Performance of $p$-Norm Detector in AWGN, Fading, and Diversity Reception

Performance analysis of the p-norm detector to date has been limited to ad hoc approximations, nonfading channels, and Rayleigh fading. To overcome these limitations, we develop several analytical/numerical solutions for detection probability Pd and false alarm probability Pf, which are necessary to specify the receiver operating characteristic (ROC) curves of the p-norm detector. First, for nonfading channels (additive white Gaussian noise (AWGN) only), the moment-generating function (mgf) of the decision variable is derived in two forms: 1) closed form for even integer p and 2) series form for arbitrary p. To evaluate Pd and Pf, a numerical method utilizing the Talbot inversion is developed for case 1, and an infinite series expansion with convergence acceleration based on the e-algorithm is derived for case 2. As an alternative to mgf-based analysis, a Laguerre polynomial series is also used to derive new Pd and Pf approximations. Second, series-form mgf-based Pd expressions are derived for κ-μ and α-μ fading channels. Third, for antenna diversity reception, new p-law combining (pLC) and p-law selection (pLS) schemes are proposed. The performance of these combiners with the p-norm detector is derived for Nakagami-m fading and is compared with that of the classical maximal ratio combining (MRC) and selection combining (SC). Interestingly, both pLC and pLS perform similarly to SC at low signal-to-noise ratio (SNR) but outperform it at relatively high SNR, with pLC performing closer to the optimal MRC. Numerical results are presented to verify the derived results and to provide further insights.

[1]  D. B. D. Costa,et al.  Highly Accurate Closed-Form Approximations to the Sum of α-μVariates and Applications , 2008, IEEE Transactions on Wireless Communications.

[2]  Anant Sahai,et al.  SNR Walls for Signal Detection , 2008, IEEE Journal of Selected Topics in Signal Processing.

[3]  R. M. A. P. Rajatheva,et al.  Energy Detection of Unknown Signals in Fading and Diversity Reception , 2011, IEEE Transactions on Communications.

[4]  H. Urkowitz.  Energy detection of unknown deterministic signals , 1967 .

[5]  M.D. Yacoub,et al.  The κ-μ distribution and the η-μ distribution , 2007, IEEE Antennas and Propagation Magazine.

[6]  Mohamed-Slim Alouini,et al.  Coded Communication over Fading Channels , 2005 .

[7]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[8]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[9]  Hai Jiang,et al.  A Mixture Gamma Distribution to Model the SNR of Wireless Channels , 2011, IEEE Transactions on Wireless Communications.

[10]  Johan Fereira,et al.  A gamma type distribution involving a confluent hypergeometric function of the second kind , 2010 .

[11]  Mohamed-Slim Alouini,et al.  On the Energy Detection of Unknown Signals Over Fading Channels , 2007, IEEE Transactions on Communications.

[12]  Mustafa M. Matalgah,et al.  Moment generating function of the generalized α - μ distribution with applications , 2009, IEEE Communications Letters.

[13]  E Hille.  On Laguerre's Series: Second Note. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[14]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .

[15]  Dennis Goeckel,et al.  Convergence of the Complex Envelope of Bandlimited OFDM Signals , 2010, IEEE Transactions on Information Theory.

[16]  Ranjan K. Mallik,et al.  Optimization of Cooperative Spectrum Sensing with an Improved Energy Detector over Imperfect Reporting Channels , 2011, 2011 IEEE Vehicular Technology Conference (VTC Fall).

[17]  Daniel Benevides da Costa,et al.  Accurate Approximations to the Sum of Generalized Random Variables and Applications in the Performance Analysis of Diversity Systems , 2009, IEEE Transactions on Communications.

[18]  Oluwatobi Olabiyi,et al.  Unified analysis of energy detection of unknown signals over generalized fading channels , 2011, 2011 7th International Wireless Communications and Mobile Computing Conference.

[19]  E Hille.  On Laguerre's Series: Third Note. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Hai Jiang,et al.  Performance of an Energy Detector over Channels with Both Multipath Fading and Shadowing , 2010, IEEE Transactions on Wireless Communications.

[21]  A. Annamalai,et al.  Performance evaluation of generalized selection diversity systems over Nakagami-m fading channels , 2003, Wirel. Commun. Mob. Comput..

[22]  P. Wynn,et al.  On the Convergence and Stability of the Epsilon Algorithm , 1966 .

[23]  Erik Ostermann,et al.  Radar Sonar Signal Processing And Gaussian Signals In Noise , 2016 .

[24]  Umberto Mengali,et al.  UWB Energy Detection Receivers with Partial Channel Knowledge , 2006, 2006 IEEE International Conference on Communications.

[25]  Gordon L. Stuber,et al.  Principles of Mobile Communication , 1996 .

[26]  D. Cabric.  Addressing feasibility of cognitive radios , 2008, IEEE Signal Processing Magazine.

[27]  Ranjan K. Mallik,et al.  Cooperative Spectrum Sensing in Multiple Antenna Based Cognitive Radio Network Using an Improved Energy Detector , 2012, IEEE Communications Letters.

[28]  Hai Jiang,et al.  Analysis of area under the ROC curve of energy detection , 2010, IEEE Transactions on Wireless Communications.

[29]  Norman C. Beaulieu,et al.  Performance of digital linear modulations on Weibull slow-fading channels , 2004, IEEE Transactions on Communications.

[30]  E Hille,et al.  On Laguerre's Series: First Note. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Masoumeh Nasiri-Kenari,et al.  Multiple antenna spectrum sensing in cognitive radios , 2010, IEEE Transactions on Wireless Communications.

[32]  J. Abate,et al.  Multi‐precision Laplace transform inversion , 2004 .

[33]  Jeffrey H. Reed,et al.  An Efficient Method for Evaluating Information Outage Probability and Ergodic Capacity of OSTBC System , 2008, IEEE Communications Letters.

[34]  M.D. Yacoub,et al.  The $\alpha$-$\mu$ Distribution: A Physical Fading Model for the Stacy Distribution , 2007, IEEE Transactions on Vehicular Technology.

[35]  Khairi Ashour Hamdi,et al.  Efficiency of energy detection for spectrum sensing in a Poisson field of interferers , 2012, 2012 IEEE Wireless Communications and Networking Conference (WCNC).

[36]  Mohamed-Slim Alouini,et al.  Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis , 2000 .

[37]  R.W. Brodersen,et al.  Implementation issues in spectrum sensing for cognitive radios , 2004, Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004..

[38]  Harry L. Van Trees,et al.  Detection, Estimation, and Modulation Theory: Radar-Sonar Signal Processing and Gaussian Signals in Noise , 1992 .

[39]  David Middleton,et al.  On the detection of stochastic signals in additive normal noise-I , 1957, IRE Trans. Inf. Theory.

[40]  Danijela Cabric,et al.  Energy Detection Based Spectrum Sensing Over $\kappa{-}\mu$ and $\kappa{-}\mu$ Extreme Fading Channels , 2013, IEEE Transactions on Vehicular Technology.

[41]  Yunfei Chen,et al.  Improved energy detector for random signals in gaussian noise , 2010, IEEE Transactions on Wireless Communications.

[42]  Robert Schober,et al.  Adaptive L_p—Norm Spectrum Sensing for Cognitive Radio Networks , 2011, IEEE Transactions on Communications.

[43]  Zhongding Lei,et al.  IEEE 802.22: The first cognitive radio wireless regional area network standard , 2009, IEEE Communications Magazine.

[44]  Georgios B. Giannakis,et al.  Space-time coding and Kalman filtering for time-selective fading channels , 2002, IEEE Trans. Commun..

[45]  Vladimir I. Kostylev,et al.  Energy detection of a signal with random amplitude , 2002, 2002 IEEE International Conference on Communications. Conference Proceedings. ICC 2002 (Cat. No.02CH37333).

[46]  Roussos G. Dimitrakopoulos,et al.  Generalized Laguerre expansions of multivariate probability densities with moments , 2010, Comput. Math. Appl..