cyclostationarity-based classification of orthogonal...
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Cyclostationarity-based classification of orthogonal frequency division multiplexing and single carrier linear digital modulations
Octavia A. Dobre and A. Punchihewa
The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of Defence R&D Canada. This work was completed in December 2006 and formally published in April 2010.
Defence R&D Canada – Ottawa
Contract Report DRDC Ottawa CR 2009-296
April 2010
Cyclostationarity-based classification of orthogonal frequency division multiplexing and single carrier linear digital modulations
Octavia A. Dobre A. PunchihewaMemorial University of Newfoundland
Prepared By: Faculty of Engineering and Applied Sciences, Memorial University of Newfoundland, 300 Prince Philip Dr., St. John's, NL, A1B 3X5, Canada
PWGSC Contract Number: W7714-050968/001/SV
Contract Scientific Authority: Mr. Robert Inkol Contract Technical Authority: Dr. Sreeraman Rajan
The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of Defence R&D Canada.
This work was completed in December 2006 and formally published in April 2010.
Defence R&D Canada – Ottawa
Contract Report
DRDC Ottawa CR 2009-296
April 2010
Technical Authority
Original signed by S. Rajan
Dr. S. Rajan
Defence Scientist
Approved by
Original signed by J.F. Rivest
Dr. J.F. Rivest
Head, REW Section
Approved for release by
Original signed by B. Eatock
B. Eatock
Head/Document Review Panel
© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2010
© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2010
DRDC Ottawa CR 2009-296 i
Abstract ……..
In recent years, new technologies for wireless communications have emerged. The wireless industry has shown great interest in orthogonal frequency division multiplexing (OFDM) technology, due to the efficiency of OFDM schemes to convey information in a frequency selective fading channel without requiring complex equalizers. On the other hand, the emerging OFDM wireless communication technology raises new challenges for the designers of intelligent radios, such as discriminating between OFDM and single-carrier modulations. In this report we investigate signal cyclostationarity to discriminate between OFDM and single carrier linear digital (SCLD) modulations. We derive the analytical expressions for the nth-order (q-conjugate) cyclic cumulant (CC) and cycle frequencies of a received baseband OFDM signal, and propose a classifier based on the second-order CC to discriminate between the two aforementioned signal classes. Simulations are carried out to confirm the theoretical developments.
Résumé ….....
Au cours des dernières années, le monde des communications sans fil s’est enrichi de nouvelles technologies. L’industrie du sans fil s’intéresse beaucoup à la technologie du multiplexage par répartition orthogonale de la fréquence (MROF) en raison de sa transmisssion efficace des données par l’intermédiaire d’un canal à évanouissement progressif de fréquences sans avoir recours à des correcteurs d’affaiblissement complexes. D’autre part, la technologie de communication sans fil MROF suscite de nouvelles difficultés dans le travail des fabricants d’appareils radio intelligents, notamment lorsqu’il faut discriminer le MROF des modulations monoporteuses. Dans le présent rapport, on examine la cyclostationnarité de signal afin de faire la discrimination entre le MROF et les modulations numériques monoporteuses de type linéraire. On dérive les expressions analytiques du cumulant cyclique (CC) et des fréquences de cycle de n-ième ordre (conjugué q) dunth o signal MROF de bande de base reçu, puis on propose un classifieur en se basant sur le CC d’ordre 2 afin de discriminer les deux classes de signal susmentionnées. Enfin, on effectue des simulations pour confirmer ces avancées théoriques.
ii DRDC Ottawa CR 2009-296
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TABLE OF CONTENTS
List of Figures iv
List of Tables v
List of Abbreviations vi
List of Symbols vii
1. Introduction 1
2. Signal Cyclostationarity: Fundamental Concepts 3
3. Cyclostationarity of Single Carrier Linearly Digitally Modulated Signals 5
3.1 Signal Model 5
3.2 Cyclostationarity of Received Single Carrier Linearly Digitally Modulated Signals 6
4. Cyclostationarity of OFDM Signals 7
4.1 Signal Model 7
4.2 Cyclostationarity of Received OFDM Signals 8
5. Classification of OFDM and Single Carrier Linear Digital Modulations
by Exploiting Signal Cyclostationarity 9
5.1 Discriminating Signal Features 9
5.2 Proposed Classification Algorithm 10
6. Simulation Results 11
6.1 Simulation Setup 11
6.2 Numerical Results 12
7. Conclusions and Ongoing and Future Work 20
References 20
Appendix A: Cyclostationarity of Received OFDM Signals 23
Appendix B: A Cyclostationarity Test 25
iv
LIST OF FIGURES
Figure Page
Fig. 1 The magnitude of second-order (one-conjugate) CC versus cycle frequency and delay (in absence of noise), for a) OFDM and b) SCLD.
13
Fig. 2 The magnitude of second-order (one-conjugate) CC versus delay (in absence of noise and at zero CF), for a) OFDM and b) SCLD.
14
Fig. 3 The estimated second-order (one-conjugate) CC magnitude versus cycle frequency and delay (at 20 dB SNR), for a) OFDM and b) SCLD.
15
Fig. 4 The estimated second-order (one-conjugate) CC magnitude of OFDM signals versus delay, at zero CF and a) 15 dB SNR, b) 5 dB SNR, and c) 0 dB SNR.
17
Fig. 5 The estimated second-order (one-conjugate) CC magnitude of SCLD signals versus delay, at zero CF and a) 15 dB SNR, b) 5 dB SNR, and c) 0 dB SNR.
18
Fig. 6 The average probability of correct classification ( | ) , OFDM, SCLDi iccP i ,
versus SNR. 19
v
LIST OF TABLES
Table Page
Table 1 Number of correct decisions for OFDM signal recognition (100 trials). 19 Table 2 Number of correct decisions for SCLD signal recognition (100 trials). 20
vi
LIST OF ABBREVIATIONS
AWGN Additive white Gaussian noise BPSK Binary Phase-Shift-Keying CC Cyclic cumulant CF Cycle frequencies CM Cycle moment FB Feature-based i.i.d. Independent and identically distributed LB Likelihood-based MC Blind modulation classification OFDM Orthogonal Frequency Division Multiplexing PSK Phase-Shift-Keying QAM Quadrature Amplitude Modulation QPSK Quadrature Phase-Shift-Keying SCLD Single carrier linear digital modulations SNR Signal-to-noise ratio
vii
LIST OF SYMBOLS
The principal symbols used in this report are listed below.
a Amplitude factor A CF for the CM of ( )r t Convolution operator A CF for the CC of ( )r u , for the case when the CC order equals twice the
number of conjugations
,( ; )r n qc t The nth-order (q-conjugate) time-varying cumulant of ( )r t
,( ; )r n qc The nth-order (q-conjugate) CC of ( )r t at CF and delay vector
,( ; )r n qc The nth-order (q-conjugate) CC of ( )r u at CF and delay vector
SCLD ,( ; )r n qc t The nth-order (q-conjugate) time-varying cumulant of SCLD ( )r t
SCLD ,( ; )r n qc The nth-order (q-conjugate) CC of SCLD ( )r t at CF and delay vector
SCLD ,( ; )r n qc The nth-order (q-conjugate) CC of SCLD ( )r u at CF and delay vector
OFDM ,( ; )r n qc t The nth-order (q-conjugate) time-varying cumulant of OFDM ( )r t
OFDM ,( ; )r n qc The nth-order (q-conjugate) CC of OFDM ( )r t at CF and delay vector
OFDM ,( ; )r n qc The nth-order (q-conjugate) CC of OFDM ( )r u at CF and delay vector
2,1( ; )wc The second-order (one-conjugate) CC of ( )w u at CF and delay vector
2,1| ( ; ) |ir
c The magnitude of the second-order (one-conjugate) CC of modulation i
2,1ˆ| ( ; ) |ir
c The estimated second-order (one-conjugate) CC magnitude of modulation i
21c Vector containing the real and imaginary parts of estimated second-order (one-conjugate) CC at CF and delay
Cum[ ] Cumulant operator
, ,s n qc The nth-order (q-conjugate) cumulant of the signal constellation
cf Carrier frequency offset
Nf Frequency separation between two adjacent subcarriers E[ ] Statistical expectation
sf Sampling frequency
2,1( ; )f u Second-order (one-conjugate) lag product of ( )r u ( )g t Overall impulse response
( )trg t Transmit pulse shape
( )recg t Receive filter impulse response A CF for the CC of ( )r t A CF for the CC of ( )r u
2,1T Test statistic used in the cyclostationarity test
viii
Threshold value used in the cyclostationarity test {} Fourier transform
i Modulation format, i OFDM, SCLD
,mn q Set of CFs for the CM of ( )r t
,cn q Set of CFs for the CC of ( )r t
,mn q Set of CFs for the CM of ( )r u
,cn q Set of CFs for the CC of ( )r u
L Number of samples available at the receive-side
,( ; )r n qm The nth-order (q-conjugate) CM of ( )r u at CF and delay vector
,ˆ ( ; )r n qm The nth-order (q-conjugate) CM estimate at CF and delay vector n Order of the statistic
N Number of subcarriers
minN Minimum number of subcarriers ( | )i i
ccP Probability of correct classification for modulation i
fP Probability of false alarm q Number of conjugations
2,0Q Components of the covariance matrix
2,1Q Components of the covariance matrix
( )r t Continuous-time received baseband signal
SCLD( )r t Continuous-time received baseband SCLD signal
SCLD( )r u Discrete-time received baseband SCLD signal
OFDM ( )r t Continuous-time received baseband OFDM signal
OFDM ( )r u Discrete-time received baseband OFDM signal Oversampling factor
ls Symbol transmitted within the lth symbol period
,k ls Symbol transmitted within the lth symbol period and on the kth subcarrier T Symbol period
cpT Cyclic prefix period
uT Useful time period Carrier phase ( )w t Continuous-time baseband Gaussian noise ( )w u Discrete-time baseband Gaussian noise
OFDM ( )x t Transmitted baseband OFDM signal
2,1 Covariance matrix
1
1. INTRODUCTION
In recent years, new technologies for wireless communications have emerged. The wireless
industry has shown great interest in OFDM, due to several advantages of OFDM, such as high
capacity data transmission, immunity to multipath fading and impulsive noise and, simplicity in
equalization [1]-[2]. OFDM has been adopted in a variety of applications, such as wireless local
area network (WLAN) IEEE 802.11a [3] and wireless metropolitan area network (WMAN) IEEE
802.16a [4]. On the other hand, the emerging OFDM wireless communication technology raises
new challenges for the designers of intelligent radios, such as discrimination between OFDM and
single-carrier modulations. Solutions to tackle such new signal recognition problems need to be
sought [5]. Blind modulation classification (MC) for single carrier signals has been studied for at
least a decade (see [5] and references herein). Algorithms for discriminating between OFDM and
single-carrier signals have been recently started to be investigated by the research community [6]-[8].
This effort explores the applicability of signal cyclostationarity to distinguish between OFDM and
the class of single carrier linear digital (SCLD) modulations.
MC is an intermediate step between signal interception and data demodulation. This is a difficult
task, especially in a non-cooperative environment, in which no prior knowledge on the detected
signal is available at the receive-side. Generally, two approaches are proposed to tackle the MC
problem, i.e., the likelihood-based (LB) and the feature-based (FB) methods (see [5] and references
herein). The LB approach is based on the likelihood function of the received signal and the
likelihood ratio test is used to decision making. This can provide an optimal solution, in the sense
that it maximizes the probability of false classification. However, a complete mathematical
representation of an optimal classifier is very complex even for simple modulation formats [5].
With the latter approach, features are extracted from the received signal, and a decision on the
modulation format is made based on their differences. Several signal features have been
investigated in the open literature, such as moments and cumulants, cyclic moments and cyclic
2
cumulants, and wavelet transform [5]. The FB approach can have the advantage of implementation
simplicity for an appropriately chosen feature set, and can provide near optimal performance. Here
we exploit cyclic cumulant-based features for distinguishing between OFDM and SCLD
modulations. In general, cyclostationary signals are present in communications, signal processing,
telemetry, radar, sonar and, control systems. Signal cyclostationarity can be exploited for several
purposes, including signal identification, blind equalization, synchronization, parameter estimation
and modulation classification [6]-[7], [9]-[14], [19]-[23]. Communication signals exhibit
cyclostationarity in connection with the symbol period, carrier frequency, chip rate and combination
of these [6]-[7], [9]-[14], [19]-[23]. First-, second- and higher-order cyclostationarity of single
carrier signals is employed for aforementioned applications in [9]-[14], [19]-[20]. Second-order
cyclostationarity of the OFDM signal is exploited for blind estimation of symbol timing and carrier
frequency offset, extraction of channel allocation information in a spectrum poling system, and
blind channel identification [21]-[23].
In this report, we exploit signal cyclostationarity for discriminating between OFDM and SCLD
modulations. We propose a classification algorithm, which employs second-order cyclic cumulant-
based features. This algorithm is based on the analytical results that we derive for the cyclic
cumulants and cycle frequencies of a received baseband OFDM signal.
The rest of the report is organized as follows. Fundamental concepts of signal cyclostationarity are
introduced in Section 2. Single carrier linearly digitally modulated and OFDM signal models, along
with corresponding signal cyclostationarity are presented in Sections 3 and 4, respectively.
The proposed cyclostationarity-based classification algorithm is introduced in Section 5, and
simulation results are discussed in Section 6. Finally, conclusions are drawn in Section 7.
In addition, results for the nth-order cyclostationarity of the OFDM signal are derived in Appendix
A, and a cyclostationarity test used for decision making is presented in Appendix B.
3
2. SIGNAL CYCLOSTATIONARITY: FUNDAMENTAL CONCEPTS
Fundamental concepts of signal cyclostationarity are presented in the following. A signal exhibits
nth-order cyclostationarity if its nth- and lower-order time-variant cumulants are almost-periodic
functions of time [16]-[19]. For a complex-valued continuous-time nth-order cyclostationary
process, ( ),r t the nth-order (q-conjugate) time-varying cumulant,
1 2 ( )( ) ( ), 1 2( ; ) Cum[ ( ), ( ), , ( )],n
r n q nc t r t r t r t (1)
is an almost periodic function of time. Here Cum[ ] represents the cumulant operator,
†1 0
[ , , ]n
n is the delay vector and ( ) ,i 1, ,i n , is a possible conjugation, with the total
number of conjugations equal to q and † as the transpose. This time-varying cumulant can be
expressed as a Fourier series [16]-[19]
,
2, ,( , ) ( ; )
cn q
j tr n q r n qc t c e , (2)
where , ,{ | ( ; ) 0}cn q r n qc represents the set of nth-order cycle frequencies (CFs) (for cyclic
cumulants) and the coefficient ,( ; )r n qc is the nth-order (q-conjugate) cyclic cumulant (CC) at CF
and delay vector , which can be expressed as [16]-[19]
/ 21 2
, ,/ 2
( ; ) lim ( ; ) .I
j tr n q r n q
II
c c t e dtI (3)
For the nth-order cyclostationarity process ( )r t , the nth-order (q-conjugate) time-varying moment
function,
1 2 ( )( ) ( ), 1 2( ; ) E[ ( ), ( ), , ( )],n
r n q nm t r t r t r t (4)
is also an almost periodic function of time of time [16]-[19]. Here E[ ] denotes the statistical
expectation. This time-varying moment can be also expressed as a Fourier series [16]-[19]
,
2, ,( , ) ( ; )
mn q
j tr n q r n qm t m e , (5)
4
where , ,{ | ( ; ) 0}mn q r n qm represents the set of nth-order CFs (for cyclic moments), and the
coefficient ,( ; )r n qm is the nth-order (q-conjugate) cyclic moment (CM) at CF and delay vector
, given by [16]-[19]
/ 21 2
, ,/ 2
( ; ) lim ( ; ) .I
j tr n q r n q
II
m I m t e dt (6)
The nth-order (q-conjugate) cumulant can be expressed in terms of the nth- and lower-order
moments by using the moment-to-cumulant formula [18],
1
( 1), ,
{ , , } 1( ; ) ( 1) ( 1)! ( ; )
z z
Z
ZZ
r n q r z n qz
c t Z m t , (7)
where 1{ , , }Z is a partition of {1, 2, , }n , with z , 1, ,z Z , as a non-empty disjoint
subset of , so that the reunion of these subsets is , Z is the number of subsets in a partition
(1 )Z n , z is a delay vector whose components are elements of 1{ }nu u , with indices specified
by z , and zn is the number of elements in the subset z , from which zq correspond to conjugate
terms, with 1
Z
zz
n n and 1
Z
zz
q q .
By combining (2), (5) and (7), the nth-order (q-conjugate) CC of ( )r t at CF and delay vector
can be expressed using the nth- and lower-order CMs as [18]
†1
( 1), ,
{ , , } 1( ; ) ( 1) ( 1)! ( ; ) ,
z z
Z
ZZ
r n q r z n qz
c Z m1
(8)
where †1, , Z is a vector of CFs and †[ , , ]1 11 is a Z -dimensional one vector. Equation
(8) is referred to as the cyclic moment-to-cumulant formula [18].
A discrete-time signal 1( ) ( )st uf
r u r t is obtained by periodically sampling the continuous-time
signal ( )r t at rate sf . The nth-order (q-conjugate) CC of the discrete-time signal, ( )r u , and the
corresponding set of CFs are respectively given by (under the assumption of no aliasing) [24]
1, ,( ; ) ( ; )r n q r s s n qc c f f , (9)
5
and
1, ,{ [ 1/ 2,1/ 2)| , ( ; ) 0}c
n q s r n qf c , (10)
where sf , with components u u sf , 1, ,u n .
A similar expression can be written for the nth-order (q-conjugate) CM of the discrete-time signal,
,( ; )r n qm , and corresponding set of CFs, ,mn q [16].
The estimator for the nth-order (q-conjugate) CM at a CF and delay vector , based on L
samples, is given by [17]
( )1 2,
1 1
ˆ ( ; ) ( ) .pnL
j ur n q p
u p
Lm r u e (11)
Furthermore, the estimator for the nth-order (q-conjugate) CC at a CF and delay vector ,
based on L samples, ,ˆ ( ; )r n qc , can be obtained by applying the cyclic moment-to-cumulant
formula given in (8), with CMs replaced by their estimates given in (11) [17].
3. CYCLOSTATIONARITY OF SINGLE CARRIER LINEARLY DIGITALLY MODULATED SIGNALS
3.1 SIGNAL MODEL
Let us assume that a single carrier linearly digitally modulated signal is transmitted through a
channel, which delays and corrupts the signal by adding white Gausian noise. The output of the
matched filter at the receive-side is a baseband waveform, given by [25]
2SCLD ( ) ( ) ( ),cj f tj
ll
r t ae e s g t lT T w t (12)
where a is the amplitude factor, is the phase, cf is the carrier frequency offset, T is the symbol
period, 0 1 is the timing offset, ls represents the symbol transmitted within the lth symbol
period, drawn either from a quadrature amplitude modulation (QAM) or phase shift keying (PSK)
constellation, ( )g t is the overall impulse response of the transmit and receive filters and ( )w t is the
zero-mean complex Gaussian noise. The overall impulse response of the transmit and receive filters
in cascade is given by ( ) ( ) ( )tr recg t g t g t , with ( )trg t and ( )recg t as the impulse response of the
6
transmit and receive filters, respectively. The data symbols { }ls are assumed to be zero-mean
independent and identically distributed (i.i.d.) random variables.
The discrete-time baseband signal SCLD ( )r u , obtained by oversampling SCLD ( )r t at rate 1sf T ,
with as a positive integer which represents the oversampling factor, is given by
2
SCLD ( ) ( ) ( ),cj f Tu
jl
l
r u ae e s g u l w u (13)
where ( )w u is wide-sense stationary zero-mean complex Gaussian noise.
3.2 CYCLOSTATIONARITY OF RECEIVED SINGLE CARRIER LINEARLY DIGITALLY MODULATED
SIGNALS
For the continuous-time baseband received signal, SCLD ( )r t , the nth-order (q-conjugate) time-
varying cumulant is given by [9]
1SCLD
2 ( )2 ( 2 )( 2 )
, , ,
( ),
1
( ; )
( ) ( ) ( ; )
n
c p pp c
p
j fj n q f tn j n q
r n q s n q
n
p w n qlp
c t a c e e e
g t t lT T c t
, (14)
where , ,s n qc is the nth-order (q-conjugate) cumulant of the signal constellation, ,( ; )w n qc t 1 is the
nth-order (q-conjugate) time-varying cumulant of ( )w t , ( ) p is the optional minus sign associated with the
optional conjugation ( ) p , 1,...,p n , ( )t is the Dirac delta function and stands for convolution.
The nth-order CC at CF and delay vector , and the set of CFs for the continuous-time signal
SCLD ( )r t are respectively given by [9], [11]
1SCLD
2 ( )1 ( 2 ) 2
, , ,
( ) 2,
1
( ; )
( ) ( ; )
n
c p pp
p
j fn j n q j T
r n q s n q
nj t
p w n qp
c a c T e e e
g t e dt c
, (15)
1 For 1n and 3n there is no additive contribution of the wide-sense stationary zero-mean Gaussian noise to the cumulant of the received signal. For n=2, the cumulant corresponding to the noise does not depend on time, due to the wide- sense stationarity of the noise.
7
with ,( ; )w n qc as the nth-order (q-conjugate) CC 2 of ( )w t , and
SCLD
1, ,{ | ( 2 ) , , integer, ( ; ) 0}.c
n q c r n qn q f lT l c (16)
For the discrete-time baseband signal, SCLD ( )r u , the nth-order (q-conjugate) CC at CF and delay
vector 2, and the set of CFs are respectively given by [9]
1SCLD
2 ( )1 ( 2 ) 2
, , ,
( ) 2,
1
( ; )
( ) ( ; )
n
c p pp
p
j f Tn j n q j
r n q s n q
nj u
p w n qu p
c a c e e e
g u e c
, (17)
and
SCLD
1 1, ,{ | ( 2 ) , , integer, ( ; ) 0}c
n q c r n qn q f T l l c . (18)
4. CYCLOSTATIONARITY OF OFDM SIGNALS
4.1 SIGNAL MODEL
The continuous-time baseband equivalent of a transmitted OFDM signal is given by [21],
OFDM
12 ( )
,0
1( ) ( ),N
Nj k f t lT tr
k lk l
x t s e g t lTN
(19)
where N is the number of subcarriers, Nf is the frequency separation between two adjacent subcarriers, T
is the OFDM symbol period, given by u cpT T T , with 1/ Nu fT as the useful symbol duration and
cpT as the length of the cyclic prefix, and ,k ls is the symbol transmitted within the lth symbol period and
on the kth subcarrier, and ( )trg t is the transmit pulse shaping window [20]. The data symbols ,{ }k ls are
assumed to be zero-mean i.i.d. random variables, drawn either from a QAM or PSK constellation.
At the receiver-side, the continuous-time baseband equivalent is given by
OFDM
12 2 ( )
,0
( ) ( ) ( ),NcN
j f t j k f t lT Tjk l
k l
r t ae e s e g t lT T w t (20)
where ( ) ( ) ( )tr recg t g t g t . 2 The cyclic cumulant corresponding to the noise is non-zero for n=2 and at zero CF.
8
A discrete-time baseband OFDM signal, OFDM ( )r u , is obtained by oversampling OFDM ( )r t at a rate
1s uf NT , where N is a positive integer which represents the oversampling factor in the useful
symbol duration. Accordingly, the expression for the discrete-time baseband OFDM signal can be
easily written as,
OFDM
2 21 ( )
,0
( ) ( ) ( ),c u
N NNj f T u j k u lD D
jk l
k l
r u ae e s e g u lD D w u (21)
where cp sD N T f .
4.2 CYCLOSTATIONARITY OF RECEIVED OFDM SIGNALS
Results derived in Appendix A for the nth-order cyclostationarity of the received OFDM signal
are presented in the following. The nth-order (q-conjugate) time-varying cumulant of the
continuous-time baseband received OFDM signal, OFDM ( )r t , is given by 1
OFDM
1 12 ( ) 2 ( )1
2 ( 2 )( 2 ), , ,
0
( )2 ( 2 ),
1
( ; )
( ) ( ) ( ; )
n n
c p p N p pp pc
pN
j f j k fNj n q f tn j n q
r n q s n qk
nj n q k f t
p w n qlp
c t a c e e e e
e g t t lT T c t
. (22)
The nth-order (q-conjugate) CC at CF and delay vector , and the set of CFs for the
continuous-time baseband received OFDM signal, OFDM ( )r t , are respectively given as 2
OFDM
1 112 ( ) 2 ( )1
( 2 ) 2, , ,
0
( )2 ( 2 ) 2,
1
( ; )
( ) ( ; )
N
N
n n
c p p p pp p
p
j f j k fNn j n q j T
r n q s n qk
nj n q k f t j t
p w n qp
c a c T e e e e
e g t e dt c
, (23)
and
1, { | ( 2 ) , , integer}c
n q cn q f lT l . (24)
The nth-order (q-conjugate) CC at CF and delay vector , and the set of CFs for the
discrete-time baseband received OFDM signal, OFDM ( )r u , are respectively given as 2
9
OFDM
1 1
2 2( ) ( )11 ( 2 ) 2
, , ,0
2 ( 2 ) ( ) 2,
1
( ; )
( ) ( ; )
n n
c u p p p pp p
p
N N
N
j f T j kNn j n q j D
r n q s n qk
nj n q kuj u
p w n qu p
c a c D e e e e
e g u e c
, (25)
and
1 1, { | ( 2 ) ( ) , , integer}c
n q c un q f T N lD l . (26)
5. CLASSIFICATION OF OFDM AND SINGLE CARRIER LINEAR DIGITAL MODULATIONS
BY EXPLOITING SIGNAL CYCLOSTATIONARITY
Results on signal cyclostationarity, presented in previous sections, are employed here to develop
an algorithm for the classification of OFDM and SCLD in additive white Gaussian noise (AWGN)
channel.
5.1 DISCRIMINATING SIGNAL FEATURE
According to (17), (18), (25) and (26), the second-order (one-conjugate) CCs for the SCLD and
OFDM signals are given respectively as 2, 3
SCLD
22 1 2 2
2,1 ,2,1 2,1( ; ) ( ) ( ) ( ; )cj f T
j j ur s w
u
c a c e e g u g u e c , (27)
and
OFDM
2 212 1 2 2
2,1 ,2,1 2,10
( ; ) ( ) ( ) ( ; )c u
N NNj f T j k
j D j ur s w
k u
c a c D e e e g u g u e c . (28)
One can easily show that 21 ( 1)
0
sin( / )( )sin( / )
N NN
N j k j N
k
e eN
, and write (28) as
OFDM
22 1 2 2
2,1 ,2,1 2,1( ; ) ( ) ( ) ( ) ( ; )c u
NN
j f Tj D j u
r s wu
c a c D e e g u g u e c . (29)
When comparing (27) and (29), one can easily notice the additional factor ( )N which appears in
the expression of the second-order (one-conjugate) CC of the OFDM signal. One can easily show
3 Note that according to (18) and (26), if n=2q (in this case n=2 and q=1), the CF is equal to . This result will be used for the CF notation throughout the report.
10
that ( )N yields significant peaks in the CC of the OFDM signal at 0, , 2 , N N .
The magnitude of the second-order (one-conjugate) CC of SCLD and OFDM signals (in the
absence of noise) is plotted versus CF and delay in Fig. 1 a) and b), respectively (for the parameter
setting see Section 6.1). For the range of the delay showed in this figure, one can notice the peaks in
the CC magnitude of the OFDM signal at N , whereas no such peaks appear for the SCLD
signals. The existence of such a peak in the magnitude of second-order (one-conjugate) CC of the
OFDM signal (at CF 0 and delay N ) is subsequently employed to develop an algorithm
for OFDM and SCLD classification.
5.2 PROPOSED CLASSIFICATION ALGORITHM
At the receive-side, the bandwidth of the incoming signal is roughly estimated, and a low-pass
filter is used to remove the out-of-band noise. Then, the signal is down-converted, normalized with
respect to received signal power (to remove any scale factor from data), and (over)sampled.
Discrimination between OFDM and SCLD signals is performed by applying the following
algorithm, which consists of two steps.
Step 1
Based on the observation interval available at the receive-side (L samples), the magnitude of the
second-order (one-conjugate) CC of normalized baseband received signal is estimated at CF 0
and for a range of delays. Over this range, we search for a local maximum in the magnitude. If the
received signal corresponds to the OFDM modulation format, then we should detect that 0 is a
CF for N (there is a significant peak at this CF and delay); otherwise not. We set the range of
delays we search over such that we can find a meaningful local maximum in the estimated CC
magnitude for the OFDM signal. With the oversampling factor, , roughly known, we choose a
minimum number of subcarriers (the number of subcarriers is unknown at the receive-side), and
search starting from min( 1)N , with minN as the minimum number of subcarriers that we consider.
11
Step 2
Once local maximum in the estimated magnitude of the second-order (one-conjugate) is found over the
investigated delay range, we test weather or not 0 is a CF for the delay corresponding to detected
local maximum. For that, we apply a cyclostationary test developed in [26]. This test is presented is
Appendix B. If 0 is a CF for the delay corresponding to the detected local maximum, then we decide
that the modulation format is OFDM; otherwise, that it belongs to the SCLD class.
6. SIMULATION RESULTS
Simulations are performed to confirm theoretical developments, and results of these simulations
are presented in the following.
6.1 SIMULATION SETUP
For SCLD modulations, we consider a pool consisting of BPSK, QPSK, 8-PSK, 16-QAM and
64-QAM. Without any loss of generality, we simulate unit variance constellations. The transmit
filter is a root-raised cosine with 0.35 roll-off factor [25], and the signal bandwidth is 40 kHz. At the
receive-side, a low-pass filter is used to eliminate the out-of-band noise, and the signal is sampled at
a rate 160 kHzsf . For the OFDM signal, we set the parameters as follows. The signal bandwidth
is set to 800 kHz, the number of subcarriers to 128, the useful time period to 160 s , and the cyclic
prefix period to 40 s . All subcarriers are modulated either using QPSK or 16-QAM. Unit variance
constellations are also used in this case. The transit pulse-shaping window is chosen as raised
cosine, with 0.025 roll-off factor [2]. At the receive-side, the signal is low-pass filtered and sampled
at a rate of 3.2 MHz. For both OFDM and SCLD, we consider an oversampling factor of 4, and an
observation interval of 0.1 s. This interval corresponds to L 320,000 and 16,000 samples for
OFDM and SCLD, respectively. In addition, a is set to one and , cf , and to zero.
The signal-to-noise ratio (SNR) is defined as the signal power to the noise power at the output of
receive filter. For the cyclostationarity test, a Kaiser window of length 61 and parameter 10 is
12
employed to compute the estimates of covariances used in the test, and a threshold of 23.0258 is
employed for decision making (see Appendix B for the description of the test and parameters
involved in it). This threshold value corresponds to a probability of false alarm 510fP [27].
The probability to correctly decide that the modulation format of the received signal is i, when
indeed the modulation format i is transmitted, ( | ) , OFDM, SCLDi iccP i , is used to evaluate the
performance of the proposed classifier. This is calculated based on 100 trials.
6.2 NUMERICAL RESULTS
The magnitude of the second-order (one-conjugate) CC of OFDM and SCLD signals is plotted in
Fig. 1 a) and b), respectively. These numerical results are obtained by using (17) and (25) in the
absence of noise, with signal parameters set as specified in Section 6.1. For the OFDM signal, one
can notice the peaks at CFs and for N (note that the range of that we use in the figure does
no allow to see peaks at 2 , 3 , N N ). No such peaks appear for SCLD signals. With zero
CF ( 0 ), the CC magnitude is plotted versus delay in Fig. 2 a) and b), for OFDM and SCLD
signals, respectively. The peak corresponding to N is to be noticed in the results presented for
OFDM; no such peak appears for SCLD.
The estimated magnitude of the second-order (one-conjugate) CC of OFDM and SCLD signals is
plotted in Fig. 3 a) and b), respectively, for 20dB SNR and 0.1s observation interval. When
comparing results presented in Figs. 1 and 2, one can notice the existence of non-zero spikes in the
estimated magnitude, at frequencies different than CFs and over the whole delay range. This is due
to the noise contribution and finite length of the observation interval.
13
Fig. 1. The magnitude of second-order (one-conjugate) CC versus cycle frequency and delay (in absence of noise), for a) OFDM and b) SCLD.
b)
a)
14
Fig. 2. The magnitude of second-order (one-conjugate) CC versus delay (in absence of noise and at zero CF), for a) OFDM and b) SCLD.
b)
a)
15
b) Fig. 3. The estimated second-order (one-conjugate) CC magnitude versus cycle frequency and delay (at 20 dB SNR), for a) OFDM and b) SCLD.
a)
16
Fig. 4 shows the estimated magnitude of the second-order (one-conjugate) CC of OFDM versus
delay, for zero CF and at different SNRs. From Fig. 4 one can notice the significant peak at delay
N . In addition, it is to be noted that its value decreases with a decrease in the SNR, such that
below a certain SNR, this will not be distinguished from statistically insignificant values of the
magnitude (these are due to estimation based on a finite observation interval and the contribution of
the noise). Fig. 5 shows the estimated magnitude of the second-order (one-conjugate) CC of SCLD
versus delay, for zero CF, and at different SNRs. From Fig. 5 one can notice that there is no
significant peak along the delay axis, even at lower SNR values. Classification performance of the
proposed algorithm is shown in Fig. 6. The probability of correct classification, ( | )i iccP , is plotted
versus SNR, for OFDM, SCLDi . It can be noticed that OFDM|OFDM( )ccP equals one up to –9 dB SNR
(the peak at N is always detected from the estimated CC magnitude), regardless the
modulation type used on subcarriers. On the other hand, SCLD|SCLD( )ccP is always one for the whole
investigated SNR range. This can be easily explained, as for SCLD there is no statistically
significant peak in the second-order (one-conjugate) CC magnitude at zero CF and over the
searched delay range. Thus, the local maximum in the CC magnitude, which is selected in Step 1 of
the classification algorithm, is due only to the finite length of the observation interval and
contribution of the noise, and does not pass the cyclostationarity test in Step 2 of the algorithm.
Hence, a correct decision is made when recognizing SCLD modulations.
17
Fig. 4. The estimated second-order (one-conjugate) CC magnitude of OFDM signals versus delay, at zero CF and a) 15 dB SNR, b) 5 dB SNR, and c) 0 dB SNR.
a)
b)
c)
18
Fig. 5. The estimated second-order (one-conjugate) CC magnitude of SCLM signals versus delay, at zero CF and a) 15 dB SNR, b) 5 dB SNR, and c) 0 dB SNR.
a)
b)
c)
19
The number of correct decisions for the recognition of an OFDM signal is given in Table 1 for
100 trials (Fig. 6 is based on these results). One can notice that similar classification results are
obtained for the OFDM signal, when using QPSK and 16-QAM modulation formats, respectively.
We can conclude that the classifier’s performance in recognizing an OFDM signal does not change
significantly with the modulation format within the OFDM signal.
Table 1. Number of correct decisions for OFDM signal recognition (100 trials).
SNR (dB) 10 5 0 -5 -6 -7 -8 -9 -10 -10.5 -11
OFDM
(QPSK/subcarrier) 100 100 100 100 100 100 100 100 94 81 62
OFDM
(16-QAM/subcarrier) 100 100 100 100 100 100 100 100 95 83 56
Fig. 6. The average probability of correct classification ( | )i iccP , OFDM, SCLDi , versus SNR.
20
Finally, the number of correct decisions when recognizing SCLD signals is given in Table 2 for
100 trials. One can notice that a probability of one is achieved at SNRs as low as –25 dB, regardless
the modulation format.
Table 2. Number of correct decisions for SCLD signal recognition (100 trials).
SNR (dB) 10 5 0 -5 -10 -15 -20 -25
BPSK 100 100 100 100 100 100 100 100
QPSK 100 100 100 100 100 100 100 100
8-PSK 100 100 100 100 100 100 100 100
16-QAM 100 100 100 100 100 100 100 100
64-QAM 100 100 100 100 100 100 100 100
7. CONCLUSIONS AND ONGOING AND FUTURE WORK
In this report we investigate signal cyclostationarity to discriminate between orthogonal frequency
division multiplexing (OFDM) and single carrier linear digital (SCLD) modulations. We derive the
analytical expressions for the nth-order cyclic (q-conjugate) cumulant (CC) of an OFDM signal, and
propose a classification algorithm based on the second-order (one-conjugate) CC. The proposed
algorithm provides a probability of correct classification of one for SNRs above –9 dB, in additive
white Gaussian noise (AWGN) channel. An extension of the classification algorithm to frequency-
selective channels is investigated as ongoing work. Exploitation of OFDM signal cyclostationarity
for parameter estimation and modulation classification within the OFDM signal will be investigated
as future work.
REFERENCES
[1] J. A. C. Bingham, “Multicarrier modulation for data transmission: an idea for whose time has come," IEEE
Commun. Mag., vol. 28, pp. 5-14, 1990. [2] R. V. Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Artech House, 2000. [3] Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High
Speed Physical Layer in the 5GHz, IEEE Standard 802.11a-1999.
21
[4] Local and Metropolitan Area Networks-Part 16, Air Interface for Fixed Broadband Wireless Access Systems, IEEE Standard IEEE 802.16a-2001.
[5] O. A. Dobre, A. Abdi, Y. Bar-Nees, and W. Su, “A survey of automatic modulation classification techniques: classical approaches and new trends," to be published in IEE Proc. Commun., March 2007.
[6] M. Oner and F. Jondral, “Cyclostationarity based air interface recognition for software radio systems," in Proc. IEEE Conf. on Radio and Wireless, 2004, pp. 263-266.
[7] M. Oner and F. Jondral, “Air interface recognition for a software radio system exploiting cyclostationarity," in Proc. IEEE PIMRC, 2004, pp. 1947-1951.
[8] D. Grimaldi, S. Rapunao, and G. Truglia, “An automatic digital modulation classifier for measurement on telecommunication networks," in Proc. IEEE IMT, 2002, pp. 957-962.
[9] O. A. Dobre, Y. Bar-Nees, and W. Su, “Higher-order cyclic cumulants for high order modulation classification," in Proc. IEEE MILCOM, 2003, pp. 112-117.
[10] P. Marchand, J. L. Lacoume, and C. Martret, “Classification of linear modulations by a combination of different orders cyclic cumulants," in Proc. Workshop on HOS, 1997, pp. 47-51.
[11] C. M. Spooner, “Classification of co-channel communication signal using cyclic cumulants," in Proc. IEEE
ASILOMAR, 1995, pp. 531-536. [12] C. M. Spooner, W. A. Brown, and G. K. Yeung, “Automatic radio frequency environment analysis," in Proc.
IEEE ASILOMAR, 2000, pp.1181-1186. [13] O. A. Dobre, Y. Bar-Nees, and W. Su, “Robust qam modulation classification algorithm based on cyclic
cumulants," in Proc. IEEE WCNC, 2004, pp. 745-748. [14] O. A. Dobre, Y. Bar-Nees, and W. Su, “Selection combining for modulation recognition in fading channels,"
in Proc. IEEE MILCOM, 2005, pp. 1-7. [15] O. A. Dobre, S. Rajan, and R. Inkol, “A Novel Algorithm for Blind Recognition of M-ary Frequency Shift
Keying Modulation,” in Proc. IEEE WCNC, 2007, Hong Kong. [16] A. V. Dandawate and G. B. Giannakis, “Nonparametric polyspectral estimators for kth-order (almost)
cyclostationary processes," IEEE Trans. Inform. Theory, vol. 40, pp. 67-84, 1994. [17] A. V. Dandawate and G. B. Giannakis, “Asymptotic theory of mixed time averages and kth-order cyclic
moment and cumulant statistics," IEEE Trans. Inform. Theory, vol. 41, pp. 216-232, 1995. [18] C. M. Spooner and W. A. Gardner, “The cumulant theory of cyclostationary time-series, part I: foundation
and part II: development and applications," IEEE Trans. Sig. Proc., vol. 42, pp. 3387-3429, 1994. [19] W. A. Gardner, Cyclostationarity in Communication and Signal Processing. IEEE Press, 1994. [20] F. Gini and G. B. Giannakis, “Frequency offset and symbol timing recovery in flat-fading channels: A
cyclostationary approach," IEEE Trans. Commun., vol. 46, pp. 400-411, 1998. [21] H. Bolcskei, “Blind estimation of symbol timing and carrier frequency offset in wireless OFDM systems,"
IEEE Trans. Commun., vol. 49, pp. 988-999, 2001. [22] M. Oner and F. Jondral, “Cyclostationary-based methods for the extraction of the channel allocation
information in a spectrum poling system," in Proc. IEEE Conf. on Radio and Wireless, 2004, pp. 279-282. [23] R. W. Heath and G. B. Giannakis, “Exploiting input cyclostationarity for blind channel identification in
OFDM systems," IEEE Trans. Sig. Proc., vol. 47, no. 3, pp. 848-856, 1999.
22
[24] A. Napolitano, “Cyclic higher-order statistics: input/output relations for discrete- and continuous-time MIMO linear almost-periodically time-variant systems," IEEE Trans. Sig. Proc., vol. 42, pp. 147-166, 1995.
[25] J. G. Proakis, Digital Communications, 4th ed. McGraw Hill, 2000. [26] A. V. Dandawate and G. B. Giannakis, “Statistical test for presence of cyclostationarity," IEEE Trans. Sig.
Proc., vol. 42, pp. 2355-2369, 1994. [27] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover Publications, 1972. [28] L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework.
Prentice Hall, 1993.
23
APPENDIX A: CYCLOSTATIONARITY OF RECEIVED OFDM SIGNALS
The expressions for the nth-order (q-conjugate) CC and set of CFs for the received baseband
OFDM signal are derived here. With the received baseband OFDM signal as in (20), and by using
the multi-linearity property of the cumulants [28], the time-varying nth-order (q-conjugate)
cumulant of OFDM ( )r t can be expressed as follows 1,
OFDM OFDM OFDM OFDM OFDM, 1 1( ; ) Cum[ ( ), , ( ), ( ), , ( )],r n q n q n q nc t r t r t r t r t (30)
1
1 1 1 1
1 11 1 1 1
1 1
2 ( )2 ( 2 )( 2 )
2 ( ), , , ,
2 ( ) 2 (
Cum[ , , , , , ] N
n
c p ppc
n q n q n q n q n n
n q n q n q n q n n
N n q n q N
j fj n q f tn j n q
k l k l k l k l
j k f t l T Tk l k l k l k l
j k f t l T T j k f t
a e e e
s s s s e
e e ) 2 ( )
1 1 1 1
,
( ) ( )
( ) ( ) ( ; )
n q n q N n nl T T j k f t l T T
n q n q
n q n q n n w n q
e
g t l T T g t l T T
g t l T T g t l T T c t
, (31)
where denotes conjugation.
In the following derivations we consider only the cumulant of the signal component 4. As the data
symbols ,{ }k ls on each subcarrier , 1, , ,k k N are i.i.d. and mutually independent for different
subcarriers, 1 1 1 1, , , ,Cum[ , , , , , ] 0
n q n q n q n q n nk l k l k l k ls s s s unless 1 nk k k and 1 nl l l .
Under such conditions, , , , ,Cum[ , , , , , ]k l k l k l k ls s s s becomes the nth-order (q-conjugate) cumulant
for the signal constellation, , ,s n qc , and (31) can be further written as
1 1
OFDM
1
2 ( ) 2 ( )12 ( 2 )( 2 )
, , ,0
( )2 ( 2 ) ( )
1
2 ( )2 ( 2 )( 2 )
, ,
( ; )
( )
N
n n
c p p p pp pc
pN
n
c p pp c
j f j k fNj n q f tn j n q
r n q s n qk
nj n q k f t lT T
pl p
j f jj n q f tn j n q
s n q
c t a c e e e e
e g t lT T
a c e e e e 12 ( )1
0
( )2 ( 2 )
1( ) ( )
N
n
p pp
pN
k fN
k
nj n q k f t
plp
e g t t lT T
. (32)
4 The cumulant of the noise component has to be added to the final result.
24
The Fourier transform of the nth-order (q-conjugate) time-varying cumulant of the received
baseband OFDM signal can be expressed as
OFDM OFDM
1 1
2, ,
2 ( ) 2 ( )12 ( 2 )( 2 )
, ,0
( )2 ( 2 ) 2
1
{ ( ; ) } ( ; )
[
( ) ( )]
n n
c p p N p pp pc
pN
j tr n q r n q
j f j k fNj n q f tn j n q
s n qk
nj n q k f t j t
plp
c t c t e dt
a c e e e e
e g t t lT T e dt
, (33)
where {} denotes the Fourier transform.
By using the convolution theorem, (33) can be written as
1 1OFDM
2 ( ) 2 ( )12 ( 2 )( 2 )
, , ,0
( )2 ( 2 ) 2
1
{ ( ; ) }
( ) ( )
n n
c p p N p pp p c
pN
j f j k fNj n q f tn j n q
r n q s n qk
nj n q k f u j t
plp
c t a c e e e e
e g u t u lT T e dudt
. (34)
With the change of variables t u T v and u u , and by using the identity
1 1{ ( )} ( )l l
t llT T T , one can easily show that
1 1OFDM
2 ( ) 2 ( )12 ( ( 2 ) )1 ( 2 )
, , ,0
( )2 ( 2 ) 2 ( ( 2 ) ) 1
1
{ ( ; ) }
( ) ( ( 2 ) )
n n
c p p N p pp pc
pN c
j f j k fNj n q f Tn j n q
r n q s n qk
nj n q k f u j n q f u
p clp
c t a c e e e e
e g u e du n q
T
f lT
.
(35)
It can be seen that OFDM ,{ ( ; ) } 0r n qc t only if 1 ( 2 ) clT n q f , with l as an integer. By using
the notations 1lT and u=t, (35) can be written as
1 1OFDM
2 ( ) 2 ( )11 ( 2 ) 2
, , ,0
( )2 ( 2 ) 2 1
1
{ ( ; ) }
( ) ( ( 2 ) )
n n
c p p N p pp p
pN
j f j k fNn j n q j T
r n q s n qk
nj n q k f t j t
p clp
c t a c T e e e e
e g t e dt n q f lT
. (36)
25
By taking the inverse Fourier transform of (36), one can easily show that OFDM ,( ; )r n qc t can be
expressed as 4
OFDM
2,
{ }( ; ) ,j t
r n qc t B e (37)
where { } denotes the set 1{ | ( 2 ) , , integer}cn q f lT l , and B is the coefficient
corresponding to frequency in the Fourier series expansion of the time-varying cumulant.
This implies that the cycle frequency domain is discrete, and the spectrum consists of a set of
finite-strength additive components. By using (2) and (37), one can easily notice that the nth-order
(q-conjugate) CC at CF 4, and the set of CFs are respectively given as
OFDM
1 12 ( ) 2 ( )1
1 ( 2 ) 2, , ,
0
( ) 2 ( 2 ) 2
1
( ; )
( )
n n
c p p N p pp p
p N
j f j k fNn j n q j T
r n q s n qk
nj n q k f t j t
pp
c a c T e e e e
g t e e dt
, (38)
and
1, { | ( 2 ) , , integer}c
n q cn q f lT l . (39)
If no aliasing occurs, the nth-order (q-conjugate) CC 4 and the set of CFs for the discrete-time OFDM
signal can be derived by using (9) and (10), which leads to the results given in (23) and (24),
respectively.
APPENDIX B: A CYCLOSTATIONARITY TEST
A cyclostationarity test, which is developed in [26], is presented here for n=2 and q=1. This is
used in Step 2 of the proposed classification algorithm, for decision making. With this test, the
presence of a CF is formulated as a binary hypothesis-testing problem, i.e., under hypothesis 0H
the tested frequency is not a CF at delay , and under hypothesis 1H the tested frequency is a
CF at delay . The cyclostationarity test consists of the following three steps.
26
Step 1:
The second-order (one-conjugate) CC at tested frequency and delay is estimated from L
samples, and a vector 2,1c is formed as
2,1 2,1 2,1ˆ ˆ ˆ[Re{ ( ; ) } Im{ ( ; ) }]r rc cc , (40)
where Re{} and Im{}are the real and imaginary parts, respectively.
Step 2:
A statistic 2,1T is computed for the tested frequency and delay ,
1 †2,1 2,1 2,1 2,1
ˆˆ ˆ L c cT , (41)
where 1 denotes the matrix inverse and 2,1ˆ is an estimate of the covariance matrix
2,0 2,1 2,0 2,12,1
2,0 2,1 2,1 2,0
Re{( ) / 2} Im{( ) / 2}Im{( ) / 2} Re{( ) / 2}
Q Q Q QQ Q Q Q
, (42)
with
2,0 2,1 2,1ˆ ˆlim Cum[ ( ; ) , ( ; ) ]r rLQ c c (43)
and
2,1 2,1 21ˆ ˆlim Cum[ ( ; ) , ( ; ) ]r rLQ c c . (44)
The covariances 2,0Q and 2,1Q are given respectively by [26] 5
11 2 2 2
2,0 2,1 2,10
lim Cum[ ( ; ), ( ; )]L
j l j
L l
Q L f l f l e e (45)
and
11 * 2 ( )
2,1 2,1 2,10
lim Cum[ ( ; ), ( ; )]L
j l
L l
Q L f l f l e , (46)
where 2,1( ; ) ( ) ( )f l r l r l is the second-order (one-conjugate) lag product.
5 These equations are valid for zero-mean processes. For the covariance estimators see, e.g. [26], eq. (48).
27
Step 3:
The test statistic 2,1T , calculated for the tested frequency and delay , is compared against a
threshold . If 2,1T , we decide that the tested frequency is a CF at delay ; otherwise not.
The threshold is set for a given probability of false alarm, fP , which is defined as the probability
to decide that the tested frequency is a CF at tested delay , when it is actually not. This can be
expressed as 2,1 0Pr{ | }fP HT . By using that the statistics 2,1T has an asymptotic chi-square
distribution with two degrees of freedom under the hypothesis 0H [24], the threshold is obtained
from the tables of the chi-squared distribution for a given probability of false alarm, fP .
28
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Unlimited
13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.)
In recent years, new technologies for wireless communications have emerged. The wirelessindustry has shown great interest in orthogonal frequency division multiplexing (OFDM)technology, due to the efficiency of OFDM schemes to convey information in a frequencyselective fading channel without requiring complex equalizers. On the other hand, the emergingOFDM wireless communication technology raises new challenges for the designers of intelligentradios, such as discriminating between OFDM and single-carrier modulations. In this report weinvestigate signal cyclostationarity to discriminate between OFDM and single carrier linear digital(SCLD) modulations. We derive the analytical expressions for the nth-order (q-conjugate) cycliccumulant (CC) and cycle frequencies of a received baseband OFDM signal, and propose aclassifier based on the second-order CC to discriminate between the two aforementioned signalclasses. Simulations are carried out to confirm the theoretical developments.
14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)
Cyclic Cumulant, Cycle Frequency, Orthogonal Frequency Division Multiplexing, Modulation Classification, Probability of Correct Classification, Signal Cyclostationarity
