Cognitive Radio Oriented Wireless Networks 2015

  Mark Weichold Mounir Hamdi Muhammad Zeeshan Shakir Mohamed Abdallah George K. Karagiannidis Muhammad Ismail (Eds.)

  Cognitive Radio Oriented Wireless Networks 10th International Conference, CROWNCOM 2015 Doha, Qatar, April 21–23, 2015 Revised Selected Papers

  156 Lecture Notes of the Institute for Computer Sciences, Social Informatics

and Telecommunications Engineering 156

Editorial Board

  Ozgur Akan Middle East Technical University, Ankara, Turkey

  Paolo Bellavista University of Bologna, Bologna, Italy

  Jiannong Cao Hong Kong Polytechnic University, Hong Kong, Hong Kong

  Falko Dressler University of Erlangen, Erlangen, Germany

  Domenico Ferrari Università Cattolica Piacenza, Piacenza, Italy

  Mario Gerla UCLA, Los Angels, USA

  Hisashi Kobayashi Princeton University, Princeton, USA

  Sergio Palazzo University of Catania, Catania, Italy

  Sartaj Sahni University of Florida, Florida, USA

  Xuemin (Sherman) Shen University of Waterloo, Waterloo, Canada

  Mircea Stan University of Virginia, Charlottesville, USA

  Jia Xiaohua City University of Hong Kong, Kowloon, Hong Kong

  Albert Zomaya University of Sydney, Sydney, Australia

  Geoffrey Coulson Lancaster University, Lancaster, UK More information about this series at

• ^• Mark Weichold Mounir Hamdi ^• Muhammad Zeeshan Shakir Mohamed Abdallah

  ^•

  George K. Karagiannidis Muhammad Ismail (Eds.)

  Cognitive Radio Oriented Wireless Networks

10th International Conference, CROWNCOM 2015 Doha, Qatar, April 21–23, 2015 Revised Selected Papers Editors Mark Weichold Mohamed Abdallah Texas A&M University at Qatar Texas A&M University at Qatar Doha Doha Qatar Qatar Mounir Hamdi George K. Karagiannidis Hamad Bin Khalifa University Aristotle University of Thessaloniki Doha Greece and Khalifa University Qatar United Arab Emirates Muhammad Zeeshan Shakir Muhammad Ismail Texas A&M University of Qatar Texas A&M University at Qatar Doha Doha Qatar Qatar

ISSN 1867-8211

  ISSN 1867-822X (electronic) Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

ISBN 978-3-319-24539-3

  ISBN 978-3-319-24540-9 (eBook) DOI 10.1007/978-3-319-24540-9 Library of Congress Control Number: 2015950861 Springer Cham Heidelberg New York Dordrecht London

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the

material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,

broadcasting, reproduction on microfilms or in any other physical way, and transmission or information

storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now

known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication

does not imply, even in the absence of a specific statement, that such names are exempt from the relevant

protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are

believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors

give a warranty, express or implied, with respect to the material contained herein or for any errors or

omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media

  

CROWNCOM 2015

Preface

  2015 marks the 10th anniversary of the International Conference on Cognitive Radio-Oriented Wireless Networks (Crowncom). Crowncom 2015 was jointly hosted by Texas A&M University at Qatar and Hamad Bin Khalifa University in Doha, Qatar, April 21–23, 2015. The event was a special occasion to look back at the contribution of Crowncom toward the advancements of cognitive radio technology since its inaugural conference in 2006 in Mykonos, Greece, as well as to look forward to the decades ahead, the ways that cognitive radio technology would like to evolve, and the ways its emerging applications and services can ensure everyone is connected everywhere.

  Evolution of cognitive radio technology pertaining to 5G networks was the theme of the 2015 edition of Crowncom. The technical program of Crowncom 2015 was structured to bring academic and industrial researchers together to identify and discuss recent developments, highlight the challenging gaps, and forecast the future trends of cognitive radio technology toward its integration with the 5G network deployment. One of the key topics of the conference was cognition and self-organization in the future networks, which are now widely considered as a striking solution to cope with the future ever-increasing spectra demands. Going beyond the theoretical development and investigation, further practical advances and standardization developments in this technology could provide potential dynamic solutions to cellular traffic congestion problems by exploiting new and underutilized spectral resources. One of the chal- lenging issues that Crowncom 2015 brought forward was to facilitate the heteroge- neous demands of users in heterogeneous-type environments — particularly in the 5G network paradigm, where the networks are anticipated to incorporate the provision of high-quality services to users with extremely low delays and consider these require- ments without explicit demand from users. Machine-type communications and Internet of Everything are now representing emerging use cases of such ubiquitous connectivity over limited spectra.

  Crowncom 2015 strongly advocated that the research community, practitioners, standardization bodies, and developers should collaborate on their research efforts to further align the development initiatives toward the evolution of emerging highly dynamic spectrum access frameworks. The biggest challenge is to design unified cross- layer new network architectures for successful aggregation of licensed and unlicensed spectra, addressing the spectrum scarcity problem for ubiquitous connectivity and preparing the ground for “The Age of the ZetaByte.”

  Crowncom 2015 received a large number of submissions, and it was a challenging task to select the best and most relevant meritorious papers to reflect the theme of the 2015 edition of Crowncom. All submissions received high-quality reviews from the Technical Program Committee (TPC) members/reviewers and eventually 66 technical

VI CROWNCOM 2015

  conference. The technical program of Crowncom 2015 is the result of the tireless efforts of 14 track chairs, and more than 200 TPC members and reviewers. We are grateful to the track chairs for handling the paper review process and their outstanding efforts, and to the reviewers/TPC for their high-quality evaluations. We offer our sincere gratitude to the Advisory Committee, local Organizing Committee (especially colleagues at Texas A&M University at Qatar), and the Steering Committee members for their insightful guidance. We would like to acknowledge the invaluable support from European Alliance for Innovation and the Qatar National Research Fund for the success of Crowncom 2015. 2015

  Mark Weichold Mounir Hamdi

  Muhammad Zeeshan Shakir Mohamed Abdallah

  George K. Karagiannidis Muhammad Ismail Organization General Chair

  Mark Weichold Texas A&M University at Qatar, Qatar Mounir Hamdi Hamad Bin Khalifa University, Qatar

  Technical Program Chair

  Muhammad Zeeshan Shakir Texas A&M University at Qatar, Qatar Mohamed Abdallah Texas A&M University at Qatar, Qatar George K. Karagiannidis Aristotle University of Thessaloniki, Greece, and Khalifa University, UAE

  Advisory Board

  Athanasios V. Vasilakos Kuwait University, Kuwait Khalid A. Qaraqe Texas A&M University at Qatar, Qatar Jinsong Wu Bell Labs, China David Grace University of York, UK Naofal Al-Dhahir University of Texas, Dallas, USA Kaushik Chowdhury Northeastern University, USA

  Special Session Chair

  Alhussein Abouzeid Rensselaer Polytechnic Institute, USA

  Panel Chair

  Maziar Nekovee Samsung, UK

  Publication Chair

  Muhammad Ismail Texas A&M University at Qatar, Qatar

  Tutorial Chair

  Nile University, Egypt Mohamed Nafie

  Exhibitions and Demos Chair VIII Organization Web Chair

  İslam Şafak Bayram Qatar Environment and Energy Research Institute, Qatar

  Local Arrangements

  Carol Nader Texas A&M University at Qatar, Qatar Mohamed Kashef Texas A&M University at Qatar, Qatar

  Track Chairs

  Track 1: Dynamic Spectrum Access/Management Mohammad Shaqfeh Texas A&M University at Qatar, Qatar Track 2: Networking Protocols for CR Tamer Khattab Qatar University, Qatar Amr Mohamed Qatar University, Qatar Track 3: Modeling and Theory Zouheir Rezki King Abdullah University of Science and Technology,

  Saudi Arabia Syed Ali Raza Zaidi University of Leeds, UK Track 4: HW Architecture and Implementations Ahmed El-Tawil University of California, Irvine, USA Fadi Kurdahi University of California, Irvine, USA Track 5: Next Generation of Cognitive Networks Muhammad Ali Imran CCSR/5G Innovation Centre University of Surrey, UK Richard Demo Souza Federal University of Technology - Paraná (UTFPR),

  Curitiba - PR - Brazil Track 6: Standards and Business Models Stanislav Fillin National Institute of Information and Communications

  Technology (NICT), Japan Stephen J. Shellhammer Qualcomm Technologies, Inc., USA Markus Dominik Mueck

  INTEL Mobile Communications, Germany Track 7: Emerging Applications for Cognitive Networks Octavia A. Dobre Memorial University, Canada Hai Lin Osaka Prefecture University, Japan

  

Contents

  Dynamic Spectrum Access/Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Hadi Hashemi, Sina Mohammadi Fard, Abbas Taherpour, and Tamer Khattab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Lokman Sboui, Hakim Ghazzai, Zouheir Rezki, and Mohamed-Slim Alouini . . .

   Mai Abdel-Malek, Karim Seddik, Tamer ElBatt, and Yahya Mohasseb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Wei Zhou, Tao Jing, Yan Huo, Jin Qian, and Zhen Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Hussein Seleem, Abdullhameed Alsanie, and Ahmed Iyanda Sulyman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Hadi Hashemi, Sina Mohammadi Fard, Abbas Taherpour, Saeid Sedighi, and Tamer Khattab

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Sina Mohammadi Fard, Hadi Hashemi, Abbas Taherpour, and Tamer Khattab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

   Hesham M. Elmaghraby, Dongrun Qin, and Zhi Ding X Contents

  Networking Protocols for CR

  Fariba Mohammadyan, Zahra Pourgharehkhan, Abbas Taherpour, and Tamer Khattab

  Mohamed Salman, Amr El-Keyi, Mohammed Nafie, and Mazen Hasna

  Anas M. Salhab, Fawaz Al-Qahtani, Salam A. Zummo, and Hussein Alnuweiri

  Anas M. Salhab

  Benjamin Drozdenko, Ramanathan Subramanian, Kaushik Chowdhury, and Miriam Leeser

  Fan Zhou, Abdulla Al Ali, and Kaushik Chowdhury

  Yuanyuan Yao, Xiaoshi Song, Changchuan Yin, and Sai Huang

  Meimei Duan, Zhimin Zeng, Caili Guo, and Fangfang Liu

  Amr Y. Elnakeeb, Hany M. Elsayed, and Mohamed M. Khairy Modeling and Theory

  Monika Jain, Vaibhav Kumar, Ranjan Gangopadhyay,

  Contents

  XI

  

  Olusegun P. Awe, Syed M. Naqvi, and Sangarapillai Lambotharan

  Sanjeev Gurugopinath, Rangarao Muralishankar, and H.N. Shankar

  Khaled M. Humadi, Ahmed Iyanda Sulyman, and Abdulhameed Alsanie

  Xue Liu, Guido H. Bruck, and Peter Jung

  Ahmed M. ElShaarany, Mohamed M. Abdallah, Salama Ikki, Mohamed M. Khairy, and Khalid Qaraqe

  

  Yi Ren, Chao Wang, Dong Liu, Fuqiang Liu, and Erwu Liu

  Ankit Kaushik, Shree Krishna Sharma, Symeon Chatzinotas, Björn Ottersten, and Friedrich Jondral

  

  Shree Krishna Sharma, Symeon Chatzinotas, and Björn Ottersten

  Faissal El Bouanani and Hussain Ben-Azza

  Anestis Tsakmalis, Symeon Chatzinotas, and Björn Ottersten

  Maha Alodeh, Symeon Chatzinotas, and Björn Ottersten XII Contents

  

  Islam A. Abdul Maksoud and Sherif I. Rabia HW Architecture and Implementations

  Ahmed Elsokary, Peter Lohmiller, Václav Valenta, and Hermann Schumacher

  Takeshi Matsumura, Kazuo Ibuka, Kentaro Ishizu, Homare Murakami, Fumihide Kojima, Hiroyuki Yano, and Hiroshi Harada

   . . . . . . . . . . . . . . . . . . . . . . . . . . . Yao-Chia Chan, Ding-Bing Lin, and Chun-Ting Chou

   Shanker Shreejith, Bhaskar Banarjee, Kizheppatt Vipin, and Suhaib A. Fahmy

  Next Generation of Cognitive Networks

  Mostafa Mohammadkarimi, Ebrahim Karami, and Octavia A. Dobre

  Audri Biswas, Sam Reisenfeld, Mark Hedley, Zhuo Chen, and Peng Cheng

  Sasa Maric and Sam Reisenfeld

  Dalia Abouelmaati, Arsalan Saeed, Oluwakayode Onireti, Muhammad Ali Imran, and Kamran Arshad

  

  Eva Lagunas, Shree Krishna Sharma, Sina Maleki, Symeon Chatzinotas, Joel Grotz, Jens Krause, and Björn Ottersten

  Contents

  XIII

  

  Ahmed Zubair, Zaid Bin Tariq, Ijaz Haider Naqvi, and Momin Uppal

  S.D. Barnes, S. Joshi, B.T. Maharaj, and A.S. Alfa

  Theodoros Spathopoulos, Oluwakayode Onireti, Ammar H. Khan, Muhammad A. Imran, and Kamran Arshad

  

  Mohammed El-Absi, Ali Ali, Mohamed El-Hadidy, and Thomas Kaiser Standards and Business Models

   Tim Esemann and Horst Hellbrück

   Takeshi Matsumura, Kazuo Ibuka, Kentaro Ishizu, Homare Murakami, Fumihide Kojima, Hiroyuki Yano, and Hiroshi Harada

  

  Seppo Yrjölä, Petri Ahokangas, Jarkko Paavola, and Pekka Talmola

  Sai Huang, Yajian Huang, Hao Zhou, Zhiyong Feng, Yifan Zhang, and Ping Zhang

  Anas Abognah and Otman Basir Emerging Applications for Cognitive Networks

  Ismail AlQerm and Basem Shihada

  Tian-Qing Wu and Hong-Chuan Yang XIV Contents

  

  Bassem Khalfi, Mahdi Ben Ghorbel, Bechir Hamdaoui, and Mohsen Guizani

  Hezerul Abdul Karim, Hafizal Mohamad, Nordin Ramli, and Aduwati Sali

  Anas Chaaban and Aydin Sezgin

  Muhammad Mahtab Alam and Elyes Ben Hamida Workshop Cognitive Radio for 5G Networks

  Fotis Foukalas and Tamer Khattab

  Rana Abbas, Mahyar Shirvanimoghaddam, Yonghui Li, and Branka Vucetic

  Konstantinos Chatzikokolakis, Alexandros Kaloxylos, Panagiotis Spapis, Nancy Alonistioti, Chan Zhou, Josef Eichinger, and Ömer Bulakci

  

  Shahriar Etemadi Tajbakhsh, Tapabrata Ray, and Mark C. Reed

  Ville Syrjälä and Mikko Valkama

  Muhammad Mahtab Alam, Dhafer Ben Arbia, and Elyes Ben Hamida

  Ammar Kabbani, Ali Ramadan Ali, Hanwen Cao, Asim Burak Güven, Yuan Gao, Sundar Peethala, and Thomas Kaiser

  

  Nafees Mansoor, A.K.M. Muzahidul Islam, Mahdi Zareei, Sabariah Baharun, and Shozo Komaki

   Ahmed H. Anwar, Karim G. Seddik, Tamer ElBatt, and Ahmed H. Zahran

   Neha Jain, Shubha Sharma, Ankush Vashistha, Vivek Ashok Bohara, and Naveen Gupta

   Mustafa Alshawaqfeh, Xu Wang, Ali Rıza Ekti, Muhammad Zeeshan Shakir, Khalid Qaraqe, and Erchin Serpedin

  Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Contents

  XV Dynamic Spectrum Access/Management

  

Fractional Low Order Cyclostationary-Based

Spectrum Sensing in Cognitive Radio Networks

  Keywords: Cognitive radio ·

  Among those, cyclostationary-based detector is one of the best way of spec- trum sensing in terms of performance and robustness against environmental parameters like ambient noise. In the context of cyclostationary-based spectrum

  tion, eigenvalues-based detection, detection based on the covariance matrix and cyclostationary based detection.

   ]. The most important methods are matched filter, energy detec-

  Increasing need for bandwidth in telecommunication and limited environmental resources lead us to take advantage of other system’s spectrum. In spectrum sensing, cognitive radio networks monitor the status of the frequency spectrum by observing their surroundings to exploit the unused frequency bands. There are several methods of spectrum sensing which need different and extra informa- tion about the primary user (PU) signal, such as accuracy and implementation complexity

  Fractional low order

  Cyclostationary signal ·

  Spectrum sensing ·

  

Abstract. In this paper, we study the problem of cyclostationary spec-

trum sensing in cognitive radio networks based on cyclic properties of lin-

ear modulations. For this purpose, we use fractional order of observations

in cyclic autocorrelation function (CAF). We derive the generalized likeli-

hood ratio (GLR) for designing the detector. Therefore, the performance

of this detector has been improved compared to previous detectors. We

also find optimum value of the fractional order of observations in additive

Gaussian noise. The exact performance of the GLR detector is derived

analytically as well. The simulation results are presented to evaluate the

performance of the proposed detector and compare its performance with

their counterpart, so to illustrate the impact of the optimum value of

fractional order over performance improvement of these detectors.

  Hadi Hashemi

  ) ₁ Department of Electrical Engineering, Imam Khomeini International University,

Qazvin, Iran

h.hashemi@edu.ikiu.ac.ir

₂ Electrical Engineering, Qatar University, Doha, Qatar

tkhattab@ieee.org

  2( B

  , and Tamer Khattab

  1

  , Abbas Taherpour

  1

  , Sina Mohammadi Fard

  1

1 Introduction

  4

H. Hashemi et al.

  sensing, in

  , this detector has been investigated for one specific cyclic fre-

  quency. The authors in

  have reviewed collaborative case and have demon-

  

   ] have used

  multiple cyclic frequencies for detection of PU signal and improvement the detec- tion performance has been shown. Furthermore, several research such as

   ]

  have been conducted where the benefit of using cyclostationary-based detectors in the collaborative systems are investigated. It is known that cyclostationary- based detectors have poor performance for situations where the environment is impulsive noisy and to compensate, the CAF with fractional order of observa-

  

   ]. In these works, the problem of fractional order of observa- tions, is investigated in Alpha stable noisy environment.

  In this paper, we provide a spectrum sensing method which benefits of PU sig- nal’s cyclostationary property and improve performance of cyclostationary-based detector in different practical cases and noise models. We suggest using fractional order of observed signals. We assume an additive Gaussian noise, thought the results could be extended for the other model of ambient noises. For this pur- pose, we formulate the spectrum sensing as a binary hypothesis testing problem and then derive the corresponding GLR detectors for the different practical sce- narios. Then we investigate the optimum value of fractional order which results in best performance in related cases.

  The remaining of the paper is organized as follows. In Section

  we introduce

  the system model and the assumptions. In Section

  we derive cyclostationary-

  based detectors in different scenarios for signal and noise prameters. In Section

  

we study the performance of the proposed detectors. The optimization of

  the performance of the proposed detectors is presented in Section

  The sim-

  ulation results are provided in Section

   summarizes the conclusions.

  Notation

  : Lightface letters denote scalars. Boldface lower-and upper-case letters denote column vectors and matrices, respectively. x(.) is the entries and x i is

  −1

  sub-vector of vector x. The inverse of matrix A is A . The M × M identity matrix is I M . Superscripts

  ∗, T and H are the complex conjugate, transpose and Hermitian (conjugate transpose), respectively. E[.] is the statistical expectation. N (m, P) denotes Gaussian distribution with mean m and covariance matrix P. ₂

  ∞

  1 −u √ Q (x) is Q-function Q(x) = exp du . _x

  2 2π

2 System Model

  Suppose a cognitive radio network in which PU and secondary user (SU) equipped with a single antenna. For presentation, it’s assumed that the PU signal is transmitted with linear modulation such that

  ∞

  s d p (t) = i (t P ), (1) _i − iT

  =−∞ Fractional Low Order Cyclostationary-Based Detector

  5

  where d i is the PU data and p(t) is shaping pulse in the PU transmitter. We suppose PU data, d i , is a random variable with zero-mean Gaussian distribu-

  2

  tion, ). For the shaping pulse, a rectangular pulse with unit amplitude _s N (0, σ and time spread T is assumed. Received signal in SU has been sampled with

  P

  1

  sampling rate of f s = . The wireless channel between PU transmitter and _{T s} SU is assumed to be a flat fading channel with additive Gaussian noise and the

  2

  channel gain. The random variable w(n) ) denotes noise samples and _w ∼ N (0, σ we assume noise and PU signal samples are mutually independent. Therefore observed signal samples in SU under two hypotheses can be shown as follows,

  : x (n) = w(n), H

  (2) : x (n) = hs(n) + w(n),

1 H

  where h is channel gain between the PU and SU antennas. It is assumed that the channel gain is constant during the sensing time. CAF for the SU observed signal samples is defined based on the correlation between samples and their complex

  < T conjugate with lag time τ i . The CAF for fractional order is defined as,

  P _N −1 _{α p}

  1

  ∗p −j2παn

  R x , _{xx ∗ (τ i ) = (n)x (n + τ i )e (3)} N _n

  =0 _k where p is fractional order 0 < p < 1, α , k = 1, 2, ...

  ∈ { T } is cyclic frequency _P for linear modulation which is assumed to be known to SU and τ i , i = 1, . . . , M s is M lag times where the CAF is calculated. _α

  We introduce vector r ∗ consisting of CAF real parts for M different lag _xx times as, _{α α α T ∗ ∗ ∗ .} r = [Re(R (τ )), ..., Re(R (τ M ))] (4) _{xx xx}

  1 xx

  By considering central limit theorem (CLT), since the CAF is summation of N random variables, according to

  , for sufficiently large number of observa-

_α

_∗

  tion samples, each member of vector r xx has Gaussian distribution . Thus, we have, , , _{α N (µ H ∗} Σ ) for r _{xx ∼}

  (5) , . Σ ) for

  1

1 H

  1 N (µ

  where µ and µ can be calculated for any given p. In Section

   for the

  1 known noise and signal variance these values are computed.

3 Cyclostationary-Based Detectors

  SUs use different detection methods in spectrum sensing to make decision about PU’s presence. In this section, we assume SU determines PUs situation based on cyclostationary properties of PU signal in which the SU has knowledge about cyclic frequency of observation signal by consideration of different scenarios.

  6

H. Hashemi et al.

  3.1 Known Signal and Noise Variance Since in

  covariance matrices under two hypotheses are unknown, we have

  to use their estimations to construct the likelihood ratio (LR) function which results in a GLR detector. Covariance matrices estimation have been calculated under two hypotheses in Appendix. It has been shown that both of the covariance matrices have same estimation. Thus, Σ = Σ = Σ. Now for the LR function,

  1

  we have, _{α T T α T H 1}

  −1 −1 −1

  LR (r ∗ ) = exp Σ µ Σ µ + 2r ∗ Σ (µ ) ≷ η. (6) _{xx xx} {µ − µ

  1

  1 1 − µ } H

  By incorporating the constant terms into threshold and taking logarithm in

  ,

  we obtain, _{α T H 1}

  

−1

  T sub = r ∗ Σ (µ ) ≷ η , (7)

  1 xx

  1 1 − µ H

  where µ and µ can be calculated. It can be seen that detector is the weighted

  1 summation of CAF real part for different lag times τ i , i = 1, 2, ..., M .

  3.2 Known Noise Variance, Unknown Signal Variance The mean of

  , when SU has just knowledge about noise variance, can be derived

  under null hypothesis according to section

  But as mentioned, signal variance

  is unknown and thus, mean of the CAF real parts under alternative hypothesis cannot be calculated. In this situation, we can use Hotelling-test

   ], because

  we definitely know that the mean under two hypotheses are different. Sup- _α pose, L > M + 1 given vector r ∗ in a vector are considered together, _{α α α xx} r = [r ∗ (1), r ∗ (2), ..., r ∗ (L)]. Statistical distribution of this vector under _{xx xx xx} , j hypothesis j = 0, 1 can be written in the form below,

  H _T

  1

  1 −1

  tr exp ([ Ψ + (r j )(r j ) ]Σ ) {− L − µ − µ j }

  2

  f ,

  (r j ) = (8)

  |H LM L _{2 2} (2π) j _{L L α α α T} |Σ |

  1 _{∗ ∗ ∗}

  where r = r xx (i) and Ψ = (r xx (i) xx (i) , under _{L i i − r)(r − r)}

  =1 =1

  alternative hypothesis, r is estimate of µ and the statement inside the bracket

  1

  of function tr(.) is the estimation covariance matrix under two hypotheses. Thus after eliminating the constants we have, _L

  1 T _{2 L}

  (Ψ + L(r )(r ) ) | L − µ − µ | T

  −1 ₂

  Λ = = (r )(r ) . (9) _L |I + LΨ − µ − µ |

  1 ₂

  Ψ | | _L

  By using the matrix determinant lemma that computes the determinant of the _T

  −1

  sum of an invertible matrix I and the dyadic product, Ψ (r )(r ) , _{T L 2} − µ − µ _L

  −1 ²

  Λ .

  = 1 + L(r ) Ψ (r ) = (1 + T sub

  2 ) (10)

  − µ − µ Fractional Low Order Cyclostationary-Based Detector

  7

  Since Λ is the strictly ascending function of T sub , therefore, T sub can be

  2

  2 considered as a statistic. _T −1

  T sub = L(r ) Ψ (r ) (11)

  2

  − µ − µ

  3.3 Unknown Signal and Noise Variance In this situation, by considering covariance matrices estimation as , we have two Gaussian distribution by same covariance matrices and different mean under two hypotheses. If estimation is used for means of CAF real parts under both hypotheses, due to equality of estimation under two hypotheses the result of GLR test does not give any information to make decision. Thus, mean of CAFs for various lag time is considered as statistic and compared with a proper threshold. _M

  H ¹

  1 _α Re ∗ η . T sub = (R (τ m )) ≷ (12)

  3 xx

  3 M _m H =1

4 Analytical Performance

  In this section, we evaluate the performance of our proposed cyclostationary- based detectors in terms of detection and false alarm probabilities, P and P ,

  d fa respectively.

  4.1 Analytical Performance of T sub ¹ We should derive statistical distribution of

  under two hypotheses. We can

  rewrite

  as follows,

_{α α}

_{T 1 1 T H 1} − − _{2 2} T sub = (r ∗ Σ )(Σ (µ ∗ w ≷ η , (13) 1 xx )) = r xx

  1 1 − µ _{1 α α 1} H − ^{2 − 2}

  where w = Σ (µ ∗ = Σ r ∗ which is distributed as Gaussian ) and r xx xx

  1 −µ

  under two hypotheses, i.e., _{α ∗ , ν ν}

  I M ), ν = 0, 1, (14) r xx ₁ |H ∼ N (m

  − ₂

  µ where m ν = Σ . As we can see in _ν , our detector is a linear combination of independent Gaussian random variables mentioned in

  . Therefore, mean

  of statistic is, _M µ m

  = ν (i)w(i), ν = 0, 1. (15)

  T _{sub |H ν} ¹ _i =1

  8

H. Hashemi et al.

  And similarly variance has been derived, _M

  2

  2

  σ = w (i), ν = 0, 1. (16)

  T _{sub 1 |H ν i =1} Then, the false alarm and detection probabilities can be calculated.

  η

  1 T ¹

  − µ sub |H P = P [T sub > η ] = Q (17)

  fa

  1

  1

  |H σ

  T _{sub 1 |H}

  If β is maximum acceptable probability false alarm, then threshold of detector

  −1 −1

  can be set, η = F (β) = Q (β) + µ . Similarly for

1 T T

  × σ sub ^{1 |H sub 1 |H}

  T _{sub 1 |H}

  probability of detection, we have, η

  1 T

  − µ sub ^{1 |H 1} P > η .

  d = P [T sub

  1

  1 1 ] = Q (18)

  |H σ

  T sub ^{1 1} |H

  4.2 Analytical Performance of T sub ² We should derive statistical distribution of

  under two hypotheses. According

  to

  under null hypothesis is central chi-

  squared with M degrees of freedom. Thus, probability of false alarm is as follows, _{M η 2} γ ,

  2

  2 P > η

  = P [T sub ] = 1 (19)

  fa

  2 2 ,

  |H − M Γ

  2

  where Γ (.) and γ(., .) are Gamma and lower incomplete Gamma function, respec- tively. The asymptotic distribution of

  under alternative hypothesis is non-

  central chi-squared with noncentrality parameter, λ. Probability of detection is as follows, √ _M √

  P = P [T sub > η ] = Q ( λ, η ), (20)

  d

  2

  2

  1

  2

  |H _{2 L} where Q(., .) is Marcum Q-function and non-centrality parameter is, λ = (µ

  1 − _T 2 −1

  µ ) Σ (µ ).

  1 1 − µ

  4.3 Analytical Performance of T sub ³ Because

  is a linear combination of Gaussian random variables, therefore,

  T sub distribution is Gaussian under two hypotheses. According to Appendix

  

  3

  mean and variance of

  can be calculated. Thus, probability of false alarm

  and detection are as follow, η

3 T

  ³

  − µ sub |H P = Q , (21)

  fa

  σ

  

T

_{sub 3 |H}

  η

  3

  − µ T sub ^{3 1}

  |H P .

  = Q (22)

  d

  σ

  

T ^{3 1} Fractional Low Order Cyclostationary-Based Detector _α

  9

  5 Calculation of r ∗ Means _{xx α}

  In this section, we have provided computations for expectation of r ∗ under _xx two hypotheses when all variables are known.

  5.1 Null Hypothesis _{α ∗} In this subsection, we investigate mean of r under null hypothesis. By consid- _xx eration of noise samples independency, expectation of

  can be easily derived

  for ith lag time as follows, _N

  −1 _{α p}

  1

  ∗p −j2παn

  E E _∗ .

  [R (τ i ) ] = [w (n)]E[w (n + τ i )]e (23) _xx |H

  N _n

  =0

  p th moment of Gaussian random variable has been calculated in Appendix, since w (n) is zero mean Gaussian random variable, therefore, _{p 2p}

  −jπα(N−1)

  e sin πσ _{α −2) n} (παN ) ( E

  [R ∗ (τ ) ] = (24) _{xx .}

  |H

  1−p

  2 N sin (πα)

  Γ

  2 Mean of for i = 1, .., M ,

2 p

  sin π (παN ) (2σ ) _n

  µ (i) = (25) cos(π(α(1

  − N) + p)).

  

1−p

2 N sin (πα)

  Γ

  

2

  5.2 Alternative Hypothesis As mentioned earlier, each of the observation samples at SU is distributed as,

  2

  

2

  2

  2

  2 X = x(n) p σ + σ ) ). (26) _{s n}

  ∼ N (0, h N (0, σ

  1 Now, we assume random variable Y to be the ith lag time of observation samples

  which is distributed same as X, i.e., Y = x(n+τ i ). It can be easily demonstrated that correlation coefficient between X and Y is,

  2

  2

  2

  2 E

  (XY ) h h p σ _s − E(X)E(Y )

  E r ,

  = = [s(t)s(t + τ i )] = (27)

  2

  2

  σ σ σ

  1

  1

  × σ

  1

  1

  which reveals that X and Y are correlated. Thus, X and Y have joint Gaussian

  2

  2

  distribution, , σ , r ). To determine the mean of CAF under alternative N (0, 0, σ

  1

  1 _{p p p}

  hypothesis, we need to calculate E[X Y ] = E[Z ] = E[T ]. First we must derive probability density function (PDF) of Z which is product X and Y . i.e.,

  ∞

  z z

  1

  1 f Z (z) = f ^{XY (x, )dx f XY (x, )dx. (28)} − x x x x

  −∞

  10

H. Hashemi et al.

  _{2 − x2 p p ∞ k − k √ 1−p} 2jrx (xσ ^{1 ))} 2(1 − r _{2σ2 r2 ) 1 (1− 2 2} e _{p 2 p √}

_1−p

_{1 k k} − × ₃ j 2σ _{1 k Γ k Γ σ k}

_{=0 !}

_{1 !} ² _{2 2 k ∞}

₂

₂ − 1 − r _{2 − x2 2} r x _{2k+p 2k+p+1 2σ2 r2 ) 1 (1−} , k, p , k, p e

  = A(r, σ ^{1 )x 1 )x} − − B(r, σ _{2 2} 2σ ) ₁ (1 − r _{k =0}

  (33) In second step, we can declare distribution of T as function of Z PDF, as follows, _{1 1}

  1 _{p p} −1 f t f _{T (t) = Z (t ). (29)} p

  And thus, for computation of T mean, we have, _{1 1 1 1}

  ∞ ∞ ∞ _{p p p p}

  t t t t E f f

  [T ] = ^{XY (x, )dtdx XY (x, )dtdx. (30)} − px x px x

  −∞ −∞ −∞

  Common part of above equation is derived in following expression, _{1 1 x 2 1}

  2 _p

  2 ∞ exp _{p p} ^{2 ∞ t}

  − ^{1 − rx} t t 2σ _{1 p} f t dt. _{XY (x, )dt = exp} √

  −

  2

  2

  2

  2

  2

  px x σ px 2x (1 ) 2πσ 1 − r

  −∞ −∞

  1 1 − r

  (31) Integral expression in equation

  is in the form of p-th moment of Gaussian

  2

  2

  2

  2

  random variable with respectively mean and variance rx and x σ (1 ) that

  1 − r

  is calculated in Appendix. Therefore, _{∞ p p p 1 1 2 2 2}

  

t t (xσ )) jrx

_{1 )x} (1 − r (2 − r f ^XY D p .

  (x, )dt = exp _{−∞ px x p 2} − √ _{2 2 2} 4σ ) σ j 2πσ ^{1 (1 − r 1} ₁ 1 − r

  (32) Result of replacement Apendix equations in

  also some calculations and

  simplifications, has led to

  , −p 2k

  2

  (σ 2(1 )) r

  1

  − r

2 A , k, p ,

  (r, σ

  1 ) = _k (34) 1 1−p

  2 p

  2 2 k

  Γ k j ! 2σ (2σ (r

  1 1 − 1))

  2

  2

_k

  √ _{p p 2k+1}

• 1 1−p 2 −2k−1

  1 (1 ))

  r 2 (σ

  − r

2 B , k, p .

  (r, σ ) = (35)

  1

_p

_k 3 p k −1

  Γ k !j (

  − −2)

  

2

  2 Finally, from ], mean of T is derived in the next page.

  Therefore, ith member of µ for i = 1, ..., M is,

  1

  sin (παN )

  µ (i) = cos (πα(N (37)

  1

  − 1))E[T ] N sin (πα)

  _∞ Fractional Low Order Cyclostationary-Based Detector

  11 _{p Γ (2k + p + 1) π}

_2k+p+1

_{2 2 2k+p+1 2} √ ) A (r, σ ^{1 , k, p )2 (2σ ))} E[T ] = (1 + (−1) ^{1 (1 − r} _{2k+p+2 k =0} Γ ( ) _{2 2k+p+2} √ Γ π _{2k+p+2 2 2 2} (2k + p + 2) _{1 )2 (2σ )) (36)} , k, p − B(r, σ ^{1 (1 − r} _2k+p+3 Γ _{2 σ = 0.01 w 2} ( ) _{2 1.5 1 σ σ = 1.00 σ = 0.10 σ = 2.00 σ = 0.05 w w 2 w 2 2 2}

  2 _{w w = 0.50}

  0.5 −0.5 _{0.2 0.4 0.6 0.8 1} Fig. 1. Normalized difference of means for ith lag time

6 Performance Optimization

  To optimize the performance of proposed detector and obtain an appropriate threshold by using the Neyman-Pearson criterion, we have to maximize the prob- ability of detection respect to fractional order of observations, p. The difference between the null and alternative is just in the mean value while their covariance _α matrix is estimated to be similar. Therefore, since r ∗ has Gaussian distribu- _xx between two hypotheses should be maximized. p = arg max (i) (i) (38)

  1

  {µ − µ } ,

  0<p<1 where i denotes ith lag time.

  Therefore, for a specific value of p, if the difference between the means of null and alternative hypotheses is maximized, it can be concluded that the perfor- mance has improved. Due to complex relations obtained for the means in

  

  and

  , differentiation and solve the result of its equation for this purpose

  is not possible, however, with the help of numerical results, we can obtain the optimal amount of fractional order, p.

  In Fig.

  difference of means under two hypotheses for a certain lag time is

  2

  12