This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 1 Characteristics of p-norm Signal Detection in Gaussian Mixture Noise V. I. Kostylev, I. P. Gres Abstract—Performance of the p-norm detector has been analyzed recently in additive white Gaussian noise and several analytical solutions were obtained. We have developed these solutions for the case of non-Gaussian noise. Expressions for the false alarm probability PF and the detection probability PD were found in series-form for an arbitrary index of power p using an analysis based on the moment generating function (MGF). Seriesform expressions for the average detection probability PD in the non-Gaussian environment for Nakagami-m, k-µ and η-µ fading channels were derived as well. Recently proposed p-law selection (pLS) and p-law combining (pLC) schemes for antenna diversity reception were analyzed. The detection characteristics of a pnorm detector were compared with a traditional energy detector and with an optimal detector. All analytical results were verified with a statistic simulation. Index Terms—Detection probability, diversity reception, energy detector, non-Gaussian noise, p-norm detector. I. D INTRODUCTION of signals corrupted by noise is an important problem for wireless communication, radar and navigation. Often these signals have to be reliably detected without any channel state information. In this case the energy detector discussed in [1] is widely used in practice, e.g. for detection of signals with random amplitude [2]. Despite the fact that it was proposed many years ago it is still exploited in the modern communication systems such as cognitive radio [3], especially for spectrum sensing [4]. As a development of the energy detector an improved energy detector was proposed in the literature [5, 6]. A decision statistic of this detector is calculated by raising an absolute value of an input mixture of noise and a signal in a positive arbitrary power p, instead of a power two for a conventional energy detector. The signal and noise under consideration in [5] are Gaussian. Since the energy detector is optimal under such conditions, it is unlikely to find any detector working better than the energy detector. The improved energy detector is also known as a p-norm and a Lpnorm detector; we will refer to it further as p-norm. In [7] the same detector is analyzed and a variety of analytical and numerical approaches was developed. However, this analysis is limited to additive white Gaussian noise (AWGN), as well, but fading channels and diversity reception schemes are also considered. Often in practice a detector has to operate in various types ETECTION of non-Gaussian noises such as atmospheric, a man-made impulsive noise, and different kinds of interference. In this case the performance of the conventional energy detector degrades significantly. In [6] the central limit theorem approach (CLT) is used for analyzing the distribution of the pnorm detector decision statistic for the non-Gaussian environment. This approach is not always precise [7], moreover, the equations are derived for a small signal-noise ratio (SNR). Nevertheless, it was shown, that the p-norm detector has a vast performance gain comparing with the energy detector. Moreover, it is easy to show, that despite the energy detector being optimal in case of the Gaussian signal detection in Gaussian noise (for independent samples), it is no longer optimal, if the noise is non-Gaussian. In this paper, exploiting the MGF-based approach from [7] and the non-Gaussian noise model (Gaussian mixture noise, GMN) [6], we derived series-form equations for the false alarm probability PF and the detection probability PD. These detection characteristics were compared with the optimal and energy detectors’ characteristics; the quasioptimal value of the power index p (the value that provides the highest detection probability if the false alarm probability is fixed) is also found under given conditions. It is shown that the quasioptimal detector can be close to optimal, but has as simple a structure as the energy detector. Fading channels are usually evaluated with the average detection probability PD , averaging over the signal-noise ratio (SNR) distribution. For such cases seriesform expressions for the average detection probability PD are obtained for the Nakagami-m, k-µ and η-µ fading channels. The pLC and pLS diversity schemes proposed in [7] are also analyzed and corresponding series-form expressions for the average detection probability PD are derived. Statistic simulation is carried out in order to validate all presented expressions. The organization of this paper is as follows. Section II describes the decision statistics and the noise model. The false alarm and detection probabilities for Gaussian mixture noise are derived in Section III. The performance analysis for Nakagami-m, k-µ and η-µ channels is carried out in Section IV. The characteristics of pLC and pLS diversity schemes are found in Section V. The numerical results and presented figures are discussed in Section VI. Section VII concludes the paper and gives directions for our future work. 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 2 II. DECISION STATISTIC AND NOISE MODEL The signal detection problem can be considered as a binary hypothesis test. It is assumed that under hypothesis H0 a signal is absent and it is present under H1. An i-th sample of a detector input signal is given by (1) yi hi si xi , where λ is an a priori unknown parameter which is defined as 0, H 0 ; (2) 1, H1 , hi is a i-th sample of the fading gain, si and xi are i-th samples of the detecting signal and the noise, i = 1,..,n, n is the sample size. The samples of fading gain, signal and noise are independent and identically distributed (i.i.d). The signal samples are assumed to be samples of real Gaussian process with zero mean and variance s 2 . A detector is intended for a decision statistic calculation. The decision statistic of the p-norm detector can be written as p n y p i , (3) i 1 n where p is an arbitrary positive power and σn is the noise standard deviation. The energy detector is a special case of the p-norm detector when p = 2. The noise samples are assumed to be Gaussian mixture noise samples (GMN). The GMN model describes man-made impulsive and atmospheric interferences [8, 9]. Also, it can be used for the Middleton class A interference approximation [9]. The probability density function (PDF) of the GMN is given by V x2 bv g ( x) exp 2 , (4) v 1 2 v2 2 v where bv > 0 is the weight of the noise component, V b 1 v 1 v , v is a standard deviation of the noise component, V is a constant representing the number of the components in the GMN. The variance of the GMN is given by V n2 bv v2 . (5) v 1 The simplest GMN model consists of V = 2 components, it is also referred as ε-mixture noise [9]. The decision statistic of the optimal detector can be derived exploiting a maximum likelihood ratio and taking into account the signal and the noise models. It can be written as P( y1 , y2 ,..., yn | H1 ) op , (6) P( y1 , y2 ,..., yn | H 0 ) where P( y1 , y2 ,..., yn | H w ) is the joint probability density function (JPDF) of the signal-noise mixture under given hypothesis; w = 0 or 1, for H0 or H1 respectively. Since samples are i.i.d. this JPDF can be written as n P( y1 , y2 ,..., yn | H w ) g ( yi | H w ) , i 1 where g ( yi | H w ) is the PDF of the i-th sample. (7) Whereas the optimal detector provides the best performance in comparison with the other detectors, calculation of the decision statistic op requires high computational complexity. Moreover, the information about the noise and the signal distributions and parameters, which might be difficult to quantify and changes with time in practice, is required. The locally optimal (LO) detector [10] does not require any information about the signal distribution and signal energy, but it exploits noise parameters and has quite a complicated decision statistic. This motivates the usage of quasioptimal detectors that have a lower performance, but they do not need channel state information and have a low implementation complexity. Detectors compare the decision statistic ((3) or (6) for pnorm or optimal respectively) with a predefined decision threshold T. The decision rule is given by H1 T, (8) H0 where is the decision statistic of the optimal detector op or p-norm detector p . According to the Neyman-Pearson criterion the threshold is defined by the value of the fixed false alarm probability PF. III. PERFORMANCE ANALYSIS FOR NON-FADING CHANNEL The false alarm probability PF and the detection probability PD are common metrics which are widely used for the detectors performance evaluation. These probabilities can be written in the form (9) PF P{ T | H0 } 1 F (T | H 0 ) , PD P{ T | H1} 1 F (T | H1 ) , (10) where P{} is the conditional probability of an event under given hypothesis ( H 0 or H1 ) and F (T | H 0 ) , F (T | H1 ) are the cumulative distribution functions (CDF) of the decision statistic under hypotheses H0 and H1. In this section we consider a non-fading channel, it means that all hi are equal to 1 and the SNR is given by s2 . n2 (11) First, using (4), (6) and (7) we can finally write the decision statistic of the optimal detector as V bv xi2 exp 2 2 2 2 n v 1 2 ( v s ) 2( v s ) op . (12) V xi2 bv i 1 exp 2 v 1 2 v2 2 v Second, utilizing the MGF-based approach [7], the conditional MGF of the decision statistic of the p-norm detector can be written as n M p ( s | H w ) exp( s ( x / n ) p ) f ( x | H w )dx , 0 (13) 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 3 where f ( x | H w ) is the PDF of the absolute value of the input sample yi. The PDF of the absolute value of the input sample is given by V bv f ( x | H w ) 2 2 Dw,v v 1 x2 exp 2D , w, v (14) D0,v , H0 ; Dw,v (15) 2 2 D1,v s v , H1 . Substituting the PDF (14) into (13) and then integrating we obtain the conditional MGF of the decision statistic p as 2 v 2 (1)k (2k 1) / p V bv n2 k 1 M p (s | H w ) , (16) k 0.5 2k k ! s (2 k 1)/ p k 0 v 1 Dw, v where ( x) is the gamma function [11, 8.310] . Using [11, 0.314], the conditional MGF can be written in the series-form as n n 2 2 M p (s | H w ) where C0 a0 n , 1 ja0 Ck s 2k n p , (17) k 0 j (kn j k ) k 1 (18) (1) (2k 1) / p C j k p 2k k ! k b 2 k 1 v k n 0.5 , v 1 Dw, v V a0 Due to multipath propagation in real wireless channels hi 1 . In this section results from the previous section for the non-fading channels are extended to the case of random SNR whose distribution depends on the fading channel type. The instantaneous SNR is given by 2 i hi2 where Cj IV. QUASIOPTIMAL DETECTOR FOR FADING CHANNELS (1 / p) V bv n . p Dw, v v 1 (19) (20) Therefore, the CDF F p (T | H w ) can be obtained by using the inverse Laplace transform on the M p ( s | H w ) / s as n Ck T (2 k n)/ p 2 2 . (21) F p (T | H w ) k 0 (2k n) / p 1 Substituting (21) into (9) and (10) we obtain the series-form expressions for the false alarm probability PF and the detection probability PD as n Ck0T (2 k n )/ p 2 2 , PF 1 k 0 (2k n) / p 1 (22) n Ck1T (2 k n )/ p 2 2 . PD 1 k 0 (2k n) / p 1 (23) where C k0 and C k1 are defined by (18)-(20) with D0,v and D1,v respectively. The detection characteristics (22) and (23) can be approximated by the partial sum. For calculation Python and mpmath library were used. This library contributes to very precise calculation of the series coefficients. s2 . n2 (24) The instantaneous detection probability can be calculated using (23). The average detection probability PD is widely used as a performance metric of the detector in the fading channel. It can be calculated by averaging the conditional MGF over the SNR distribution. Exploiting (15) and (11) we rewrite (16) in the form 2 (1) k (2k 1) / p M p ( s, | H w ) 2k k !s(2k 1)/ p k 0 (25) n V bv . 2 2 k 0.5 v 1 ( n v ) Therefore, the average conditional MGF over the SNR distribution is given by 2 (1) k (2k 1) / p M p ( s | H w ) 2k k !s(2k 1)/ p k 0 (26) n V n2 k 1bv f ( )d , 2 2 k 0.5 v 1 0 ( v / n ) where f ( ) is the SNR distribution. Further we derive the average detection probabilities for different fading channels such as Nakagami-m, k-µ and η-µ [12]. It should be pointed out that the k-µ and η-µ distributions include Nakagami-m as a special case, but we analyze it separately because resulting expressions for this model has a lower complexity (the equations for the coefficients do not include an infinite sum) and can be calculated faster. A. Nakagami-m distribution The PDF of the Nakagami-m distribution is given by m f Nak ( ) 1 m m 1 , e ( m) m (27) where m and are distribution parameters. Substituting (27) into (26) and calculating the integral, we obtain 2 m m (1) k (2k 1) / p M p (s | H w ) 2k k ! s (2 k 1)/ p k 0 (28) n m k 0.5 V v2 m v2 bv 2 U m, m k 0.5, , n2 v 1 n 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 4 Fig. 1. Detection probability PD as a function of the power index p of the Fig. 2. Detection probability PD as a function of the SNR for different p-norm detector for GMN with V = 2, V = 3 components and for Gaussian noise, the SNR 3 and the various sample sizes n. values of the power index p. The same characteristics of the optimal and LO detectors are included for comparison. The noise parameters are following: V = 2, b1 0.9985 , b2 0.0015 , 22 / 12 100 ; the sample size n = 15. Fig. 3. Detection probability PD as a function of the sample size n for the 1- Fig. 4. Average detection probability PD vs. average SNR norm and energy detectors for the different values of the SNR. The noise parameters are following: V = 2, b1 0.9985 , b2 0.0015 , 22 / 12 100 . (curves at left) and the energy (curves at right) detectors in the different fading channels and noise environments, namely: k-µ with k = 1 and µ = 3, V = 3, n = 15; Nakagami with m = 1, V = 3, n = 15; η-µ with η = 0.99 and µ = 2.5, V = 2, n = 20. where U(a,b,z) is the confluent hypergeometric function of the second kind [13, 13.1.3]. The series-form expression for the MGF can be written as C0Nak (a0Nak )n , n 2 2 m M p (s | H w ) where mn C k 0 Nak k s 2k n p , (29) C jNak 1 ja0Nak bv v 1 V 2 v 2 n j k 1 for the 1-norm (30) (kn j k ) m k 0.5 (1) (2k 1) / p Nak C j k p 2k k ! k m v2 U m, m k 0.5, n2 , (31) 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 5 Fig. 5. Average detection probability PD vs. average SNR for the 1-norm (curves at left) and the energy (curves at right) detectors in the different fading channels, namely: Rayleigh, V = 3, n = 15; Rice with k = 1, V = 3, n = 15; Rice with k = 2, V = 3, n = 15. m 0.5 m v2 (1 / p ) V v2 bv 2 U m, m 0.5, (32) . p v 1 n n2 Therefore, the average detection probability after the inverse Laplace transform application is given by a0Nak PD 2 2 m 1 mn CkNak T p . k 0 (2k n) / p 1 (33) B. k-µ distribution The k-µ distribution describes small scale variations of a fading signal with line-of-sight conditions [12]. The k-µ distribution includes Nakagami-m and Rice distributions as special cases. The PDF of the k-µ distribution can be written as f k ( ) (1 k ) 1 1 k 2 1 1 (1 k ) 2 e k (1 k ) I 1 2 , (34) k 2 2 e k where I t ( x) is the modified Bessel function of the first kind and order t [13, 9.6], is the average SNR. Exploiting representation of I t ( x) [13, 9.6.10], substituting it into (34) and using (26), we obtain the average series-form MGF. Then, using the inverse Laplace transform we derive the average detection probability PDk as n 2 2 (1 k ) PDk 1 k e where C0k (a0k )n , n 2l n Clk T p , l 0 (2l n) / p 1 (35) (36) for the 1-norm b2 0.002 , b3 0.0015 , 32 / 12 100 , 22 / 12 50 , n = 15) for no diversity case, pLC and pLS with L = 2 branches. C kj 1 ja0k j (un j u ) u 1 (1)u (2u 1) / p p 2u u ! and the energy detectors in the Nakagami fading channel ( b1 0.9965 , 2k n n Nak Fig. 6. Average detection probability PD vs. average SNR lk i 0 k u C kj u (37) , ( 2 k (k 1))i ii! 2 bv v2 v 1 n V (1 k ) v2 U i, i l 0.5, n2 (1/ p) k a0k 0 . p l i 0.5 , (38) (39) C. η-µ distribution The η-µ distribution is another general fading model which is in contrast to the k-µ distribution describes small scale variations of a signal with non-line-of-sight conditions [12]. The η-µ distribution includes Hoyt (Nakagami-q) and Nakagami-m distributions as special cases. The PDF of the ηµ distribution can be written as 2 h 2 0.5 h 0.5 f ( ) e I 0.5 2 H , (40) 0.5 0.5 ( ) H where 2 1 , (41) h 4 1 . (42) H 4 By analogy with the derivation of the average detection probability for the k-µ fading channel PDk we can obtain 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 6 TABLE I DETECTION PROBABILITIES FOR DIFFERENT VALUES OF INDEX OF POWER AND FIXED SNR Index of power p Detection Probability PD 0.7 1 2 2.5 0.985 0.946 0.605 0.342 PD C0 Cj 2 h 1 2 2 2 ( ) (a0 )n , 1 ja0 n j n Cl p 2u u ! µ; and W , Ck for η-µ. These coefficients are given by T Ln 2l n p (43) W Nak l 0 2 2 m Lmn , Ln k 2 2 (1 k ) k e (44) W (45) W 2 2 u , (50) L n , (51) Ln Ln 2 h , 2 ( ) (52) C0Nak (a0Nak ) Ln , V ( H ) 2i bv i 0 i !( k 0.5) v 1 l (49) channel type: W Nak , CkNak for Nakagami; W k , Ckk for k- (2l n) / p 1 . j u (1)u (2u 1) / p PDpLC CkfcT p 1 W fc , k 0 (2k Ln) / p 1 where W fc and Ckfc are the coefficients for the given fading (un j u )C u 1 2 k Ln 2 v 2 n 2i 2 2 i l 0.5 C jNak h (2 2i)U 2 2i, 2 2i l 0.5, 2 , (1/ p) a0 0 . p (46) 2 v 2 n C kj V. DIVERSITY RECEPTION It is apparent that multipath propagation decreases the performance of a detector, but modern wireless communication systems require a reliable signal detection and high performance. Diversity-combining schemes facilitate to mitigation of the performance loss. Two recently proposed non-coherent diversity-combining techniques (pLS and pLC) showed better performance than the classical maximal ratio combining and selection combining [7]. So we analyze these schemes for the fading channels in GMN in order to assess the performance gain in comparison with the no diversity reception. A. p-Law Combining The p-law combining scheme consists of L branches each of them includes the p-norm detector. The outputs of the detectors then added together and form the pLC decision statistic p L n y i ,l , (48) pLC n l 1 i 1 where yi,l is an i-th sample form a l-th branch. The decision rule is still described by (8). pLC The false alarm probability PF can be found using expressions (18-20, 22) and replacing n by Ln into them. The pLC D average detection probability P scheme is given by for the pLC diversity ( Lkn j k ) k 1 (1) (2k 1) / p Nak C j k p 2k k ! k m k 0.5 m v2 bv U m, m k 0.5, , n2 v 1 C0k (a0k ) Ln , 2 v 2 n V (47) 1 ja0Nak (53) j 1 ja0k j ( Lun j u )C u 1 (1)u (2u 1) / p p 2u u ! k j u (55) uk (56) , C0 (a0 ) Ln , Cj 1 ja0 (57) j ( Lun j u )C j u u 1 (1)u (2u 1) / p p 2u u ! (54) u (58) , where a0Nak , uk , a0k , u , a0 are defined by (32), (38), (39), (46) and (47) respectively. B. p-Law Selection The p-law selection scheme also consists of L branches, but only the largest branch output is selected. The decision statistic of the pLS diversity scheme is given by . (59) For independent and identically distributed branches expressions for the false alarm PF pLS and the average pLS n y p n y i ,1 i ,2 , max i 1 n i 1 n p n y i,L ,..., i 1 n p pLS detection PD probabilities are given by PFpLS 1 (1 PF ) L , (60) PDpLS 1 (1 PD ) L , (61) where PF is given by (22) and PD is the average detection probability for the Nakagami-m, k-µ or η-µ fading channels 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 (equations (33), (35) and (43) respectively). VI. NUMERICAL RESULTS AND DISCUSSION In this section, the analysis of the derived expressions is done. For all the presented curves we set the false alarm probability PF = 0.01 and found a corresponding detection threshold. For the false alarm probability PF, the detection probability PD and the average detection probability PD calculations Python and mpath library were used. Since expressions for these probabilities are written in series-form, it is required to truncate the infinite sums. We carried out a sufficient number of numerical experiments and found out that these infinite sums are readily converging and can be assessed by the partial sums. Furthermore, all results were proven by the statistic simulation with 106 iterations. A. Non-fading channel performance In Fig. 1, the detection probability PD vs. the index of power p curves are presented. The curves are plotted using (22) and (23) for the following parameters: GMN with V = 2 components ( b1 0.9985 , b2 0.0015 , 22 / 12 100 ), the sample sizes n = 8, 10, 15; and with V = 3 components ( 32 / 12 50 , b1 0.9965 , b3 0.0015 b2 0.002 , 22 / 12 30 ), the sample size n = 15. Relying on the achieved results it can be said that a quasioptimal value of pqo (value providing the maximal detection probability PD with the fixed false alarm probability PF) depends on the number of the GMN components, the sample size and the noise parameters. It has been found that the value pqo is contained in [0.5, 2]. The quasioptimal index of power pqo is close to 2 if the noise is close to Gaussian (the noise pulses are weak or absent), otherwise if the noise has powerful pulses it tends to be close to 0.5. For the presented curves (Fig. 1, 1-5) the indices of power are pqo1 = 1.7, pqo2 = 1.2, pqo3 = 0.7, pqo4 = 0.7, pqo5 = 2. In order to evaluate the performance of the quasioptimal detector in comparison with the p-norm, optimal and energy detectors, the detection probability PD vs. the SNR curves are plotted in Fig. 2. These results (curves 1-5) are obtained for GMN with the V = 2 components ( b1 0.9985 , b2 0.0015 , 22 / 12 100 ) and the sample size n = 15. The curve 1 in Fig. 2 corresponds to the optimal detector and establishes an upper bound for the performance, there is no a detector which operates better. The quasioptimal p-norm detector (curve 2, p = 0.7) has lower performance, but its detection probability is sufficiently higher in comparison with the energy detector (curve 4, p = 2), for the SNR 4 the detection probabilities are given by table I. As is shown by Fig. 2, the detection probabilities decrease if the index of power grows, e.g. the detection probability of the 2.5-norm detector is lower than detection probability of the energy detector. As it was said in Section II the GMN model describes impulse noise, which can be considered as a train of randomamplitude and randomly occurring narrow pulses in a 7 background of Gaussian noise. The amplitude of a pulse raised in the power p > 1 increases if it more than 1. In other words, the energy detector (the index of power p = 2) increases the amplitude of powerful pulses, but the 1-norm detector does not. Therefore, the detection threshold T grows rapidly for the p-norm detectors with p > 1. High detection threshold limits the detection probability. It means that the p-norm detectors with p < 1 are more resistant to the powerful noise pulses as it was shown by Fig. 1-2. It is noteworthy that the 1-norm detector (Fig. 2, curve 3) has almost equal performance with the 0.7-norm detector (Fig. 2, curve 2), but has a less complex structure. Its decision statistic has the lowest complexity (no power operation required), so it can be easily implement in practice. Another argument in favor of the 1-norm detector is a slight decrease in the detection probability when p growing for the curve 5 in Fig. 1, which is plotted for Gaussian noise and has maximum at p = 2. Noise parameters tend to change in time, so let us assume, that during the first period of time the noise is close to Gaussian and during the second period is non-Gaussian. The detection probability of the 1-norm detector during the first period will be slightly lower than the energy detector, but it will be considerably higher during the second period. Thus, the 1-norm detector can be a kind of a versatile solution for the GMN environment with parameters that change in time. In Fig 3, the detection probability PD vs. the sample size n dependences are presented for the 1-norm and the energy detectors (separate dots are connected to form curves for illustration purposes). The curves are plotted using (22) and (23) for the following parameters: GMN with V = 2 components ( b1 0.9985 , b2 0.0015 , 22 / 12 100 ) and the SNR 3 and 6 . As is shown by Fig 3, the detection probabilities PD of the 1-norm detector increase if the sample size n grows (curves 3 and 4 for SNR 3 and 6 correspondingly). Concerning the energy detector, as it can be seen from Fig. 3 the curves 1 and 2 have a peak at n = 10 and further growth of the sample size does not lead to the detection performance increase for the given noise parameters. The growth of the sample size n increases the probability to find the noise pulse in the sample. It means that for the sample size n > 10 and given noise parameters the detection threshold of the energy detector starts to grow rapidly, that is why the detection probability decreases in spite of the sample size growth. The PD vs. n curves start to grow again after their minimum. This happens because a contribution of one pulse energy decreases with the sample size growth. B. Fading channels performance The performances of the 1-norm detector and the energy detector are compared in the different fading channels exploiting expressions (33), (35) and (43). The curves presented in Fig. 4 and Fig 5 (the 1-norm detector curves at left and the energy detector curves at right) characterize the performance with the average detection probability PD vs. the average SNR for the two noise environments (1. V = 2 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 8 TABLE II AVERAGE DETECTION PROBABILITIES FOR DIFFERENT FADING CHANNEL TYPES AND TWO VALUES OF POWER INDEX Nakagami Channel k-µ η-µ Power index k = 1, µ = 3 η = 0.99, µ = 2.5 1 0.848 0.896 0.97 2 0.462 0.488 0.443 TABLE IV AVERAGE DETECTION PROBABILITIES FOR D IFFERENT DIVERSITY SCHEMES AND TWO VALUES OF POWER INDEX No diversity Scheme pLC pLS Power index 1 0.874 0.989 0.984 2 0.539 0.645 0.787 TABLE III AVERAGE DETECTION PROBABILITIES FOR DIFFERENT FADING CHANNEL TYPES AND TWO VALUES OF POWER INDEX Rayleigh Rice Channel Rice Power index k = 1, µ = 1 k = 2, µ = 1 1 0.848 0.867 0.884 2 0.462 0.473 0.482 channels are found. Obtained results reveal that the p-norm detector having almost as simple structure as the energy detector has characteristics which are very close to the optimal detector ones and are significantly higher than the energy detector characteristics. Exploiting the GMN model, the components b1 0.9985 , b2 0.0015 , 22 / 12 100 ; 2. V = 3, b1 0.9965 , b2 0.002 , b3 0.0015 32 / 12 50 , 22 / 12 30 ). The following parameters of the fading channels are set for Fig. 4: 1. k-µ distribution, k = 1, µ = 3 (solid line); 2. Nakagami distribution m = 1 (coincides with Rayleigh distribution, dashed line), 3. η-µ distribution, η = 0.99, µ = 2.5 (dotted line) and for Fig. 5: 1. Rayleigh distribution (solid line), 2. Rice distribution (a special case of the k-µ distribution when µ = 1), k = 1 (dotted line), 3. Rice distribution, k = 2 (dashed line). As is illustrated by the figures the 1-norm detector has higher average detection probability than the energy detector for the all considered fading conditions. For the average SNR 4 the average detection probabilities are given by table II and III for the given noise and channel parameters. C. Diversity-combining schemes performance Two diversity schemes are compared using the average detection probability PD vs. the average SNR curves plotted in Fig. 6. The curves were obtained for the 1-norm and energy detectors for the following noise parameters: V = 3, 32 / 12 100 , b1 0.9965 , b3 0.0015 b2 0.002 , / 50 , for the Nakagami fading channel with m = 1 using (49), (50), (54), (61), and (33). In this example, the impulse components of the noise are more powerful than in the previous examples, that is why the average detection probabilities for the no diversity case is quite low. In order to improve the performance, the pLC and pLS diversity schemes with L = 2 branches were analyzed. The results of the calculations for the average SNR 8 are given by table IV. So, it is shown for the given noise parameters that, firstly, the 1-norm detector provides the higher average detection probability than the energy detector and, secondly, the pLC scheme yields slightly better performance than pLS for the 1norm detector, but the pLS scheme is better than the pLC for the energy detector. 2 2 average detection probability PDk for Nakagami-m, k-µ, η-µ fading channels and antenna diversity schemes (pLS and pLC) were found for an arbitrary index of power p in order to assess the detection performance in more realistic environment. It has been found that the p-norm detector with p ≤ 1 has significant performance gain comparing with the energy detector for the considered fading channels and diversity schemes. The p-norm detector with p > 1 in GMN rapidly increases the amplitudes of the powerful noise pulses and, therefore, has an extremely high detection threshold. This leads to low detection probability in non-fading channels and low average detection probability in fading conditions regardless of the fading channel type. The p-norm detector with p ≤ 1 appears to be more resistant to powerful noise pulses and provides higher performance in the GMN environment. Future work directions may include investigations of the pnorm detector performance for non-Gaussian models which are different from the GMN model. REFERENCES [1] [2] [3] 2 1 [4] [5] [6] [7] [8] [9] [10] VII. CONCLUSION [11] In this paper, the performance analysis of the p-norm signal detection is carried out for non-Gaussian noise described by the GMN model. Analytical expressions for the false alarm probability PF and the detection probability PD for non-fading [12] [13] H. Urkowitz, “Energy Detection of Unknown Deterministic Signals,” Proc. IEEE, vol. 55, pp. 523–531, April 1967. V. Kostylev, “Energy detection of a signal with random amplitude,” in Proc. IEEE Int. Conf. Commun. (ICC), 2002, pp. 1606–1610. S. Haykin, “Cognitive Radio: Brain-Empowered Wireless Communications,” IEEE J. Select. Areas Commun., vol. 23, pp. 201– 220, Feb. 2005. S. Atapattu, C. Tellambura, and H. Jiang, Energy Detection for Spectrum Sensing in Cognitive Radio. Springer, New York, 2014. Y. Chen, “Improved energy detector for random signals in Gaussian noise,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 558–563, Feb. 2010. F. Moghimi, A. Nasri, and R. Schober, “Adaptive Lp-norm spectrum sensing for cognitive radio networks,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1934–1945, July 2011. V. Sharma Banjade, C. Tellambura, and H. Jiang, “Performance of pnorm detector in AWGN, fading, and diversity reception,” IEEE Trans. Veh. Technol., vol. 63, no. 7, pp. 3209–3222, Sep. 2014. D. Middleton, “Statistical-physical Models of Man–made Radio Noise – Parts I and II,” U.S. Dept. Commerce Ofﬁce Telecommun., Apr. 1974 and 1976. S. A. Kassam, Signal Detection in Non-Gaussian Noise. Springer. New York, 1988. I. Song, J. Bae, S. Y. Kim, Advanced Theory of Signal Detection: Weak Signal Detection in Generalized Observations, Berlin, Heidelberg, New York:Springer-Verlag, 2002. I. S. Gradshteyn and I. Ryzhik, Tables of Integrals, Series and Products, 7th ed. London: Academic, 2007. M. D. Yacoub, “The κ−µ distribution and the η−µ distribution,” IEEE Antennas Propag. Mag., vol. 49, no. 1, pp. 68–81, Feb. 2007 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards, Nov. 1970. 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE Transactions on Vehicular Technology VT-2017-00673.R1 Professor Vladimir Kostylev graduated from Voronezh State University (VSU), Voronezh, Soviet Union, in 1981, with an honor MSc in Radiophysics. His academic career continued at VSU with the awarding of a Ph.D. in 1985, Doctor of Science in 2002 and Professor in 2007. Now he is a professor of the Electronics department of VSU. Professor Kostylev is a co-author of the book “Bistatic Radar: Principles and Practice”, Wiley, 2007. His research interests include cognitive radio, signal processing and radar theory. 9 Ivan Gres received the Bachelor of Radiophysics Degree in Computer Electronics from the Voronezh State University, Voronezh, Russian Federation in 2012 and Master of Radiophysics Degree in Information Systems from the Voronezh State University, Voronezh, Russian Federation in 2014. He is currently working towards the Ph.D. degree in System Analysis and Information Processing at the Voronezh State University. His research interests include wireless signal detection and digital signal processing. 0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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