Antenna Selection and Detection Performance on Correlation Based Detection Systems

— The increase data size transmitted in digital communication systems necessitates the use of high-capacity communication channels for the transmission of these data. To increase the channel capacity, MIMO systems are widely used. Antenna selection in MIMO systems is very important for efficient use of communication sources. Cognitive Radio systems generally aim to increase spectrum efficiency by using MIMO systems. Therefore, Cognitive Radio must find empty spectrum regions with spectrum detection methods. Spectrum detection performance is related to the antenna numbers in MIMO systems. Therefore, this study examines the effect of antenna numbers on detection performance in correlation-based detection method. Thanks to this study, the relationship between channel capacity, detection performance and antenna number for MIMO systems was investigated.

that MIMO systems can provide [6]. One of these applications is Cognitive Radio (CR) applications. The use of multiple antennas in multi-antenna CR systems is indispensable for some spectrum sensing models. For example, in eigenvalue and covariance based detection models, it is necessary to use multi-antenna in both transmitter and receiver [7]. However, there is no such requirement in the energy based sensing model. Basically, spectrum detection applications in CR systems are divided into classes such as single user-single antenna, single user-multi antenna or multi-user single antenna. Which method will be chosen to detect spectrum should be selected depending on some hardware requirements (number of users, antenna numbers) and communication environment. The main aim in spectrum sensing applications is to realize the most successful detection in the presence of the highest noise. Nevertheless , this process is not easy due to noise, fading and other disruptive effects in the wireless communication environment [8]. For this reason, various methods are proposed to ensure the most successful detection with the least user or antenna by using collaboration detection or some grouping algorithms [9]. The correlation-based detection method is very advantageous in terms of both computational cost and detection performance. In addition, the detection method used in this study detects as blind. That is, it does not need any prior knowledge for detection(for example: noise variance, modulation scheme).
Increasing the antenna numbers in spectrum sensing applications may increase detection performance, but increasing the antenna numbers is difficult in spectrum sensing applications. Because the antennas must be far enough from each other to not correlate with each other. On the other hand, the increase in the antenna numbers both increases the channel capacity and positively affects the detection performance.
Therefore, in this study, channel capacity-sensing performances are evaluated for correlation-based spectrum sensing in MIMO-OFDM systems. Thus firstly, theoretical information is given by channel capacity and correlation based spectrum sensing method for MIMO systems. Following, the relationship between antenna number and sensing performance in MIMO -OFDM based spectrum sensing method was investigated.
The organization of the study is as follows. Section 2 describes the antenna selection and channel capacity for MIMO systems. Cognitive radio and correlation-based spectrum detection is explained in chapter 3 and necessary Antenna Selection and Detection Performance on Correlation Based Detection Systems F. YAVUZ ILGIN  theoretical information is given. The results of the simulation studies made in section 4 are given. Chapter 5 is reserved for discussion and conclusion. Some of the notation we use is as follows: superscript (. ) and stands for transpose and the identity matrix of order L respectively. And (. ) denotes expectation operation.

II. CHANNEL CAPACITY MIMO SYSTEMS
Suppose there is MIMO communication system, where and indicate the number of transmit and receive antennas, respectively. With this communication system, the maximum throughput achieved by transmitting min( ) number of parallel data stream, is given by [10] where, x N r and N t x N r are the identity matrix and channel coefficient matrix, respectively. and is the power spectral density of additive Gaussian noise with η~N(0, 2 ). In addition denotes the transmitted signal power.
Antenna selection technique aims to reduce the number of transmitting antennas without changing the system efficiency by using the channel information on the receiver. Optimum throughput is acquired through selection of best set of antennas out of combinations. Where defines the number of transmit antennas to be used for transmission. Thus, the relationship between optimum system efficiency and antenna number is defined by Channel State Information (CSI) as follows [11].
where, is the channel capacity for optimally selected antenna number. The traditional MIMO system is given in  As it is known, in addition to increasing the channel capacity, MIMO systems also reduce the negative effects of noise thanks to the spatial diversity of the antennas.

III. COGNITIVE RADIO AND BASIC SPECTRUM SENSING
Suppose there is a Primary Base Station (PBS) with antenna and CR users within this coverage area. The purpose of CR users is to determine whether the PBS is active / passive. Thus, if PBS is passive, this communication band is used opportunistically by CR users. In this detection model, if the PBS has antenna and M CR users, this system can be defined as a MIMO system [12]. Basically the signal detection problem is to determine whether there is a embedded communication signal in the noise. In detection theory, this decision mechanism is explained by binary hypothesis testing. In binary hypothesis test, 0 indicates that there is only a noise signal, 1 indicates that it is a noise + communication signals. Mathematical decision making process is given below [13] .
Where ( ) is the form of the signal received from CR users (M numbers) stored in a matrix. For example, if M is the number of CR users and the number of samples is K, the size of the ( ) matrix will be MxK. In addition to this where ( ) denotes the independent and identically distributed (i.i.d.) circularly symmetric Gaussian (CSCG) white noise. ( ) and ( ) represents channel coefficient vector and primary user signal, respectively. In multi-antenna BR systems, spectrum detection is given by comparing the test statistic that varies according to the method with the threshold. The spectrum detection model for CR systems is given below.
Where = 1 under 1 and = 0 under 0 , M represents the number of CR user. In order to make a spectrum decision, the threshold and test statistics are compared in correlationbased spectrum detection. This phenome can be expressed mathematically as follows [12].
Where and define test statistics and threshold for correlation based detector, respectively.

A. Correlation Based Spectrum Sensing
Let ( ) = ( ) + ( ) be the continuous-time received signal. Assume that we are interested in the frequency band with central frequency W and bandwidth . We sample the received signal at a sampling rate . Thus, consecutive signal vectors for ( ), ( ) and ( ) are thus defined as follows.
Where L is defined as the smoothing factor. Specifies how many samples the spectrum detection process will be done with. Thus, the statistical covariance matrices of the received signal ( ( )) are defined as follows. .
Where denotes LxL identity matrix. According to Equation 12, if ( ) = 0, 1 = 0. Therefore, all non-diagonal components of 0 are equal to 0. If there is a signal samples and the signal are correlated, 1 is not a diagonal matrix. Hence, some of the off-diagonal elements of 0 should be nonzeros. Denote as the element of matrix 0 at the m.th row and k.th column. Then the following test statistics can be written.

B. Determination threshold for Correlation based Spectrum Sensing
To obtain the threshold value (under asymptotic conditions) the following equation can be written.

IV. SIMULATION STUDIES
In this section, we will give some simulation results for spectrum sensing performance and channel capacity. The simulation studies in this study were simulated in MATLAB environment. During the simulation studies, is selected as 0,1. Because this value is the limit value determined by the International Communication Committee. Additionally, noise uncertainty factor was not included in simulation studies. Because the Correlation Based Spectrum detection method is not affected by noise uncertainty factor [14], [15]. Looking at the threshold given with Equation 24, it is seen that there is no noise variance component. The absence of noise power when calculating the threshold indicates that this method is not affected by noise uncertainty.
In Fig. 2, only channel capacity is given for MIMO systems without detection performance. As can be seen, the highest channel capacity is reached when the antenna numbers is highest and the noise level is lowest. Although the increase in the antenna numbers in MIMO systems affects the channel capacity positively, it may not be easy to use more antennas. As explained in the previous chapters, the increase in the antenna numbers increases the spatial diversity, so the MIMO system is more resistant to noise. The antenna correlation should be at a minimum level in order to get maximum benefit from the spatial diversity. However, in many communication applications except CR applications, this may cause difficulties in practice, since the receiving antennas are located on the same receiver. In multi-user-single antenna CR applications, since each antenna is on a different user, maximum efficiency can be obtained from the spatial correlation.
One of the result graphics that constitute the main purpose of this study is given in Fig. 4. Where, the channel capacity for 4x4 and 6x6 MIMO systems is given along with the detection performance (the red numbers in the boxes describe the detection performance). When calculating , Monte Carlo analysis was applied. Thus, according to the detection scenario given in Figure 2, spectrum detection is performed by creating 500 times random PBS signal and channel coefficients. The average of 500 detection results is calculated for . First of all, it is seen that the detection performance for the same noise levels in 4x4 and 6x6 MIMO systems is directly proportional to the antenna numbers. For example, in the presence of 1 dB noise, detection performances are measured as 0.95 and 1 for 4x4 and 6x6 MIMO. Moreover, at noise levels higher than 0 dB, the 4x4 MIMO system has not made any accurate sensing at all. However, in a 6x6 MIMO system, it is seen that under -2db noise, 0.4 probability correct detection is made.
The antenna numbers and detection performance are given in Figure 5 for 2x2 and 3x3 MIMO systems. Unlike Figure 4, the most striking difference is that both channel capacity and detection performance are significantly reduced. For example, in the presence of 10 dB noise, the detection probability was 1 for 4x4 and 6x6 MIMO systems. However, in Figure 5, it is observed that these values decrease to 0,7.   Table 1 and Table 2.    SNR  4x4  5x5  6x6  8x8  10 dB  14  15  22  28  5 dB  11  13  16  18  0 dB  11  13  16  18  -5 Db  11  13  16  18 As can be seen from the tables, the 8x8 MIMO system is the most successful antenna combination in both channel capacity and detection performance.

V. CONCLUSION
In this study, correlation based spectrum sensing and channel capacity are investigated for MIMO systems. In the simulation studies, the effects of the antenna numbers on detection performance and channel capacity in MIMO systems containing different antenna numbers are examined. Thus, simulation studies were carried out by applying Monte Carlo analysis according to the given detection scenario. As a result of these simulation studies, the increase in the antenna numbers has positive effects on both detection performance and channel capacity.