Robust QAM Modulation Classification Algorithm Using Cyclic Cumulants Octavia A. Dobre and Yeheskel Bar-Ness
Wei Su
New Jersey Institute of Technology Newark, USA (octavia.a.dobre, yeheskel.barness)@njit.edu
U.S. Army CECOM Fort Monmouth, USA
[email protected] threshold. Moreover, GLRT can lead to equal likelihoods for nested constellations (e.g., 4-QAM, 16-QAM and 64-QAM) [4]. In the PR approach, on the other hand, several distinct features of a modulation, normally chosen in an ad-hoc way, are used to distinguish between different modulation types. Although a PR-based method may not be optimal, it is usually simple to implement, with near-optimal performance, when designed properly. Furthermore, it can be robust w.r.t. model mismatches. Relevant AMC work employing cyclic cumulantbased features is briefly reviewed: - Marchand et al. [5] proposed a combination of fourth- and second-order CC magnitudes for QPSK and QAM signal classification,- Spooner [6] employed CC based features with order up to sixth for PSK and QAM signal classification, - In [7] eight-order CCs were employed to discriminate real- and complex-valued signal constellations, respectively. In the afore-mentioned works complete model match was assumed. Hence, performance degradation due to mismatches was not studied.
Abstract—In this paper we develop an algorithm based on higherorder cyclic cumulants for the automatic recognition of QAM signals. The method is robust to the presence of carrier phase and frequency offsets. Theoretical arguments are verified with simulations performed for 4-QAM and 16-QAM signals. Keywords-automatic modulation cumulants; model mismatches.
I.
classification;
cyclic
INTRODUCTION
Automatic modulation classification (AMC) has obvious relevance to military communication systems. In a noncooperative environment the recognition of the modulation format is a difficult task, as no prior knowledge of the incoming signal is available. The design of a classifier essentially involves two steps: preprocessing of the incoming signal and proper selection of the classification algorithm. The required accuracy in preprocessing depends on the classification algorithm chosen in the second step. For the classification part there are two approaches: the decisiontheoretic with maximum-likelihood (ML) [1]-[4] and pattern recognition (PR) [5]-[8] methods. The former casts AMC as a multiple composite hypothesis-testing problem and involves likelihood ratio tests. When the actual probability density function (pdf) of the unknown parameters is identical to the hypothesized one, the average likelihood ratio test (ALRT) provides an optimal solution, in the sense that it minimizes the probability of false classification. Wei and Mendel [1] investigated ALRT to identify linearly modulated signals under ideal conditions, i.e., additive white Gaussian noise (AWGN) and preprocessing tasks perfectly accomplished. The method suffers from lack of robustness with respect to (w.r.t) model mismatches, e.g., carrier frequency and phase offsets, nonGaussian noise distribution, etc. A close form of the optimal solution is difficult to be obtained if the number of unknown parameters taken into consideration when designing the classifier increases. Polydoros et al. [2]-[4] proposed quasioptimal classification algorithms with the carrier phase and timing epoch as unknown parameters. However, this method is inefficient to identify QAM signals [3]. With the carrier phase as unknown parameter, the generalized likelihood ratio test (GLRT) and hybrid likelihood ratio test (HLRT) were investigated for QAM signal recognition in [4]. Both algorithms perform a discrete search over the carrier phase to compute the likelihood function and employ an empirically set
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In this paper robust higher-order CC based features are proposed for the automatic recognition of QAM signals, and classification performance degradation due to carrier phase and frequency offsets, and impulsive noise are examined. It is shown that the proposed classifier outperforms that proposed in [7] under a non-ideal scenario. The paper is organized as follows. Section II introduces the signal model and corresponding statistical characterization. In addition, robust discriminating features for QAM signals are derived from CC analysis. The AMC system and simulation results are presented respectively in Section III and IV and conclusions are drawn in Section V. II.
SIGNAL CHARACTERIZATION AND FEATURE SELECTION
A. Signal model For an M-QAM modulated signal transmitted through an impulsive environment, the baseband waveform at the receiver can be written as
r (t ) = sPN (t ) + wN (t ) + wI (t ) where sPN (t ) = ae j (2 π∆fc t +Φ ( t )) ∑ sk p (t − kT − t0 ) = s (t )e jΦ ( t ) ,
(1)
k
a is the amplitude factor, ∆fc is the residual carrier frequency or frequency offset, Φ(t) is the phase noise, sk=skI+jskQ
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The statistics of wN(t) are time independent due to stationarity and non-zero only for the second order. In considering higher-order (n≥3) CCs the nominal noise contribution was omitted in (3).
represents the symbol transmitted within the kth period, T is the symbol period, t0 is the propagation delay, p(t) is the overall signaling pulse, wN(t) is wide-sense stationary complex additive Gaussian low-pass noise and wI(t) is impulsive correlated noise. The symbol sequence {sk} is a zero-mean independently identically distributed (i.i.d.) sequence, with values drawn from a finite-alphabet constellation, with variance σ 2S . Phase noise (PN) is a result of carrier frequency fluctuation caused by imperfections in the transmitter and receiver oscillators, which is modeled as a Wiener-Lévy process with zero mean and variance 2πBL|t|, where BL represents the 3-dB bandwidth of the Lorentzian power spectrum of the oscillators. BL is commonly referred to as the oscillator linewidth [9]-[10]. Impulsive noise (IN) is a class of non-Gaussian noise, typically characterized by a symmetric first-order probability density function (pdf) with tails heavier than Gaussian, causing occasional impulsive events of a magnitude that may be considerably greater than the background or nominal noise wN(t). The mixture pdf [11] and Poisson impulsive noise [10], [12] are commonly used models for IN. Here we employ the latter model. The limited bandwidth of the receiver filter spreads the impulse events over multiple time samples, hence inducing correlation on this noise. The correlated impulsive noise wI(t) is modeled as a filtered Poisson process [10], [12] wI ( t ) =
CC magnitudes for a zero delay vector τ =0 and β1 =1/T were proved to be suitable signal features for identifying linearly modulated signals [7]. By choosing the CC magnitude the rotation caused by a fixed phase shift and signal delay disappears. From (4) one can notice that the choice of q=n/2 counteracts the effect of frequency offset on CFs. Subsequently we concentrate on separately analyzing the effect of impulsive and phase noises on the CCs of r(t). As a result of this analysis, a new set of features arises, which outperforms that proposed in [7] for QAM signal classification under model mismatches. A steady-state model for impulsive noise is reached when t>τp, where τp is the width of p(rec)(t). Since for a root raised cosine pulse τp can be taken to be infinite, the steady state behavior is approached as t → ∞ . In other words wI(t) is not a stationary process; but asymptotically stationary, with mean, autocorrelation function and nth-order cumulant given by [10], [12]
mwI = mY λP( rec ) (0) 2
N I (t )
∑ Yi δ ( t − ti ) ∗ p( rec) (t ) , t ≥ 0
RwI (τ) = mY λP( rec ) (0) + λ(σY2 + mY2 ) R p( rec ) (τ)
(2)
∞
i =1
cr ( γ k ; τ) n, q = a ∞
n
cs ,n ,q
where E{.} denotes statistical expectation, P(rec)(f) is the transfer function of the receiver filter and R p( rec ) (τ) is the autocorrelation function of p(rec)(t). Theoretically, the computation of the nth-order CCs is based on infinite-time averaging and the steady-state behavior of impulsive noise is reached. Therefore, the impulsive noise contribution is cwI ( τ ) n δ( γ ) , which shows that impulsive noise
×∫ ∏ p
T (*)m
affects the nth-order CCs of the received signal only at CF γ = 0. Using the fact that data symbols and PN are statistically 2 independent and that E{e jx } = e −σx 2 for a zero-mean Gaussian random variable (r.v.) [9], it can be easily proven that the nth-order time-varying moment msPN (t ; 0)n , q is given by
n
e
− j 2 πβk t0
(t + τm )e
e
jθ( n − 2 q )
− j 2 πtβk
e
j 2 π∆fc
dt ,
∑ ( − )m τ m m =1
msPN (t; 0) n , q = ms (t; 0)n ,q e −( n − 2 q )
where τ = [ τ1 ,..., τn −1 , τn ] τ
n
=0
π BL t
(6)
Hence, for q=n/2 the effect of PN on msPN ( t ; 0 )n , q disappears.
(4)
For q ≠ n 2 the Fourier transform of (6) w.r.t. to t can be easily obtained as;
is the delay-vector, cs,n,q is the
F{msPN ( t ; 0 )n , q }
nth-order cumulant (q conjugations) for signal constellation, (*)m is the optional conjugation of the mth term so that the number of conjugations equals q and (-)m is the optional minus sign associated with the optional conjugation (*)m, m=1,…,n.
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2
(3)
−∞ m =1
γ k = βk + (n − 2q)∆f c , βk = k T , k integer
(5)
−∞ m =1
B. Feature selection In the absence of phase and impulsive noises, but taking into account a fixed phase shift θ, the received signal r(t) was proved to be (almost) cyclostationary, with the nth-order (n≥ 3) cyclic cumulants (CCs) and cycle frequencies (CFs) given respectively by [6]-[7]; n
n
cwI ( τ ) n = λE{Yi n } ∫ ∏ p( rec ) (u + τm )du
where Yi is the random area of the ith unit impulse which occurs at a random time instant ti, NI(t) is the number of events during (0,t), p(rec)(t) is the impulse response of the receiver filter and ∗ denotes convolution. The sequence of random event times {ti} constitute a Poisson point process, {NI(t), t≥ 0} is a homogeneous Poisson counting process of intensity λ and the areas {Yi} form an i.i.d. sequence, with mean mY and variance σY2 , independent of the sequence of occurring times {ti}.
= ∑ ms ( γ k ; 0) n, q δ( γ − γ k ) ∗ k
746
1
( n − 2q )
2
πBL + j 2πγ
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= ∑ ms ( γ k ; 0) n , q k
ms ( γ k ; 0)n , q
where
is
1
( n − 2q ) the
2
πBL + j 2π ( γ − γ k )
nth-order
cyclic
III.
(7)
We assume to operate in a coherent, synchronous environment with single-tone signaling. The AMC system is a pattern classifier, which involves three steps; data acquisition and preprocessing, feature extraction, and decision-making. Preprocessing tasks include carrier, timing and waveform recovery, as well as signal power estimation. At first the classifier is trained, i.e., the nth-order warped CC magnitudes
moment
(q conjugations) of s(t) at CF γk and zero-delay vector. Equation (7) shows that for q ≠ n 2 the nth-order time-varying moment is not an (almost) periodic function of t and PN causes a frequency spread around each CF.
cr (1 T ; 0) n , n / 2
The nth-order time-varying cumulant can be expressed as a function of nth- and lower-order time-varying moments using the moment-to-cumulant (M-C) formula [13]; csPN (t; τ ) n , q =
Q
∑ ( −1) ( Q − 1)!∏ m Q −1
Q
p =1
∪ Ip =I
s PN
(t ; τ I p ) nI
p
, qI p
AUTOMATIC MODULATION CLASSIFICATION SYSTEM
n/2
n=4,6,8, are chosen as components of the
catalog feature-vector employed for classification. In the second step the baseband waveform r(t) is (over)sampled and the measured feature-vector is estimated from data [14]. The modulation type is selected by comparing the measured feature-vector with catalog patterns using the criterion
(8)
Q
where I={1,2,…,n} is the set of indices,
i =1,.., N mod
∪ I p = I denotes
summation over all partitions of the set I, Q is the number of the subsets of a partition, msPN (t ; τ I p ) nI , qI is the nI p th-order p
time-varying moment, nI p is the number of elements of the subset Ip, qI p is the number of conjugations and τ I p is a vector of lags from τ whose indices belong to Ip. Using the M-C formula written for the nth-order time-varying cumulants, n=2,4,6,8, q=n/2, at τ =0 and taking into account the particular values of the nth-order moments for QAM signal constellations it can be proven that PN has no effect on csPN (t ; 0) n, q for
IV.
signal classification. Their value and variance range are not the same for different orders; both increase with the CC order [14]. A warping operation, which consists in taking the 2/nth power of the nth-order CC magnitudes, is performed to mitigate this problem [6]. Theoretical values of the nth-order cumulants (n=2,4,6,8, q=n/2) for QAM signal constellations are given in 1 Table I . These values are computed with the M-C formula, wherein the nth-order moments are calculated as ensemble averages over the ideal noise-free constellation under the constraint of unity variance and the assumption of equiprobable symbols.
cs,n,q cs,2,1 cs,4,2 cs,6,3 cs,8,4
THEORETICAL CUMULANTS FOR SIGNAL CONSTELLATIONS UNDER THE CONSTRAINT OF UNITY VARIANCE. 4QAM 1 -1 4 -34
16QAM 1 -0.68 2.08 -13.981
64QAM 1 -0.619 1.797 -11.502
256QAM 1 -0.604 1.734 -10.97
1
The fourth-, sixth- and eight-order cumulants for the constellations considered in Table I are computed respectively in [8], [6] and [7].
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SIMULATION RESULTS
the new-proposed feature-vector, denoted by FV1 c (1 T ; 0 ) 1/ 2 , c (1 T ; 0 ) 1/ 3 , c (1 T ; 0 ) 1/ 4 r r 4,2 6,3 8,4 r and the feature-vector proposed in [7], denoted by FV2 cr (1 T ; 0 ) , cr (1 T ; 0 ) ,..., cr (1 T ; 0 ) . 8,0 8,2 8,8 The classifier performance is defined by the probability of correct classification Pcc, averaged over 500 trials. A raised cosine pulse shape with 25% excess bandwidth was considered, an oversampling factor P=9 is chosen to eliminate cycle aliasing [7], and PN was generated using the Wiener process property of independent stationary and Gaussian increments [10]. An asymptotic relationship between the Poisson model and the binomial model with the success parameter p=λT/P was used to develop an algorithm for generating realizations of {ti, NI(t)} over the time interval (0, NT), where N is the number of processed symbols [12]. The amplitudes {Yi} are i.i.d. complex Gaussian variables, Yi∈N(0, σY2 ). The correlated background and impulsive noise components were generated by passing zero-mean complex Gaussian deviates and Poisson impulses through the receiver square-root raised cosine filter to yield autocorrelation sequences σ 2N p (τ) and λσY2 p(τ) , respectively. With these parameters Fig. 1 shows performance of the proposed classifier (FV1) under an ideal scenario (no model mismatches) for 5dB, 7dB and 10dB SNR, wherein the signal-to-noise ratio was defined as SNR= σ 2S σ 2N . We note
csPN ( γ k ; 0) n , n / 2 , n=2,4,6,8.
Therefore, we will confine our attention to employing the CC magnitudes cr (1 T ; 0) n, n / 2 n=4,6,8, as features for QAM
TABLE I.
(9)
Simulations have been performed to distinguish between 4-QAM and 16-QAM signals under a non-ideal scenario by using two feature-vectors:
n=2,4,6 and a weak effect on csPN (t ; 0)8,4 . The same behavior holds for the nth-order CCs
)
where the integers i=1,…, Nmod enumerate the candidate modulation types, iˆ represents the decision on the modulation type of the incoming signal, fi is the catalog feature-vector for the ith modulation, f is the measured feature-vector and d(.,.) is a similarity metric. The warping operation ensures the feature-vector homogeneity and hence the classic Euclidian distance is used as a similarity metric.
p =1
p
(
iˆ = arg min d fi , f
p =1
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4-QAM vs. 16-QAM
that classification performance improves with an increase in SNR for a specific N and with the number of processed symbols for a specific SNR. Frequency and phase offsets cause a spread of the symbol points along arcs and hence classification performance may degrade. Figs. 2 and 3 show respectively that FV1 is robust to the presence of residual carrier frequency and classification performance does not degrade as a result of adding phase noise. Moreover, it is shown that FV1 outperforms FV2 under such model mismatches. Theoretically, both FV1 and FV2 are insensitive to a fixed phase offset. Simulations were also performed adding a fixed phase offset, θ ∈U [-π,π), to a residual carrier frequency offset. With N=900, 7dB SNR and ∆fcT=5e-5 an average Pcc of 0.971 and 0.958 was respectively obtained for FV1 and FV2, which is in agreement with the theoretical result. Simulations were performed to investigate the robustness of the FV1-based classifier to the presence of outliers (impulsive noise). The SNR was set based on the total noise variance, σ 2N + λσY2 . With p=10-2 and σY2 = 100σ2N , N=900 and 5, 7 and 10dB SNR a Pcc of 0.845, 0.95 and 0.99 was respectively obtained. By comparing these results with those under the ideal scenario, one can notice that there is a loss in performance as the SNR decreases.
1
0.9
Average Pcc
0.55
0.5
Average Pcc
5dB
[2]
0.8
[3] FV1, SNR=5dB FV1, SNR=7dB FV1, SNR=10dB
800
850
[4] 900
Number of symbols
[5]
Figure 1. Classification performance under ideal scenario. 4-QAM vs. 16-QAM 1
[6]
FV1
0.95
[7]
0.9
FV2
[8]
Average Pcc
0.85 0.8 0.75
[9]
0.7
[10] [11]
0.65 0.6
FV2, N=900, SNR=7dB FV1, N=900, SNR=7dB
0.55 0.5
0
0.2
0.4
0.6
0.8
Normalized residual carrier frequency , ∆fcT
[12] [13]
1 x 10
-3
Figure 2. Performance degradation due to frequency offset.
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0.6
0.8
1 x 10
-3
CONCLUSION
REFERENCES [1]
0.85
750
0.4
Normalized 3-dB bandwidth, BLT
The proposed algorithm is suitable for classifying QAM signals. The advantage of this algorithm is its robustness to carrier phase and frequency offsets. A hierarchical classification scheme, which identifies the signal class format first (M-QAM, M-PSK, etc.), and then chooses the modulation order (M) within the selected class can be used to increase the classifier generality.
0.9
700
0.2
V.
7dB
650
0
Figure 3. Performance degradation due to phase noise.
10dB
600
0.7
0.6
0.95
550
0.8
0.75
0.65
4-QAM vs. 16-QAM
0.7 500
FV1, FV2,
FV2
∆fcT=1e-3, N=900, SNR=7dB ∆fcT=5e-5, N=900, SNR=7dB
0.85
1
0.75
FV1
0.95
[14]
748
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