Sensors 2015, 15, 16105-16135; doi:10.3390/s150716105 OPEN ACCESS sensors ISSN 1424-8220 www.mdpi.com/journal/sensors Article SNR Wall Effect Alleviation by Generalized Detector Employed in Cognitive Radio Networks Modar Safir Shbat † and Vyacheslav Tuzlukov †,* School of Electronics Engineering, College of IT Engineering, Kyungpook National University, 1370 Sankyuk-dong, Buk-gu, Daegu 702-701, Korea; E-Mail: [email protected] † These authors contributed equally to this work. * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +82-53-950-5509; Fax: +82-53-950-5506. Academic Editor: Assefa M. Melesse Received: 8 April 2015 / Accepted: 18 June 2015 / Published: 3 July 2015 Abstract: The most commonly used spectrum sensing techniques in cognitive radio (CR) networks, such as the energy detector (ED), matched filter (MF), and others, suffer from the noise uncertainty and signal-to-noise ratio (SNR) wall phenomenon. These detectors cannot achieve the required signal detection performance regardless of the sensing time. In this paper, we explore a signal processing scheme, namely, the generalized detector (GD) constructed based on the generalized approach to signal processing (GASP) in noise, in spectrum sensing of CR network based on antenna array with the purpose to alleviate the SNR wall problem and improve the signal detection robustness under the low SNR. The simulation results confirm our theoretical issues and effectiveness of GD implementation in CR networks based on antenna array. Keywords: cognitive radio (CR); spectrum sensing; generalized detector (GD); energy detector (ED); noise uncertainty; sample complexity; antenna array; SNR wall 1. Introduction The main aim of the cognitive radio (CR) network is to improve the spectrum utilization efficiency, by introducing an opportunistic use of unemployed frequency band by the primary user (PU) (see Figure 1). The spectrum sensing is needed to define the frequency holes that could be allocated for Sensors 2015, 15 16106 the secondary user (SU). The spectrum sensors search continuously an availability of frequency holes and assign them to SU without causing harmful interference to the PU. Fundamental limitations in practice are involved in spectrum sensing process [1–3]. The sensitivity to noise power uncertainty, for example, variations in the noise variance as a function of real time, is one of the most common and serious problems among the well-known spectrum sensors such as the energy detector (ED), matched filter (MF), and even the cyclostationary detector under some conditions at the low signal-to-noise ratio (SNR) [4]. The impact of noise power uncertainty is quantified by SNR wall location, i.e., if the SNR value is less than the SNR wall, the PU signal detector will fail to achieve the desired performance and maintain a robustness against power noise uncertainty independently of how long the sensing time is [3–5]. Both theoretical and experimental analysis confirmed the SNR wall phenomenon existence under the noise power uncertainty conditions. This phenomenon negatively effects the receiver operation characteristic (ROC). Other uncertainties also can be considered as SNR wall generators, for example, the noise power estimation error, assumptions made under the white and stationary noise, fading process, shadowing, non-ideal filters, non-precise analog-to-digital (A/D) converters, quantization noise, aliasing effect caused by imperfect front-end filters, and interference between the PU and SU. An alternative presentation for the SNR wall is given by the number of samples N as a function of SNR, the probability of false alarm P and probability of miss P , i.e., N = f(SNR,P ,P ). The PU signal FA miss FA miss detector should minimize the number of samples N required to achieve the desired detection performance. The lowest SNR satisfying the probability of false alarm P and the probability of miss FA P constraints is called the detector sensitivity [3]. miss Figure 1. CR systems and SU with M antenna array elements. In general, the ideal ED does not have the SNR wall, but owing to the noise power uncertainty the ED suffers from the SNR wall phenomenon making the ED non-robust under the low SNR [6,7]. In many published papers, the ED spectrum sensing performance is investigated under the noise uncertainty conditions. Different solutions are presented in the form of the dynamic detection threshold [8], log-normal approximation of the noise uncertainty [9], falling the SNR wall using the cross-correlation [10], improving the noise power estimation using the maximum likelihood (ML) estimator [6], SNR Sensors 2015, 15 16107 estimation based on the pseudo bit error rate (BER) for the modified ED [11], and algebraic spike detection method introduced in [12,13]. In fact, the best non-coherent detector is non-robust as the ED under the noise power uncertainty. In the coherent detector case, the SNR wall is pushed back only to a limited value and for a large channel coherence time K →∞. In the MF case, the SNR wall location is c proportional to 1/K [3], and in the case of feature detector, the SNR wall value is less in c comparison with the ED one and scales only as 1/ K with the relevant channel coherence time [3]. c An interesting new four-level hypothesis blind detector for spectrum sensing in CR systems is presented in [14]. The proposed detector in [14] reduces the negative effects on the CR system performance, which are forming under the in-phase and quadrature-phase (I/Q) imbalance, based on the orthogonal frequency division multiplexing (OFDM) multiple access scheme, and presents a promising solution for any noise power uncertainties or SNR wall problem that could be caused by this I/Q imbalance. Cooperative spectrum sensing, in the course of which the multiple sensors are involved in cooperative spectrum sensing, demonstrates an effective approach to improve the spectrum sensing performance under several problems such as the noise power uncertainty, multipath fading, shadowing, and receiver uncertainties issues. The cooperative spectrum sensing can also solve the critical energy efficiency issue as shown in [15] where the energy efficient cooperative spectrum sensing is proposed and the optimal scheduling of active time for each spectrum sensor helps to extend the network lifetime. Selective grouping based on the cooperative sensing is discussed in [16] where during the sensing time each sensors group senses different radio channels while sensors in the same group perform the joint detection by the targeted channel. This process assures obtaining the more robust and efficient sensing performance comparing with the individual spectrum sensor case under the noise power uncertainty. To mitigate the negative effects of noise power uncertainty at the low SNR, an implementation of the generalized detector (GD), which is constructed based on the generalized approach to signal processing (GASP) in noise, for the spectrum sensing in CR networks based on antenna array is proposed. The GD represents a combination of the correlation detector, which is optimal in the Neyman-Pearson (NP) criterion sense when there is a priori information about the PU signal parameters, and ED, which is optimal in the NP criterion sense if there is no any a priori information about the PU signal parameters that are random [17–19]. The GD likelihood ratio test, based on which we can make a decision about the PU signal presence or absence in the process incoming at the SU input, demonstrates a definition of the jointly sufficient statistics of the mean and variance of the likelihood ratio and does not require any information about the PU signal and its parameters [17], ([18], Chapter 3). As was discussed in detail in ([18], Chapter 7, pp. 685–692), the main function of GD energy detector (GD ED) is to detect the PU signal and the main function of the GD correlation detector is to define the detected PU signal parameters and make a decision: the detected signal is the expected PU signal with the required parameters or not. Note that the conventional correlation detector makes a decision about the PU signal presence or absence in the incoming process based on definition of the mean only of the process incoming at the SU input. The conventional ED defines the decision statistics with respect to PU signal presence or absence at the SU input based on determination of the variance only of the process incoming at the SU input. Definition of the jointly sufficient statistics of the mean and variance based of the incoming process at the SU input allows us to make more accurate decision in favor of the PU signal presence or absence and Sensors 2015, 15 16108 obtain more information about the PU signal parameters under GD employment in CR networks in comparison with the conventional MF, ED, correlation receiver and so on. A great difference between the GD ED and conventional ED is a presence of the additional linear system (the additional bandpass filter at the GD input) considered as the secondary data or reference noise source. The PU signal bandwidth is mismatched with the additional linear system bandwidth. The PU signal bandwidth is matched only with another linear system bandwidth at the GD front-end. Thus, the GD has two input linear systems, namely, the preliminary filter (PF) and the additional filter (AF). The last is considered as the reference noise source ([18], Chapter 3), [19]. The GD PF central frequency is detuned relatively to the GD AF central frequency to ensure, firstly, the PU signal passing only through GD PF and, secondly, the independence and uncorrelatedness between the stochastic processes at the GD PF and AF outputs. Thus, it is possible to obtain the PU signal plus noise at the GD PF output in the case of “a yes” PU signal at the GD input and only the noise in the opposite case. Consequently, only the noise is obtained at the GD AF output for both cases of “a yes” and “a no” PU signal at the GD input, in other words, under the hypotheses H and H . The case when there is the PU signal generated 1 0 by another source with the frequency content within the limits of the GD AF bandwidth, and considered as the additional interference, is discussed in [20]. The GD employment in wireless communications [21,22], radar sensor systems [20,23], and CR networks for spectrum sensing [24] allows us to improve the signal detection performance of these systems in comparison with implementation of widely used conventional detectors. This work differs from the previously published paper [24] by introducing a new advantage of GD employing in CR network systems based on antenna array, which is the SNR wall problem alleviation under the noise power uncertainty. Additionally, the GD optimal detection threshold is defined based on the minimal probability of error criterion under the noise power uncertainly at the low SNR condition. Intuitive approach to reduce the noise power uncertainty at run time by employing the GD in CR network is to define the noise power at the GD AF output, i.e., the another narrow band closed to the PU signal frequency band, with the purpose to calibrate the noise power in the PU signal frequency band. Even if we believe that the noise power forming at the GD PF and AF outputs is not the same, the noise calibration error can be much lower than the noise power uncertainty itself. The noise power calibration in real time improves the immunity against the SNR wall phenomenon [3]. In this paper, we investigate the GD noise power calibration effects on the SNR wall problem in coarse spectrum sensing for CR network systems based on antenna array and we define the GD sample complexity under the noise power uncertainty. The complementary receiver operating characteristic (ROC) and sample complexity of the ED, MF, and GD are compared under the same initial conditions for different uncertainty parameters. The real scenario of simulation demonstrates that the GD is able to alleviate the SNR wall problem and achieve the low probability of error in comparison with the conventional ED. The reminder of this paper is organized as follows. Section 2 presents the system model and the GD test statistics. Section 3 delivers the GD signal detection performance under the noise power uncertainty. The real scenario simulation results are discussed in Section 4. The concluding remarks are presented in Section 5. Sensors 2015, 15 16109 2. System Model and GD Test Statistics 2.1. System Model The spectrum sensor has an antenna array with the number of elements equal to M and each antenna array element receives N samples during the sensing time. The spectrum sensing problem can be modeled as the conventional binary hypothesis test: H  z[k]=w[k], i =1,...,M; k =0,...,N −1,  0 i i (1) H  z[k]=h[k]s[k]+w[k], i =1,...,M; k =0,...,N −1, 1 i i i where z[k] is the discrete-time received signal at the spectrum sensor input; w[k] is the discrete-time i i circularly symmetric complex Gaussian noise with zero mean and variance σ2, i.e., w w[k]~CN(0,σ2); h[k] is the discrete-time channel coefficients obeying the circularly symmetric i w i complex Gaussian distribution with zero mean and variance equal to σ2, i.e., h[k]~CN(0,σ2); and h i h s[k] is the discrete-time PU signal, i.e., the signal to be detected. We consider the same initial conditions with respect to s[k] as in [3]. The channel parameters are not varied during the sensing time and the channel coefficients h[k] are spatially correlated between each other. Throughout this paper, i the PU signal s[k], the channel coefficients h[k] , and the noise w[k] are independent and i i uncorrelated between each other. The same channel model is widely used in [25–27]. In general, the ED does not require channel state information (CSI) for spectrum sensing [28] and the GD shares this property with ED because the ED is a constituent of the GD. It is well known that information about the CSI allows us to obtain better spectrum sensing performance in comparison with unknown CSI case. The knowledge about CSI can be more useful and effective in the cooperative spectrum sensing case. Under the low SNR and noise power uncertainty conditions, we can claim that we have imperfect CSI [29]. When the noise power estimation is applied, we have partial knowledge about the CSI. In this paper, we assume that the coarse spectrum sensing is performed without knowledge about the CSI. Owing to its simplicity, the exponential matrix model is widely used to describe the spatial correlation between the adjacent antenna array elements [30]. The components of the M ×M antenna array element correlation matrix C can be presented in the following form: C ={ρi−j} , i≤ j, i, j =1,...,M (2) ij where ρ is the coefficient of spatial correlation between the adjacent antenna array elements (0≤ρ≤1, the real values). Applying the results presented in [30], the coefficient of spatial correlation ρ can be given as ρ=exp{−23Λ2(d /λ)2} (3) where Λ is the angular spread, an important propagation parameter defining a distribution of multipath power of radio waves coming in at the receiver input from a number of azimuthal directions with respect to the horizon; λ is the wavelength; and d is the distance between two adjacent antenna array elements (the antenna array element spacing). The correlation matrix of antenna array elements C given by Equation (2) is the symmetric Toeplitz matrix [25]. Sensors 2015, 15 16110 We define the NM ×1 signal vector Z that collects all the observed signal samples during the sensing time using the following form: Z=[z[0],...,z [0], ... ,z[N −1],...,z [N −1]]T (4) 1 M 1 M where T denotes a transpose. The data distribution in the complex matrix Z can be expressed as: CN(0,σ2I) ,  H Z ~  0 (5) CN(0,E σ2I+σ2I) ,  H s h 1 where E is the average energy of transmitted signal at the spectrum sensor input, and I is the s MN×MN identity matrix. We consider a situation when the primary signaling scheme is unknown (the PU has a total freedom of choosing the signaling strategy). Thus, the detector should be able to detect a presence of any possible PU signal satisfying the power and bandwidth constraints. The received signal vector Z has a complex Gaussian distribution with the covariance matrices Cov 0 and Cov under the hypotheses H and H,respectively. If z[k]=w[k], the received signals z[k] 1 0 1 i i i are independent between each other. Under the hypothesis H, when z[k]=h[k]s[k]+w[k], the 1 i i i received signals are spatially correlated. The covariance matrices Cov and Cov can be determined 0 1 in the following form:  Cov = E[ZZHH]=σ2I 0 0 w  (6)  Cov1 = E[ZZHH1]= Esσh2A+σ2wI where E[⋅] is the mathematical expectation; H denotes the Hermitian conjugate (conjugate transpose); I is the MN×MN identity matrix; E is the PU signal energy at the SU input; and A is theMN×MN s matrix defined based on the correlation matrix C given by Equation (2) [30]:  C 0 ... 0  M M   0    A=  M  (7)     0  M   0 ... 0 C   M M MN×MN where 0 is the M ×M zero matrix. M 2.2. GD Statistics The GD has been constructed based on the (GASP) in noise discussed in detail in [17–19]. The GD is considered as a linear combination of the correlation detector, which is optimal in the Neyman-Pearson criterion sense under detection of signals with a priori known parameters, and the ED, which is optimal in the Neyman-Pearson criterion sense under detection of signals with a priori unknown or random parameters. The main functioning principle of GD is a complete matching between the model signal generated by the local oscillator in GD and the information signal, in particular, the PU signal at the GD input by whole range of parameters. In this case, the noise component of the GD correlation detector caused by interaction between the model signal generated by the local oscillator in GD and the input noise and the random component of the GD ED caused by interaction between the incoming information Sensors 2015, 15 16111 signal (the PU signal) and input noise are cancelled in the statistical sense. This GD feature allows us to obtain the better detection performance in comparison with other classical receivers or detectors. The specific feature of GASP is introduction of the additional noise source that does not carry any information about the incoming signal with the purpose to improve a qualitative signal detection performance. This additional noise can be considered as the reference noise without any information about the PU signal [17]. The jointly sufficient statistics of the mean and variance of the likelihood ratio is obtained in the case of GASP implementation, while the classical and modern signal processing theories can deliver only a sufficient statistics of the mean or variance of the likelihood ratio. Thus, the implementation of GASP allows us to obtain more information about the input process or received information signal (the PU signal). Owing to this fact, an implementation of receivers constructed based on the GASP basis allows us to improve the spectrum sensing performance of CR wireless networks in comparison with employment of other conventional receivers at the sensing node. Figure 2. GD flowchart. The GD flowchart is presented in Figure 2. As we can see from Figure 2, the GD consists of three channels: • The GD correlation channel—the PF, multipliers 1 and 2, model signal generator MSG; • The GD ED channel—the PF, AF, multipliers 3 and 4, summator 1; • The GD compensation channel—the summators 2 and 3 and accumulator Σ. As follows from Figure 2, under the hypothesis H (a “yes” PU signal), the GD correlation channel 1 generates the signal component smod[k]s[k] caused by interaction between the model signal, the i i reference signal at the GD model signal generator (MSG) output, and the incoming information signal (the PU signal) and the noise component 2smod[k]ξ[k] caused by interaction between the model signal i i smod[k] and the noise ξ[k] (the PF output). Under the hypothesis H, the GD ED generates the i i 1 information signal energy s2[k] and the random component 2s[k]ξ[k] caused by interaction i i i Sensors 2015, 15 16112 between the information signal s[k] and the noise ξ[k]. The main purpose of the GD compensation i i channel is to cancel in the statistical sense the GD correlation channel noise component 2smod[k]ξ[k] i i and the GD ED random component 2s[k]ξ[k] between each other based on the same nature of the i i noise ξ[k]. i To describe the GD flowchart we consider the discrete-time processes without loss of any generality. Evidently, the cancelation in the statistical sense between the GD correlation channel noise component 2smod[k]ξ[k] and the GD ED random component 2s[k]ξ[k] is possible only based on the i i i i same nature of the noise ξ[k] satisfying the condition of equality between the signal model smod[k] i i and incoming PU signal s[k] over the whole range of parameters. The condition i smod[k]= s[k] (8) i i is the main GD functioning condition. Naturally, in practice, the signal parameters are random. The complete matching between the model signal smod[k] and the incoming signal s[k] (the PU i i signal), especially by amplitude, is a very hard problem in practice and only in the ideal case the complete matching is possible. How the GR sensing performance can be deteriorated under mismatching between the model signal smod[k] and the incoming (PU) signal s[k] is discussed in this paper. i i Under the hypothesis H , i.e., a “no” information signal (the PU signal), satisfying the GD main 0 functioning condition given by (8), we obtain only the background noise η2[k]−ξ2[k] at the GD i i output. The GD PF bandwidth is matched with the bandwidth of the information signal (the PU signal) s[k]. The threshold apparatus (THRA) device defines the GD threshold. i The GD PF and AF can be considered as the linear systems, for example, as the bandpass filters, with the impulse responses h [m] and h [m], respectively. For simplicity of analysis, we assume that PF AF these filters have the same amplitude-frequency characteristics or impulse responses by shape. Moreover, the GD AF central frequency is detuned with respect to the GD PF central frequency on such a value that the information signal (the PU signal) cannot pass through the GD AF. Thus, the PU signal and noise can appear at the GD PF output and the only noise is appeared at the GD AF output (see Figure 3). If a value of detuning between the GD AF and PF central frequencies is more than 4 or 5Δf , where Δf is the PU signal bandwidth, the processes at the GD AF and PF outputs can be s s considered as the uncorrelated and independent processes and, in practice, under this condition, the coefficient of correlation between GD PF and AF output processes is not more than 0.05 that was confirmed experimentally [31,32]. In the present paper, we consider the spectrum sensing problem of a single radio channel where the GD AF bandwidth is always idle and cannot be used by the SU because it is out of the useful spectrum of the PU network. There is a need to note that in a general case, the GD AF portion of the spectrum may be occupied by the PU signals from other networks and can be not absolutely unoccupied. In this case, the PU signals from other networks can be considered as interferences or interfering signals. Investigation and study of GD under this case is discussed in [20]. The processes at the GD AF and PF outputs present the input stochastic samples from two independent frequency-time regions. If the noise w[k] at the GD PF and AF inputs is Gaussian, the noise at the GD PF and AF outputs is Gaussian, too, because the GD PF and AF are the linear systems, and we believe that these linear systems do not change the statistical parameters of the input process. We Sensors 2015, 15 16113 use this assumption for simplicity of theoretical analysis. Thus, the GD AF can be considered as a reference noise source with a priori knowledge a “no” signal (the reference noise sample). Detailed discussion of the GD AF and PF can be found in [18,19]. The noise at the GD PF and AF outputs can be presented in the following form:  M ∞ w [k]=ξ[k]=  h [m]w[k−m]  PF i PF i  i=1 m=−∞ (9) M ∞ w [k]=η[k]=  h [m]w[k−m]  AF i AF i  i=1 m=−∞ Under the hypothesis H, the signal at the GD PF output can be defined as x[k]=s[k]+ξ[k] (see 1 i i i Figure 2), where ξ[k] is the observed noise at the GD PF output and i s[k]=h[k]×s[k] (10) i i Figure 3. Signals at the GD PF and AF outputs: (a) GD AF response and noise; (b) GD PF response and noise; (c) GD PF response and PU signal. Under the hypothesis H, and for all i and k, the process x[k]=ξ[k] at the GD PF output is 0 i i subjected to the complex Gaussian distribution and can be considered as the independent and identically distributed (i.i.d.) process. The process at the GD AF output is the reference noise η[k] with the same i statistical parameters as the noise ξ[k] in the ideal case. We make this assumption for simplicity. In i practice, the statistical parameters of the noise ξ[k] and η[k] are different, as a rule. We consider i i this case below. The decision statistics at the GD output presented in [17,18] is extended to the case of antenna array employment when an adoption of multiple antennas and antenna arrays is effective to mitigate the negative attenuation and fading effects [20,24]. The GD decision statistics can be presented in the following form: N−1 M N−1 M N−1 M TGD(X)=2xi[k]simod[k]−xi2[k]+ηi2[k]><H1THRGD (11) k=0 i=1 k=0 i=1 k=0 i=1 H0 Sensors 2015, 15 16114 where X=[x[0],,x [0],  ,x[N −1],,x [N −1]]T (12) 1 M 1 M is the stochastic process vector at the GD PF output and THR is the GD detection threshold. We can GD rewrite Equation (11) in the vector form: T (X)=2SmodX−X2 +η2 >H1 THR GD < GD (13) H0 where X=[x(0),...,x(N −1)] (14) is the M ×1 vector of the random process at the GD PF output with elements defined as x[k]=[x[k],,x [k]]T (15) 1 M Smod =[smod(0),,smod(N −1)] (16) is the M ×1 vector of the process at the MSG output with the elements defined as smod[k]=[smod[k],,smod[k]]T (17) 1 M η=[η(0),,η(N −1)] (18) is the M ×1 vector of the random process at the AF output with the elements defined as η[k]=[η[k],,η [k]]T (19) 1 M and THR is the GD detection threshold. According to GASP and GD structure shown in Figure 2 and GD the main GD functioning condition (8), the GD test statistics takes the following form under the hypotheses H and H , respectively: 1 0 N−1 M N−1 M N−1 M si2[k]+ηi2[k]−ξi2[k] H1  T (X)=k=0 i=1 k=0 i=1 k=0 i=1 (20) GD N−1 M N−1 M η2[k]−ξ2[k] H  i i 0 k=0 i=1 k=0 i=1 The term N−1M s2[k] is the average energy of received signal and the term k=0 i=1 i N−1M η2[k]− N−1M ξ2[k] presents the background noise at the GD output that is a difference k=0 i=1 i k=0 i=1 i between the noise power at the GD PF and AF outputs. It is important to mention that the GD main functioning condition is the equality between parameters of the model signal smod[k] and the PU signal i s[k] (see Equation (8)) over all range of parameters and, in particular, by amplitude. How we can i satisfy this condition in practice is discussed in detail in [17] and ([18], Chapter 6, pp. 611–621 and Chapter 7, pp. 631–695) when there is no a priori information about the signal s[k]. Additionally, a i practical implementation of the GD decision statistics requires an estimation of the noise variance σ2 w using the reference noise η[k] at the GD AF output. i The mean mGD and variance VarGD of the test decision statistics T (X) under the hypothesis H H0 H0 GD 0 are given in the following form ([19], Chapter 3):