Maximum Likelihood Based Modulation Classification for ...
Maximum Likelihood Based Modulation Classification for Unsynchronized QAMs Qinghua Shi and Y. Karasawa Department of Electronic Engineering University of Electro-Communications 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan E-mails: {qhshi, karasawa}@ee.uec.ac.jp
Abstract—We consider modulation classification for quadrature amplitude modulation (QAM) formats. The received signal is assumed to be unsynchronized in both time and frequency, since in practice the receiver has little prior knowledge about the transmitted signal. To tackle this challenging problem, we propose a classifier that is based on a combination of blind time synchronization, differential processing, and maximum likelihood (ML) detection. A computationally efficient scheme is then developed. Numerical results are provided to justify our approach.
I. I NTRODUCTION Modulation classification (MC) is traditionally used in military applications such as signal monitoring and interception. It has recently received renewed interest and found applications in areas including adaptive modulation, software-defined radios, cognitive radios, and wireless sensor networks. MC can be carried out by estimating statistical properties of a received signal [1]–[3]. This method is relatively simple but offers mediocre performance. In contrast, maximum likelihood (ML) based approach [4]–[8] is optimal in terms of performance but is much more complex to implement. A comprehensive and up-to-date review of MC is provided in [9]. As a modulation classifier typically operates in a noncooperative environment, the receiver has little a priori knowledge about the transmitted signal and also no training is available. Consequently, a practical modulation classifier should be able to deal with signals unsynchronized in both time and frequency. To the best of the authors’ knowledge, however, such a totally unsynchronized case has rarely been studied and perfect synchronization is widely assumed in the literature. Note that in [1], the effects of imperfect synchronization are investigated, but the time and frequency synchronization errors are assumed to be very small. In this work, we assume synchronization in both time and frequency is not achieved. The received signal is thus subject to unknown time delay, frequency offset, and phase shift. A hybrid classification scheme is developed to tackle the challenging problem. Specifically, • blind time synchronization is performed to handle time delay; • differential processing is used to provide robustness against frequency offset and phase shift;
•
ML principle is applied to complete the MC task and cope with high-order QAMs. II. S YSTEM M ODEL
Consider a modulation classifier operating in a practical environment, i.e., synchronization in both time and frequency is not achieved. As illustrated in Fig. 1, the received signal is affected by unknown time delay ε, frequency offset Δf , and phase shift φ. We assume ε, Δf , and φ are all fixed within one frame of N symbols but can vary independently from one frame to another according to a uniform distribution on [−0.5T, 0.5T ] (T denotes the symbol period), [−0.5/T, 0.5/T ], and [−π, π], respectively. The received signal can be expressed as y(t) =
∞
exp (j2πΔf t + φ) s(l)p(t−lT −εT )+η(t) (1)
l=−∞
where s(l) is the transmitted symbol sequence drawn from QAM constellations, p(t) represents a raised cosine filter used for spectral shaping, and η(t) is a complex AWGN with zero mean and variance 2σ 2 . We assume there are in total I types of QAMs, whose constellations can be represented by {Ci }Ii=1 ,
Ci {ci,1 , ci,1 , · · · , ci,Mi }
where Mi is the number of constellation points in Ci .
Fig. 1.
Block diagram of modulation classifier.
The received signal is then oversampled at a rate P/T (P is
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
By considering r(l) instead of z(l), the frequency offset problem is entirely solved. It follows from (3) that r(l) has a n exp j2πΔf T + φ s(l)p(n−lP −εP )+η(n). conditional Rician probability density function (PDF) y(n) = 2 P l=−∞ r(l) r(l)a(l) r (l) + a2 (l) I (5) exp − (2) f (r(l)) = 0 2σ 2 4σ 2 2σ 2 a(l)
the oversampling factor), yielding a discrete-time signal model ∞
III. P ROPOSED C LASSIFICATION S CHEME We propose a hybrid scheme based on a combination of blind time synchronization, differential processing, and ML detection to combat the unknown time delay, frequency offset, and phase shift. A. Blind Time Synchronization The square timing recovery technique [10] shown in Fig. 2 is adopted to blindly estimate the time delay from the oversampled sequence y(n). Then the transmitted symbol sequence s(l) is recovered via a digital interpolator. It is worth noting that the square timing recovery technique is general and robust, regardless of the frequency offset and phase shift, since it uses only the power of the received signal.
where I0 (·) is the zero-order modified Bessel function of the first kind and a(l) |ˆ s(l)ˆ s∗ (l + 1)|. ML classification of I types of QAMs is essentially an Iary hypothesis testing problem. Under the ith hypothesis, Hi , the unconditional PDF of r(l) can be written as r(l) r(l)|s(l)s∗ (l + 1)| f (r(l)) = I0 2σ 2 2σ 2 Hi s(l)∈Ci s(l+1)∈Ci 2 r (l) + |s(l)s∗ (l + 1)|2 . (6) exp − 4σ 2 The corresponding log-likelihood function of a frame of N symbols is
N f (r(l)) Li ln l=1
=
N l=1
Fig. 2.
Square timing recovery based blind delay estimation.
ln
r(l) r(l)|s(l)s∗ (l + 1)| I0 2σ 2 2σ 2 s(l)∈Ci s(l+1)∈Ci 2
r (l) + |s(l)s∗ (l + 1)|2 , exp − 4σ 2
i = 1, 2, · · · , I.
B. Differential Processing It remains to deal with the frequency offset and phase shift. A straightforward strategy is that the frequency offset 1 can be blindly estimated [11] and then compensated. Our investigation suggests that most classification schemes are very sensitive to even a small residual frequency offset. The main reason is that signal distortions caused by a residual frequency offset increase progressively with time. To effectively combat the catastrophic impact of frequency offset, a differential operation [1] is used to transfer the frequency offset into a phase shift, whose influence can be eliminated by taking only the signal amplitude. As a by-product, the original phase shift φ is automatically ruled out by the differential processing. The resulting signal can be written as z(l) = sˆ(l)ˆ s∗ (l + 1) exp (−2πΔf T ) + η (l)
(3)
where ∗ denotes complex conjugate and η (l) represents a complex AWGN with zero mean and variance 4σ 2 . Note that the differential operation incurs a loss of 3dB in SNR. C. ML Classification–Rician Based After differential processing, ML classification is performed on the amplitude of the sequence z(l), defined as r(l) |z(l)|. 1 The
Hi
(4)
phase shift can be easily handled by considering only the signal amplitude.
(7)
The classifier makes a decision, according to the ML principle, ˆi = max{L1 , L2 , · · · , LI , }.
(8)
D. Improved ML Classification–Nakagami Based From (7) we can see that the Rician based scheme requires to compute the modified Bessel function I0 (·) for each received signal r(l). This is computationally intensive and determines the implementation complexity of the classifier. To simplify the above ML classification scheme, we propose an improved alternative that is much more efficient in terms of computational complexity. The motivation is that, as a wellbehaved and versatile PDF, Nakagami distribution can be used to approximate Rician distribution to greatly simplify related treatment. According to (55) of [12], the Rician PDF (5) can be replaced by the following Nakagami PDF m(l) 2[r(l)]2m(l)−1 m(l) m(l) 2 f (r(l)) = r (l) exp − Γ(m(l)) Ω(l) Ω(l) a(l) (9) where Γ(·) is the Gamma function, and Ω(l) = 2σ 2 + |ˆ s(l)ˆ s∗ (l + 1)|2 , Ω2 (l) m(l) = . 2 Ω (l) − |ˆ s(l)ˆ s∗ (l + 1)|4
(10)
We emphasize that Ω(l), m(l), and Γ(m(l)) are only determined by σ 2 and the QAM constellation (e.g., Ci under hypothesis Hi ), regardless of the received signal r(l). This
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
separation from r(l) makes (9) much more appealing than (5) in computation and implementation as well, since Ω(l), m(l), and Γ(m(l)) can be predetermined. For each received signal r(l), (9) only requires to calculate power and exponential functions (elementary functions), whereas (5) additionally requires to calculate a modified Bessel function (special function). Under hypothesis Hi , the PDF of r(l) becomes m(l) 2 m(l) f (r(l)) = Γ(m(l)) Ω(l) Hi s(l)∈Ci s(l+1)∈Ci m(l) 2 [r(l)]2m(l)−1 exp − r (l) (11) Ω(l)
TABLE I N UMBERS OF DIFFERENT AMPLITUDES OF QAM CONSTELLATIONS WITH / WITHOUT DIFFERENTIAL PROCESSING . 16QAM (square) 64QAM (square) 256QAM (square) 32QAM (cross) 128QAM (cross) 16QAM (star) 32QAM (star)
with
Without 3 9 32 5 16 2 3
With Differential Processing 6 44 478 15 126 3 6
Constellations of Square QAMs
1.5
1
(12)
The corresponding log-likelihood function of a frame of N symbols is m(l) N 2 m(l) ln Li = Γ(m(l)) Ω(l) l=1 s(l)∈Ci s(l+1)∈Ci
m(l) 2 2m(l)−1 , r (l) [r(l)] exp − Ω(l) i = 1, 2, · · · , I.
Imaginary
Ω(l) = 2σ 2 + |s(l)s∗ (l + 1)|2 , Ω2 (l) m(l) = . 2 Ω (l) − |s(l)s∗ (l + 1)|4
16 Square QAM 64 Square QAM 256 Square QAM
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
Real
(13)
Constellations of Cross QAMs
1.5
Similarly, a classification output is produced via (8).
32 Cross QAM 128 Cross QAM
1
Imaginary
E. A New Look at Differential Processing
0.5
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
Real Constellations of Star QAMs 1.5
16 Star QAM 32 Star QAM
1
Imaginary
We will consider most commonly used 7 types of QAM constellations with unity average power [13]. As illustrated in Fig. 3, they can be divided into three groups: square QAMs (16QAM, 64QAM, and 256QAM), cross QAMs (32QAM and 128 QAM), and star QAMs (16QAM and 32QAM). In particular, the ratio between the outer and inner circles for star 16QAM is 1:2.5 and the ratios among the three circles for star 32QAM is 1:2.5:4.3. We have already known that differential processing yields robustness against frequency offset and phase shift but at the cost of 3dB loss in SNR. Here we show that it can do more than this. Fig. 4 shows a comparison between the amplitude histograms of square QAMs with and without differential processing. It can be seen that differential processing significantly changes the amplitude PDFs of QAM constellations. In particular, the number of different amplitudes for QAM constellations is clearly increased due to use of differential processing, especially for high-order QAMs. This phenomenon is further elaborated in Table I. This observation allows us to gain new insight into the advantages of differential processing. Intuitively, we can view differential processing as a nonlinear mapping/transform, with which the differences among QAMs are increased in the transform domain. In other words, assuming a constellation Ci is used at the transmitter, differential processing creates a
0.5
0.5
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
Real
Fig. 3.
Commonly used QAM constellations.
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
new constellation Ci , which is related (in a nonlinear manner) to but different from the original Ci . We can infer from Fig. 4 and Table I that the differences among the newly generated constellations {Ci }Ii=1 appears to be larger than those in the original constellations {Ci }Ii=1 . As a result, the classification performance should be improved accordingly. This inference will be examined by numerical results.
Without differential processing 20 16 QAM
64 QAM
6 4
256 QAM
10
Frequency
Frequency
Frequency
8
5
2 0
0
2
1
0
3
0
1
2
15 10 5 0
3
0
1
2
3
Amplitude
Amplitude
Amplitude
With differential processing 1200
300
80
64 QAM
40 20 0
256 QAM
1000 Frequency
60
Frequency
Frequency
16 QAM
200
100
800 600 400 200
0
1
2
Amplitude
Fig. 4.
3
0
0
1
2
3
0
0
1
2
3
Amplitude
Amplitude
Amplitude histograms of square QAMs.
IV. N UMERICAL R ESULTS AND D ISCUSSION In this section, we examine the performance of the differential processing based ML classifier, termed as Diff-ML, and its counterpart proposed in [8], terms as NonDiff-ML. Referring to Fig. 1, only the differential processing block is not used in NonDiff-ML [8]. Due to its clear advantage in computational complexity, the improved ML scheme (i.e., Nakagami approach) detailed in Section. III.D. is applied unless otherwise specified. In the numerical results, the probability of correct classification (Pcc) is adopted as the performance measure and SNR is defined as SN R = 2σ1 2 . The classifier makes a decision for each frame of N symbols. A total of 100 independent frames of symbols for each QAM constellation (7 QAMs in total) is used to obtain the Pcc. We assume that the raised cosine filter has a rolloff factor of 0.25 and the oversampling factor is P = 4. We compare the Pcc performance of the proposed (i.e., Diff-ML) and existing (i.e., NonDiff-ML) ML classifiers in Fig. 5. Diff-ML is shown to significantly outperform NonDiffML for SNR of practical importance. Only when SNR is high enough (close to 20dB), Diff-ML shows a marginal performance disadvantage, which may be caused by the 3dB SNR loss inherent to differential processing. In fact, this small performance loss in the high SNR region can be reduced by further increasing the number of symbols N used to make a
decision. From Fig. 6 we see that there is still room for DiffML to enhance its performance by using larger N . However, the computational complexity involved is also increased. In Fig. 7, we compare the performance of the proposed DiffML and existing NonDiff-ML with different frame length N . It can be seen that for Diff-ML, increasing the frame length N can further improve the performance, especially when SNR is relatively large. In contrast, a frame length N larger than 1000 yields virtually no performance enhancement for NonDiff-ML. This can be explained by the fact that for NonDiff-ML the maximum number of constellation points is 256, whereas for Diff-ML the maximum number is 256 × 256. In other words, NonDiff-ML only needs N = 256 symbols to approximately reflect all constellation points in {Ci }Ii=1 , whereas Diff-ML requires roughly 256 × 256 symbols to achieve the same goal for the constellations {Ci }Ii=1 . In Fig. 8, we investigate the performance of Rician and Nakagami based approaches. It is interesting to see that the Nakagami approach produces similar performance, compared with the more computationally intensive Rician approach. The observation that the Nakagami model can possibly outperform the Rician model (supposed to be a more accurate PDF) can be understood by noting that the Nakagami and Rician models essentially correspond to two related but different nonlinear mappings/transforms. In this example, the Nakagami approximation method is more than one order of magnitude faster (in simulation time), compared with the Rician method. Finally, we comment that SNR is assumed to be known in this work. In practice, SNR should be estimated. One blind SNR estimation method is proposed in [14]. This method is promising since it relies on noise and signal subspace estimation without knowing other parameters. V. C ONCLUSION In this paper, we have considered a practical MC scenario, in which synchronization in both time and frequency is not accomplished. We developed a hybrid classification scheme to tackle this challenging MC problem. The proposed classifier is based on a combination of blind time synchronization, differential processing, and ML detection. We found that differential processing can not only enable MC to be robust against frequency mismatch, but also generate new constellations. Interestingly, the differences among the newly generated constellations are larger than those in the original constellations, thereby resulting in better MC performance. A computationally efficient scheme was also proposed. R EFERENCES [1] A. Swami and B. m. Sadler, “Hierarchical digital modulation classification using cumulants,” IEEE Trans. Commun., vol. 48, pp. 416-429, March 2000. [2] S. S. Soliman and S.-Z. Hsue, “Signal classification using statistical moments,” IEEE Trans. Commun., vol. 40, pp. 908-916, May 1992. [3] A. K. Nandi and E. E. Azzouz, “Algorithms for automatic modulation recognition of communication signals,” IEEE Trans. Commun., vol. 46, pp. 431-436, April 1998. [4] Wen Wei and J. M. Mendel, “Maximum-likelihood classification for digital amplitude-phase modulations,” IEEE Trans. Commun., vol. 48, pp. 189-193, Feb. 2000.
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
[5] Chung-Yu Huan and A. Polydoros, “Likelihood methods for MPSK modulation classification,” IEEE Trans. Commun., vol. 43, pp. 14931504, Feb-Mar-Apr 1995. [6] Yawpo Yang and Ching-Hwa Liu, “An asymptotic optimal algorithm for modulation classification,” IEEE Commun. Lett., vol. 2, pp. 117-119, May 1998. [7] Yawpo Yang and S. S. Soliman, “A suboptimal algorithm for modulation classification,” IEEE Trans. Aerospace and Electronic Systems, vol. 33, pp. 38-45, Jan. 1997. [8] Yawpo Yang, Ching-Hwa Liu, and Ta-Wei Soong “A log-likelihood function-based algorithm for QAM signal classification,” Signal Processing, vol. 70, no. 1, pp. 61-71, Oct. 1998. [9] O. A. Dobre, A. Abdi, Y. Bar-Ness, and W. Su, “Survey of automatic modulation classification techniques: classical approaches and new trends,” IET Commun., vol. 1, pp. 137-156, April 2007. [10] M. Oerder and K. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun., vol. 36, pp. 605-612, May 1988. [11] 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, March 1998. [12] M. Nakagami, “The m-distribution—a general formular of intensity distribution of rapid fading,” in Statistical Methods in Radio wave Propagation, W. C. Hoffman, Ed. New York: Pergamon, 1960. [13] L. Hanzo, W. Webb, and T. Keller, Single- and Multi-carrier Quadrature Amplitude Modulation, John Wiley & Sons, 2000. [14] D. Ramakrishna, N. B. Mandayam, and R. D. Yates, “Subspace-based SIR estimation for CDMA cellular systems,” IEEE Trans. Vehi. Techno., vol 49, pp. 1732-1742, Sept. 2000.
Fig. 6. Performance of the proposed (Diff-ML) scheme with different numbers of symbols.
Fig. 7. Comparison of the proposed (Diff-ML) and existing (NonDiff-ML) schemes with different SNR values.
Fig. 5. Performance of the proposed (Diff-ML) and existing (NonDiff-ML) schemes as a function of SNR.
Fig. 8.
Comparison of Nakagami and Rician based approaches.
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Abstract—We consider modulation classification for quadrature amplitude modulation (QAM) formats. The received signal is assumed to be unsynchronized in both time and frequency, since in practice the receiver has little prior knowledge about the transmitted signal. To tackle this challenging problem, we propose a classifier that is based on a combination of blind time synchronization, differential processing, and maximum likelihood (ML) detection. A computationally efficient scheme is then developed. Numerical results are provided to justify our approach.
I. I NTRODUCTION Modulation classification (MC) is traditionally used in military applications such as signal monitoring and interception. It has recently received renewed interest and found applications in areas including adaptive modulation, software-defined radios, cognitive radios, and wireless sensor networks. MC can be carried out by estimating statistical properties of a received signal [1]–[3]. This method is relatively simple but offers mediocre performance. In contrast, maximum likelihood (ML) based approach [4]–[8] is optimal in terms of performance but is much more complex to implement. A comprehensive and up-to-date review of MC is provided in [9]. As a modulation classifier typically operates in a noncooperative environment, the receiver has little a priori knowledge about the transmitted signal and also no training is available. Consequently, a practical modulation classifier should be able to deal with signals unsynchronized in both time and frequency. To the best of the authors’ knowledge, however, such a totally unsynchronized case has rarely been studied and perfect synchronization is widely assumed in the literature. Note that in [1], the effects of imperfect synchronization are investigated, but the time and frequency synchronization errors are assumed to be very small. In this work, we assume synchronization in both time and frequency is not achieved. The received signal is thus subject to unknown time delay, frequency offset, and phase shift. A hybrid classification scheme is developed to tackle the challenging problem. Specifically, • blind time synchronization is performed to handle time delay; • differential processing is used to provide robustness against frequency offset and phase shift;
•
ML principle is applied to complete the MC task and cope with high-order QAMs. II. S YSTEM M ODEL
Consider a modulation classifier operating in a practical environment, i.e., synchronization in both time and frequency is not achieved. As illustrated in Fig. 1, the received signal is affected by unknown time delay ε, frequency offset Δf , and phase shift φ. We assume ε, Δf , and φ are all fixed within one frame of N symbols but can vary independently from one frame to another according to a uniform distribution on [−0.5T, 0.5T ] (T denotes the symbol period), [−0.5/T, 0.5/T ], and [−π, π], respectively. The received signal can be expressed as y(t) =
∞
exp (j2πΔf t + φ) s(l)p(t−lT −εT )+η(t) (1)
l=−∞
where s(l) is the transmitted symbol sequence drawn from QAM constellations, p(t) represents a raised cosine filter used for spectral shaping, and η(t) is a complex AWGN with zero mean and variance 2σ 2 . We assume there are in total I types of QAMs, whose constellations can be represented by {Ci }Ii=1 ,
Ci {ci,1 , ci,1 , · · · , ci,Mi }
where Mi is the number of constellation points in Ci .
Fig. 1.
Block diagram of modulation classifier.
The received signal is then oversampled at a rate P/T (P is
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
By considering r(l) instead of z(l), the frequency offset problem is entirely solved. It follows from (3) that r(l) has a n exp j2πΔf T + φ s(l)p(n−lP −εP )+η(n). conditional Rician probability density function (PDF) y(n) = 2 P l=−∞ r(l) r(l)a(l) r (l) + a2 (l) I (5) exp − (2) f (r(l)) = 0 2σ 2 4σ 2 2σ 2 a(l)
the oversampling factor), yielding a discrete-time signal model ∞
III. P ROPOSED C LASSIFICATION S CHEME We propose a hybrid scheme based on a combination of blind time synchronization, differential processing, and ML detection to combat the unknown time delay, frequency offset, and phase shift. A. Blind Time Synchronization The square timing recovery technique [10] shown in Fig. 2 is adopted to blindly estimate the time delay from the oversampled sequence y(n). Then the transmitted symbol sequence s(l) is recovered via a digital interpolator. It is worth noting that the square timing recovery technique is general and robust, regardless of the frequency offset and phase shift, since it uses only the power of the received signal.
where I0 (·) is the zero-order modified Bessel function of the first kind and a(l) |ˆ s(l)ˆ s∗ (l + 1)|. ML classification of I types of QAMs is essentially an Iary hypothesis testing problem. Under the ith hypothesis, Hi , the unconditional PDF of r(l) can be written as r(l) r(l)|s(l)s∗ (l + 1)| f (r(l)) = I0 2σ 2 2σ 2 Hi s(l)∈Ci s(l+1)∈Ci 2 r (l) + |s(l)s∗ (l + 1)|2 . (6) exp − 4σ 2 The corresponding log-likelihood function of a frame of N symbols is
N f (r(l)) Li ln l=1
=
N l=1
Fig. 2.
Square timing recovery based blind delay estimation.
ln
r(l) r(l)|s(l)s∗ (l + 1)| I0 2σ 2 2σ 2 s(l)∈Ci s(l+1)∈Ci 2
r (l) + |s(l)s∗ (l + 1)|2 , exp − 4σ 2
i = 1, 2, · · · , I.
B. Differential Processing It remains to deal with the frequency offset and phase shift. A straightforward strategy is that the frequency offset 1 can be blindly estimated [11] and then compensated. Our investigation suggests that most classification schemes are very sensitive to even a small residual frequency offset. The main reason is that signal distortions caused by a residual frequency offset increase progressively with time. To effectively combat the catastrophic impact of frequency offset, a differential operation [1] is used to transfer the frequency offset into a phase shift, whose influence can be eliminated by taking only the signal amplitude. As a by-product, the original phase shift φ is automatically ruled out by the differential processing. The resulting signal can be written as z(l) = sˆ(l)ˆ s∗ (l + 1) exp (−2πΔf T ) + η (l)
(3)
where ∗ denotes complex conjugate and η (l) represents a complex AWGN with zero mean and variance 4σ 2 . Note that the differential operation incurs a loss of 3dB in SNR. C. ML Classification–Rician Based After differential processing, ML classification is performed on the amplitude of the sequence z(l), defined as r(l) |z(l)|. 1 The
Hi
(4)
phase shift can be easily handled by considering only the signal amplitude.
(7)
The classifier makes a decision, according to the ML principle, ˆi = max{L1 , L2 , · · · , LI , }.
(8)
D. Improved ML Classification–Nakagami Based From (7) we can see that the Rician based scheme requires to compute the modified Bessel function I0 (·) for each received signal r(l). This is computationally intensive and determines the implementation complexity of the classifier. To simplify the above ML classification scheme, we propose an improved alternative that is much more efficient in terms of computational complexity. The motivation is that, as a wellbehaved and versatile PDF, Nakagami distribution can be used to approximate Rician distribution to greatly simplify related treatment. According to (55) of [12], the Rician PDF (5) can be replaced by the following Nakagami PDF m(l) 2[r(l)]2m(l)−1 m(l) m(l) 2 f (r(l)) = r (l) exp − Γ(m(l)) Ω(l) Ω(l) a(l) (9) where Γ(·) is the Gamma function, and Ω(l) = 2σ 2 + |ˆ s(l)ˆ s∗ (l + 1)|2 , Ω2 (l) m(l) = . 2 Ω (l) − |ˆ s(l)ˆ s∗ (l + 1)|4
(10)
We emphasize that Ω(l), m(l), and Γ(m(l)) are only determined by σ 2 and the QAM constellation (e.g., Ci under hypothesis Hi ), regardless of the received signal r(l). This
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
separation from r(l) makes (9) much more appealing than (5) in computation and implementation as well, since Ω(l), m(l), and Γ(m(l)) can be predetermined. For each received signal r(l), (9) only requires to calculate power and exponential functions (elementary functions), whereas (5) additionally requires to calculate a modified Bessel function (special function). Under hypothesis Hi , the PDF of r(l) becomes m(l) 2 m(l) f (r(l)) = Γ(m(l)) Ω(l) Hi s(l)∈Ci s(l+1)∈Ci m(l) 2 [r(l)]2m(l)−1 exp − r (l) (11) Ω(l)
TABLE I N UMBERS OF DIFFERENT AMPLITUDES OF QAM CONSTELLATIONS WITH / WITHOUT DIFFERENTIAL PROCESSING . 16QAM (square) 64QAM (square) 256QAM (square) 32QAM (cross) 128QAM (cross) 16QAM (star) 32QAM (star)
with
Without 3 9 32 5 16 2 3
With Differential Processing 6 44 478 15 126 3 6
Constellations of Square QAMs
1.5
1
(12)
The corresponding log-likelihood function of a frame of N symbols is m(l) N 2 m(l) ln Li = Γ(m(l)) Ω(l) l=1 s(l)∈Ci s(l+1)∈Ci
m(l) 2 2m(l)−1 , r (l) [r(l)] exp − Ω(l) i = 1, 2, · · · , I.
Imaginary
Ω(l) = 2σ 2 + |s(l)s∗ (l + 1)|2 , Ω2 (l) m(l) = . 2 Ω (l) − |s(l)s∗ (l + 1)|4
16 Square QAM 64 Square QAM 256 Square QAM
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
Real
(13)
Constellations of Cross QAMs
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Similarly, a classification output is produced via (8).
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E. A New Look at Differential Processing
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We will consider most commonly used 7 types of QAM constellations with unity average power [13]. As illustrated in Fig. 3, they can be divided into three groups: square QAMs (16QAM, 64QAM, and 256QAM), cross QAMs (32QAM and 128 QAM), and star QAMs (16QAM and 32QAM). In particular, the ratio between the outer and inner circles for star 16QAM is 1:2.5 and the ratios among the three circles for star 32QAM is 1:2.5:4.3. We have already known that differential processing yields robustness against frequency offset and phase shift but at the cost of 3dB loss in SNR. Here we show that it can do more than this. Fig. 4 shows a comparison between the amplitude histograms of square QAMs with and without differential processing. It can be seen that differential processing significantly changes the amplitude PDFs of QAM constellations. In particular, the number of different amplitudes for QAM constellations is clearly increased due to use of differential processing, especially for high-order QAMs. This phenomenon is further elaborated in Table I. This observation allows us to gain new insight into the advantages of differential processing. Intuitively, we can view differential processing as a nonlinear mapping/transform, with which the differences among QAMs are increased in the transform domain. In other words, assuming a constellation Ci is used at the transmitter, differential processing creates a
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Commonly used QAM constellations.
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
new constellation Ci , which is related (in a nonlinear manner) to but different from the original Ci . We can infer from Fig. 4 and Table I that the differences among the newly generated constellations {Ci }Ii=1 appears to be larger than those in the original constellations {Ci }Ii=1 . As a result, the classification performance should be improved accordingly. This inference will be examined by numerical results.
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IV. N UMERICAL R ESULTS AND D ISCUSSION In this section, we examine the performance of the differential processing based ML classifier, termed as Diff-ML, and its counterpart proposed in [8], terms as NonDiff-ML. Referring to Fig. 1, only the differential processing block is not used in NonDiff-ML [8]. Due to its clear advantage in computational complexity, the improved ML scheme (i.e., Nakagami approach) detailed in Section. III.D. is applied unless otherwise specified. In the numerical results, the probability of correct classification (Pcc) is adopted as the performance measure and SNR is defined as SN R = 2σ1 2 . The classifier makes a decision for each frame of N symbols. A total of 100 independent frames of symbols for each QAM constellation (7 QAMs in total) is used to obtain the Pcc. We assume that the raised cosine filter has a rolloff factor of 0.25 and the oversampling factor is P = 4. We compare the Pcc performance of the proposed (i.e., Diff-ML) and existing (i.e., NonDiff-ML) ML classifiers in Fig. 5. Diff-ML is shown to significantly outperform NonDiffML for SNR of practical importance. Only when SNR is high enough (close to 20dB), Diff-ML shows a marginal performance disadvantage, which may be caused by the 3dB SNR loss inherent to differential processing. In fact, this small performance loss in the high SNR region can be reduced by further increasing the number of symbols N used to make a
decision. From Fig. 6 we see that there is still room for DiffML to enhance its performance by using larger N . However, the computational complexity involved is also increased. In Fig. 7, we compare the performance of the proposed DiffML and existing NonDiff-ML with different frame length N . It can be seen that for Diff-ML, increasing the frame length N can further improve the performance, especially when SNR is relatively large. In contrast, a frame length N larger than 1000 yields virtually no performance enhancement for NonDiff-ML. This can be explained by the fact that for NonDiff-ML the maximum number of constellation points is 256, whereas for Diff-ML the maximum number is 256 × 256. In other words, NonDiff-ML only needs N = 256 symbols to approximately reflect all constellation points in {Ci }Ii=1 , whereas Diff-ML requires roughly 256 × 256 symbols to achieve the same goal for the constellations {Ci }Ii=1 . In Fig. 8, we investigate the performance of Rician and Nakagami based approaches. It is interesting to see that the Nakagami approach produces similar performance, compared with the more computationally intensive Rician approach. The observation that the Nakagami model can possibly outperform the Rician model (supposed to be a more accurate PDF) can be understood by noting that the Nakagami and Rician models essentially correspond to two related but different nonlinear mappings/transforms. In this example, the Nakagami approximation method is more than one order of magnitude faster (in simulation time), compared with the Rician method. Finally, we comment that SNR is assumed to be known in this work. In practice, SNR should be estimated. One blind SNR estimation method is proposed in [14]. This method is promising since it relies on noise and signal subspace estimation without knowing other parameters. V. C ONCLUSION In this paper, we have considered a practical MC scenario, in which synchronization in both time and frequency is not accomplished. We developed a hybrid classification scheme to tackle this challenging MC problem. The proposed classifier is based on a combination of blind time synchronization, differential processing, and ML detection. We found that differential processing can not only enable MC to be robust against frequency mismatch, but also generate new constellations. Interestingly, the differences among the newly generated constellations are larger than those in the original constellations, thereby resulting in better MC performance. A computationally efficient scheme was also proposed. R EFERENCES [1] A. Swami and B. m. Sadler, “Hierarchical digital modulation classification using cumulants,” IEEE Trans. Commun., vol. 48, pp. 416-429, March 2000. [2] S. S. Soliman and S.-Z. Hsue, “Signal classification using statistical moments,” IEEE Trans. Commun., vol. 40, pp. 908-916, May 1992. [3] A. K. Nandi and E. E. Azzouz, “Algorithms for automatic modulation recognition of communication signals,” IEEE Trans. Commun., vol. 46, pp. 431-436, April 1998. [4] Wen Wei and J. M. Mendel, “Maximum-likelihood classification for digital amplitude-phase modulations,” IEEE Trans. Commun., vol. 48, pp. 189-193, Feb. 2000.
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[5] Chung-Yu Huan and A. Polydoros, “Likelihood methods for MPSK modulation classification,” IEEE Trans. Commun., vol. 43, pp. 14931504, Feb-Mar-Apr 1995. [6] Yawpo Yang and Ching-Hwa Liu, “An asymptotic optimal algorithm for modulation classification,” IEEE Commun. Lett., vol. 2, pp. 117-119, May 1998. [7] Yawpo Yang and S. S. Soliman, “A suboptimal algorithm for modulation classification,” IEEE Trans. Aerospace and Electronic Systems, vol. 33, pp. 38-45, Jan. 1997. [8] Yawpo Yang, Ching-Hwa Liu, and Ta-Wei Soong “A log-likelihood function-based algorithm for QAM signal classification,” Signal Processing, vol. 70, no. 1, pp. 61-71, Oct. 1998. [9] O. A. Dobre, A. Abdi, Y. Bar-Ness, and W. Su, “Survey of automatic modulation classification techniques: classical approaches and new trends,” IET Commun., vol. 1, pp. 137-156, April 2007. [10] M. Oerder and K. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun., vol. 36, pp. 605-612, May 1988. [11] 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, March 1998. [12] M. Nakagami, “The m-distribution—a general formular of intensity distribution of rapid fading,” in Statistical Methods in Radio wave Propagation, W. C. Hoffman, Ed. New York: Pergamon, 1960. [13] L. Hanzo, W. Webb, and T. Keller, Single- and Multi-carrier Quadrature Amplitude Modulation, John Wiley & Sons, 2000. [14] D. Ramakrishna, N. B. Mandayam, and R. D. Yates, “Subspace-based SIR estimation for CDMA cellular systems,” IEEE Trans. Vehi. Techno., vol 49, pp. 1732-1742, Sept. 2000.
Fig. 6. Performance of the proposed (Diff-ML) scheme with different numbers of symbols.
Fig. 7. Comparison of the proposed (Diff-ML) and existing (NonDiff-ML) schemes with different SNR values.
Fig. 5. Performance of the proposed (Diff-ML) and existing (NonDiff-ML) schemes as a function of SNR.
Fig. 8.
Comparison of Nakagami and Rician based approaches.
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