Identification of SM-OFDM and AL-OFDM Signals Based on Their Second-Order Cyclostationarity

Identification of SM-OFDM and AL-OFDM Signals Based on Their Second-Order Cyclostationarity

Ebrahim Karami,  and Octavia Dobre,

Department of Engineering and Applied Sciences, Memorial University of Newfoundland, St. John’s, Canada Email: {ekarami,odobre}@mun.ca
Abstract

Automatic signal identification (ASI) has important applications to both commercial and military communications, such as software defined radio, cognitive radio, spectrum surveillance and monitoring, and electronic warfare. While ASI has been intensively studied for single-input single-output systems, only a few investigations have been recently presented for multiple-input multiple-output systems. This paper introduces a novel algorithm for the identification of spatial multiplexing (SM) and Alamouti coded (AL) orthogonal frequency division multiplexing (OFDM) signals, which relies on the second-order signal cyclostationarity. Analytical expressions for the second-order cyclic statistics of SM-OFDM and AL-OFDM signals are derived and further exploited for the algorithm development. The proposed algorithm provides a good identification performance with low sensitivity to impairments in the received signal, such as phase noise, timing offset, and channel conditions.

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Automatic signal identification (ASI), multiple-input multiple-output (MIMO), space-time block code (STBC), orthogonal frequency division multiplexing (OFDM), cyclostationarity, cyclic correlation function (CCF), cycle frequency (CF).

Ii System Model

The baseband equivalent block diagram of a MIMO-OFDM transmitter is presented in Fig. 1. The input signal is a stream of data blocks, , where each block contains independent and identically distributed (i.i.d.) symbols drawn from either an -ary quadrature amplitude modulation (QAM) or phased-shift-keying (PSK) signal constellation, . Two transmit antenna () are considered, and, accordingly, the data stream is demultiplexed into two sub-streams. Such sub-streams are fed into the MIMO encoding block, which in this work is either SM or AL. Hence, the th group of two data blocks, , is encoded according to a code matrix of size , in order to be transmitted during block instants [38]. Note that for SM and for AL. The code matrices and corresponding to the SM111Note that is considered for SM; however, the identification algorithm is applicable for , as well. and AL encoders are respectively given by [36, 38]

 (1)

and

 (2)

where represents the data block to be transmitted from the th antenna, , at block instant , , and * denotes complex conjugation.

The output of the MIMO encoder is fed into the inverse fast Fourier transform (IFFT) block, yielding the OFDM symbol as

 x(f)Uk+u(n)=1√NN−1∑n1=0c(f)Uk+u(n1)ej2πnn1N,n=0,1,...,N−1, (3)

where is the th element of , .

The cyclic prefix (CPR), which represents a copy of the last samples of the OFDM symbol, is then added. Windowing is also applied; this increases the CPR to , where is the number of samples in the transition time between two consecutive OFDM symbols [39]. Furthermore, the first samples of the OFDM symbol are transmitted after the effective part of the symbol, during the next transition time, as a cyclic postfix (CPO) [39]. By taking into account the CPR, CPO, and windowing, the OFDM symbol is expressed as , with

 z(f)Uk+u(n)=Wnx(f)Uk+u(~n),n=−ν,...,N+NW−1,f=0,1,u=0,...,U−1, (4)

where , with , represent the window coefficients222Note that the commonly used raised-cosine window is considered in this work. [39] and .

Finally, the transmit sequence from the th antenna, , is expressed as

 s(f)(m)=+∞∑k=−∞U−1∑u=0N+NW−1∑n=−νz(f)Uk+u(n)δ(m−(Uk+u)(N+ν)−n), (5)

where is the Kronecker delta function equal to one if and zero otherwise. The transmit sequence from the th antenna propagates through an unknown frequency-selective wireless channel. Hence, the th sample of the signal received at the th receive antenna, , can be expressed as

 r(v)(m)=1∑f=0Lp−1∑p=0hvf(p)s(f)(m−ϑ(p))+w(v)(m), (6)

where is the number of propagation paths, is the channel coefficient corresponding to the th path between the th transmit and th receive antenna, is the propagation delay corresponding to the th path, and is the additive white Gaussian noise (AWGN) with variance . Subsequently, we will develop an algorithm to identify the SM-OFDM and AL-OFDM signals from the received sequences , , where is the number of receive antennas.

Iii Second-Order Cyclostationarity-Based SM-OFDM and AL-OFDM Signal Identification

In this section, the CCF and its corresponding CFs are derived for the SM-OFDM and AL-OFDM signals and employed to develop a novel feature-based signal identification algorithm.

Iii-a Cyclostationarity Concept

The received sequences and 333Note that the case of two receive antennas () is considered; later in the paper, will be also studied. exhibit second-order cyclostationarity if their first444Note that due to the symmetry in the signal constellations, the first-order statistics equal zero. and second-order time-varying correlation functions are periodic in time [40]. Here we consider the non-conjugate second-order time-varying cross-correlation function, defined as

 c(m,τ)=E[r(0)(m)r(1)(m+τ)], (7)

where is the statistical expectation and is the delay. If is periodic in with the fundamental period , then it can be expressed by a Fourier series [40],

 c(m,τ)=∑{α}C(α,τ)ej2πmα, (8)

where the coefficients

 C(α,τ)=1M0∑mc(m,τ)e−j2πmα, (9)

are referred to as the CCF at CF and delay , and the set of CFs is given by .

Iii-B Analytical Expressions for the CCF of the SM-OFDM and AL-OFDM Signals

The analytical expressions for the CCF and its corresponding CFs are derived here for the SM-OFDM and AL-OFDM signals. Results are obtained by following the commonly used assumptions that [29, 30, 31, 32, 33, 34, 35, 36]: a) the transmitted sequences are uncorrelated with the noise: , , and ; b) the noise in each channel is uncorrelated with that of the other channels: , , , and ; c) the data symbols are uncorrelated with each other: , , , , and , where is the transmit signal power; and d) the channel gains for each transmit-receive antenna link remain constant over the observation interval.

SM-OFDM

By using (1), (3), and (4), one can easily show that

 E[z(f0)k0(n0)z(f1)k1(n1)]=0, (10)

. Furthermore, based on (6), (7), and (10), it can be obtained that the time-varying cross-correlation function of the SM-OFDM signals is zero, i.e.,

 c(SM)(m,τ)=0,∀m,τ. (11)

Consequently, from (9) and (11), it can be seen that

 C(SM)(α,τ)=0,∀α,τ. (12)

AL-OFDM

By using (2), (3), (4), the complex conjugation property of the Fourier transform, and following [36], one can obtain

 E[z(f0)2k0+u0(n0)z(f1)2k1+u1(n1)]=
 (13)

In other words, this correlation is non-zero for adjacent OFDM symbols within the same AL block (due to the structure of the AL coding matrix), and for samples within such OFDM symbols which satisfy the condition . By using (6) and (7), the time-varying cross-correlation function of the received AL-OFDM signal is expressed as (the proof is provided in Appendix A)

 c(AL)(m,τ)=Lp−1∑p0,p1=0(h00(p0)h11(p1)−h01(p0)h10(p1))+∞∑k=−∞δ(m−2k(N+ν)−ϑ(p0))⊗ N+NW−1∑n0,n1=−νσ2sWn0Wn1δ(mod(n0+n1,N))(δ(m−n0−ϑ(p0)) (14) δ(τ−(N+ν)+n0−n1+ϑ(p0)−ϑ(p1)) −δ(m−(N+ν)−n0−ϑ(p0))δ(τ+(N+ν)+n0−n1+ϑ(p0)−ϑ(p1))),

where is the convolution operator. From (14), one can see that the time-varying cross-correlation function is periodic in with the fundamental period , which proves that the AL-OFDM signal exhibits second-order cyclostationarity. Furthermore, also according to the results in Appendix A, the time-varying cross-correlation function of the received AL-OFDM signal for the special case of flat fading channel can be easily obtained as

 c(AL)(m,τ)=(h00h11−h01h10)+∞∑k=−∞δ(m−2k(N+ν))⊗N+NW−1∑n0,n1=−νσ2sWn0Wn1 δ(mod(n0+n1,N))(δ(m−n0)δ(τ−(N+ν)+n0−n1) (15) −δ(m−(N+ν)−n0)δ(τ+(N+ν)+n0−n1)).

By calculating the Fourier coefficients of in (15), one can easily show that the CCF of the received AL-OFDM signal affected by flat fading channel is expressed as

 C(AL)(α,τ)=h00h11−h01h102(N+ν)σ2sN+NW−1∑n0,n1=−νWn0Wn1δ(mod(n0+n1,N)) (16) (δ(τ−(N+ν)+n0−n1)e−j2παn0−δ(τ+(N+ν)+n0−n1)e−j2πα(N+ν+n0)),

where the corresponding CFs are given as . By considering the conditions imposed by the three Kronecker delta functions on the right-hand side of (16) and after some mathematical manipulations, the CCF can be further expressed as (the proof is provided in Appendix B)

 C(AL)(α,τ)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩sgn(τ)g1(τ)e−jπα(2N+ν−τ),|τ|∈I1∩Ic0,sgn(τ)1∑q=0gq(τ)e−jπα((q+1)N+ν−τ),|τ|∈I0∩Ic2,sgn(τ)2∑q=0gq(τ)e−jπα((q+1)N+ν−τ),|τ|∈I2,0,otherwise, (17)

where is the signum function, is the absolute value, represents the intersection operator, the superscript denotes the set complement, , and , , and are defined as

 I0={N−ν,N−ν+2,...,N+3ν−2,N+3ν}, (18)
 I1={ν−2NW+2,ν−2NW+4,...,2N+ν+2NW−4,2N+ν+2NW−2}, (19)
 I2={N+ν−2NW+2,N+ν−2NW+4,...,N+ν+2NW−4,N+ν+2NW−2}. (20)

From (17), one can notice that there are three regions of for which the CCF is non-zero; in these regions, CCF consists of one term (when ), two terms (when ), and three terms (when ), respectively. Based on the results in (12) and (17), it is clear that CCF represents a discriminating feature for the SM-OFDM and AL-OFDM signals; in the sequel, this will be exploited to develop a signal identification algorithm.

Iii-C Proposed Algorithm for SM-OFDM and AL-OFDM Signal Identification

Two Receive Antennas () Case: The block diagram of the proposed identification algorithm is presented in Fig. 2. In the first step, the CCF is estimated at CFs , with , and different values of for which CCF is non-zero for the AL-OFDM signals (the details on the delay values are provided later on in the paper). The estimate of the CCF at CF and delay is [41]

 ^C(α,τ)=1MrMr−1∑m=0r(0)(m)r(1)(m+τ)e−j2παm, (21)

where is the number of received samples, equal to , with as the number of OFDM symbols.

In the second step, the estimated CCF magnitude is compared with a threshold set up based on a constant false alarm criterion. The probability of false alarm is defined as the probability of identifying the received signal as AL-OFDM while it is SM-OFDM. An analytical closed form expression of the false alarm probability is obtained based on the distribution of the CCF magnitude estimate for the SM-OFDM signals. According to [41], the CCF estimate has an asymptotic complex Gaussian distribution. Consequently, based on (12), one can further infer that the CCF magnitude estimate of the SM-OFDM signal has an asymptotic Rayleigh distribution. Hence, if the CCF for a single CF and delay is used as a discriminating feature, the probability of false alarm is calculated using the complementary cumulative density function of the Rayleigh distribution as

 Pf=exp(−Γ2σ2), (22)

where is the threshold and is the variance of the CCF magnitude estimate for the SM-OFDM signal555Note that the variance can be estimated based on the CCF magnitude estimate of the received signal at any arbitrary CF and delays . In such cases, the distribution of the CCF magnitude estimate is the same regardless of the received signal type, CF, and delay.. When the CCF at various and values is used for identification, the -out-of- rule is employed for decision making, i.e., if out of estimated CCF magnitudes exceed the threshold, the signal is identified as AL-OFDM; otherwise, it is identified as SM-OFDM (see the third block in Fig. 2, where is the final decision made by the algorithm). In this case, the probability of false alarm is666Note that (23) is written under the assumption that the estimated CCF of the SM-OFDM signal is uncorrelated for different CFs and delays. The validity of this assumption was verified through extensive simulations.

 PF=ζ∑ℓ=κ(ζℓ)Pℓf(1−Pf)ζ−ℓ. (23)

In this case, for a constant false alarm rate , is calculated using (23), and then the threshold value is obtained from (22). The remaining problem is the selection of the parameters and . From (17), one can notice that for each value of , the CCF has a larger magnitude when (the second and third branches of (17)). When a single CF is used, we consider 777Note that the closed-form expression for the CCF at CF and delay in (17) was obtained under the assumption of flat fading channel. For the frequency-selective fading channel, in addition to the set of delays , we also include the set . For the reason of additionally considering this set of delays, the reader is referred to ”Analytical and simulation results for the CCF magnitude,” as well as Figs. 4 and 5 in Section IV., which leads to . Furthermore, when three CFs are used, . After extensive simulations run under various scenarios, we selected , as providing a good performance. A summary of the proposed algorithm is provided as follows.

Computational complexity: We evaluate the computational complexity of the proposed identification algorithm through the number of floating point operations (flops) [42], where a complex multiplication and addition require six and two flops, respectively. According to the algorithm description, CCF is estimated for combinations of CFs and delays. Based on (21), one can easily see that the number of complex multiplications and additions required to calculate the CCF at a certain CF and delay equals to and , respectively. By considering that the thresholding does not require additional complexity, it is straightforward that the number of flops needed by the algorithm equals to . It is worth noting that with an average Intel Core i750, the identification process takes sec for , , and , whereas with an IBM Sequoia supercomputer, this time reduces to sec.

Number of Receive Antennas Case: Here we extend the proposed algorithm to the case of . For each pair of receive antennas, , , , we define the corresponding CCF. Consequently, for each pair of receive antennas, the CCF is estimated using (21) at the CFs and delays considered for the case of . Then, with the values of and in (23) scaled by , the same steps are applied as for the case of . Note that represents the number of different combinations of two received antennas. It is easy to notice that the complexity of the proposed algorithm for the case is also scaled by .

Iv Simulation Results

In this section, we compare the analytical and simulation results for the CCF magnitude, investigate the performance of the proposed algorithm, and compare it with that of the algorithm in [36].

Simulation setup: Unless otherwise indicated, we consider an OFDM signal with quadrature phase-shift-keying (QPSK) modulation, , , a raised-cosine window with , and . The carrier frequency is GHz and the duration of the OFDM symbol is sec. The probability of false alarm is , and the number of OFDM symbols is . The received signal is affected by AWGN with variance and a frequency-selective fading channel consisting of statistically independent taps, each being a zero-mean complex Gaussian random variable. The channel is characterized by an exponential power delay profile, where and is chosen such that the average power is normalized to unity [36]. The SNR is defined as . The probability of correct identification, , with as the decided signal type and , is considered as a performance measure and is evaluated using Monte Carlo simulations with trials.

Analytical and simulation results for the CCF magnitude: Figs. 3 and 4 present analytical and simulation results for the CCF magnitude versus delay , for and at CFs , with . An OFDM signal with , , , and flat Rayleigh fading channel are considered. For simulation results, SNR=10 dB and OFDM symbols. As can be noticed, the analytical and simulation results are in agreement, and as expected, a larger CCF magnitude is observed for (corresponding to the second and third branches on the right-hand side of (17)). Furthermore, Fig. 5 presents the CCF magnitude estimate versus delay for the frequency-selective Rayleigh fading channel. One can see that, when compared to the flat fading channel, there is a dispersion in the CCF magnitude which results in non-zero values for odd delays. Hence, for identification purposes, we considered the CCF magnitudes for the delay range as discriminating signal features (see the formal description of the algorithm in Section III).

Identification performance of the proposed algorithm: Figs. 6 and 7 show the probability of correct identification, , , versus SNR for different numbers of OFDM symbols, , and probability of false alarm, , respectively. As expected, results obtained for are close to regardless of the SNR and . improves as SNR and increase and decreases. This can be easily explained, as the accuracy of the CCF magnitude estimate enhances when a larger SNR and observation period are available, and a lower threshold corresponds to a reduced . According to Fig. 6, approaches to one at 0 dB SNR with , while 8 dB SNR is required for when . On the other hand, only 2 dB SNR is required to reach such a performance for when , as shown in Fig. 7. Additionally, the behaviour of , , as a function of is provided in Fig. 8. Note that since is the same for different values of SNR, in Fig. 8, one curve with solid line is used to show .

Fig. 9 shows the probability of correct identification, , , versus SNR for different numbers of OFDM symbols, , and receive antenna, . According to Fig. 9, one can see that by increasing the number of receive antennas, the performance improves. As such, a certain probability of correct identification is achieved at lower SNR and/or with shorter observation time. For example, is achieved with at SNR = 5.6 dB and -1.8 dB for and , respectively. In other words, by increasing the number of receive antennas from two to three, a 7.4 dB performance gain is obtained. From Fig. 9, one can further see that with and , does not reach , whereas is obtained at SNR= 5.3 dB and -0.2 dB for and , respectively.

Performance comparison with the algorithm in [36]: Fig. 10 shows a performance comparison with the algorithm in [36], for different values of and . One can observe that while both algorithms provide a similar performance in terms of , the proposed algorithm outperforms the one in [36] for , especially at lower SNR.

In the following, we investigate the robustness of the proposed algorithm and the one in [36] to diverse impairments, i.e., phase noise, frequency offset, timing offset, and channel conditions.

Effect of phase noise: The phase noise is modeled as a Wiener process with rate , where is the two-sided 3 dB bandwidth of the Lorenzian distribution spectrum [43]. Fig. 11 shows the probability of correct identification, 888As the impairments in the received signal do not affect the results for the SM-OFDM signal identification, we only show the effect of the signal impairments on the probability of correct identification of AL-OFDM signals., versus SNR for different values of the phase noise rate. As can be seen, the proposed algorithm is relatively robust for , and its performance starts degrading for . This can be explained based on the results obtained in Appendix B for the dependency of the CC magnitude estimate on the phase noise. According to these results, for , the CCF magnitude estimate is scaled with a factor of , , and for , , and , respectively. Clearly, a reduction in the CCF magnitude leads to a performance degradation. Additionally, from Fig. 11, one can see that the proposed algorithm is more robust to the phase noise when compared with the algorithm in [36].

Effect of frequency offset: Fig. 12 shows the probability of correct identification, , versus SNR for different values of the normalized frequency offset, , with as the frequency offset. As one can notice, both the proposed algorithm and the one in [36] are robust for , with the former exhibiting a better performance when compared with the latter.

Effect of timing offset: Fig. 13 shows the probability of correct identification, , versus SNR for different values of the timing offset, . By following [30, 35], the effect of the timing offset was obtained by passing the signal through an two path filter, when the pulse shape is rectangular. As it can be seen from Fig. 12, both the proposed algorithm and the one in [36] are relatively robust to the timing offset, with a better performance provided by the former. As expected, the performance degrades as reaches , and in the lower SNR range.

Effect of channel conditions: We investigate the performance of the proposed algorithm and the one in [36] in the pedestrian and vehicular A fading channels [44]. The maximum Doppler frequency for the pedestrian channel was Hz, while the maximum Doppler frequency was Hz for the vehicular channel. With the proposed algorithm, the channel dispersion is beneficial for identification, as introducing additional CCF peaks (see results showed in Figs. 4 and 5). As such, as can be seen from Fig. 14, despite a larger , the identification performance for the vehicular A channel is slightly better than that for the pedestrian A case. Also, according to Fig. 14, both algorithms provide a good and similar performance under both channel conditions.

V Conclusions

In this paper, we proposed a second-order cyclostationarity-based discriminating feature for SM-OFDM and AL-OFDM signals, along with a signal identification algorithm. The proposed algorithm provides a reasonable performance at relatively low SNR and with a short observation period. Furthermore, it is relatively robust to the phase noise, timing offset, and channel conditions, and outperforms the algorithm existing in the literature. As part of future work, the analysis and identification algorithm presented in this paper are planned to be extended to additional STBCs.

Acknowledgment

The authors would like to acknowledge the constructive comments and suggestions of the Editor, Professor Jia-Chin Lin, as well as the anonymous reviewers.

Appendix A

Here, we present the proof of (14). By replacing (6) into (7), the time-varying cross-correlation function of the received AL-OFDM signal is expressed as

 c(AL)(m,τ)=Lp−1∑p0,p1=0h00(p0)h10(p1)E[s(0)(m−ϑ(p0))s(0)(m+τ−ϑ(p1))] +h00(p0)h11(p1)E[s(0)(m−ϑ(p0))s(1)(m+τ−ϑ(p1))] +h01(p0)h10(p1)E[s(1)(m−ϑ(p0))s(0)(m+τ−ϑ(p1))] +h01(p0)h11(p1)E[s(1)(m−ϑ(p0))s(1)(m+τ−ϑ(p1))]. (24)

One can easily find that the first and last terms on the right-hand side of (24) are zero. Additionally, by using (5) and (13), it can be shown that . Consequently, if we define as the time-varying cross-correlation function of the transmitted signal, (24) can be re-written as

 c(AL)(m,τ)=Lp−1∑p0,p1=0(h00(p0)h11(p1)−h01(p0)h10(p1))c(AL)s(m−ϑ(p0),τ+ϑ(p0)−ϑ(p1)), (25)

where

 c(AL)s(m,τ)=+∞∑k0,k1=−∞1∑u0,u1=0N+NW−1∑n0,n1=−νE[z(0)2k0+u0(n0)z(1)2k1+u1(n1)] (26) δ(m−(2k0+u0)(N+ν)−n0)δ(m+τ−(2k1+u1)(N+ν)−n1).

Furthermore, based on (13), (26) becomes

 c(AL)s(m,τ)=+∞∑k=−∞N+NW−1∑n0,n1=−νσ2sWn0Wn1δ(mod(n0+n1,N))(δ(m−2k(N+ν)−n0) (27) δ(m+τ−(2k+1)(N+ν)−n1)−δ(m−(2k+1)(N+ν)−n0)δ(m+τ−2k(N+ν)−n1)),

which can be further expressed as

 c(AL)s(m,τ)=+∞∑k=−∞δ(m−2k(N+ν))⊗N+NW−1∑n0,n1=−νσ2sWn0Wn1δ(mod(n0+n1,N)) (28) (δ(m−n0)δ(m+τ−(N+ν)−n1)−δ(m−(N+ν)−n0)δ(m+τ−n1)),

where is the convolution operator. Finally, by using the properties of the Kronecker delta function, (28) can be re-written as

 c(AL)s(m,τ)=+∞∑k=−∞δ(m−2k(N+ν))⊗N+NW−1∑n0,n1=−νσ2sWn0Wn1δ(mod(n0+n1,N)) (29) (δ(m−n0)δ(τ−(N+ν)+n0−n1)−δ(m−(N+ν)−n0)δ(τ+(N+ν)+n0−n1)).

Finally, from (25) and (29), (14) can be easily obtained.

Appendix B

Here, we derive the expression of the CCF for the AL-OFDM signals, which is given in (17). Due to the three Kronecker delta functions on the right-hand side of (16), the two summations over and are taken over a few non-zero terms only. For non-zero terms in (16), and either or .

if and only if , where . Since , , and , one can easily see that , and consequently, . As such, we have the following set of constraint linear equations to solve

 ⎧⎪⎨⎪⎩n0+n1=qN,n1−n0=τ−ρ(N+ν),subject to−ν≤n0,n1≤N+NW−1,q=0,1,2,ρ=−1,+1, (30)

where and correspond to and , respectively. For given values of and , and are simply obtained from (30) as

 n0=qN−τ+ρ(N+ν)2, (31)
 n1=qN+τ−ρ(N+ν)2. (32)

For each value of , the values of are obtained as follows.

A. Values of for : For , from the first two equations of (30), one can easily see that

 τ=−2n0+ρ(N+ν). (33)

Since and considering that , it is straightforward to find that takes integer values in the range . Consequently, based on (33), when , and when , . We can compactly write these results as , where .

B. Values of for : For , from the first two equations of (30), one can easily see that

 τ=N−2n0+ρ(N+ν). (34)

Since and considering that , it is straightforward to find that takes integer values in the range . Consequently, based on (34), when , and when , . We can compactly write these results as , where