Moment-based Spectrum Sensing Under Generalized Noise Channels
aa r X i v : . [ c s . I T ] S e p Moment-based Spectrum Sensing UnderGeneralized Noise Channels
Nikolaos I. Miridakis,
Senior Member, IEEE , Theodoros A. Tsiftsis,
Senior Member, IEEE and Guanghua Yang,
Senior Member, IEEE
Abstract —A new spectrum sensing detector is proposed andanalytically studied, when it operates under generalized noisechannels. Particularly, the McLeish distribution is used to modelthe underlying noise, which is suitable for both non-Gaussian(impulsive) as well as classical Gaussian noise modeling. Theintroduced detector adopts a moment-based approach, whereasit is not required to know the transmit signal and channel fadingstatistics (i.e., blind detection). Important performance metricsare presented in closed forms, such as the false-alarm probability,detection probability and decision threshold. Analytical andsimulation results are cross-compared validating the accuracyof the proposed approach. Finally, it is demonstrated that theproposed approach outperforms the conventional energy detectorin the practical case of noise uncertainty, yet introducing acomparable computational complexity.
Index Terms —Blind estimation, cognitive radio, impulsive non-Gaussian noise, spectrum sensing.
I. I
NTRODUCTION S IGNAL detection and spectrum sensing represent twowell-known complementary research topics that have at-tracted a vast research interest over the last decades. Someof the most popular spectrum sensing strategies includethe coherent, cyclostationary and energy detection (ED) [1],[2]. The former two approaches are typically optimal undernoisy channels; yet with the cost of either requiring perfectsynchronization and information of signal statistics or highcomputational complexity. ED, on the other hand, reflectson a simple implementation as well as it does not requireany prior information regarding the transmitted signal andunderlying channel fading. However, ED is quite sensitive innoise power uncertainty, which is typically the case in realisticconditions. To this end, another signal detection type has beenproposed; namely the moment-based detection (MD), whichproduces a similar complexity as ED but preserving robustnessin the presence of noise uncertainty at the same time [3],[4]. However, thus far, MD has been studied under additivewhite Gaussian noise (AWGN) channels and in the absenceof channel fading.In practice, there are various types of wireless commu-nication channels where signal transmission is subjected tonon-Gaussian (i.e., impulsive) noise. Typical examples in-clude urban and indoor wireless channels, ultra-wide band
N. I. Miridakis, T. A. Tsiftsis and G. Yang are with the Institute of PhysicalInternet and School of Intelligent Systems Science & Engineering, JinanUniversity, Zhuhai Campus, Zhuhai 519070, China. N. I. Miridakis is also withthe Dept. of Informatics and Computer Engineering, University of West Attica,Aegaleo 12243, Greece (e-mails: [email protected], theo [email protected],[email protected]). communications, frequency/time-hopping with jamming, mil-limeter wave communications, and wireless transmissionsunder strong interference conditions (see [5], [6] and ref-erences therein). Impulsive effects that introduce additivenon-Gaussian noise can also be found in cognitive radio(CR); e.g., due to the simultaneous spectrum access undermiss-detection events [7]. Unfortunately, detectors designedfor AWGN do not perform well in non-Gaussian noise [8],[9]. The most widely-used circularly symmetric non-Gaussiannoise models are the Gaussian mixture, Middleton’s ClassA, ǫ -mixture, Laplacian, generalized Gaussian, and α -stable[5]. The latter model (although denotes an accurate modelfor impulsive noise) does not have finite moments and thusit is not amenable for further processing. Generally, all theformer noise models cannot be suitably interpreted as the sumof a large number of independent and identically distributedimpulsive noise sources with small power. Doing so, the lackof matching the real-world phenomena of impulsive noisesources from a Gaussian to a non-Gaussian distribution definesa crucial weakness of the aforementioned non-Gaussian noisedistributions. Besides, focusing on the aforementioned non-Gaussian models, existing art studied the generalized energydetection and fractional low-order moment approaches [5],[6]. Nonetheless, these works introduced a considerably highcomputational complexity and presented critical performancemetrics (such as the detection probability or optimal settingparameters) in integral-only forms or by utilizing numerical-search methods.On another front, the McLeish distribution (also known asgeneralized symmetric Laplace or Bessel function distribution)represents an alternative noise model, appropriate for bothGaussian and non-Gaussian noise channels. It was originatedby D. Mcleish in [10] and quite recently it was revisitedand thoroughly analyzed in [11], [12]. McLeish distributionresembles the Gaussian distribution; it is unimodal, symmetric,it has all its moments finite, and has tails that are at leastas heavy as those of Gaussian distribution. Moreover, theevolution of its impulsive nature from Gaussian distributionto non-Gaussian distribution is explicitly parameterized in arigorous way with psychical meaning (please, see the detailedanalysis in [11, § IV.B.]); especially than those of Laplacian, α -stable and generalized Gaussian distributions.In this letter, for the first time, we study the spectrumsensing and signal detection under McLeish noise channels.A moment-based approach is adopted, aiming to a simpleimplementation. Also, the considered MD does not haveany knowledge of the transmitted signal and channel fading statistics (i.e., blind detection), while it operates on a fast-faded channel. It is only aware of the underlying noisestatistics. For sufficiently large number of channel samples,which is usually the practical case, new analytical closed-formexpressions are derived regarding some important performancemetrics; the false-alarm and detection probabilities as wellas the decision threshold. Most importantly, it is explicitlydemonstrated that the considered test statistic based on the MDapproach is independent of noise power (for both Gaussianand non-Gaussian noise channels), which in turn reflects ona rather robust and accurate detection strategy. In fact, MD iscross-compared to the benchmark ED under noise uncertaintyconditions, while the superiority of the former against the latterapproach is clearly manifested. Notation:
Matrices and vectors are represented by uppercaseand lower bold typeface letters, respectively. The coefficient atthe i th row and j th column of A is defined as A ( i,j ) . Super-script ( · ) T denotes vector transposition and |·| represents abso-lute (scalar) value. E [ · ] is the expectation operator and symbol d = means equality in distribution. Var[ · ] and Kurt[ · ] are thevariance and kurtosis operators, respectively. µ x ( n ) = E [ x n ] represents the n th moment function of a random variable(RV) x , while y | z denotes that y is conditioned on z event. CN ( µ, σ ) and N ( µ, σ ) define, respectively, a complex andcircularly symmetric (CCS) Gaussian RV as well as a real-valued Gaussian RV with mean µ and variance σ . Also, CML ( µ, σ , v ) and ML ( µ, σ , v ) denote, respectively, a CCSand real-valued RV following the McLeish distribution withmean µ , variance σ and non-Gaussianity parameter v . Further, Q ( · ) and Q − ( · ) are the Gaussian Q -function and inverse Q -function, respectively. Γ( · ) denotes the Gamma function [13,Eq. (8.310.1)], ( · )!! is the double factorial operator [13, p. xliii]and K v ( · ) denotes the v th order modified Bessel function ofthe second kind [13, Eq. (8.432)]. Finally, Re { x } and Im { x } denote the real and imaginary part of a complex-valued x ,respectively. II. S YSTEM AND S IGNAL M ODEL
Consider the classical binary hypothesis problem applied ona wireless spectrum sensing device, which reads as H : y [ u ] = w [ u ] , no signal is present, H : y [ u ] = h [ u ] s [ u ] + w [ u ] , signal transmission, (1)where y [ u ] ∈ C , h [ u ] ∈ C , s [ u ] ∈ R and w [ u ] ∈ C denotethe received signal, channel fading coefficient, transmitteddiscrete-time baseband signal and additive noise, respectively,at the u th sample. The transmitted signal samples, s [ · ] , arecaptured by a given constellation with transmit power s p ,whereby can be efficiently modeled as discrete uniformlydistributed RVs. In most practical wireless digital applications,such a constellation may be either an M -ary phase shift keying( M -PSK) or quadrature amplitude modulation ( M -QAM).Further, it is assumed that the channel fading coefficient, h [ · ] ,may follow an arbitrary distribution. Also, h [ i ] = h [ j ] and w [ i ] = w [ j ] ∀ i = j , while h [ · ] and w [ · ] remain unchangedduring a sample duration, whereas they may change betweenconsecutive samples. Additionally, w [ · ] d = CML (0 , σ w , v ) with σ w ∈ R + and v ∈ R + standing for the noise variance and non-Gaussianityparameter, respectively, with a symmetric and unimodal prob-ability density function defined as [11, Eq. (85)] f w ( w ) = 2 √ v | w | v − p σ w π Γ( v ) K v − s vσ w | w | ! . (2)Some special cases of f w ( · ) are obtained for v = 1 , v → + ∞ and v → + resulting to the CCS Laplacian, Gaussianand Dirac’s distribution, respectively [11]. It turns out that theMcLeish distribution is a generalized and versatile distributionmodel, which is suitable for both Gaussian and non-Gaussian(impulsive) noise channels. In fact, the noise statistics can becomputed and fitted to the McLeish distribution model using σ w , Var[ w ] , and v , w ] − . (3)Moreover, it is assumed that the spectrum sensing deviceis fully unaware of both the instantaneous and statisticalchannel gains as well as the signal statistics (i.e., neither thevariance of channel gains nor the transmit signal power andthe utilized modulation scheme are available); reflecting ona blind spectrum sensing. Yet, it is assumed that the noisestatistics are known. Thereby, a moment-based estimator isadopted for the test statistic including the 4 th and 2 nd absolutemoments of the received signal, which reads as [3] T , − µ | y | (4) µ | y | (2) . (4)Note that Kurt[ | y | ] = − T . For the H hypothesis, we simplyget T |H = − E [ | w | ] E [ | w | ] = − (cid:18) v (cid:19) . (5)The proof of (5) is provided in Appendix A with the aidof (A.4) and (A.5) and after some simple manipulations.It is noted that T → − as v → + ∞ (i.e., for AWGNchannels), which is in accordance to [3], while T = − / for Laplacian noise. It turns out that the sensing problem canbe formulated by setting that the considered test is equal orgreater than − − / (2 v ) , reflecting the signal absence orpresence, correspondingly, yielding T H ≧ H − (cid:18) v (cid:19) . (6)III. P ERFORMANCE M ETRICS
In realistic conditions, T is obtained by estimating themoments function of the received signal via a given numberof samples, N , such that ˆ T = − ˆ µ | y | (4)ˆ µ | y | (2) , (7) Illustratively, IEEE 802.22 and ECMA 392 standards utilize sporadic longsensing periods for fine sensing and more frequent short sensing periods inwhich a variety of signal-free samples can be collected and further processedfor noise estimation [14]. where ˆ µ | y | ( n ) , N N X u =1 | y [ u ] | n . (8)Obviously, ˆ T → T as N → + ∞ . To obtain the exact teststatistics and to evaluate the mismatch between actual andestimated moments, we introduce the RV √ N ( ˆ T − T ) , whichhas the following asymptotic property based on the centrallimit theorem (CLT): √ N ( ˆ T − T ) d = N (cid:0) , σ (cid:1) , (9)where σ |H = 16 v + 120 v + 294 v + 1894 v , (10)and σ |H = (cid:20) µ x (2) − µ x (4) µ x (2) + (cid:0) µ x (4) + µ x (8) (cid:1) µ x (2) − µ x (4) µ x (6) µ x (2) + 4 µ x (4) (cid:21) (cid:0) µ x (2) (cid:1) − , (11)with symbol x standing for a shorthand notation for Re { y } and µ x ( n ) = µ Re { y } ( n ) is given in (A.3). The proof is relegatedin Appendix B. It is noteworthy that σ |H = 4 in the AWGNcase (when v → + ∞ ), as it should be [3]. Also, an insightfulremark is the fact that (5) and (10) are independent of noisepower σ w under hypothesis H ; reflecting on quite an efficientand robust test statistic in the presence of detrimental yetrealistic uncertain noise power estimation [15].The scenario of a false-alarm probability, namely, P f ( · ) , isformulated as P f ( λ ) , Pr [ ˆ T > λ |H ] with λ denoting thedecision threshold. Hence, since ˆ T |H d = N (0 , σ |H ) , it holdsthat P f ( λ ) = Q (cid:0) λ/σ |H (cid:1) = Q λ √ v √ v + 120 v + 294 v + 189 ! . (12)As it is obvious from (12), the false-alarm probability is an offline operation, i.e., it is independent of the instantaneouschannel gain, the presence of signal transmission as well asthe noise power σ w . For known N , the common practice ofsetting the decision threshold is based on the constant false-alarm probability. Also, this is a reasonable assumption sincefor various practical spectrum sensing applications, the highestpriority is to satisfy a predetermined false-alarm rate (e.g.,underlay CR). Doing so, the desired threshold, λ ⋆ , yields as λ ⋆ , Q − (cid:16) P ( τ ) f (cid:17) (16 v + 120 v + 294 v + 189)4 v , (13)where P ( τ ) f represents the predetermined target on the maxi-mum attainable false-alarm probability.In the case of signal transmission, modeled by the H hypothesis, the estimated test statistic is distributed as ˆ T d = N (cid:18) √ N (cid:18) T + 2 + 32 v (cid:19) , σ |H (cid:19) . (14) Thus, the detection probability, P d ( · ) , is directly obtained ina closed form as P d ( λ ) , Pr [ ˆ T > λ |H ]= Q λ ⋆ − √ N (cid:0) T |H + 2 + v (cid:1) σ |H ! , (15)where T |H is the estimation test during the H hypothesisexpressed as T |H = − µ | y | (4) µ | y | (2) = − µ Re { y } (4)2 µ { y } (2) − , (16)with µ Re { y } ( n ) provided by (A.3).IV. N UMERICAL R ESULTS AND D ISCUSSION
In this section, the derived analytical results are verifiedvia numerical validation where they are cross-compared withcorresponding Monte-Carlo simulations. For the signal trans-mission, it is assumed that both the transmitter and receiveremploy a square-root raised cosine filter with roll-off factor . , oversampling factor F = 4 and filter length F + 1 . Also,zero-mean CCS McLeish noise instances, i.e., CML (0 , σ w ) ,are generated as per [11, Thm. 10], while it is assumed that thenon-Gaussianity parameter v is known at the receiver. In whatfollows, and without loss of generality, h [ u ] d = CN (0 , forthe u th sample; reflecting on unit-scale Rayleigh fast-fadedchannels. Hence, the received signal-to-noise ratio (SNR) isdefined as SNR , s /σ w . All the simulation results are con-ducted by averaging independent trials. Hereinafter, line-curves and cross-marks denote the analytical and simulationresults, respectively.In Fig. 1, the receiver operating characteristic (ROC) curveis illustrated for the case when BPSK (i.e., M = 2 ) or 16-QAM (i.e., M = 4 ) modulation scheme is applied for datatransmission. Also, the CCS Laplacian ( v = 1 ) and Gaussian( v → + ∞ ) noise models are included as the two extremescenarios. Interestingly, the detection performance is beingenhanced for lower-order modulation and/or when the noisebecomes non-Gaussian and thus more impulsive (reflecting ona reduced value of v ). This is an insightful outcome becausethe increased robustness on non-Gaussian noise is similar tothe performance of the detectors based on fractional low-order moment statistics [6]; yet utilizing considerably lowercomputational efforts.In Fig. 2, MD is compared to ED in low SNR regions.For a fair comparison, the decision threshold of the energydetector as well as its corresponding performance metricsshould be carefully designed in McLeish noise channels;i.e., see Appendix C. Additionally, the practical scenario ofuncertain estimation of noise power is considered by assuminga uniformly distributed (in dB) uncertainty factor β , ˆ σ w σ w ,where ˆ σ w is the estimated noise power, while β ∈ [ − L, L ] with L ≤ dB [15]. Clearly, the performance difference be-tween the moment-based detector and the conventional energydetector is emphatic in the presence of noise uncertainty.Although the ED performance gets worse for impulsive non-Gaussian noise (as expected; and in accordance to [6, Fig. 6]), P f ( λ * ) P d ( λ * ) SNR = −10dB; N = 2 v = 1, BPSK v = 3, BPSKAWGN, BPSK v = 1, 16−QAM v = 3, 16−QAMAWGN, 16−QAM Fig. 1. Detection probability vs. false-alarm probability for different modu-lation schemes and noise channels. −15 −12 −9 −6 −3 00.10.20.30.40.50.60.70.80.91
SNR (dB) P d ( λ * ) N = 2 ; L = 2dB; P f = 0.1 MD, BPSK, v = 1MD, BPSK, AWGNMD, 16−QAM, v = 1MD, 16−QAM, AWGNED, BPSK, v = 1ED, BPSK, AWGNED, 16−QAM, v = 1ED, 16−QAM, AWGN Fig. 2. Detection probability vs. various SNR values for different modulationschemes and noise channels. MD is cross-compared to ED. the MD performance is being enhanced which verifies theaforementioned outcomes from Fig. 1.In a similar basis, the ROC performance of three differentdetectors is illustrated in Fig. 3. In particular, the consideredMD is compared with ED as well as the locally optimal(LO) detector [5, Eq. (9)] which is numerically computed.Note that the LO detector approaches the global optimal inlow SNR regions and, hence, may serve as a performancebenchmark. In accordance to the aforementioned discussion,MD outperforms ED in the presence of noise uncertainty,while the said performance difference increases for moreimpulsive noise channels (with a reduced v ). It is also obviousthat LO outperforms both MD and ED under non-Gaussiannoise; yet, at the cost of considerably higher computationalefforts as compared to MD and ED.V. C ONCLUDING R EMARKS
Blind spectrum sensing and signal detection was studiedunder CCS McLeish noise, fast-faded channels and discrete M -ary PSK or M -ary QAM signals. A moment-based ap-proach was considered to implement the test statistic, which P f P d N = 2 ; L = 2dB; BPSK; SNR = −10dB MD, v = 1MD, v = 5ED, v = 1ED, v = 5LO, v = 1LO, v = 5 Fig. 3. Detection probability vs. false-alarm probability for different detectorsand noise channels. generalizes the AWGN case to the impulsive non-Gaussiannoise environments. Insightfully, the discussed test statisticbased on MD is independent of noise power (and hence noiseuncertainty), which corresponds to a relatively simple yet quiteaccurate and robust detection strategy; especially in channelswith potentially many unpredicted impulsive sources. Finally,the superiority of MD against ED was demonstrated, whilethe performance of the former approach is being enhanced formore impulsive noise channels.A
PPENDIX
A. Derivation of Moments Functions
The n th moment function of a discrete uniform RV with M possible values in the range [ − s p , s p ] is given by µ s ( n ) = M− X l =0 ( − n ( M − l − n s n p M ( M − n , n ∈ R + . (A.1)For a CCS h , Re { h } d = Im { h } d = N (0 , σ h / . Then, the n th moment function of Re { h } reads as µ Re { h } ( n ) = ( n − − n ) σ nh , n ∈ R + . (A.2)Further, Re { y } = Re { hs } + Re { w } d = Im { y } . The n th moment function of Re { y } is derived by [11, Eq. (30)] µ Re { y } ( n ) = n X k =0 (cid:18) nk (cid:19) (1 + ( − k )Γ (cid:0) v + k (cid:1) Γ (cid:0) k (cid:1)
2Γ ( v ) Γ (cid:0) (cid:1) × (cid:18) σ w v (cid:19) k µ Re { h } ( n − k ) µ s ( n − k ) , n ∈ N + . (A.3)For the H hypothesis (i.e., signal transmission), while uti-lizing (A.1)-(A.3), the nd absolute moment of the receivedsignal, y , is computed in a closed form by µ | y | (2) = E (cid:2) | hs + w | (cid:3) = E h (Re { hs } + Re { w } ) + (Im { hs } + Im { w } ) i = 2 µ Re { y } (2) . (A.4)Likewise, we have that µ | y | (4) = E (cid:2) | hs + w | (cid:3) = E (cid:20)(cid:16) (Re { hs } + Re { w } ) + (Im { hs } + Im { w } ) (cid:17) (cid:21) = 2 (cid:16) µ Re { y } (4) + µ { y } (2) (cid:17) . (A.5)In a similar basis, we get µ | y | (6) = 2 µ Re { y } (6) + 6 µ Re { y } (4) µ Re { y } (2) , (A.6)and µ | y | (8) = 2 µ Re { y } (8) + 8 µ Re { y } (6) µ Re { y } (2) + 6 µ { y } (4) . (A.7)Regarding the H hypothesis (i.e., absence of signal s ), thelatter absolute moments of y are directly computed by setting k = n in (A.3). B. Derivation of σ According to [16, Thm. 3.3.A], the RV defined in (9) is azero-mean Gaussian RV with variance σ , cΣc T , such that c = " ∂ ˆ T∂ ˆ µ | y | (2) , ∂ ˆ T∂ ˆ µ | y | (4) ˆ µ | y | (2)= µ | y | (2) , ˆ µ | y | (4)= µ | y | (4) = " µ | y | (4) µ | y | (2) , − µ | y | (2) , (B.1)and Σ = " µ | y | (4) − µ | y | (2) µ | y | (6) − µ | y | (2) µ | y | (4) µ | y | (6) − µ | y | (2) µ | y | (4) µ | y | (8) − µ | y | (4) . (B.2)It follows that σ =4 µ | y | (4) Σ (1 , − µ | y | (4) µ | y | (2) Σ (1 , + µ | y | (2) Σ (2 , µ | y | (2) . (B.3)Consequently, utilizing the relevant moments formulae fromAppendix A and after performing some tedious yet straight-forward manipulations, we arrive at (10) and (11), correspond-ingly. C. Statistics for Energy Detection
The normalized test statistic of the typical energy detectoris defined as [5], [14] T ED , N σ w N X u =1 | y [ u ] | . (C.1) For sufficiently large number of mutually independent sam-ples, N , while invoking CLT, the latter statistic is approxi-mately distributed as N ( E [ | y | ] /σ w , Var[ | y | ] / ( N σ w )) . Then,it is straightforward to show that P f ( λ ED ) = Q λ ED − E [ | y | |H ] /σ w q Var[ | y | |H ] / ( N σ w ) , (C.2)and P d ( λ ED ) = Q λ ED − E [ | y | |H ] /σ w q Var[ | y | |H ] / ( N σ w ) , (C.3)where λ ED is the decision threshold under energy detection,which is specified in a similar basis as per (13) by λ ⋆ ED , q Var[ | y | |H ] / ( N σ w ) Q − ( P ( τ ) f ) + E [ | y | |H ] /σ w . Note that E [ | y | ] = µ | y | (2) and Var[ | y | ] = µ | y | (4) − µ | y | (2) for either H or H . Without delving into details, the above expressionscan be easily computed in closed forms via the relevantformulae of Appendix A, namely, (A.4), (A.5) in conjunctionwith (A.3). R EFERENCES[1] A. Mariani, A. Giorgetti, and M. Chiani, “Effects of noise powerestimation on energy detection for cognitive radio applications,”
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