SER Performance of Large Scale OFDM-SDMA Based Cognitive Radio Networks

Abstract

This paper proposes an adaptive large scale multiple input multiple output-beamforming scheme (LSMIMO-BF) for uplink (UL) access in broadband wireless cognitive networks with multiple primary users (PUs) and secondary users (SUs) sharing the same spectrum and employing orthogonal frequency-division multiplexing (OFDM). The proposed algorithm seeks the optimal transmit/receive weight vectors that maximize the UL MIMO channel capacity for each cognitive user while controlling the interference levels to PUs. Under the assumption of very large number of antennas at the base station, a closed-form expression for the symbol error rate (SER) performance of the cognitive LSMIMO-OFDM system is derived and compared with the one based on conventional beamforming schemes such as MIMO-maximum ratio combining (MIMO-MRC). The analysis and simulation show that when SUs are transmitting with the proposed constrained capacity-aware (CCA) scheme, the total interference level at the primary base station is reduced and the SER of PUs is improved compared to the case when the secondary network is using conventional MIMO-MRC. It was also shown that, as the number of base station antennas becomes larger, the constraints imposed by the primary network could be relaxed and the symbol error rate (SER) of SUs is improved without harming the PUs performance.

1. Introduction

Recently, we are observing a huge interest in cognitive radio networks (CRNs), from both the research and policy/regulation communities [1, 2]. These CRNs can intelligently share spectrum and extract more bandwidths via “opportunistic use” of spectrum resources. They will be the essential technology needed to make significantly better use of available spectrum and to address the real issue of fixed spectrum access approaches. However, for CRNs to discover underutilized spectrum and adapt their transmission settings accordingly without causing interference to licensed users, their physical layer needs to be highly flexible and adaptable. Among many possible technologies, orthogonal frequency-division multiplexing (OFDM) has been widely recognized as a versatile modulation technique that has the potential of fulfilling the requirements of CRNs. Using OFDM it is possible to disable the set of subcarriers used by the primary network and to adaptively change the modulation order according to the channel conditions. However, opportunistic spectrum sharing may not be reliable and may limit the system capacity since it suffers from the interruptions imposed by the primary network (PN) on the secondary network (SN) which must leave the licensed channel when primary users (PUs) emerge. Also, with opportunistic spectrum sharing, secondary users (SUs) can still cause interference to PUs due to their imperfect spectrum sensing. One way to overcome these limitations is to incorporate space division multiple access (SDMA) into OFDM-based CRNs. SDMA can help in achieving higher spectral efficiency, by multiplexing multiple users on the same time-frequency resources. OFDM-SDMA techniques have been successfully deployed in 3G/4G cellular systems based on traditional static spectrum access approach [3–9] and a vast number of multiuser detection algorithms, such as maximum ratio combining (MRC) and minimum mean-squared error (MMSE), are presently being tailored towards solving the SDMA processing in MIMO cognitive networks [10–15], where additional constraints to protect licensed users’ QoS are imposed. Within this context, we are extending our recently developed capacity-aware-based OFDM-SDMA schemes [9] to cognitive radio by constraining the transmitted power of SUs and controlling the interference level to PUs while enhancing the overall system capacity for each SU. Moreover, to relax the power and interference constraints imposed on SUs without affecting the performance of PUs, we are considering the deployment of a large number of base station (BS) antennas in comparison to the served users’ antennas. This concept was initially investigated for cellular networks [16–18], and there is a clear potential in the area of cognitive radio where interference between PN and SN is even more problematic. With this large number of antennas, the central limit theorem and the law of large numbers can be applied and close to optimal performances can be achieved with the simplest forms of user detection and beamforming, that is, MRC, and eigen-beamforming (EBF). Due to the nonconvexity of the constrained channel capacity, we propose using the steepest ascent gradient of the cognitive OFDM-SDMA channel capacity to iteratively seek the transmitting weight and derive a closed-form expression for the SER of the system. Using the derived expression, we compare the performance of the proposed cognitive capacity-aware (CCA) scheme and the conventional nonconstrained MIMO-MRC (eigen-beamforming is performed at the mobile transmitters and MRC at the receiving base station) in the presence of secondary and primary multiuser access interferences as well as the correlated fading encountered in antenna array systems. Through the analysis and simulation, the impacts of the constraints imposed by the PN on the SN are also investigated.

2. System Model

We consider the UL multiuser access scenario shown in Figure 1, where 𝐿_𝑠SUs and one secondary base station (SBS) coexist with 𝐿_𝑝 PUs and one primary base station (PBS) via concurrent spectrum access. In both networks, the users and the base stations are equipped with multiple antennas. It is also assumed that both the SBS and the PBS receivers detect independent OFDM data streams from multiple SUs and PUs simultaneously on the same time-frequency resources. Let 𝐱^𝐬[𝑘]={𝑥^𝑠₁,𝑥^𝑠₁,…,𝑥^𝑠_𝐿𝑠} and 𝐱^𝐩[𝑘]={𝑥^𝑝₁,𝑥^𝑝₁,…,𝑥^𝑝_𝐿𝑝} denote, respectively, the set of 𝐿_𝑠 SUs signals and 𝐿_𝑝 PUs signals transmitted on each subcarrier, 𝑘=1,…,𝑁_𝑐, where 𝑁_𝑐 denotes the number of subcarriers per OFDM symbol in the system. It is assumed that 𝑥^𝑠_𝑖 and 𝑥^𝑝_𝑖 are complex-valued random variables with unit power; that is, 𝐸[‖𝑥^𝑠_𝑖‖²]=𝐸[‖𝑥^𝑝_𝑖‖²]=1. The expression for the array output of the SBS in Figure 1 can be written for each subcarrier as 𝐲_SBS[𝑘]=^𝐿𝑠_𝑙𝑠=1𝐇_{𝑠𝑠,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘]𝐱^𝑠_𝑙𝑠[𝑘]+𝐧[𝑘]+𝐈_PU[𝑘], (1) where 𝐲_SBS[𝑘]=[𝑦^𝑠₁[𝑘],𝑦^𝑠₂[𝑘],…,𝑦^𝑠_𝑁𝑟𝑠[𝑘]]^𝑇 is the 𝑁^𝑟_𝑠×1 vector containing the outputs of the 𝑁^𝑟_𝑠-element array at the SBS, with (⋅)^𝑇 denoting the transpose operation, 𝐇_{𝑠𝑠,𝑙𝑠}[𝑘] is the 𝑁^𝑟_𝑠×𝑁^𝑡_𝑠 frequency-domain channel matrix representing the transfer functions from secondary user 𝑙_𝑠’s 𝑁^𝑡_𝑠-element antenna array to the SBS’s 𝑁^𝑟_𝑠-element antenna array, 𝐰^𝑡_𝑙𝑠[𝑘]=[𝑤^𝑡_𝑙𝑠1[𝑘],𝑤^𝑡_𝑙𝑠2[𝑘],…,𝑤^𝑡_{𝑙𝑠𝑁𝑡𝑠}[𝑘]]^𝑇 is the 𝑁^𝑡_𝑠×1 complex transmit weight vector for SU 𝑙_𝑠, 𝑙_𝑠=1,…,𝐿_𝑠, 𝐧[𝑘]=[𝑛₁[𝑘],𝑛₂[𝑘],…,𝑛_𝑁𝑟[𝑘]]^𝑇 is the 𝑁^𝑟_𝑠×1 complex additive white Gaussian noise vector, and 𝐈_PU[𝑘] represents the interference introduced by PUs at the SBS, given by 𝐈_PU[𝑘]=^𝐿𝑝_𝑙𝑝=1𝐇_{𝑙𝑝𝑠,𝑝}[𝑘]𝐰^𝑡_𝑙𝑝[𝑘]𝐱^𝑝_𝑙𝑝[𝑘], (2) where 𝐇_{𝑝𝑠,𝑙𝑝}[𝑘] is the 𝑁^𝑟_𝑠×𝑁^𝑡_𝑝 channel matrix representing the fading coefficients from PUs to the SBS’s 𝑁^𝑟_𝑠-element antenna array. On the other hand, the interference seen by the primary base station due to secondary transmission is given by 𝐲^𝑠𝑝[𝑘]=^𝐿𝑠_𝑙𝑠=1𝐇_{𝑠𝑝,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘]𝐱^𝑠_𝑙𝑠[𝑘] (3) and its corresponding power level can be expressed as 𝐽_𝑠𝑝[𝑘]=^𝐿𝑠_𝑖=1𝐇_{𝑠𝑝,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘]𝐰_𝑙^𝑡,𝐻_𝑠[𝑘]𝐇^𝐻_{𝑠𝑝,𝑙𝑠}[𝑘], (4) where (⋅)^𝐻 denotes the Hermitian transpose and 𝐇_{𝑠𝑝,𝑙𝑠}[𝑘] is the 𝑁^𝑟_𝑝×𝑁^𝑡_𝑠 channel matrix representing the fading coefficients from the 𝑙_𝑠th SU to the PBS’s 𝑁^𝑟_𝑝-element antenna array. Assuming the well-known Kronecker correlation structure [19–21], 𝐇_{𝑠𝑠,𝑙𝑠}[𝑘], 𝐇_{𝑠𝑝,𝑙𝑠}[𝑘], 𝐇_{𝑝𝑠,𝑙𝑝}[𝑘], and 𝐇_{𝑝𝑝,𝑙𝑝}[𝑘] can be modeled as 𝐇_{𝑠𝑠,𝑙𝑠}[𝑘]=𝐑^1/2_𝑟𝑠𝐇^𝜔_{𝑠𝑠,𝑙𝑠}[𝑘]𝐑^1/2_𝑡𝑠,  𝐇_{𝑠𝑝,𝑙𝑠}[𝑘]=𝐑^1/2_𝑟𝑝𝐇^𝜔_{𝑠𝑝,𝑙𝑠}[𝑘]𝐑^1/2_𝑡𝑠, 𝐇_{𝑝𝑠,𝑙𝑝}[𝑘]=𝐑^1/2_𝑟𝑠𝐇^𝜔_{𝑝𝑠,𝑙𝑝}[𝑘]𝐑^1/2_𝑡𝑝, and 𝐇_{𝑝𝑝,𝑙𝑝}[𝑘]=𝐑^1/2_𝑟𝑝𝐇^𝜔_{𝑝𝑝,𝑙𝑠}[𝑘]𝐑^1/2_𝑡𝑝, respectively, where 𝐑_𝑟𝑠∈ℂ^{𝑁𝑟𝑠×𝑁𝑟𝑠},  𝐑_𝑟𝑝∈ℂ^{𝑁𝑟𝑝×𝑁𝑟𝑝}, 𝐑_𝑡𝑠∈ℂ^{𝑁𝑡𝑠×𝑁𝑡𝑠}, and 𝐑_𝑡𝑝∈ℂ^{𝑁𝑡𝑝×𝑁𝑡𝑝} are positive definite Hermitian matrices specifying the receive and transmit fading correlations.𝐇^𝜔_{𝑠𝑠,𝑙𝑠}[𝑘]∈ℂ^{𝑁𝑟𝑠×𝑁𝑡𝑠},  𝐇^𝜔_{𝑠𝑝,𝑙𝑠}[𝑘]∈ℂ^{𝑁𝑟𝑝×𝑁𝑡𝑠}, 𝐇^𝜔_{𝑝𝑠,𝑙𝑝}[𝑘]∈ℂ^{𝑁𝑟𝑠×𝑁𝑡𝑝}, and 𝐇^𝜔_{𝑝𝑝,𝑙𝑠}[𝑘]∈ℂ^{𝑁𝑟𝑝×𝑁𝑡𝑝} consist of iid (independent and identically distributed) complex Gaussian entries, with zero means and unit variance. In this paper we assume that the mobile transmit antennas and the base station antennas have sufficient interelement spacing such that spatial decorrelation between simultaneously transmitted data streams on different transmit channels is feasible. For mobile devices, MIMO-beamforming can usually be accommodated for frequency bands higher than 2.5 GHz. However, a new approach is needed to support multiple antennas in the lower frequency ranges due to the highly correlated channels. The transfer functions from the 𝑙_𝑠th SU device to the SBS antenna array (the cascade of 𝐇_{𝑠𝑠,𝑙𝑠}[𝑘] and 𝐰^𝑡_𝑙𝑠[𝑘]) result in a unique spatial signature for each SU, which can be exploited to affect the separation of the user data at the SBS using appropriate multiuser detection techniques. The SBS detects all 𝐿_𝑠 SUs, simultaneously at the multiuser detection module of the SDMA system, by multiplying the output of the array with the 𝑁^𝑟_𝑠×1 receiving weight vectors, 𝐰^𝑟_𝑙𝑠[𝑘]=[𝑤^𝑟_𝑙𝑠1[𝑘],𝑤^𝑡_𝑙𝑠2[𝑘],…,𝑤^𝑟_𝑙𝑠𝑁^𝑟_𝑠[𝑘]]^𝑇, for each SU 𝑙_𝑠. The detection of secondary user 𝑙_𝑠 out of 𝐿−1 interfering users (𝐿=𝐿_𝑠+𝐿_𝑝) can thus be depicted as 𝐱_𝑙𝑠[𝑘]=𝐰_𝑙^𝑟,𝐻_𝑠[𝑘]𝐲_SBS[𝑘]=𝐒_𝑑[𝑘]+𝐒_𝐼𝑠[𝑘]+𝐒_𝐼𝑝[𝑘]+𝐍[𝑘], (5) where 𝐍[𝑘]=𝐰_𝑙^𝑟,𝐻_𝑠[𝑘]𝐧[𝑘] is the noise signal at the array output of the SBS, 𝐒_𝑑[𝑘] is the desired signal for the detection of user 𝑙_𝑠’s signal, 𝐒_𝐼𝑠[𝑘] is the multiple-access interference (MAI) contributed by the 𝐿_𝑠−1 other SUs, and 𝐒_𝐼𝑝[𝑘] is the MAI from 𝐿_𝑝 PUs. 𝐒_𝑑[𝑘], 𝐒_𝐼𝑠[𝑘], and 𝐒_𝐼𝑝[𝑘] are, respectively, given by 𝐒_𝑑[𝑘]=𝐰_𝑙^𝑟,𝐻_𝑠[𝑘]𝐇_{𝑠𝑠,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘]𝐱^𝑠_𝑙𝑠[𝑘],𝐒_𝐼𝑠[𝑘]=𝐰_𝑙^𝑟,𝐻_𝑠[𝑘]^𝐿𝑠_{𝑖=1,𝑖≠𝑙𝑠}𝐇^𝑡_{𝑠𝑠,𝑖}[𝑘]𝐱^𝑠_𝑖[𝑘],𝐒_𝐼𝑝[𝑘]=𝐰_𝑙^𝑟,𝐻_𝑠[𝑘]^𝐿𝑝_𝑙𝑝=1𝐇^𝑡_{𝑝𝑠,𝑙𝑝}[𝑘]𝐱^𝑝_𝑙𝑝[𝑘], (6) where 𝐇^𝑡_{𝑠𝑠,𝑙𝑠}[𝑘]=𝐇_{𝑠𝑠,𝑙𝑝}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘] and 𝐇^𝑡_{𝑝𝑠,𝑙𝑝}[𝑘]=𝐇_{𝑝𝑠,𝑙𝑝}[𝑘]𝐰^𝑡_𝑙𝑝[𝑘].

Figure 1 

System model: primary and secondary OFDM-SDMA systems.

During the analysis, perfect channel estimation is assumed. This assumption is justified by the fact that when the number of antennas grows towards infinity, effects of noise, interference, and imperfect channel state information (CSI) disappear [16–18]. However, when pilot-based CSI is used with time-division duplex (TDD) MIMO systems, pilot contamination remains (PC) a limiting factor [22–25]. PC is a well-known problem in pilot-aided channel estimation and it happens when other users in the system are reusing the same set of pilot signals due to the limitation imposed by the number of available orthogonal pilots. This PC problem is accentuated even more with massive MIMO since it causes the interference rejection performance to quickly saturate with the number of antennas. Within the context of cognitive radio, if there is no cooperation between the SN and the PN, it is likely that the pilots used in both networks do not satisfy orthogonality, which may lead to pilot contamination. However, with some cooperation between the two networks and under the assumption that the pilots of each network are orthogonal, the SBS and the PBS can efficiently mitigate pilot contamination from PUs and SUs, respectively, by assigning different time to the training phase in both networks. Another alternative, if there is no cooperation between the two networks, is to use blind channel estimation techniques that require no or a minimal number of pilot symbols. One particular class of blind methods that works well with massive MIMO is the one based on subspace estimation techniques [26–29] that estimate the channel from the eigenvector of the covariance matrix of the received signal. This class requires unused degrees of freedom, which is the case of massive MIMO where the number of antennas at the base station is much larger than the number of users.

3. Capacity-Aware MIMO-Beamforming in Cognitive OFDM-SDMA Systems

Our objective is to find the optimal beamforming vector, (𝐰^𝑡_𝑙𝑠[𝑘])_opt, that maximizes the Ergodic capacity of the cognitive OFDM-SDMA channel for each SU 𝑙_𝑠 of the SN imposing the following two sets of constraints: (1) each secondary user 𝑙_𝑠 has a limited maximum transmission power equal to 𝑃^𝑡_max,𝑙 and (2) the total maximum interference power at the PBS from the SUs does not exceed the maximum power constraint of 𝐽^max_𝑠𝑝. In mathematical terms, these two constraints are expressed as follows [30]: max_{𝐰𝑡𝑙𝑠[𝑘]}𝐸log₂𝜌𝐈+^𝑙𝑠𝑁^𝑡_𝑠𝐵^−1/2_𝑙𝑠[𝑘]𝐇^𝑡_{𝑠𝑠,𝑙𝑠}[𝑘]²𝐰Subjectto∶^{𝑙𝑡,𝐻𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘]≤𝑃^𝑡_max,𝑙𝑠𝐽_𝑠𝑝=^𝐿𝑠_𝑖=1𝐇^𝑡_{𝑠𝑝,𝑙𝑠}[𝑘]𝐇^𝑡,𝐻_{𝑠𝑝,𝑙𝑠}[𝑘]≤𝐽^max_𝑠𝑝, (7) where 𝐸[⋅] denotes the expectation operator, 𝐇^𝑡_{𝑠𝑠,𝑙𝑠}[𝑘]=𝐇_{𝑠𝑠,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘], 𝐇^𝑡_{𝑠𝑝,𝑙𝑠}[𝑘]=𝐇_{𝑠𝑝,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘], 𝐁_𝑙𝑠[𝑘]=𝐁_𝑠𝑠[𝑘]+𝐁_𝑝𝑠[𝑘]+𝜎²_𝑛𝐈_𝑁^𝑟_𝑠[𝑘],  𝐁_𝑠𝑠[𝑘]=∑^𝐿𝑠_{𝑖=1,𝑖≠𝑙𝑠}𝐇^𝑡_{𝑠𝑠,𝑙𝑠}[𝑘]𝐇^𝑡,𝐻_{𝑠𝑠,𝑙𝑠}[𝑘], and 𝐁_𝑝𝑠[𝑘]=∑^𝐿𝑝_𝑙𝑝=1𝐇^𝑡_{𝑝𝑠,𝑙𝑠}[𝑘]𝐇^𝑡,𝐻_{𝑝𝑠,𝑙𝑠}[𝑘].

This problem is a constrained optimization problem which is highly nonconvex and complicated to solve. However, a suboptimal solution can be obtained by exploiting the method of Lagrange multipliers as follows: ℒ_CCA𝐰^𝑡_𝑙𝑠,𝜈_𝑙𝑠,𝜆_𝑙𝑠=𝐸log₂𝜌𝐈+^𝑙𝑠𝑁^𝑡_𝑠𝐵^−1/2_𝑙𝑠[𝑘]𝐇^𝑡_{𝑠𝑠,𝑙𝑠}[𝑘]²−𝜈_𝑙𝑠∑^𝐿𝑠_𝑙𝑠=1𝐇^𝑡_{𝑠𝑝,𝑙𝑠}[𝑘]𝐇^𝑡,𝐻_{𝑠𝑝,𝑙𝑠}[𝑘]𝐽^max_𝑠𝑝−1−𝜆_𝑙𝑠𝐰^𝑡,𝐻_{𝑠,𝑙𝑠}[𝑘]𝐰^𝑡_𝑙𝑠[𝑘]𝑃^𝑡_max,𝑙𝑠−1, (8) where 𝜈_𝑙𝑠 and 𝜆_𝑙𝑠 are the Lagrange multipliers associated with the 𝑙_𝑠th SU transmission power and the PBS received interference, respectively. In the proposed cognitive capacity-aware (CCA) algorithm, the weight vector for user 𝑙_𝑠 is updated at each iteration 𝑛, according to 𝐰^𝑡,CCA_𝑙𝑠(𝑛+1)=𝐰^𝑡,CCA_𝑙𝑠(𝑛)+𝜇∇_𝐰^𝑡,CCA_𝑙𝑠ℒ_CCA𝐰^𝑡,CCA_𝑙𝑠,𝜂_𝑙𝑠,𝜆_𝑙𝑠, (9) where ∇_𝐰^𝑡,CCA_𝑙𝑠 is the gradient of ℒ_CCA(𝐰^𝑡,CCA_𝑙𝑠,𝜂_𝑙𝑠,𝜆_𝑙𝑠) with respect to 𝐰^𝑡,CCA_𝑙𝑠 and 𝜇 is an adaptation constant to be chosen relatively small in order to achieve convergence [31]. Notice that SDMA is subcarrier parallel and that the update is done separately on each subcarrier. For brevity, therefore, we drop the frequency index [𝑘] and concentrate on the iteration index (𝑛) in this recursion. Using the matrix derivative formula 𝜕log|𝐀+𝐁𝐗𝐂|/𝜕𝐗^∗=[𝐂(𝐀+𝐁𝐗𝐂)^−𝟏𝐁]^𝐻 [32, 33], the gradient of the Lagrangian can be expressed as ∇_𝐰^𝑡_𝑙𝑠ℒ_CCA𝐰^𝑡_𝑙𝑠,𝜂_𝑙𝑠,𝜆_𝑙𝑠=1𝜌ln2^𝑙𝑠𝐵_𝑙⁻¹_𝑠(𝑛)𝐰^𝑡_𝑙𝑠(𝑛)1+𝜌_𝑙𝑠𝐰_𝑙^𝑡,𝐻_𝑠(𝑛)𝐵_𝑙⁻¹_𝑠(𝑛)𝐰^𝑡_𝑙𝑠(𝑛)−𝜆_𝑙𝑠(𝑛)𝑃^𝑡_max,𝑙𝑠𝐰^𝑡_𝑙𝑠(𝑛)−𝜈_𝑙𝑠(𝑛)𝐽^max_𝑠𝑝𝐇^𝐻_{𝑠𝑝,𝑙𝑠}(𝑛)𝐇_{𝑠𝑝,𝑙𝑠}(𝑛)𝐰^𝑡_𝑙𝑠(𝑛). (10)

The Lagrange multipliers, 𝜈_𝑙𝑠(𝑛) and 𝜆_𝑙𝑠(𝑛), are updated iteratively using the subgradient based method as described in [31] 𝜆_𝑙𝑠(𝑛+1)=𝜆_𝑙𝑠(𝑛)−𝛼_𝜆𝑙𝑠𝐰_𝑙^𝑡,𝐻_𝑠(𝑛+1)𝐰^𝑡_𝑙𝑠(𝑛+1)−𝑃^𝑡_max,𝑙𝑠,𝜈_𝑙𝑠(𝑛+1)=𝜈_𝑙𝑠(𝑛)−𝛼_𝜈𝑙𝑠^𝐿𝑠_𝑙𝑠=1𝐇^𝑡_{𝑠𝑝,𝑙𝑠}(𝑛)𝐇^𝑡,𝐻_{𝑠𝑝,𝑙𝑠}(𝑛)−𝐽^max_𝑠𝑝, (11) where the parameters 𝛼_𝜆𝑙𝑠 and 𝛼_𝜈𝑙𝑠 are the subgradients’ stepsizes, whose values are to be chosen relatively small in order to achieve convergence [31].

In our optimization procedure we consider that the initial value of 𝐰^𝑡,CCA_𝑙𝑠(𝑛) and 𝐰^𝑡,CEBF_𝑙𝑠(𝑛) at iteration index 𝑛=0 is given by the eigen-beamforming (EBF) weight; that is, 𝐰^𝑡,CCA_𝑙𝑠(0)=𝐰^𝑡,CEBF_𝑙𝑠(0)=𝑃^𝐭_max,𝑙𝑠𝐮_max,𝑙𝑠, where 𝐮_max,𝑙𝑠 denotes the eigenvector corresponding to 𝝀_max,𝑙𝑠, the maximum eigenvalue of 𝐇^𝐻_{𝑠𝑠,𝑙𝑠}𝐇_{𝑠𝑠,𝑙𝑠}. This initial value is then used to compute the initial value of the received beamforming vector at iteration index 𝑛=0. In our case we assume MRC at the receiving SBS; that is, 𝐰^𝑟_𝑙𝑠(0)=𝐁_𝑙𝑠(0)⁻¹𝐇^𝑡_{𝑠𝑠,𝑙𝑠}(0). (12)

4. Symbol Error Rate (SER) Performance of Cognitive OFDM-SDMA Systems

The symbol error rate, SER_{𝑘,𝑙𝑠}, associated with 𝐾th subcarrier of user 𝑙_𝑠, can be expressed as [34] SER_{𝑘,𝑙𝑠}=𝐸_{𝛾𝑠𝑘,𝑙}𝑎𝑄2𝑏𝛾_{𝑘,𝑙𝑠}, (13) where 𝐸[⋅] denotes the expectation operator, 𝑄(⋅) denotes the Gaussian Q-function, 𝛾_{𝑘,𝑙𝑠} is the signal-to-interference-plus-noise ratio (SINR) associated with the 𝐾th subcarrier of user 𝑙_𝑠, and 𝑎 and 𝑏 are modulation-specific constants. For binary phase shift keying (BPSK), 𝑎=1 and 𝑏=1, and, for binary frequency shift keying (BFSK) with orthogonal signaling, 𝑎=1 and 𝑏=0.5, while, for M-ary phase shift keying (M-PSK), 𝑎=2 and 𝑏=sin²(𝜋/𝑀). The SINR for SU 𝑙_𝑠 at iteration 𝑛, 𝛾_{𝑘,𝑙𝑠}(𝑛), is given by 𝛾_{𝑘,𝑙𝑠}(𝑛)=𝐰^𝑟_𝑙𝑠(𝑛)^𝐻𝐇^𝑡_{𝑠𝑠,𝑙𝑠}(𝑛)𝐇^𝑡_{𝑠𝑠,𝑙𝑠}(𝑛)^𝐻𝐰^𝑟_𝑙𝑠(𝑛)𝐰^𝑟_𝑙𝑠(𝑛)^𝐻𝐁_𝑙𝑠(𝑛)𝐰^𝑟_𝑙𝑠(𝑛). (14)

The average SER_𝑙𝑠 performance for user 𝑙_𝑠 can be estimated as [6] SER_𝑙𝑠=1𝑁_𝑐^𝑁𝑐−1_𝑘=0SER_{𝑘,𝑙𝑠}. (15)

We observe that, in general, the off-diagonal elements of 𝐁_𝑙𝑠(𝑛) are nonzero, reflecting the colour of the interference. However, in the asymptotic case of large 𝑁^𝑟_𝑠-element array and given equal power transmitted by all users (𝑃_𝑙𝑠=𝑃_𝑠 and 𝑃_𝑙𝑝=𝑃_𝑝), the central limit theorem (CLT) can be invoked to show that [11] 𝐁_𝑙𝑠(𝑛)=𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝𝜎²_𝑛+1𝜎²_𝑛𝐈_𝑁^𝑟_𝑠. (16)

Thus, assuming MRC at the receiving SBS, we can express 𝛾_𝑙𝑠(𝑛) as 𝛾^CCA_𝑙𝑠(𝑛)=1𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛×𝐰_𝑙^𝑡,𝐻_𝑠,CCA(𝑛)𝐇^𝐻_{𝑠𝑠,𝑙𝑠}(𝑛)𝐇_{𝑠𝑠,𝑙𝑠}(𝑛)𝐰^𝑡_𝑙𝑠,CCA(𝑛). (17)

For the case 𝑛=1, 𝐰^𝑡,CCA_𝑙𝑠(1)=𝐰^𝑡,EBF_{𝑠,𝑙𝑠}+𝜇×𝜌ln2_𝑙𝑠𝐰^𝑡,EBF_𝑙𝑠𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛+𝜌_𝑙𝑠𝐰_𝑙^𝑡,𝐻_𝑠,EBF𝐰^𝑡_𝑙𝑠,EBF𝜆−𝜇^𝑙𝑠(0)𝑃^𝑡_max,𝑙𝑠𝐰^𝑡_𝑙𝑠,EBF𝜈−𝜇^𝑙𝑠(0)𝐽^max_𝑠𝑝𝐇^𝐻_{𝑠𝑝,𝑙𝑠}(𝑛)𝐇_{𝑠𝑝,𝑙𝑠}(𝑛)𝐰^𝑡_𝑙𝑠,EBF. (18) Substituting (18) into (17), we can express 𝛾^CCA_𝑙𝑠(𝑛) at iteration index 𝑛=1 as 𝛾^CCA_𝑙𝑠(1)=1+𝛼+𝛽+𝜂𝛾^EBF_𝑙𝑠, (19) where 𝛼=2𝜇𝜅₀𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛+2𝜇²𝜅²₀𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛²,𝜅₀=1𝜌ln21+𝑃_𝑠/𝐿_𝑠−1𝑃^𝑙𝑠_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛,𝛽=𝜇⋅𝜆_𝑙𝑠(0)𝑃^𝑡_max,𝑙𝑠²,𝜂=𝜇⋅𝜈_𝑙𝑠(0)𝐽^max_𝑠𝑝², (20) and 𝛾^EBF_𝑙𝑠 represents the SINR obtained when eigen-beamforming is used and can be expressed as 𝛾^EBF_𝑙𝑠=1𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛𝐰_𝑙^𝑡,𝐻_𝑠,EBF𝐇^𝐻_{𝑠𝑠,𝑙𝑠}𝐇_{𝑠𝑠,𝑙𝑠}𝐰_𝑙^𝑡,𝐻_𝑠,EBF=𝑃_𝑠𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛𝜆_max,𝑙𝑠. (21) Substituting (21) into (19), then we can write 𝛾_𝑙^𝐶𝐶𝐴_𝑠(1) as 𝛾^CCA_𝑙𝑠(1)=1+𝛼+𝛽+𝜂𝑃_𝑠𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛𝜆_max,𝑙𝑠. (22)

Therefore, an alternate expression for (13) can be written as [8] SER_{𝑘,𝑙𝑠}=𝑎√𝑏2√𝜋×^∞₀𝑒^−𝑏𝑢√𝑢𝐹_{𝜆𝑠max,𝑙}𝐿_𝑠−1𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛𝑢𝑃_𝑠1+𝛼+𝛽+𝜂𝑑𝑢, (23) where 𝐹_{𝜆𝑠max,𝑙}(𝑢) is the cumulative distribution function (cdf) of 𝜆_max,𝑙𝑠. For the case 𝑁^𝑡_𝑠=2 and 𝑁^𝑡_𝑠≤𝑁^𝑟_𝑠, 𝐹_{𝜆𝑠max,𝑙}(𝑢) is given by [35] 𝐹_{𝜆𝑠max,𝑙}(𝑢)=det𝐑^𝑡_𝑠Δ₂𝐑^𝑡_𝑠Δ_𝑁^𝑟_𝑠𝐑^𝑟_𝑠^{𝑁𝑟𝑠𝑁}_𝑝=1^𝑟𝑠_{𝑡=1,𝑡≠𝑝}(−1)^{𝑝+𝜙(𝑡)}⋅𝛽_𝑝𝛽_𝑡^{𝑁𝑟𝑠−1}Δ_𝑁^𝑟_𝑠−2(𝐴)Φ_𝑝,𝑡(𝑢), (24) where Δ_𝑚(𝐑) is a Vandermonde determinant in the eigenvalues {𝛽_𝑖}^𝑚_𝑖=1 of the 𝑚-dimensional matrix argument, given by Δ_𝑚(𝐑)=Π^𝑚_𝑖<𝑗(𝛽_𝑗−𝛽_𝑖), while 𝜙(𝑡)=𝑡,for𝑡<𝑝𝑡−1,for𝑡>𝑝, (25) and 𝐴={𝛽_𝑖;𝑖∈{1,…,𝑁^𝑟_𝑠}⧵{𝑝,𝑡}}. Also Φ_𝑝,𝑡(𝑢)=(1/𝑢)𝑒^{−𝑢/𝜔2𝛽𝑝}Ψ(𝑁^𝑟_𝑠;−𝑢/𝜔₂𝛽_𝑝)𝑒^{−𝑢/𝜔1𝛽𝑡}Ψ(𝑁^𝑟_𝑠;−𝑢/𝜔₁𝛽_𝑡), where Ψ(𝑙;𝑦)=1−𝑒^−𝑦∑^𝑙−1_𝑘=0𝑦^𝑘/𝑘! is the regularized lower incomplete gamma function. Next, we substitute (24) into (23) and we consider high SNR case; then the SER can be expressed asymptotically as [8] _SER^∞_{𝑘,𝑙𝑠}=^{𝑎4𝑁𝑟𝑠𝑃−1!!𝑠/𝜎2𝑛−2𝑁𝑟𝑠}_{𝑏2𝑁𝑟𝑠22𝑁𝑟𝑠+1𝑁𝑟𝑠!𝑁𝑟𝑠𝑅+1!det𝑡𝑠𝑁𝑟𝑠𝑅det𝑟𝑠2}^{𝐿𝑠𝑃−1𝑠+𝐿𝑝𝑃𝑝}_𝜎2𝑛+1^{2𝑁𝑟𝑠}_{MIMO-MRCinMAI}11+𝛼+𝛽+𝜂^{2𝑁𝑟𝑠}. (26)

The first term represents the MIMO-MRC system, where the SER performance is mainly dominated by the multiuser access interference (MAI), under transmitted power constraints. It is noted that the presence of PUs’ interferences contributes asymptotically an SER degradation of (𝐿_𝑝𝑃_𝑝/[(𝐿_𝑠−1)𝑃_𝑠+𝜎²_𝑛]+1)^{2𝑁𝑟𝑠}compared to the system without PUs. This degradation is for each SU and in each OFDM subcarrier. On the other hand, we observe that, in the asymptotic case of large 𝑁^𝑟_𝑠-element, the additional factor in the SER expression of the proposed scheme, (1/[1+𝛼+𝛽+𝜂])^{2𝑁𝑟𝑠}, will approach zero, since 𝛼≥0,𝛽≥0,and𝜂≥0. Therefore, the proposed adaptive MIMO-beamforming scheme, in addition to protecting the primary network, can improve the SER performance of cognitive OFDM-SDMA multiuser access systems compared to conventional MIMO-MRC that suffers an error floor in the SER performance due to MAI. It is also noted that at the asymptotic case of large number of base station antennas and high SNR the SER performances are almost independent of the number of transmitting antennas. Next, we examine the total interference power at the PBS from the SUs when using CCA. Substituting (18) into (4) and after some manipulations, we can express 𝐽^CCA_𝑠𝑝 at iteration index 𝑛=1 as 𝐽^CCA_𝑠𝑝=𝐿_𝑠𝑃_𝑠(𝐶−𝐾)𝐶𝛽_𝑙𝑠,max−𝐾=𝐽^EBF_𝑠𝑝(𝐶−𝐾)𝐶𝛽_𝑙𝑠,max−𝐾, (27) where 𝐾=(𝜇/ln2)[(𝜌_𝑙𝑠𝐰^𝑡,EBF_{𝑠,𝑙𝑠})/((𝐿_𝑠−1)𝑃_𝑠+𝐿_𝑝𝑃_𝑝+𝜎²_𝑛+𝑃_𝑠𝜌_𝑙𝑠)]−𝜇(𝜆_𝑙𝑠(0)/𝑃^𝑡_max,𝑙𝑠)−1, 𝐶=𝜇(𝜈_𝑙𝑠(0)/𝐽^max_𝑠𝑝), 𝛽_𝑙𝑠,max is the maximum eigenvalue of 𝐇^𝐻_{𝑠𝑝,𝑙𝑠}𝐇_{𝑠𝑝,𝑙𝑠}, and 𝐽^EBF_𝑠𝑝=𝐿_𝑠𝑃 is the total interference power at the PBS from the SUs when using EBF. Since the term (𝐶−𝐾)(𝐶𝛽_𝑙𝑠,max−𝐾)≤1, the total interference power at the PBS from the SUs with the transmit weights obtained using the proposed CCA is better controlled and less than the one based on conventional eigen-beamforming.

5. Simulation Results

In our simulation setups we consider a CR-based OFDM-SDMA system with 𝑁^𝑡_𝑠=𝑁^𝑡_𝑝=2 transmit antennas. The number of antennas at the PBS and at the SBS is the same, 𝑁^𝑟_𝑝=𝑁^𝑟_𝑠, and varies from 8 up to 24. 𝐿_𝑠=4 SUs and 𝐿_𝑝=4 PUs. We assume BPSK modulation. We impose 𝑃^𝑡_max,𝑙=0 dB and 𝐽^max_𝑠𝑝=−5,−10, and −15 dB on the SUs. For OFDM configurations, we assume the 256-OFDM system (𝑁_𝑐=256) which is widely deployed in broadband wireless access services and we ran for 50 iterations with 𝜇=0.001. For the PN we assume an OFDM-SDMA system with the conventional MIMO-MRC, that is, EBF at the transmitter and MRC at the receiving PBS.

Figure 2 compares the SER performance of SUs achieved by the proposed CCA and the conventional MIMO-MRC for 𝐽^max_𝑠𝑝=−15 dB and 𝑁^𝑟_𝑝=𝑁^𝑟_𝑠=8,16, and 24. It is observed from the results that when 𝑁^𝑟_𝑠 increases, the SER of both methods decreases, but the capacity-aware-based method achieves significantly better performance, as expected from the analysis. It is also noted that as 𝑁^𝑟_𝑠 increases the performance gap between the two schemes increases.

Figure 2 

SER performance of SN using CCA and MIMO-MRC schemes for 𝐽𝑠𝑝=−15dB.

Figure 3, on the other hand, shows the impact of the interference power constraints, 𝐽^max_𝑠𝑝, on the SER performance of PUs. For both cases, 𝑁^𝑟_𝑝=𝑁^𝑟_𝑠=16 and 𝑁^𝑟_𝑝=𝑁^𝑟_𝑠=24, it is noted that, as 𝐽^max_𝑠𝑝 becomes larger, the SER of PUs is improved due to the decreased interferences. It is also noted that the most significant interference at the PBS is caused when the SN is using the nonconstrained MIMO-MRC scheme, because it is designed without considering interference to PBS. This shows that when SN is using the proposed cognitive capacity-aware scheme, it could efficiently control the interference to PBS. It is also noted that when 𝑁^𝑟_𝑝=𝑁^𝑟_𝑠=24, the SER performance of PUs varies slightly with 𝐽^max_𝑠𝑝 variations. This means that when the number of base station antennas becomes large, the interference constrained imposed on SUs could be relaxed without impacting the SER performance of PUs.

Figure 3 

Impact of 𝐽max𝑠𝑝 on the SER of PUs when the SN is using CCA and MIMO-MRC schemes.

6. Conclusion

This paper presents a new adaptive beamforming algorithm for multiuser access in large scale cognitive OFDM-SDMA systems. The proposed algorithm iteratively seeks the optimal transmit weight vectors that maximize the channel capacity of each secondary user in the network while protecting PUs from SUs’ interferences. It is shown that the proposed large scale CCA scheme could efficiently control the interference to PBS and could relax the constraints imposed on conventional CR networks.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.