Interference Alignment in Multi-Input Multi-Output Cognitive Radio-Based Network

Abstract

This study investigates the interference alignment techniques for cognitive radio networks toward 5G to meet the demand and challenges for future wireless communications requirements. In this context, we examine the performance of the interference alignment in two parts. In the first part of this chapter, a multi-input multi-output (MIMO) cognitive radio network in the presence of multiple secondary users (SUs) is investigated. The proposed model assumes that linear interference alignment is used at the primary system to lessen the interference between primary and secondary networks. Herein, we derive the closed-form mathematical equations for the outage probability considering the interference leakage occurred in the primary system. The second part of this study analyzes the performance of interference alignment for underlay cognitive two-way relay networks with channel state information (CSI) quantization error. Here, a two-way amplify-and-forward relaying scheme is considered for independent and identically distributed Rayleigh fading channel. The closed-form average pairwise error probability expressions are derived, and the effect of CSI quantization error is analyzed based on the bit error rate performance. Finally, we evaluate the instantaneous capacity for both primary and secondary networks*.

1. Introduction

The rapidly growing number of mobile devices, higher data rates and cellular traffic, and quality of service requirements trigger the development of mobile communications. It is expected that the next-generation cellular networks (5G and beyond) will meet the advanced technology requirements. 4G networks are not powerful enough to support massively connected devices with low latency and high spectral efficiency, which is critical for next-generation networks. 5G networks are characterized by three fundamental functions in general: connectivity for everywhere, low latency for communication, and very high-speed data transmission [1].

In the near future, a large number of mobile devices will connect to one another in everywhere and provide a seamless mobile user experience. Real-time applications and critical systems and services (medical applications, traffic flow, etc.) with zero latency are expected to be offered over 5G cellular networks. Besides, the fast data transmission and reception will be ensured by supporting zero latency using a high-speed link. For this reason, the scope of 5G cellular networks bring the emerging advantages, new architectures, methodologies, and technologies on telecommunications such as energy-efficient heterogeneous networks, software-defined networks (SDN), full-duplex radio communications, device-to-device (D2D) communications, and cognitive radio (CR) networks. An increasing number of mobile devices and the bandwidth requirement for large amounts of data require the development of the new technologies and infrastructures in addition to the existing technology. It is inevitable that the number of smart phones, high-definition televisions, cameras, computers, transport systems, video surveillance systems, robots, sensors, and wearable devices produces a huge amount of voice-data traffic in the near future. To meet the growth and to provide fast and ubiquitous Internet access, several promising technologies have been developed. Regarding the deployment of the 5G wireless communication systems, the corresponding growth in the demand for wireless radio spectrum resources will appear. The capacity of the communication networks will be increased by using the energy-efficiency techniques with the evolving technology in 5G networks [2, 3, 4, 5].

One of the candidates for solving the problem of spectrum shortage is the CR network which will be a key technology for 5G networks. CR has attracted considerable interest as it can cope with the spectrum underutilization phenomenon. Performing spectrum sharing using a CR network is an important issue in wireless communication networks. There are three main ways for a primary network user to share the frequency spectrum with a cognitive user: underlay, overlay, and interweave. In the underlay method, the secondary user (SU) transmits its information simultaneously with the primary user (PU) as long as the interference between SU and PU receivers is within a predefined threshold. In the overlay approach, SU helps PU by sharing its resources, and in return, PU allows SU to communicate. In the interweave technique, SU can use the bandwidth of PU if PU is not active. In this model, SU should have perfect spectrum-sensing features to analyze the spectrum [6, 7, 8, 9].

Among the various methods of solving the interference problem, interference alignment (IA) is one of the most promising ways to achieve it. IA is an important approach for CR to recover the desired signal by utilizing the precoding and linear suppression matrices which consolidates the interference beam into one subspace in order to eliminate it [10, 11, 12, 13]. In the literature, linear IA is adopted in CR interference channels in [14, 15, 16, 17, 18, 19, 20] and the references therein. In [14], adaptive power allocation schemes are considered for linear IA-based CR networks where the outage probability and sum rate were derived. In [15], adaptive power allocation was studied for linear IA-based CR using antenna selection at the receiver side. Ref. [16] enhances the security of CR networks by using a zero-forcing precoder. Moreover, in [17], a similar work was proposed to improve the overall outage performance of the interference channel by using power allocation optimization. These studies have shown that interference management is a critical issue to be handled in all multiuser wireless networks.

CR technology can be capable of utilizing the spectrum efficiently as long as the interference between PU and SU is perfectly aligned as shown in Figure 1. A set of studies discussing IA is presented in the literature [21, 22, 23, 24, 25, 26, 27, 28, 29].

Figure 1.

Illustration of the primary link between PU pair and interference links generated by the SUs.

Motivated by the above works, in the first part of this study, we examine the impact of interference leakage on multi-input multi-output (MIMO) CR networks with multiple SUs. Specifically, a closed-form outage probability expression is derived to provide the performance of the primary system. Then, in the second part of our work, we investigate the performance of IA in underlay CR networks for Rayleigh fading channel. Moreover, unlike the mentioned papers, the effect of CSI quantization error is taken into account in our analysis. Then, a two-way relaying scheme with amplify-and-forward (AF) strategy is studied. Finally, the effects of the relay location and the path loss exponent on the BER performance and system capacity and CSI quantization on the average pairwise error probability (PEP) performance for this two-way AF system are presented.

The main simulation parameters and their descriptions used in this study are summarized in Table 1.

Symbol	Description
$P_1$ and $P_2$	Transmitted powers of the PU and SU
${\sigma }_N^2$	Variance of the circularly symmetric additive white Gaussian noise vector
$R_{\mathit{th}}$	Data rate threshold
$\alpha$	Interference-leakage parameter
$M_p$ and $N_p$	Number of transmit-and-receive antennas of PU
$M_s$ and $N_s$	Number of transmit-and-receive antennas of SU
$K$	Number of SU
$d_{j,i}$	Distance between the $i$ th transmitter and the $j$ th receiver nodes
${\tau }_{j,i}$	Path loss exponent between the $i$ th transmitter and the $j$ th receiver nodes
$B_{j,i}$	Channel state information exchange amount between the $i$ th transmitter and the $j$ th receiver nodes

Table 1.

The simulation symbols and their descriptions.

2. The impact of interference leakage on MIMO CR networks

In this study, MIMO interference alignment-based CR network with a PU and multiple SUs is considered under Rayleigh fading channel.

2.1. System model

In the system model as it is shown in Figure 2, the number of transmit-and-receive antennas of the PU is given by $M_p$ and $N_p$. The transmit antennas at each SU are given as $M_s$. The received signal, ${\mathbf{y}}_p$, implementing the IA technique is given as

$${\mathbf{y}}_p={\mathbf{U}}_p^{\mathrm{H}}{\mathbf{H}}_{\mathit{pp}}{\mathbf{V}}_p{\mathbf{x}}_p+\sqrt{\alpha }\sum _{i=1}^K{\mathbf{U}}_s^{\mathrm{H}}{\mathbf{H}}_{{\mathit{ps}}_i}{\mathbf{V}}_s{\mathbf{x}}_{s_i}+{\mathbf{U}}_p^{\mathrm{H}}\mathbf{n},$$E1

where $x_p$ and $x_{s_i}$ are the transmitted signals from PU and the $i$ th SU $\left(\text{for}i=1,2,\dots ,K\right)$, respectively. Herein, ${\mathbf{H}}_{\mathit{pp}}$ is the matrix of channel coefficients between the PU pair, and ${\mathbf{H}}_{{\mathit{ps}}_i}$ denotes the channel matrix between the primary receiver and the $i$ th secondary transmitter. The interference leakage is modeled similar to the one in [30]. The interference-leakage parameter $\alpha$ $\left(0\le \alpha \le 1\right)$ represents the status of the alignment, i.e., $\alpha =0\text{and}1$ corresponds to perfect alignment and perfect misalignment cases, respectively. $\mathbf{V}$ and $\mathbf{U}$ are the precoding- and interference-suppression matrices. The superscript ${\left(\cdot \right)}^{\mathrm{H}}$ denotes the Hermitian operator, and $\mathbf{n}$ is the zero-mean unit variance (${\sigma }_N^2=1$) circularly symmetric additive white Gaussian noise (AWGN) vector.

Figure 2.

IA-based CR network with single PU and $K$ SUs sharing the spectrum.

The following conditions must be satisfied for perfect interference alignment between PU and SUs:

$${\mathbf{U}}_s^{\mathrm{H}}{\mathbf{H}}_{{\mathit{ps}}_i}{\mathbf{V}}_s{\mathbf{x}}_{s_i}=0,$$E2

$$\text{Rank}\left({\mathbf{U}}_s^{\mathrm{H}}{\mathbf{H}}_{{\mathit{ps}}_i}{\mathbf{V}}_s{\mathbf{x}}_{s_i}\right)=d.$$E3

Each user transmits $d$ data streams. Using the ideal linear IA technique, (1) can be re-expressed as

$${\mathbf{y}}_p={\mathbf{U}}_p^{\mathrm{H}}{\mathbf{H}}_{\mathit{pp}}{\mathbf{V}}_p{\mathbf{x}}_p+{\mathbf{U}}_p^{\mathrm{H}}\mathbf{n}.$$E4

2.2. Outage probability analysis

The channel capacity and outage probability are the most important impairments which affect the quality of service (QoS) in wireless communication systems. When no CSI conditions are given, MIMO channel capacity is expressed as in [31]. The channel capacity of the considered MIMO system in PU can be expressed as

$$C={\text{log}}_2\text{det}\left|\mathbf{I}+\frac{{\gamma }_1}{\left(1+{\gamma }_2\right)N_p}{\mathbf{H}}_{\mathit{pp}}{\mathbf{H}}_{\mathit{pp}}^H\right|,$$E5

where ${\gamma }_1=P_1{‖{\mathbf{H}}_{\mathit{pp}}‖}^2/{\sigma }_N^2$ is the signal-to-noise ratio (SNR) of the primary link. ${\gamma }_2$ can be expressed as ${\gamma }_2=\left(P_2/{\sigma }_N^2\right){\mathbf{\text{∑}}}_{i=1}^K{‖{\mathbf{H}}_{{\mathit{ps}}_i}‖}^2$. Note that ${‖.‖}^2$ demonstrates the squared Frobenius norm of the channel matrix, $\mathbf{I}$ denotes for identity matrix, and $P_1$ and $P_2$ are the transmitted powers of the PU and SUs, respectively. If linear IA perfectly eliminates the interference between SU and PU, then SNR of the interference channel, ${\gamma }_2$, becomes zero. It is important to note that precoding and linear suppression vectors are assumed as ${\left|{\mathbf{U}}_p^{\mathrm{H}}\right|}^2={\left|{\mathbf{V}}_p\right|}^2={\left|{\mathbf{U}}_{s_i}^{\mathrm{H}}\right|}^2={\left|{\mathbf{V}}_{s_i}\right|}^2=1$. In the presence of interference-free communication, primary system works in the single-input and single-output (SISO) fashion [14]. Hence, the probability density function (PDF) of ${\gamma }_1$ can be written as $f_{{\gamma }_1}\left(\gamma \right)=\frac{1}{{\overline{\gamma }}_1}\text{exp}\left(-\gamma /{\overline{\gamma }}_1\right)$, and the outage probability of the system can be obtained as

$$P_{\mathit{\text{out}}}={\int }_0^{2^{R_{\mathit{th}}}-1}{f}_{{\gamma }_1}\left(\gamma \right)\mathit{d\gamma },$$E6

where $R_{\mathit{th}}$ is the data rate threshold and ${\overline{\gamma }}_1=P_1/{\sigma }_N^2$ denotes the average SNR of the primary system. By substituting $f_{{\gamma }_1}\left(\gamma \right)$ into (6), the outage probability can be obtained as

$$P_{\mathit{\text{out}}}=1-\text{exp}\left(\frac{2^{R_{\mathit{th}}}-1}{{\overline{\gamma }}_1}\right).$$E7

In the presence of interference, the primary system works in MIMO fashion, and leakages may occur due to fast-fading Rayleigh channel. To improve the performance of the primary system, we adopt maximum ratio transmission and maximum ratio combining at the transmitter and receiver, respectively. Thereby, the end-to-end signal-to-interference-plus-noise ratio (SINR) of the primary system can be written as ${\gamma }_{\tau }={\gamma }_1/\left(1+{\gamma }_2\right)$. In the proposed system, all channels are modeled as independent and identically distributed Chi-squared distribution, and the PDF of ${\gamma }_1$ can be expressed as

$$f_{{\gamma }_1}\left(\gamma \right)=\frac{{\gamma }^{M_p{N}_p-1}\text{exp}\left(-\gamma /\left({\overline{\gamma }}_1/M_p\right)\right)}{{\left(\frac{{\overline{\gamma }}_1}{M_p}\right)}^{M_p{N}_p}\left(M_p{N}_p-1\right)!}.$$E8

In addition, the PDF of ${\gamma }_2$ can be defined as

$$f_{{\gamma }_2}\left(\gamma \right)=\frac{{\gamma }^{{\mathit{KM}}_s{N}_p-1}\text{exp}\left(-\gamma /\left(\alpha {\overline{\gamma }}_2/M_s\right)\right)}{{\left(\frac{\alpha {\overline{\gamma }}_2}{M_s}\right)}^{{\mathit{KM}}_s{N}_p}\left({\mathit{KM}}_s{N}_p-1\right)!},$$E9

where ${\overline{\gamma }}_2=P_2/{\sigma }_N^2$ is the average SNR of the secondary system. Finally, the PDF of ${\gamma }_{\tau }$ can be written as

$$f_{{\gamma }_{\tau }}\left(\gamma \right)={\int }_0^{\infty }\left(x+1\right)f_{{\gamma }_1}\left(\left(x+1\right)\gamma \right)f_{{\gamma }_2}\left(x\right)\mathit{dx}.$$E10

By substituting (8) and (9) into (10), then with the help of [32, Eq. 3.351.3] and after few manipulations, PDF expression of $f_{{\gamma }_{\tau }}\left(\gamma \right)$ is given as

$$f_{{\gamma }_{\tau }}\left(\gamma \right)=\mathrm{\Delta }\sum _{m=0}^{M_p{N}_p}\left(\begin{array}{l}{M}_p{N}_p\\ m\end{array}\right)\left({\mathit{KM}}_s{N}_p+m-1\right)!{\left(\frac{\gamma M_p}{{\overline{\gamma }}_1}+\frac{M_s}{\alpha {\overline{\gamma }}_2}\right)}^{-{\mathit{KM}}_s{N}_p+m}.$$E11

Furthermore, collecting constant terms in (11), $\mathrm{\Delta }$ is defined by

$$\mathrm{\Delta }={\mathit{\beta \gamma }}^{M_p{N}_p-1}\text{exp}\left(-\frac{M_p\gamma }{{\overline{\gamma }}_1}\right).$$E12

Hereby, $\beta$ is constituted as

$$\beta =\frac{{\left(\frac{{\overline{\gamma }}_1}{M_p}\right)}^{-M_p{N}_p}{\left(\frac{\alpha {\overline{\gamma }}_2}{M_s}\right)}^{-{\mathit{KM}}_s{N}_p}}{\left(M_p{N}_p-1\right)!\left({\mathit{KM}}_s{N}_p-1\right)!}.$$E13

To achieve the closed-form expression of (11), binomial expression of ${\left(\frac{\gamma M_p}{{\overline{\gamma }}_1}+\frac{M_s}{\alpha {\overline{\gamma }}_2}\right)}^{-{\mathit{KM}}_s{N}_p+m}$ term must be completed. The binomial expansion of this negative exponential term is given as

$${\left(\frac{\gamma M_p}{{\overline{\gamma }}_1}+\frac{M_s}{\alpha {\overline{\gamma }}_2}\right)}^{-\zeta }=\sum _{t=0}^{\infty }{\left(-1\right)}^t\left(\begin{array}{l}\zeta +t-1\\ t\end{array}\right){\left(\frac{\gamma M_p}{{\overline{\gamma }}_1}\right)}^t{\left(\frac{\mathit{Ms}}{\alpha {\overline{\gamma }}_2}\right)}^{\zeta +t},$$E14

where $\zeta$ is given as $\zeta ={\mathit{KM}}_s{N}_p+m$. Besides, the validation of (14) is restricted via $\mid \frac{\gamma M_p}{{\overline{\gamma }}_1}\mid <\frac{M_s}{\alpha {\overline{\gamma }}_2}$ condition. Under these conditions, the closed-form expression of $f_{{\gamma }_{\tau }}$ is given below:

$$f_{{\gamma }_{\tau }}\left(\gamma \right)=\mathrm{\Delta }\sum _{m=0}^{M_p{N}_p}\sum _{t=0}^{\infty }{\left(-1\right)}^t\left(\begin{array}{l}{M}_p{N}_p\\ m\end{array}\right)\left(\zeta -1\right)!\left(\begin{array}{l}\zeta +t-1\\ t\end{array}\right){\left(\frac{\gamma M_p}{{\overline{\gamma }}_1}\right)}^t{\left(\frac{\mathit{Ms}}{\alpha {\overline{\gamma }}_2}\right)}^{\zeta +t}.$$E15

Outage probability function of the proposed MIMO system with respect to $f_{{\gamma }_{\tau }}$ can be expressed as

$$P_{\mathit{\text{out}}}={\int }_0^{2^{R_{\mathit{th}}}-1}{f}_{{\gamma }_{\tau }}\left(\gamma \right)\mathit{d\gamma }.$$E16

The closed-form expression for (16) can be validated with the numerical integral operation [33].

2.3. Performance evaluation

Herein, the system performance of the MIMO CR network is studied in the presence of interference leakage for Rayleigh fading channel by comparing the analytical results with computer simulations. We assumed $P_1$ = $P_2$ = $\rho$ while ${\sigma }_N^2=1$ in the performance evaluation.

In Figure 3, the $P_{\mathit{\text{out}}}$ performance for different $R_{\mathit{th}}$ values is presented. We take $\alpha =-20$ dB, $M_p=2$, $N_p=2$, $K=5$, and $M_s=1$. It can be seen from Figure 3 that when $R_{\mathit{th}}$ is increased from $1$ to $4$ bits/channel, the $P_{\mathit{\text{out}}}$ performance is degraded.

Figure 3.

$P_{\mathit{\text{out}}}$ performance for different data rate threshold $R_{\mathit{th}}$.

In Figure 4, the impact of the leakage coefficient, $\alpha$, on the outage probability performance is depicted for $M_p=2$, $N_p=2$, $K=1$, $M_s=1$, and $R_{\mathit{th}}=3$ bits/channel. As can be seen from the figure, when $\alpha$ is changed from $-10$ dB to $-30$ dB, the performance of the primary system is enhanced.

Figure 4.

$P_{\mathit{\text{out}}}$ performance with varying SNR for different interference-leakage values.

In Figure 5, $\alpha$, $M_p$, $N_p$, $M_s$, and $R_{\mathit{th}}$ are taken as $-20$ dB, $2$, $2$, $1$, and $1$ bits/channel, respectively. It can be observed from the figure that increasing the number of SUs decreases the outage probability performance of the primary system considerably.

Figure 5.

$P_{\mathit{\text{out}}}$ vs. SNR for different numbers of SUs.

In Figure 6, the impact of antenna diversity on the $P_{\mathit{\text{out}}}$ performance is investigated for $\alpha =-10$ dB, $K=2$, and $R_{\mathit{th}}=1$ dB. It is observed from the figure that, when the number of antennas at the primary transmitter and receiver increases, the system performance enhances. Besides, the receiver diversity effect on the system performance is greater than the transmitter diversity, as expected.

Figure 6.

The effect of antenna diversity on the outage probability performance.

3. The effect of CSI quantization on interference alignment in CR networks

In this section, we investigate a cognitive two-way relaying network composed of a primary network (PN) with one pair of PU and a secondary network (SN) with two source terminals and a relay terminal ($R$).

3.1. System model

We consider a MIMO interference network shown in Figure 7, where the transmitter, $T_x$, and receiver, $R_x$, are equipped with $M_1$ and $N_1$ antennas in PN, respectively. Each PN transmitter transmits to its corresponding receiver by interfering with the SN nodes, namely, two source terminals ($S_1$ and $S_2$) and a relay terminal. That means $T_x$ transmitter sends messages to its intended receiver $R_x$, whereas it also causes interference to the unintended receivers in the SN. The SN consists of two source terminals and a relay terminal. We assume that all nodes in SN operate in an AF half-duplex mode with the help of information relaying from each source terminal to $R$ in two phases. All nodes in SN are assumed to have MIMO antennas, and there is no direct transmission between $S_1$ and $S_2$ [34, 35, 36]. We consider a scenario where the source terminals and a relay terminal are equipped with $N_{S_1}$, $N_{S_2}$, and $N_R$ antennas, respectively. In the system model based on IA for cognitive two-way relay network, the received signal at $R_x$ in PN can be written as

$${\mathbf{y}}_{R_x}=\sqrt{\frac{P_{T_x}}{d_{R_x,T_x}^{{\tau }_{R_x,T_x}}}}{\mathbf{U}}_{R_x}^H{\mathbf{H}}_{R_x,T_x}{\mathbf{V}}_{T_x}{\mathbf{s}}_{T_x}+\mathtt{\Upsilon }+{\tilde{\mathbf{n}}}_{R_x},$$E17

where $\mathtt{\Upsilon }$ is the interference term generated from SN to $R_x$ defined as follows:

$$\mathtt{\Upsilon }=\left\{\begin{array}{llll}& \sqrt{\frac{P_{S_1}}{d_{R_x,S_1}^{{\tau }_{R_x,S_1}}}}{\mathbf{U}}_{R_x}^H{\mathbf{H}}_{R_x,S_1}{\mathbf{V}}_{S_1}{\mathbf{s}}_{S_1}+\sqrt{\frac{P_{S_2}}{d_{R_x,S_2}^{{\tau }_{R_x,S_2}}}}{\mathbf{U}}_{R_x}^H{\mathbf{H}}_{R_x,S_2}{\mathbf{V}}_{S_2}{\mathbf{s}}_{S_2},& & \text{first}\text{phase}\\ & \sqrt{\frac{P_R}{d_{R_x,R}^{{\tau }_{R_x,R}}}}{\mathbf{U}}_{R_x}^H{\mathbf{H}}_{R_x,R}{\mathbf{V}}_R{\mathbf{s}}_R,& & \text{second}\text{phase}.\end{array}\right.$$E18

Figure 7.

System model for interference alignment-based cognitive two-way relay network with primary network and secondary network.

The effective additive white Gaussian noise (AWGN) term with zero mean and unit variance, ${\tilde{\mathbf{n}}}_{R_x}$ at $R_x$ in PN, is defined by ${\mathbf{U}}_{R_x}^H{\mathbf{n}}_{R_x}$, where ${\mathbf{n}}_{R_x}$ is the AWGN vector with E $\left[{\mathbf{n}}_{R_x}{\mathbf{n}}_{R_x}^H\right]$ = ${\sigma }_{R_x}^2\mathbf{I}$ in which $\mathbf{I}$ is the unitary matrix, ${\sigma }_{R_x}^2$ is the noise variance, and E $\left[.\right]$ is the expectation operator. The transmit powers at the terminals $T_x$, $S_1$, $S_2$, and $R$ are denoted by $P_i$, for $i$ = $T_x$, $S_1$, $S_2$, and $R$, respectively. Each receive node employs the interference-suppression matrix, ${\mathbf{U}}_j$, (for $j$ = $R_x$, $R$, $S_1$, $S_2$), while each transmit node employs a precoding matrix ${\mathbf{V}}_i$ [37]. The conjugate transpose of the matrix is associated with the Hermitian operator ${\left(.\right)}^H$ [38]. The transmit signal vector for the $i$ th user is defined by ${\mathbf{s}}_i$. The channel between the $i$ th transmitter and the $j$ th receiver nodes is denoted by ${\mathbf{H}}_{j,i}$ for both PN and SN. The quantized CSI is passed to the transmitter by the corresponding receiver. Because of limited feedback, the transmitters have imperfect CSI causing certain performance loss. To clarify the effect of CSI quantization error on the performance of interference alignment in underlay cognitive two-way relay networks, we investigate the BER performance, instantaneous capacity, and average PEP of the considered system. Based upon the accuracy parameter, the relation between perfect CSI (${\rho }_{j,i}=0$) and imperfect CSI ($0<{\rho }_{j,i}\le 1$) can be given as

$${\mathbf{H}}_{j,i}=\sqrt{1-{\rho }_{j,i}}{\widehat{\mathbf{H}}}_{j,i}+\sqrt{{\rho }_{j,i}}{\mathbf{E}}_{j,i},$$E19

where ${\mathbf{H}}_{j,i}$ is the real channel matrix and ${\widehat{\mathbf{H}}}_{j,i}$ is the estimated channel matrix. The quantization error, ${\mathbf{E}}_{j,i}1\mathit{mm}$, can be expressed with the upper bound of $2^{-B_{j,i}/\left(M_1{N}_1-1\right)}$, where $B_{j,i}$ is the CSI exchange amount and $M_1$ and $N_1$ are the numbers of transmit-and-receive antennas, successively [21, 39]. It is assumed that both ${\widehat{\mathbf{H}}}_{j,i}$ and ${\mathbf{E}}_{j,i}$ are independent of ${\mathbf{H}}_{j,i}$. Besides, each channel link is also modeled by two additional parameters: the distance between $i$ th transmitter and the $j$ th receiver nodes $d_{j,i}$ and the path loss exponent for the corresponding link, ${\tau }_{j,i}$, regarding for different radio environments, respectively.

In the first phase of the transmission (multiple-access phase) in SN, both $S_1$ and $S_2$ transmit their signals simultaneously to the relay terminal, $R$. Then the received signal at $R$ can be written as

$${\mathbf{y}}_R=\sqrt{\frac{P_{S_1}}{d_{R,S_1}^{{\tau }_{R,S_1}}}}{\mathbf{U}}_R^H{\mathbf{H}}_{R,S_1}{\mathbf{V}}_{S_1}{\mathbf{s}}_{S_1}+\sqrt{\frac{P_{S_2}}{d_{R_x,S_2}^{{\tau }_{R_x,S_2}}}}{\mathbf{U}}_R^H{\mathbf{H}}_{R,S_2}{\mathbf{V}}_{S_2}{\mathbf{s}}_{S_2}+\sqrt{\frac{P_{T_x}}{d_{R,T_x}^{{\tau }_{R,T_x}}}}{\mathbf{U}}_R^H{\mathbf{H}}_{R,T_x}{\mathbf{V}}_{T_x}{\mathbf{s}}_{T_x}+{\tilde{\mathbf{n}}}_R,$$E20

where ${\tilde{\mathbf{n}}}_R={\mathbf{U}}_R^H{\mathbf{n}}_R$ at the relay terminal in SN is expressed as zero-mean AWGN vector with E $\left[{\mathbf{n}}_R{\mathbf{n}}_R^H\right]$ = ${\sigma }_R^2\mathbf{I}$ in which the noise variance at the relay terminal is depicted with ${\sigma }_R^2$. Besides, the received signal at $S_1$ and $S_2$ terminals in SN is defined, respectively, as

$${\mathbf{y}}_{S_1}=\sqrt{\frac{P_R}{d_{S_1,R}^{{\tau }_{S_1,R}}}}{\mathbf{U}}_{S_1}^H{\mathbf{H}}_{S_1,R}{\mathbf{V}}_R{\mathbf{s}}_R+\sqrt{\frac{P_{T_x}}{d_{S_1,T_x}^{{\tau }_{S_1,T_x}}}}{\mathbf{U}}_{S_1}^H{\mathbf{H}}_{S_1,T_x}{\mathbf{V}}_{T_x}{\mathbf{s}}_{T_x}+{\tilde{\mathbf{n}}}_{S_1},$$E21

$${\mathbf{y}}_{S_2}=\sqrt{\frac{P_R}{d_{S_2,R}^{{\tau }_{S_2,R}}}}{\mathbf{U}}_{S_2}^H{\mathbf{H}}_{S_2,R}{\mathbf{V}}_R{\mathbf{s}}_R+\sqrt{\frac{P_{T_x}}{d_{S_2,T_x}^{{\tau }_{S_2,T_x}}}}{\mathbf{U}}_{S_2}^H{\mathbf{H}}_{S_2,T_x}{\mathbf{V}}_{T_x}{\mathbf{s}}_{T_x}+{\tilde{\mathbf{n}}}_{S_2}.$$E22

Here, ${\tilde{\mathbf{n}}}_{S_1}$ and ${\tilde{\mathbf{n}}}_{S_2}$ are the AWGN vector with E $\left[{\mathbf{n}}_{S_k}{\mathbf{n}}_{S_k}^H\right]$ = ${\sigma }_{S_k}^2\mathbf{I}$, for $k=1,2$ and the noise variance of ${\sigma }_{S_k}^2$. In addition to that, in the second phase of the signal transmission (broadcast phase), $R$ broadcasts the combined signal ${\mathbf{y}}_R$ after multiplying with an ideal amplifying gain, $G$, which is expressed as

$$\begin{array}{l}\raisebox{1ex}{$G=1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{\frac{P_{S_1}\left(1-{\rho }_{R,S_1}\right)}{d_{R,S_1}^{{\tau }_{R,S_1}}}\parallel {\mathbf{U}}_R^H{\widehat{\mathbf{H}}}_{R,S_1}{\mathbf{V}}_{S_1}{\parallel }^2+\frac{P_{S_2}\left(1-{\rho }_{R,S_2}\right)}{d_{R,S_2}^{{\tau }_{R,S_2}}}\parallel {\mathbf{U}}_R^H{\widehat{\mathbf{H}}}_{R,S_2}{\mathbf{V}}_{S_2}{\parallel }^2.}.$}\right.\\ ..+\frac{P_{T_x}\left(1-{\rho }_{R,T_x}\right)}{d_{R,T_x}^{{\tau }_{R,T_x}}}\parallel {\mathbf{U}}_R^H{\widehat{\mathbf{H}}}_{R,T_x}{\mathbf{V}}_{T_x}{\parallel }^2,\end{array}$$E23

where ${\mathbf{s}}_{\mathbf{R}}=G{\mathbf{y}}_R$. We assume that both $S_1$ and $S_2$ have knowledge about their own information and can remove back-propagating self-interference from the imposed signals. We also assume that all interference at the receive terminals are perfectly aligned and the following feasible conditions are satisfied for the receive nodes:

$${\mathbf{U}}_j^{\mathrm{H}}{\mathbf{H}}_{j,i}{\mathbf{V}}_i{\mathbf{s}}_i=0,$$E24

$$\text{rank}\left({\mathbf{U}}_j^{\mathrm{H}}{\mathbf{H}}_{j,i}{\mathbf{V}}_i{\mathbf{s}}_i\right)=f_i,$$E25

where $f_i$ is the degree of freedom and rank (.) denotes the rank operation of a matrix. By assuming that the interference is perfectly aligned by the proposed IA algorithm, and the channel matrices are constant during the transmission, we ensure that there is no interference from the unintended transmitters and guarantee that received signal achieves $f_i$ degrees of freedom [39]. The corresponding signal-to-interference-plus-noise ratio (SINR) for the links $T_x\to R_x$, $S_1\to R$, and $R\to S_2$ can be derived by

$${\gamma }_{T_x\to R_x}=\frac{\frac{P_{T_x}\left(1-{\rho }_{R_x,T_x}\right)}{d_{R_x,T_x}^{{\tau }_{R_x,T_x}}}\parallel {\mathbf{U}}_{R_x}^H{\widehat{\mathbf{H}}}_{R_x,T_x}{\mathbf{V}}_{T_x}{\parallel }^2}{\Psi +{\sigma }_{{\tilde{n}}_{R_x}}^2},$$E26

$${\gamma }_{S_1\to R}=\frac{\frac{P_{S_1}\left(1-{\rho }_{R,S_1}\right)}{d_{R,S_1}^{{\tau }_{R,S_1}}}\parallel {\mathbf{U}}_R^H{\widehat{\mathbf{H}}}_{R,S_1}{\mathbf{V}}_{S_1}{\mathbf{s}}_{S_1}{\parallel }^2}{\frac{P_{T_x}{\rho }_{R,T_x}}{d_{R,T_x}^{{\tau }_{R,T_x}}}\parallel {\mathbf{U}}_R^H{\mathbf{E}}_{R,T_x}{\mathbf{V}}_{T_x}{\parallel }^2+{\sigma }_{{\tilde{n}}_R}^2},$$E27

$${\gamma }_{R\to S_2}=\frac{\frac{P_R\left(1-{\rho }_{S_2,R}\right)}{d_{S_2,R}^{{\tau }_{S_2,R}}}\parallel {\mathbf{U}}_{S_2}^H{\widehat{\mathbf{H}}}_{S_2,R}{\mathbf{V}}_R{\mathbf{s}}_R{\parallel }^2}{\frac{P_{T_x}{\rho }_{S_2,T_x}}{d_{S_2,T_x}^{{\tau }_{S_2,T_x}}}\parallel {\mathbf{U}}_{S_2}^H{\mathbf{E}}_{S_2,T_x}{\mathbf{V}}_{T_x}{\parallel }^2+{\sigma }_{{\tilde{n}}_{S_2}}^2},$$E28

$$\Psi =\left\{\begin{array}{llll}& \frac{P_{S_1}{\rho }_{R_x,S_1}}{d_{R_x,S_1}^{{\tau }_{R_x,S_1}}}\parallel {\mathbf{U}}_{R_x}^H{\mathbf{E}}_{R_x,S_1}{\mathbf{V}}_{S_1}{\parallel }^2+\frac{P_{S_2}{\rho }_{R_x,S_2}}{d_{R_x,S_2}^{{\tau }_{R_x,S_2}}}\parallel {\mathbf{U}}_{R_x}^H{\mathbf{E}}_{R_x,S_2}{\mathbf{V}}_{S_2}{\parallel }^2,& & \text{first}\text{phase}\\ & \frac{P_R{\rho }_{R_x,R}}{d_{R_x,R}^{{\tau }_{R_x,R}}}\parallel {\mathbf{U}}_{R_x}^H{\mathbf{E}}_{R_x,R}{\mathbf{V}}_R{\parallel }^2,& & \text{second}\text{phase}\end{array}\right.$$E29

where ${\mathbf{E}}_{j,i}$ is the quantization error and $‖.‖$ is the Euclidean norm. In here, ${\gamma }_{S_2\to R}$ and ${\gamma }_{R\to S_1}$ can be found by changing the subscript $S_1$ with $S_2$ of (27) and $S_2$ with $S_1$ of (28). Assuming the channels are reciprocal over SN direct links, thus the channel gains for $S_1\to R$ and $R\to S_1$ and $S_2\to R$ and $R\to S_2$ links are identical, respectively.

3.2. Performance analysis

This section starts by the instantaneous capacity analysis of the proposed system with interference alignment in underlay cognitive two-way relay networks with CSI quantization. We then study the BER and average PEP performance.

The capacity is expressed as the expected value of the mutual information between the transmitting terminal and receiving one. In light of this fact, we consider the method developed in [29]; the instantaneous capacity in PN can be expressed as

$$C_{R_x}={\text{log}}_2\left(1+{\gamma }_{T_x\to R_x}\right),$$E30

where ${\gamma }_{T_x\to R_x}$ is the instantaneous SINR for the corresponding link of $T_x\to R_x$. On the other hand, end-to-end capacity for the SN, based on the least strong link over two-hop transmission, is denoted as follows:

$$C_R=\frac{1}{2}{\text{log}}_2\left(1+\text{min}\left({\gamma }_{S_1\to R},{\gamma }_{R\to S_2}\right)\right)+\frac{1}{2}{\text{log}}_2\left(1+\text{min}\left({\gamma }_{S_2\to R},{\gamma }_{R\to S_1}\right)\right).$$E31

${\gamma }_{S_1\to R}$ and ${\gamma }_{R\to S_2}$ are the instantaneous SINR for the $S_1\to R$ and $R\to S_2$ links, respectively.

Average BER for binary phase shift keying (BPSK) modulation can be expressed as

$${\mathrm{BER}}_j=Q\left(\sqrt{{\gamma }_j}\right)$$E32

where $Q\left(x\right)$ is the Gaussian Q-function and defined by $Q\left(x\right)=\left(1/\sqrt{2\pi }\right){\int }_x^{\infty }{e}^{-t^2/2}\mathit{dt}$ [37].

Average pairwise error probability ($\overline{\mathrm{PEP}}$) can be computed as averaging the Gaussian Q-function over Rayleigh fading statistics [40], $f_{{\gamma }_{T_x\to R_x}}\left(\gamma \right)=\left(e^{-\gamma /{\overline{\gamma }}_{T_x\to R_x}}\right)/{\overline{\gamma }}_{T_x\to R_x}1\mathit{mm}$, where ${\overline{\gamma }}_{T_x\to R_x}$ $=P_{T_x}\left(1-{\rho }_{R_x,T_x}\right)/d_{R_x,T_x}^{{\tau }_{R_x,T_x}}{\sigma }_{{\tilde{n}}_{R_x}}^2$

$$\overline{\mathrm{PEP}}={\int }_0^{\infty }Q\left(\sqrt{{\gamma }_{T_x\to R_x}}\right)f_{{\gamma }_{T_x\to R_x}}\left(\gamma \right)\mathit{d\gamma }.$$E33

Finally, this integral can be evaluated with the help of Mathematica and average PEP under Rayleigh fading channel can be derived in a closed form as follows:

$$\overline{\mathrm{PEP}}=\frac{1}{2}\left(1-\sqrt{\frac{{\overline{\gamma }}_{T_x\to R_x}}{2+{\overline{\gamma }}_{T_x\to R_x}}}\right).$$E34

3.3. Numerical results

In this section the numerical results are provided with various scenarios to evaluate the performance analysis for IA in underlay cognitive two-way relay networks with CSI quantization. BER performance for direct transmission links of the proposed system is illustrated in Figure 8 over Rayleigh distribution for different amounts of CSI exchange with varying SNR. For convenience, we set $d_{j,i}=3m$ and $\tau =2.7$, and 3 $\times$ 3 MIMO configuration is studied in this figure. Because of the number of interfering links, the quantization error for the $T_x\to R_x$ transmission is greater than the other links ($S_1\rightleftarrows R\rightleftarrows S_2$). Even if the analyzed BER performance of the SN seems better than the PN, it should not be forgotten that SN operates in half-duplex mode. Performance loss in BER due to imperfect CSI ($B_{j,i}=4$, for instance) becomes larger as SNR increases compared to the perfect CSI (for $B_{j,i}=\infty$) case.

Figure 8.

BER performance for different amounts of CSI exchange with varying SNR.

In Figure 9, the average PEP versus SNR is plotted for $d_{j,i}=3m$ and $\tau =2.7$ over Rayleigh fading channel in PN. It can be noticed from the figure that as SNR increases, average PEP decreases, as expected. To reach the perfect CSI case, we take $B_{j,i}=\infty$, and the average PEP performance noticeably enhances. We also consider the case of imperfect CSI ($B_{j,i}=4$) for the comparison purposes in the same figure.

Figure 9.

Average PEP performance for different amounts of CSI exchange with varying SNR over Rayleigh fading channel in primary network.

Figure 10 examines the capacity analysis with perfect and imperfect CSI for different direct links in PN and SN. The results clearly show that, examining the capacity with perfect CSI, performance improvement becomes larger as the SNR increases.

Figure 10.

Capacity vs. SNR of the primary network and secondary network nodes under different CSI scenarios.

Figure 11 demonstrates the effects of $B_{j,i}$ and $d_{j,i}$ parameters on the BER performance for the SN with varying SNR when $\tau =2.7$ and 3 $\times$ 3 MIMO scheme is used. The results clearly show that for a fixed SNR value, the performance of the considered system increases with the decrease of the $d_{j,i}$. It can be seen from the same figure that the increase on the amount of CSI exchange $B_{j,i}$ positively affects the BER performance.

Figure 11.

BER performance for different amounts of CSI exchange and distances with varying SNR over Rayleigh fading channel for secondary network.

Figure 12 shows the capacity performance of PU in the underlay cognitive two-way relay network over Rayleigh fading channel with varying path loss exponent, $\tau$. The results show a performance improvement while the value of $\tau$ decreases. In this plot, $B_{j,i}$ = $8$, $d_{j,i}$ = $3m$, and the 3 $\times$ 3 MIMO scheme are considered. Depending on the environmental conditions for mobile communications, typical $\tau$ values, ranging from 1.6 to 5, are used to plot this figure. First, for the line of sight in a building, the environment is considered with the $\tau$ values of 1.6 and 1.8. Second, capacity is computed for the free-space environment with $\tau =2$. Then, the capacity performance is presented with $\tau$ values of 2.7 and 3.3 for urban area cellular radio environment. Finally, the shadowed urban cellular radio environment is associated with two different $\tau$ values of 3 and 5 to analyze the capacity performance with varying SNR [41].

Figure 12.

Capacity changes with SNR for the environmental conditions having different path loss exponents.

4. Conclusion

In this chapter, the system performance of linear interference alignment on the MIMO CR network is investigated under interference leakage. To quantify the performance of the primary system under a certain level of interference leakage, the closed-form outage probability expression is derived for Rayleigh fading channel. In all analyses, the theoretical results closely match with the simulations which confirm the accuracy of the derived expressions.

In the second part of this work, considering a practical issue, we investigate the performance of interference alignment in underlay cognitive radio network with CSI quantization error over general MIMO interference channel. Amplify-and-forward scheme for two-way relay network under Rayleigh fading is considered. The impact of the CSI exchange amount, the distance between the $i$ th transmitter and the $j$ th receiver nodes, and the path loss exponent on the BER performance, system capacity, and average PEP for the proposed system model are analyzed. We provide the exact closed-form expression for the average PEP in primary network over Rayleigh distribution, while IA algorithm perfectly eliminates the interference. The present performance analysis can be extended to the multiple secondary user pairs, and this approach will be another subject of our future work.

It would be interesting to study on various scenarios, including single-hop, multi-hop, and multi-way networks in future work to analyze the system performance over the recently developed interference alignment algorithms for next-generation 5G wireless communication systems.

Acknowledgments

The authors wish to express their special thanks to Seda Ustunbas (Wireless Communication Research Laboratory, Istanbul Technical University, Turkey) for useful discussions of this chapter.