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

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 Mpand Np. The transmit antennas at each SU are given as Ms. The received signal, yp, implementing the IA technique is given as

where xpand xsiare the transmitted signals from PU and the ith SU fori=12…K, respectively. Herein, Hppis the matrix of channel coefficients between the PU pair, and Hpsidenotes the channel matrix between the primary receiver and the ith secondary transmitter. The interference leakage is modeled similar to the one in [30]. The interference-leakage parameter α0≤α≤1represents the status of the alignment, i.e., α=0and1corresponds to perfect alignment and perfect misalignment cases, respectively. Vand Uare the precoding- and interference-suppression matrices. The superscript ⋅Hdenotes the Hermitian operator, and nis the zero-mean unit variance (σN2=1) circularly symmetric additive white Gaussian noise (AWGN) vector.

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

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

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

where γ1=P1Hpp2/σN2is the signal-to-noise ratio (SNR) of the primary link. γ2can be expressed as γ2=P2/σN2∑i=1KHpsi2. Note that .2demonstrates the squared Frobenius norm of the channel matrix, Idenotes for identity matrix, and P1and P2are 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, γ2, becomes zero. It is important to note that precoding and linear suppression vectors are assumed as UpH2=Vp2=UsiH2=Vsi2=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 γ1can be written as fγ1γ=1γ¯1exp−γ/γ¯1, and the outage probability of the system can be obtained as

where Rthis the data rate threshold and γ¯1=P1/σN2denotes the average SNR of the primary system. By substituting fγ1γinto (6), the outage probability can be obtained as

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 γτ=γ1/1+γ2. In the proposed system, all channels are modeled as independent and identically distributed Chi-squared distribution, and the PDF of γ1can be expressed as

In addition, the PDF of γ2can be defined as

where γ¯2=P2/σN2is the average SNR of the secondary system. Finally, the PDF of γτcan be written as

By substituting (8) and (9) into (10), then with the help of [32, Eq. 3.351.3] and after few manipulations, PDF expression of fγτγis given as

Furthermore, collecting constant terms in (11), Δis defined by

Hereby, βis constituted as

To achieve the closed-form expression of (11), binomial expression of γMpγ¯1+Msαγ¯2−KMsNp+mterm must be completed. The binomial expansion of this negative exponential term is given as

where ζis given as ζ=KMsNp+m. Besides, the validation of (14) is restricted via ∣γMpγ¯1∣<Msαγ¯2condition. Under these conditions, the closed-form expression of fγτis given below:

Outage probability function of the proposed MIMO system with respect to fγτcan be expressed as

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 P1= P2= ρwhile σN2=1in the performance evaluation.

In Figure 3, the Poutperformance for different Rthvalues is presented. We take α=−20dB, Mp=2, Np=2, K=5, and Ms=1. It can be seen from Figure 3 that when Rthis increased from 1to 4bits/channel, the Poutperformance is degraded.

In Figure 4, the impact of the leakage coefficient, α, on the outage probability performance is depicted for Mp=2, Np=2, K=1, Ms=1, and Rth=3bits/channel. As can be seen from the figure, when αis changed from −10dB to −30dB, the performance of the primary system is enhanced.

In Figure 5, α, Mp, Np, Ms, and Rthare taken as −20dB, 2, 2, 1, and 1bits/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.

In Figure 6, the impact of antenna diversity on the Poutperformance is investigated for α=−10dB, K=2, and Rth=1dB. 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.

3.1. System model

We consider a MIMO interference network shown in Figure 7, where the transmitter, Tx, and receiver, Rx, are equipped with M1and N1antennas in PN, respectively. Each PN transmitter transmits to its corresponding receiver by interfering with the SN nodes, namely, two source terminals (S1and S2) and a relay terminal. That means Txtransmitter sends messages to its intended receiver Rx, 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 Rin two phases. All nodes in SN are assumed to have MIMO antennas, and there is no direct transmission between S1and S2[34, 35, 36]. We consider a scenario where the source terminals and a relay terminal are equipped with NS1, NS2, and NRantennas, respectively. In the system model based on IA for cognitive two-way relay network, the received signal at Rxin PN can be written as

where ϒis the interference term generated from SN to Rxdefined as follows:

The effective additive white Gaussian noise (AWGN) term with zero mean and unit variance, n˜Rxat Rxin PN, is defined by URxHnRx, where nRxis the AWGN vector with E nRxnRxH= σRx2Iin which Iis the unitary matrix, σRx2is the noise variance, and E .is the expectation operator. The transmit powers at the terminals Tx, S1, S2, and Rare denoted by Pi, for i= Tx, S1, S2, and R, respectively. Each receive node employs the interference-suppression matrix, Uj, (for j= Rx, R, S1, S2), while each transmit node employs a precoding matrix Vi[37]. The conjugate transpose of the matrix is associated with the Hermitian operator .H[38]. The transmit signal vector for the ith user is defined by si. The channel between the ith transmitter and the jth receiver nodes is denoted by Hj,ifor 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 (ρj,i=0) and imperfect CSI (0<ρj,i≤1) can be given as

where Hj,iis the real channel matrix and Ĥj,iis the estimated channel matrix. The quantization error, Ej,i1mm, can be expressed with the upper bound of 2−Bj,i/M1N1−1, where Bj,iis the CSI exchange amount and M1and N1are the numbers of transmit-and-receive antennas, successively [21, 39]. It is assumed that both Ĥj,iand Ej,iare independent of Hj,i. Besides, each channel link is also modeled by two additional parameters: the distance between ith transmitter and the jth receiver nodes dj,iand the path loss exponent for the corresponding link, τj,i, regarding for different radio environments, respectively.

In the first phase of the transmission (multiple-access phase) in SN, both S1and S2transmit their signals simultaneously to the relay terminal, R. Then the received signal at Rcan be written as

where n˜R=URHnRat the relay terminal in SN is expressed as zero-mean AWGN vector with E nRnRH= σR2Iin which the noise variance at the relay terminal is depicted with σR2. Besides, the received signal at S1and S2terminals in SN is defined, respectively, as

Here, n˜S1and n˜S2are the AWGN vector with E nSknSkH= σSk2I, for k=1,2and the noise variance of σSk2. In addition to that, in the second phase of the signal transmission (broadcast phase), Rbroadcasts the combined signal yRafter multiplying with an ideal amplifying gain, G, which is expressed as

where sR=GyR. We assume that both S1and S2have 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:

where fiis 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 fidegrees of freedom [39]. The corresponding signal-to-interference-plus-noise ratio (SINR) for the links Tx→Rx, S1→R, and R→S2can be derived by

where Ej,iis the quantization error and .is the Euclidean norm. In here, γS2→Rand γR→S1can be found by changing the subscript S1with S2of (27) and S2with S1of (28). Assuming the channels are reciprocal over SN direct links, thus the channel gains for S1→Rand R→S1and S2→Rand R→S2links 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

where γTx→Rxis the instantaneous SINR for the corresponding link of Tx→Rx. 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:

γS1→Rand γR→S2are the instantaneous SINR for the S1→Rand R→S2links, respectively.

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

where Qxis the Gaussian Q-function and defined by Qx=1/2π∫x∞e−t2/2dt[37].

Average pairwise error probability (PEP¯) can be computed as averaging the Gaussian Q-function over Rayleigh fading statistics [40], fγTx→Rxγ=e−γ/γ¯Tx→Rx/γ¯Tx→Rx1mm, where γ¯Tx→Rx=PTx1−ρRx,Tx/dRx,TxτRx,Txσn˜Rx2

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:

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 dj,i=3mand τ=2.7, and 3 ×3 MIMO configuration is studied in this figure. Because of the number of interfering links, the quantization error for the Tx→Rxtransmission is greater than the other links (S1⇄R⇄S2). 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 (Bj,i=4, for instance) becomes larger as SNR increases compared to the perfect CSI (for Bj,i=∞) case.

In Figure 9, the average PEP versus SNR is plotted for dj,i=3mand τ=2.7over 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 Bj,i=∞, and the average PEP performance noticeably enhances. We also consider the case of imperfect CSI (Bj,i=4) for the comparison purposes in the same figure.

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 11 demonstrates the effects of Bj,iand dj,iparameters on the BER performance for the SN with varying SNR when τ=2.7and 3 ×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 dj,i. It can be seen from the same figure that the increase on the amount of CSI exchange Bj,ipositively affects the BER performance.

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, τ. The results show a performance improvement while the value of τdecreases. In this plot, Bj,i= 8, dj,i= 3m, and the 3 ×3 MIMO scheme are considered. Depending on the environmental conditions for mobile communications, typical τvalues, 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 τvalues of 1.6 and 1.8. Second, capacity is computed for the free-space environment with τ=2. Then, the capacity performance is presented with τvalues of 2.7 and 3.3 for urban area cellular radio environment. Finally, the shadowed urban cellular radio environment is associated with two different τvalues of 3 and 5 to analyze the capacity performance with varying SNR [41].