Generalized Energy Detection Under Generalized Noise Channels
aa r X i v : . [ c s . I T ] S e p Generalized Energy Detection Under GeneralizedNoise Channels
Nikolaos I. Miridakis,
Senior Member, IEEE , Theodoros A. Tsiftsis,
Senior Member, IEEE and Guanghua Yang,
Senior Member, IEEE
Abstract —Generalized energy detection (GED) is analyticallystudied when operates under fast-faded channels and in thepresence of generalized noise. For the first time, the McLeishdistribution is used to model the underlying noise, which issuitable for both non-Gaussian (impulsive) as well as classicalGaussian noise channels. Important performance metrics arepresented in closed forms, such as the false-alarm and detectionprobabilities as well as the decision threshold. Analytical andsimulation results are cross-compared validating the accuracy ofthe proposed approach in the entire signal-to-noise ratio regime.Finally, useful outcomes are extracted with respect to GEDsystem settings under versatile noise environments and whennoise uncertainty is present.
Index Terms —Cognitive radio, generalized energy detection,impulsive non-Gaussian noise, spectrum sensing.
I. I
NTRODUCTION S IGNAL detection and spectrum sensing have attracted avast research interest over the last decades, whereas theyare considered as essential counterparts of various practicalapplications. The most representative ones include cognitiveradio (CR) transmissions, radar communications as well asnetwork slicing and dynamic frequency resource allocation in5G networks [1]–[3]. Among the available signal detectionschemes, energy detection (ED) is one of the most popularones because it provides an efficient tradeoff between com-putational complexity and performance [4]. In fact, ED isa relatively simple implementation approach while it is theoptimum detector in the presence of Gaussian signals andadditive white Gaussian noise (AWGN) channels [5]. Further,it does not require any knowledge regarding the signal andchannel fading statistics.Nonetheless, in realistic conditions, neither the transmittedsignals are always Gaussian distributed nor the underlyingnoise is AWGN. Particularly, there are various types ofwireless communication channels where signals are subjectedto non-Gaussian (i.e., impulsive with heavy-tailed distribu-tional behavior) noise. Typical examples include urban andindoor wireless channels, ultra-wide band communications,frequency/time-hopping with jamming, millimeter wave com-munications, and wireless transmissions under strong inter-ference conditions (e.g., see [6], [7] and relevant references
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]). therein). Accordingly, a modified type of ED has been thor-oughly analyzed and tested, which is entitled as generalizedED (GED) or p -norm detector, where p is a tunable parameterso as to enhance the detector performance. GED includes asspecial cases the absolute value detector when p = 1 [8],ED when p = 2 and fractional low order detector when < p < [6], [7]. The performance of GED was studied in[9] for a certain popular type of non-Gaussian impulsive noisechannels; namely, additive white Laplacian noise (AWLN).Also, GED was studied in the presence of generalized noisein [10], by using the Gaussian mixture distribution model.However, the derived results were quite complex, i.e., definedin an infinite series representation. On a similar basis, [6] and[7] studied GED under non-Gaussian noise channels; yet, theirdetection performance results were tightly accurate in the casewhen the noise power is much higher than the signal power.In this Letter, for the first time, we analytically study theGED performance under McLeish noise channels. McLeishdistribution represents a generalized model, appropriate forboth Gaussian and non-Gaussian noise channels. It was orig-inated by D. Mcleish in [11] and quite recently it was re-visited and thoroughly analyzed in [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 [12, § IV.B.]); especially than those of Laplacian, α -stable and generalized Gaussian distributions. It models anykind of impulsive noise between the two extreme cases, i.e.,Dirac’s distribution (highly impulsive noise) and AWGN (non-impulsive noise).For sufficiently large number of samples, which is usuallythe practical case, analytical closed-form expressions are de-rived for key performance metrics, namely, the false-alarmand detection probabilities as well as the decision thresholdof GED. The mentioned expressions are sharp in the entiresignal-to-noise (SNR) ratio regime. In addition, the case whenthe received signal undergoes fast-faded Rician channels is in-cluded as well as the detrimental effect of uncertain/imperfectnoise power estimation. Finally, the enclosed analytical andnumerical results reveal some useful engineering insights. Notation: | · | represents absolute (scalar) value. E [ · ] isthe expectation operator, Var[ · ] is the variance operator andsymbol d = means equality in distribution. f x ( · ) denotes the probability density function (PDF) of random variable (RV) x . Also, 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 . Moreover, CML ( µ, σ , v ) denotes a CCS RV following the McLeishdistribution with mean µ , variance σ and non-Gaussianityparameter v . Further, Q ( · ) and Q − ( · ) are the Gaussian Q -function and inverse Q -function, respectively, while csc( · ) stands for the cosecant function. Γ( · ) denotes the Gammafunction [13, Eq. (8.310.1)] and Γ( · , · ) is the upper incompleteGamma function [13, Eq. (8.350.2)]. I ( · ) is the th ordermodified Bessel function of the first kind [13, Eq. (8.445)]; K v ( · ) denotes the v th order modified Bessel function of thesecond kind [13, Eq. (8.432)]; F ( · , · ; · ) is the Kummerconfluent hypergeometric function [13, Eq. (9.210.1)]; and G m,np,q [ ·|· ] represents the Meijer’s G-function [13, Eq. (9.301)].Finally, Re { x } and Im { x } denote the real and imaginary partof a complex-valued x , respectively.II. S YSTEM AND S IGNAL M ODEL
Consider the binary hypothesis problem, 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, transmittedbaseband signal and additive noise, respectively, at the u th sample. The signal samples, s [ · ] , are being transmitted withpower s , and are subjected to an arbitrary continuous ordiscrete distribution. Further, it is assumed that the channelfading coefficient, h [ · ] , follows a non zero mean CCS Gaussiandistribution, such that Re { h [ · ] } d = N (cos( θ ) α, σ h / and Im { h [ · ] } d = N (sin( θ ) α, σ h / for arbitrary θ ∈ [0 , π ) and α ∈ R + , while σ h stands for the variance of h [ · ] .Consequently, | h [ · ] | follows the Rice distribution with Ricianfactor K , α /σ h , which sufficiently models both line-of-sight (LoS) and non-LoS channel fading conditions; note that | h [ · ] | becomes Rayleigh distributed for α = K = 0 . It is alsoassumed that h [ · ] remains fixed during a sample time whereasit may change between different samples.In addition, 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 PDFdefined as [12, 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 [12]. It turns out that theMcLeish distribution is a generalized and versatile distributionmodel, which is suitable for both Gaussian and non-Gaussian(impulsive) noise channels. Moreover, GED is fully unaware of channel gains as well asthe signal and noise statistics; reflecting on a blind spectrumsensing. The considered test statistic for GED reads as T , N X u =1 | y [ u ] | p , (3)where N represents the number of samples and p ≥ isa tunable exponent that provides flexibility to the detector.When p = 2 , GED coincides with the conventional ED, whileit becomes the fractional low order detector for < p < .III. P ERFORMANCE M ETRICS
The scenario of a false-alarm probability, namely, P f ( · ) ,is modeled by P f ( λ ) , Pr [ T > λ |H ] with λ denoting thedecision threshold. For sufficiently large number of samples,which is usually the practical case, the PDF of T closelyapproaches a Gaussian distribution even if the underlying noise(having finite moments) is non-Gaussian. Thus, for arbitrary p , the false-alarm probability is presented in a simple closedform as P f ( λ ) = Q λ − N µ p N σ ! , (4)where µ , E [ | y | p |H ] = E [ | w | p ] , (5)and σ , Var[ | y | p |H ] = E [ | w | p ] − E [ | w | p ] , (6)with E [ | w | n |H ] = Γ (cid:0) n + v (cid:1) Γ (cid:0) n + 1 (cid:1) Γ ( v ) v n/ σ nw , n ∈ R . (7)The proof of (4) is relegated in Appendix A.As it is obvious from (4), the false-alarm probability is an offline operation, i.e., it is independent of channel gains andsignal statistics. For known N and σ w , the common practiceof setting 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 ⋆ , stems as λ ⋆ , Q − (cid:16) P ( τ ) f (cid:17) q N σ + N µ , (8)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 detection probability, P d ( · ) , is directly obtainedin a closed form as P d ( λ ⋆ ) , Pr [ T > λ |H ]= Q λ ⋆ − N µ p N σ ! , (9) Since consecutive samples are mutually independent and for notationalsimplicity, hereinafter we drop sample indexing. where µ , E [ | y | p |H ] = E [ | hs + w | p ] , (10)and σ , Var[ | y | p |H ] = E [ | hs + w | p ] − E [ | hs + w | p ] , (11)with E [ | hs + w | n |H ] = Γ (cid:0) n (cid:1) (cid:16) s σ h α (cid:17) n/ Γ ( v ) Γ (cid:0) − n (cid:1) F (cid:16) − n , − α (cid:17) × G , , (cid:20) σ w s σ h v − v, n + 10 (cid:21) , (12)for arbitrary n excluding any even integer n ≥ . For the lattercase (where n = 2 , , , . . . ), (12) relaxes to E [ | hs + w | n |H ] = Γ (cid:0) n (cid:1) Γ( v )(1 + α ) n/ F (cid:16) − n , − α (cid:17) × n/ X k =0 (cid:18) n/ k (cid:19) (cid:0) s σ h (cid:1) n/ − k σ kw Γ( k + v ) v − k . (13)The proof of (9) is provided in Appendix B. Note that forRayleigh faded channels, F ( − n/ ,
1; 0) = 1 in (12) and(13). In addition, for the special case of CCS Laplacian noise(i.e., when v = 1 ), the Meijer’s G-function in (12) reduces to G , , (cid:20) σ w s σ h , n + 10 (cid:21) = − π exp (cid:18) − σ w s σ h (cid:19) (cid:18) σ w s σ h (cid:19) n/ × csc (cid:16) πn (cid:17) Γ (cid:16) n + 1 , s σ h σ w (cid:17) Γ (cid:0) n + 1 (cid:1) . (14)In the presence of detrimental yet unavoidable effect ofuncertain noise power estimation, the corresponding uncer-tainty factor can be modeled such that σ w ∈ { ˆ σ w /ρ, ρ ˆ σ w } with ˆ σ w standing for the estimated noise power and ρ ≥ [14]. In practice, the distribution of the actual uncertaintyfactor is quite difficult to obtain. However, the bound on thementioned uncertainty (i.e., ρ ) is measurable and thus can beconsidered as known. To evaluate the worst-case scenario (i.e.,the lower bound on P d ( · ) given a fixed P ( τ ) f ), (7) and (12) aredirectly computed by substituting σ w with ρ ˆ σ w . Doing so andaccording to (8), it turns out that λ ⋆ ( ρ ) , ρ p/ λ ⋆ , reflectingon a corresponding deviation on the decision threshold (thus,on the detector performance) which is proportional to p/ .Hence, it is verified that detectors based on fractional loworder statistics are indeed more robust to noise uncertainty.Yet, setting p → + is not always effective since (9) is anon-concave function with respect to the exponent p or thenon-Gaussianity parameter v . Hence, to obtain the optimum p ⋆ , it is required to solve arg max p P d ( λ ⋆ ) , subject to { v, p } > , (15) As an illustrative example, IEEE 802.22 and ECMA 392 standards utilizesporadic long sensing periods for fine sensing and more frequent short sensingperiods in which a variety of signal-free samples can be collected and furtherprocessed for noise estimation [15]. P ( τ ) f = 0.01; SNR = −10dB v P d SNR = −10dB P f P d p = 1 p = 2 p = 4 p = 1, AWLN p = 1, AWGN p = 2, AWLN p = 2, AWGN Fig. 1. GED performance for various system parameters and noise channels. which can be numerically computed by a simple line searchvia (8) and (9).IV. N
UMERICAL R ESULTS AND D ISCUSSION
In this section, the derived analytical results are verifiedvia numerical validation, whereas they are cross-comparedwith corresponding Monte-Carlo simulations. Subsequently,in Figs. 1 and 2, h [ u ] d = CN (0 , for the u th sample;reflecting on unit-scale Rayleigh fast-faded channels. In Fig. 3, h [ u ] d = CN ( α, denoting Rician channel fading with a corre-sponding factor K = α . Hence, the received SNR is definedas SNR , s /σ w . All the simulation results are conductedby averaging independent trials. Hereinafter, line-curvesand square-marks denote the analytical and simulation results,respectively, while the number of samples is set to be N = 2 .At the left-hand side (LHS) of Fig. 1, the impact ofnon-Gaussianity parameter v on the detection probability isdepicted. When the noise becomes more impulsive (i.e., areduced v ), a lower p exponent is beneficial. In fact, this effectgets even more emphatic as v → + . On the other hand, asthe noise tends to approach the Gaussian type (i.e., for anincreased v ), the detection performance of a low-order p isdegraded. At the right-hand side (RHS) of Fig. 1, the receiveroperating characteristic curve of GED is shown for two certainnoise types, namely, AWLN ( v = 1 ) and AWGN ( v → + ∞ ),while comparing the absolute value detector ( p = 1 ) and ED( p = 2 ). Obviously, the former detector outperforms the latterone in the case of Laplacian noise (highly impulsive noise),whereas quite the opposite outcome arises from the Gaussiannoise scenario.In Fig. 2, the GED performance is illustrated for thepractical case of noise uncertainty. Particularly, the worst-case scenario of the detection performance is shown for agiven/known uncertainty bound ρ (perfect noise power esti-mation is defined as ρ = 0 dB). The LHS and RHS of Fig. 2correspond to the AWLN and AWGN, respectively. A numberof useful engineering insights can be drawn from Fig. 2.Reducing the p exponent of GED in AWLN channels enhancesthe detection performance both in the presence and absence −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 000.10.20.30.40.50.60.70.80.91 SNR (dB) P d P ( τ ) f = 0.1; AWLN −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 000.10.20.30.40.50.60.70.80.91 SNR (dB) P d P ( τ ) f = 0.1; AWGN p = 0.1, ρ = 0dB p = 0.1, ρ = 0.1dB p = 1, ρ = 0dB p = 1, ρ = 0.1dB p = 2, ρ = 0dB p = 2, ρ = 0.1dB p = 0.1, ρ = 0dB p = 0.1, ρ = 0.1dB p = 1, ρ = 0dB p = 1, ρ = 0.1dB p = 2, ρ = 0dB p = 2, ρ = 0.1dB Fig. 2. Detection probability of GED vs. various SNR regions for differentnoise channels and system settings. of noise uncertainty. Indicatively, setting p = 0 . providesapproximately a dB gain against the classical ED ( p = 2 ).Nevertheless, an entirely different behavior is observed inAWGN channels, where ED presents a better detection per-formance whilst the corresponding performance of the low-order detector with p = 0 . degrades. It is also noteworthythat the detection performance of all the considered GEDcases almost coincide to each other in the case of imperfectnoise estimation; thus revealing the detrimental effect of noiseuncertainty regardless of the value of p exponent in Gaussiannoise.In Fig. 3, the case of Rician faded channels is illustrated fortwo additive noise models; namely, AWLN and AWGN. Thepractical scenario of noise uncertainty is also included (i.e., ρ = 0 . dB) as well as the ideal case of perfect noise estimation( ρ = 0 dB). Moreover, K = 0 denotes the classical Rayleighfading, whereas K = 10 indicates the presence of a dominant(LoS) factor. The detection performance is being enhanced forthe ideal noise estimation case and when the received signalundergoes Rician fading. This is a reasonable outcome sincethe presence of a strong dominant signal power factor makesthe actual received signal more distinguishable than additivenoise. From an engineering standpoint, the non-concavity ofthe detection performance with respect to exponent p is evidentunder various types of channel fading and/or noise environ-ments. Obviously, under Rayleigh faded channels, energy-typedetectors (i.e., p ∝ ) are suitable for both ideal and non-idealnoise estimation. Nevertheless, fractional low order detectorsare much more beneficial for an increased Rician K factoror when the noise becomes impulsive, since the detectionperformance is being enhanced as p is reduced. Finally, it isworthy to state that an integer-valued exponent p produces lesscomputational complexity than its fractional-order counterpart[9]. Thereby, whenever the complexity reduction is of primeimportance, absolute value detector with p = 1 representsquite an effective detector (as compared to ED or higher orderdetectors) in the presence of Rician channel fading, impulsivenoise and/or noise uncertainty. p P d P ( τ ) f = 0.01; SNR = −10dB K = 0; ρ = 0.1dB; AWGN K = 10; ρ = 0.1dB; AWGN K = 0; ρ = 0dB; AWGN K = 10; ρ = 0dB; AWGN K = 0; ρ = 0.1dB; AWLN K = 10; ρ = 0.1dB; AWLN K = 0; ρ = 0dB; AWLN K = 10; ρ = 0dB; AWLN p * Fig. 3. Detection probability of GED vs. various exponent p values fordifferent fading channels and noise conditions. V. C
ONCLUSION
The GED (also known as p -norm detector) was analyticallystudied under the presence of Rician faded channels andMcLeish noise, thus capturing a wide range from highlyimpulsive to non-impulsive noise conditions. Important systemperformance metrics were derived in straightforward closed-form expressions; namely, the decision threshold, detectionand false-alarm probabilities. Capitalizing on these expres-sions, the optimum exponent of GED can be numericallycomputed quite easily. Finally, some useful outcomes havebeen manifested including the case when the detrimental yetunavoidable effect of noise uncertainty is present.A PPENDIX
A. Derivation of the n th -moment function for hypothesis H We commence by decomposing w to the product of thesquared-root of a Gamma distributed RV and a CCS GaussianRV, which are mutually independent [12, Thm. 10], i.e., w = √ GX , where X d = CN (0 , σ w ) and f G ( g ) = v v Γ( v ) g v − exp( − vg ) , g ∈ R + . (A.1)Then, E [ | w | n ] = E [ G n/ ] E [ | X | n ] . Since | X | is Rayleighdistributed, it is straightforward to show that E [ | X | n ] = Γ (cid:16) n (cid:17) σ nw . (A.2)Further, it holds that E [ G n ] = v v Γ( v ) Z + ∞ g n/ v − exp( − vg ) dg = Γ (cid:0) n + v (cid:1) Γ ( v ) v n . (A.3)Combining (A.2) and (A.3), we arrive at (7). B. Derivation of the n th -moment function for hypothesis H Following the same lines of reasoning as in Appendix A,we get y = hs + w = hs |{z} , z + √ GX | {z } , z , (B.1) where G and X are defined in Appendix A. Conditioned on s and G and utilizing the linear properties of Gaussian RVs, itholds that z d = CN ( α, s σ h ) and z d = CN (0 , Gσ w ) . Recallthat h, s, G and X are all mutually independent RVs. Thereby,by introducing the auxiliary variable r , z + z , we have that r | s,G d = CN ( α, s σ h + Gσ w )= ( s σ h + Gσ w ) × CN ( α, | {z } , z . (B.2)Thus, the absolute moments of r are given by E [ | r | n ] = E (cid:2) ( s σ h + Gσ w ) n (cid:3) E [ | z | n ] . (B.3)It follows that | z | is Rice distributed with PDF [16, Eq.(2.17)] f | z | ( z ) =2(1 + α ) z exp( − (1 + α ) z − α ) × I (2 αz p α ) , z ≥ , (B.4)and, hence, with the aid of [17, Eq. (2.15.5.4)], we get E [ | z | n ] = Γ (cid:0) n + 1 (cid:1) (1 + α ) n/ F (cid:16) − n , − α (cid:17) . (B.5)Regarding the remaining factor of (B.3), it is required toaverage out the Gamma distributed G parameter. We firstutilize the following transformation [18, Eq. (8.4.2.5)] ( s σ h + Gσ w ) n = ( s σ h ) n Γ (cid:0) − n (cid:1) G , , (cid:20) σ w Gs σ h n + 10 (cid:21) , (B.6)which is valid for any n except those that are positiveeven numbers greater or equal to . This is due to the factthat, whenever n = 2 , , , . . . , the Gamma function in thedenominator of (B.6) returns singularity. Then, using (A.1),(B.6) and utilizing [18, Eq. (2.24.3.1)], it yields E (cid:2) ( s σ h + Gσ w ) n (cid:3) = ( s σ h ) n Γ( v )Γ (cid:0) − n (cid:1) × G , , (cid:20) σ w s σ h v − v, n + 10 (cid:21) . (B.7)Therefore, combining (B.5) and (B.7), we reach (12). For thealternative case where n ≥ is an even integer, while using the binomial expansion in the left-hand side of (B.6), we arriveat (13) after some straightforward manipulations.R EFERENCES[1] X. Foukas, G. Patounas, A. Elmokashfi, and M. K. Marina, “Networkslicing in 5G: Survey and challenges,”
IEEE Commun. Mag. , vol. 55,no. 5, pp. 94–100, May 2017.[2] L. Jiang, Y. Li, Y. Ye, Y. Chen, M. Jin, and H. Zhang, “Unilateralleft-tail Anderson Darling test-based spectrum sensing with Laplaciannoise,”
IET Commun. , vol. 13, no. 6, pp. 696–705, 2019.[3] Y. Ye, G. Lu, Y. Li, and M. Jin, “Unilateral right-tail Anderson-Darlingtest based spectrum sensing for cognitive radio,”
Electron. Lett. , vol. 53,no. 18, pp. 1256–1258, 2017.[4] H. Urkowitz, “Energy detection of unknown deterministic signals,”
Proc.IEEE , vol. 55, no. 4, pp. 523–531, May 1967.[5] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms forcognitive radio applications,”
IEEE Commun. Surveys Tuts. , vol. 11,no. 1, pp. 116–130, 2009.[6] F. Moghimi, A. Nasri, and R. Schober, “Adaptive L p − norm spectrumsensing for cognitive radio networks,” IEEE Trans. Commun. , vol. 59,no. 7, pp. 1934–1945, 2011.[7] X. Zhu, T. Wang, Y. Bao, F. Hu, and S. Li, “Signal detection ingeneralized Gaussian distribution noise with Nakagami fading channel,”
IEEE Access , vol. 7, pp. 23 120–23 126, 2019.[8] R. Gao, Z. Li, H. Li, and B. Ai, “Absolute value cumulating basedspectrum sensing with Laplacian noise in cognitive radio networks,”
Wireless Pers. Commun. , vol. 83, no. 2, pp. 1387–1404, Mar. 2015.[9] Y. Ye, Y. Li, G. Lu, and F. Zhou, “Improved energy detection withLaplacian noise in cognitive radio,”
IEEE Syst. J. , vol. 13, no. 1, pp.18–29, 2019.[10] V. Kostylev and I. Gres, “Characteristics of p -norm signal detection inGaussian mixture noise,” IEEE Trans. Veh. Technol. , vol. 67, no. 4, pp.2973–2981, Apr. 2018.[11] D. L. Mcleish, “A robust alternative to the normal distribution,”
Can. J.Statist. , vol. 10, no. 2, pp. 89–102, Jun. 1982.[12] F. Yilmaz, “McLeish distribution: Performance of digital communica-tions over additive white McLeish noise (AWMN) channels,”
IEEEAccess , vol. 8, pp. 19 133–19 195, 2020.[13] I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series, andProducts . Academic Press, 2007.[14] R. Tandra and A. Sahai, “SNR walls for signal detection,”
IEEE J. Sel.Topics Signal Process. , vol. 2, no. 1, pp. 4–17, Feb. 2008.[15] R. Senanayake, P. J. Smith, P. A. Dmochowski, A. Giorgetti, and J. S.Evans, “Mixture detectors for improved spectrum sensing,”
IEEE Trans.Wireless Commun. , to be published, 2020.[16] M. K. Simon,
Probability Distributions Involving Gaussian RandomVariables . Springer, 2006.[17] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,
Integrals andSeries. Vol. 2: Special Functions . Taylor & Francis Ltd, 1998.[18] ——,