t)]\_{N\_T \times \mathbf{1}} + [n(t)]\_{N\_R \times \mathbf{1}} \tag{5}$$
