On the Cyclostationarity of OFDM and Single Carrier Linearly Digitally Modulated Signals in Time Dispersive Channels with Applications to Modulation Recognition O. A. Dobre and A. Punchihewa Memorial University of Newfoundland St. John’s, NL A1B 3X5, Canada (dobre, anjana@engr.mun.ca) S. Rajan and R. Inkol Defence Research and Development Canada Ottawa, Ontario, K1A 0Z4, Canada (sreeraman.rajan, robert.inkol@drdc-rddc.gc.ca) Abstract—This paper studies the nth-order cyclostationarity of orthogonal frequency division multiplexing (OFDM) and single carrier linearly digitally modulated (SCLD) signals affected by a time dispersive channel, additive Gaussian noise, carrier phase, and frequency and timing offsets. The analytical closed-form expressions for the nth-order cyclic cumulants (CCs) and cycle frequencies (CFs) of OFDM and SCLD signals are derived. Furthermore, a second-order CC-based algorithm is developed to recognize OFDM against SCLD signals under the afore- mentioned conditions. This algorithm obviates the need for signal preprocessing tasks, such as symbol timing estimation, carrier and waveform recovery, and signal and noise power estimation. Simulation experiments confirm the theoretical analysis. Keywords: Cyclic cumulants, Cycle frequencies, Cyclostationarity test, Modulation recognition. I. INTRODUCTION Signal cyclostationarity represents a powerful tool, which finds applications to different areas of communications, such as frequency and timing recovery [1]-[3] and modulation recognition [3]-[6]. Cyclostationarity of diverse communication signals was studied in [2]-[4], [7]-[8]; however, these studies were performed either for simplified signal and channel models or for second-order cyclostationarity. A contribution of this paper is the theoretical analysis of the nth-order cyclostationarity of orthogonal frequency division multiplexing (OFDM) and single carrier linearly digitally modulated (SCLD) signals affected by time dispersive channel, additive Gaussian noise, carrier phase, and frequency and timing offsets. Closed-form expressions are derived for the nth-order cyclic cumulants (CCs) and cycle frequencies (CFs) of such signals, and, furthermore, the applicability to modulation recognition (MR) is illustrated. MR is of importance in military and commercial applications, such as electronic warfare, and spectrum monitoring and management. Although the MR topic has been extensively studied (see the comprehensive survey [9]), the recognition of the OFDM signal has been investigated only in recent years. Algorithms for discriminating between OFDM and SCLD signals have been reported in [6], [10]- [12]. The algorithms proposed in [10] and [11]-[12] require estimation of signal-to-noise ratio and symbol period, respectively. In [6], we proposed a cyclostationarity-based algorithm which does not require such preprocessing tasks. However, this was developed under the simplified assumption of additive white Gaussian noise (AWGN) channel. Based on the new theoretical developments presented in this paper, we extend the applicability of the algorithm to recognize OFDM versus SCLD signals under more realistic conditions. The rest of the paper is organized as follows. The OFDM and SCLD signal models and analytical closed-form expressions for the n th-order CCs and CFs are presented in Sections II and III, respectively. The proposed recognition algorithm is introduced in Section IV, and simulation results are discussed in Section V. Finally, conclusions are drawn in Section VI. Fundamental concepts of signal cyclostationarity and derivations of the analytical closed-form expressions for the n th-order CCs and CFs of OFDM and SCLD signals under afore-mentioned conditions are presented in Appendices A and B, respectively. A cyclostationarity test used for decision making with the MR algorithm is introduced in Appendix C. II. SIGNAL MODELS The continuous-time baseband equivalents of received OFDM and SCLD signals affected by time dispersive channel, additive Gaussian noise, phase, and carrier frequency and timing offsets as given in [13]-[14], are OFDM 1 2 ∆ 2 ∆ ( ε ) θ , 0 1 () ( ) c K m K M j ft j k f t lT T j kl m k l m r t ae e s h e - ∞ π π -ζ - - = =-∞ = = ζ ∑∑∑   ( ε ) ( ), m gt lT T wt × -ζ - - +  (1) and SCLD 2 ∆ θ 1 () ( )( ε ) ( ), c M j ft j l m m l m r t ae e sh gt lT T wt ∞ π =-∞ = = ζ -ζ - - + ∑∑   (2) where a is the amplitude factor, θ is the phase, ∆ c f is the carrier frequency offset, T is the symbol period, 0 ε 1 ≤ ≤ is the timing offset, () g t is the impulse response of the transmit and receive filters in cascade, ( ) m h ζ  is the channel coefficient at time m ζ  , 1, , m M = … , K is the number of subcarriers, ∆ K f is the frequency separation between two adjacent subcarriers, l s and , kl s represent the symbols transmitted within the l th period and the l th period and kth subcarrier, respectively, and () wt is the zero-mean complex Gaussian noise. The data symbols {} l s and , { } kl s are assumed to be zero-mean independent and identically distributed (i.i.d.) random variables, with values drawn either from a quadrature amplitude modulation (QAM) or phase shift keying (PSK) constellation. For the OFDM signal, the symbol period is given by u cp T T T = + , with 1/∆ u K T f = as the useful symbol duration and cp T as the length of the cyclic prefix. One can easily see that (2) is a particular case of (1), obtained for 1 K = and 0 cp T = . A discrete-time baseband signal, SCLD () r u , is obtained by oversampling SCLD () r t at a rate 1 s f T - =ρ , where ρ is an integer representing the number of samples per symbol (oversampling factor). Similarly, a discrete-time baseband OFDM signal, OFDM () r u , is obtained by oversampling OFDM () r t at a rate 1 ρ s u f KT - = , with ρK as a positive integer which represents the number of samples in the useful symbol duration, and ρ as the number of samples per symbol per subcarrier 1525-3511/08/$25.00 ©2008 Crown Copyright This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.