Analysis of Minimum Hellinger Distance Identification for Digital Phase Modulation

Transcription

1 Analysis of Minimum Hellinger Distance Identification for Digital Phase Modulation Kenta Umebayashi, Janne Lehtomäki and Keijo Ruotsalainen Centre for Wireless Communications, University of Oulu P. O. BOX 4500 FIN University of Oulu, Finland Abstract This paper presents a modulation identification (MOD ID) method for digital M-ary phase-shift keying (M- PSK) under additive white Gaussian channel, using the minimum Hellinger distance (MHD). When compared with the maximum likelihood (ML) method, the MHD has two significant advantages, low complexity and robustness against perturbations. In addition, the MHD is implementable for MOD ID by partitioning the I- Q plane. Our contribution is as follows. We obtain theoretical results for the modulation identification error probability of the MHD. Employing the Hellinger distance criteria, we present a sub-optimal partition design. Computer simulation experiments compare the MHD and the ML methods regarding the MOD ID performance and show that difference in performance is around 2dB in ideal channel. I. INTRODUCTION Modulation identification (MOD ID) is a challenging and important topic that has been studied for military, surveillance, software defined radio (SDR), cognitive radio and noncooperative communications. A classical approach to the problem of MOD ID is to employ the maximum likelihood (MLD) method. In [1], Wei and Mendel presented an analysis of the optimum modulation classification method based on Bayesian decision theory to achieve the minimum error-rate classification using the ML criteria. In [2], Yang and Soliman also proposed a modulation classification algorithm based on a sub-optimal algorithm employing the Tikhonov function to obtain an approximation of the likelihood function. In [3], Yang and Liu proposed a modulation classification algorithm of M-ary PSK employing the asymptotical phase distribution instead of the Tikhonov function. Actually it is difficult to present unified discussion and comparison since there are a wide variety of methods, e.g., likelihood based methods, high order statistics (HOS) and neural network, and they discussed the MOD ID problem under different assumptions, e.g., channel models, applications, length of observation period, preprocessing, and modulation types. This paper employs the minimum Hellinger distance (MHD) to identify M-ary phase-shift keying (M-PSK) modulation. The MHD has been investigated for estimation problems and shown to have less complexity and more robustness than the ML method [7], [8]. Although several other estimators that have more robustness than the ML have been proposed, most This research was supported by the National Technology Agency of Finland, Nokia, the Finnish Defence Forces and Elektrobit. of them entail a loss of asymptotic efficiency [7]. On the other hand, the MHD has not only robustness but also asymptotic efficiency under the right model. In [9] and [10], Donoho and Huo proposed a MOD ID algorithm based on the MHD, and described several properties of this algorithm. However, there was no conclusive proof regarding these properties. Considering the issues mentioned above, we will investigate the following three topics. First we present novel theoretical analysis of modulation identification error probability. Previously the modulation identification error probability based on theoretical analysis have been presented for ML [1], [2]. Secondly we study the sub-optimum partition design regarding the MHD. In the MHD based MOD ID, the I-Q plane is partitioned into L cells which are used to compute the discrete empirical distribution that is an approximation of the true distribution as shown in [9] and [10]. The partitioning of the I-Q plane has a great impact on the MOD ID performance. There are two criteria for optimizing the partition. The optimal criteria is minimum error probability. However, it is difficult to use. We use the partition obtained by the HD criteria that is asymptotically optimal [8]. Finally, we compare the performance of the ML based MOD ID with the MHD based MOD ID and discuss the complexity of the proposed MHD based MOD ID algorithm. The rest of this paper is organized as follows. In Section II, the signal and channel models and other assumptions are described. In Section III, the MOD ID algorithms based on the ML and on the MHD are introduced. Next, in Section IV, we present a theoretical analysis of the MHD based MOD ID. In Section V, we investigate the optimal partition design in terms of the MHD. In Section VI, numerical results comparing the ML and the MHD are shown. Finally, in Section VII, we give the conclusions. II. SYSTEM DESCRIPTIONS In our analysis we assume that the received PSK signal has been ideally filtered, already equalized and sampled at the optimum sampling instant. Moreover, carrier recovery and relative gain control have been established. Under the above assumptions, the baud-rate samples of the output of a matched filter are described by [11] r B (k) = s B (k)+n B,I (k)+jn B,Q (k), (1) 1 k N /06/$20.00 (c) 2006 IEEE

2 where s B (k) is k-th symbol of the transmitted signal at time kt S, T S is the symbol duration, NT S is observation period defined by the number of symbols N and n B,I (k) and n B,Q (k) are the real and imaginary components of the noise. We assume that the noise samples n are zero-mean with variance σ 2. The transmitted M-ary PSK symbol s B (k) can be represented as s B (k) = exp[j(θ i )] (2) where θ i {2πi/M; i =0, 1,...M 1} is the information component. In addition, E[s B (k)s B (k) ]=1(a unit average energy constellation), where E[] and denote an expectation operator and complex conjugate, respectively. Let γ =1/2σ 2 denote the energy per symbol to noise power density ratio (Es/No). We assume an AWGN channel without any distortion due to the fading. As mentioned in [1], the analysis and evaluations in an AWGN channel will provide an upper bound for the performance. Moreover, to develop the MOD ID method in a more realistic situation e.g. consideration of the fading factors, we believe that the basic analysis shown in this paper is useful and necessary as a first step. In other words, if an evaluations in a fading channel is carried out without this basic analysis, it is not clear which factor deteriorates the performance. A contribution of this paper is to show the basic analysis and a more realistic analysis is the subject of future research. III. MODULATION IDENTIFICATION ALGORITHMS In this section we present MOD ID algorithms based on the ML and on the MHD. The MOD ID problem corresponds to a hypothesis-testing problem. The H m hypothesis corresponds to M-ary PSK where m = log 2 M, e.g., H 2 corresponds to QPSK. In addition, m is a set {1, 2,,m MAX } and m m. Before explaining the MOD ID algorithms, the probability density function (PDF) of the received signal, represented in (1), is derived firstly. A. Phase probability density function of M-PSK In the case of AWGN channel the PDF depends only on the phase component. The extracted phase φ of the received signal (1) is given by φ(k) = arctan(im{r B (k)}/re{r B (k)}) π<φ π. (3) Given phase component of the received signal φ, the PDF P P (φ(k); M) can be written as [12] P P (φ(k); M) = 1 M 1 2πM exp[ γ] {1+ πγ cos(φ(k) θ i ) i=0 [1 + erf( γ cos(φ(k) θ i ))] exp[γ cos 2 (φ(k) θ i )]}, (4) where the error function is defined in [13] as erf(x) = 2 x exp[ t 2 ]dt. (5) π 0 B. MOD ID algorithm based on the ML The posteriori probability for each hypothesis is denoted by P (H m φ) where φ = [φ(1),φ(2) φ(n)] denotes a extracted phase sequence whose size is N. Using the maximum a posteriori (MAP) rule, the modulation type can be identified by choosing the hypothesis with the largest a posteriori probability as ˆm = arg max m m P (H m φ), (6) where ˆm corresponds to identified modulation type. We assume that all possible modulation types have the same a priori probability, i.e. P (H m ) = 1/m MAX. Hence, the MOD ID problem presented in (6) can be written as ˆm = arg max m m l(φ H m), (7) which corresponds to ML and l(r B φ) is the log-likelihood function given by l(φ H m )= N ln(p P (φ(k); H m )). (8) k=1 C. MOD ID algorithm based on the MHD The HD between the PDFs f and g is defined by D(f,g) ={ (f 1/2 g 1/2 ) 2 } 1/2. (9) HD shows a disparity between functions f and g. Assuming f is a empirical function derived by the observed data and g is a adjustable model function by a estimator λ, λ can be estimated by minimizing HD, namely this is the MHD based estimation [7], [8]. On the application of the MHD to the MOD ID problem, the I-Q plane is partitioned into L cells. The number of symbols, which lie in l-th cell, is denoted as Y l, and the empirical distribution, denoted as ˆf r,is ˆf r = [Y 1 /N, Y 2 /N,,Y L /N ]. The reference PDF of the M-PSK modulation is denoted by f m =[f m (1),f m (2),,f m (L) ], where f m (j) = P I,Q (r B (x); H m )dx (10) j th cell where P I,Q is joint probability density function of the received symbol on I-Q plane. The f m (j) give us the probability that the received symbol lies in the j-th cell. The HD between the empirical PDF ˆf r and the reference PDF of the M-PSK modulation f m is now 1/2 L D( ˆf r, f m ) = ( Y j /N f m (j) ) 2 = L 2 2 Y j /N f m (j) 1/2. (11) The MHD based MOD ID chooses the hypothesis H ˆm where ˆm = arg min H m D( ˆf r, f Hm ), (12) 2953

3 αk Q 10 0 ϕαk ϕα1 αnline α1 I Error rate BPSK QPSK 10 6 Computer simulation Fig. 1. Boundary π+ϕαk Structure of partition; boundary and phase components where ˆm corresponds to identified modulation type. In the MHD based modulation identification we choose the modulation for which the HD from the empirical distribution is smaller. In the next section, we will study, how the optimum design of the partition can be derived. Since we assume a M- ary PSK modulation, a reasonable way to design the partition is as follows; all boundaries are straight-line passing through the origin of I-Q plane. All straight lines are assumed that they do not coincide with horizontal axis. Suppose that there are N LINE lines, thus L =2N LINE. In Fig. 1, the structure of this partition is shown. Each boundary is denoted by α k, where k =1, 2, N LINE. In addition, there are two phase components for each boundary, which are ϕ αk and π + ϕ αk and therefore there are 2N LINE phase components. We redefine the new phase components =[ 1,, 2NLINE ] obtained by rearranging the phase components ϕ αj in ascending order as k = { π + ϕαk (1 k N LINE ) ϕ αk NLINE (1 + N LINE k 2N LINE ). (13) Since we assume the modulation type is M-ary PSK and the boundary is a straight line, each cell β j,j =1, 2,, 2N LINE can be defined by using k as { {ϕβj, β j = j ϕ βj < j+1 } (1 j 2N LINE 1) {ϕ βj, j ϕ βj π π<ϕ βj < 1 } (j=2n LINE). (14) If an M-ary PSK modulation is used the reference PDF of the received signal is f m (j) = P P (φ; M)dφ (15) β j IV. PERFORMANCE ANALYSIS The modulation identification error probability, P (ˆm m H m ), or success probability, 1 P (ˆm m H m ), when the Theoretical analysis Es/No [db] Fig. 2. Performance of MOD ID based on the MHD. The circle and dot lines indicate Monte Carlo simulation and the straight lines indicate the analysis. Upper curves: QPSK, lower curves: BPSK. ML based MOD ID is employed, have been derived in [1] and [2]. The derivation of the corresponding MHD based MOD ID is not as straightforward. Therefore, we will introduce the following approach. Let us denote by X the set of possible values of [Y 1,Y 2, Y L ]. An element X k of X is [Y1 k,y2 k, YL k], where k indicates that Yj k is an element of X k.nowthe probability P (X k H m ) can be expressed as where P (X k H m )= L 1 l=1 N ξ l C Y k l L { 0 (l =1) ξ l = l 1 p=1 Y p k (l >1) {f m (j) } Y k j, (16) i! and i C k = k!(i k)! is the binomial coefficient. Once f m and X k are determined, the identified modulation type ˆM is given by (12) by replacing ˆf Xk by ˆf r where ˆf Xk denotes the empirical PDF based on X k. Finally, assuming X, the modulation identification error probability is given by P (ˆm m H m )= P (X k H m ), (17) X k, ˆm m where the summation is with respect to X k satisfying ˆm m. To confirm the validity of this analysis, we compare our theoretical results with the one obtained by Monte Carlo simulation. The comparison is shown in Fig. 2. The assumptions for this comparison are as follows; N =50, L =4, ϕ α1 = π/4, ϕ α2 =3π/4, and the candidate modulation types are BPSK and QPSK, i.e. M =2or 4. The circles denote the modulation identification error rate obtained by the Monte Carlo simulation and the continuous line denotes the theoretical error probability. We can observe that the experimental and theoretical results coincide very well justifying our approach. 2954

4 V. THE OPTIMIZATION OF THE PARTITION DESIGN In this section we describe how to obtain an optimum partition. The key parameters to optimize the partition are the phase component of the boundary k,es/no(γ), the number of symbols (N) in the observation period and the number of partitions (L). A controllable parameter in the receiver is k. The optimum partition to minimize the error probability is m MAX OPT = arg min P (error H m ). (18) m=1 Whereas, if we use the alternative criteria of minimizing the HD, we get the phase boundaries as [10] OPT,MHD = arg max D(f mi, f mj ). (19) From the literature it is known that the optimum partition obtained by the HD criteria is asymptotically equivalent with the ML approach, e.g. OPT,MHD OPT as N [8]. In that sense, (19) expresses a sub-optimal solution. We consider only m = {1, 2}. The case with m MAX > 2 leads to a multiple hypothesis testing and our approach can be extended to cover this case as well. In the case with m MAX =2, (11) and (19) lead to minimizing a cost function C( ) = L i=1 i j f (i) 1 and the optimization OPT,MHD is given by f (i) 2, (20) OPT,MHD = arg min C( ). (21) In addition, assuming L =4and due to the symmetry of the constellations, we can assume that =( 3 π, 3, 3,π 3 ), that is to say, the cost function depends on the parameter 3.InFig.3weshowC( 3 ) versus 3 in terms of each Es/No. Each arrow shown in Fig. 3 indicates the minimum of the cost function, and they are located around ϕ =55. In the high Es/No region, where 15 γ 20, the cost function approaches a lower bound corresponding to 1/ 2 since f 1 Es/No = [1/2, 0, 1/2, 0] and f 2 Es/No = [1/4, 1/4, 1/4, 1/4]. Next, Fig. 4, where π/4 < 3 <π/2 and N =50,shows the MOD ID error rate obtained by computer simulations and analysis respectively, as function of the partition phase 3 for different value of the Es/No γ =3, 4 and 5. The reason that the error rate function is not smooth depends on the number of observations. As we mentioned above, the HD criteria is asymptotically equivalent with the ML approach, namely, the error rate function becomes smoother as N. Although the results show that all of the curves are not smooth in Fig. 4, it can be observed that the minimums are located around 55 as shown in Fig. 3. Cost function Es/No=1dB 8dB 15dB dB 20dB 10dB Bound Partition Phase Fig. 3. Cost function C(M MAX =4, 3 ) for each Es/No. Arrow indicates the minimum of cost function. Error rate Es/No=3dB, analysis and simulation Es/No=4dB, analysis and simulation Partition phase 2dB 3dB 4dB 5dB 6dB 7dB Es/No=5dB, analysis and simulation Fig. 4. MOD ID error rates and probabilities versus partition phase 3 in terms of Es/No=3,4 and 5. N=50 symbols. VI. NUMERICAL RESULTS The MHD and the ML based MOD ID algorithms are compared in this section. The purpose of this evaluation is to measure a disparity with respect to accuracy of MOD ID between the MHD and the ML methods. The MHD cannot outperform the ML under the ideal conditions, i.e. AWGN channel without fading factors. Fig. 5 shows the MOD ID error rate in terms of Es/No. The partition for the MHD is optimized based on (21), and the other assumptions are same as the one in the simulation shown in Fig. 2. Solid lines and dot lines denote the MHD and the ML, respectively and cross and circle indicate the modulation types, BPSK and QPSK respectively. Comparing with the result in Fig. 2 where the partition is not optimized, MOD ID error performances of the MHD are improved for both modulation types. The improvement level in terms of Es/No are about 0.5dB and 1dB for QPSK and BPSK, respectively. In the ML, the performances of BPSK and QPSK are almost equivalent, on the other hand, the performance of the MHD 2955

5 10 0 MHD QPSK MHD BPSK ML QPSK ML BPSK ACKNOWLEDGMENT The authors thank Carlos Pomalaza-Raez at Indiana University-Purdue University Fort Wayne for helpful comments. Error Rate Es/No [db] Fig. 5. Comparison between the ML and the MHD. The MOD ID error rate versus Es/No. MOD type is BPSK and QPSK. is determined by QPSK, in other words, an upper bound of MOD ID error rate is determined by QPSK. When the MOD ID error rate is, the disparity between the MHD and the ML can be seen to be about 2dB. Considering the implementation of the MHD, calculations of f m and for each observation period are not reasonable. Instead, concerning the complexity, it is acceptable that the receiver has a table where f m and are memorized for each Es/No. For the cases studied the optimum is around 55 for all Es/No values. Using the above idea, the necessary calculations are the counting the Y j, (11) and a choice of the minimum Hellinger distance (12). It implies that the MHD based MOD ID has much less complexity than the ML approach. VII. CONCLUSION The MHD based MOD ID algorithm has been presented. Its theoretical analysis regarding the MOD ID error probability was derived, and the optimal partition design was discussed. Monte Carlo simulation showed that the theoretical analysis is valid. To obtain the sub-optimal design of the partition, the MHD was applied. In addition, the sub-optimal partition approximately approaches the optimal solution based on the MOD ID error probability, when N is large enough. A comparison between the ML and the MHD was presented and we confirmed that a disparity is about 2dB in terms of MOD ID performance. Regarding a implementation of the MOD ID algorithm, this paper indicates that the MHD is more suitable than the ML method. Our future subjects are as follows; an evaluation in realistic channel model, e.g. fading channel and an implementation on adaptive modulation system. REFERENCES [1] W. Wei and J. M. Mendel, Maximum-likelihood classification for digital amplitude-phase modulations, IEEE Trans. Commun., vol. 48, no. 2, pp , Feb [2] Y. Yang and S. S. Soliman, A suboptimal algorithm for modulation classification, IEEE Trans. Commun., vol. 33, no. 1, pp , Jan [3] Y. Yang and C. H. Liu, An asymptotic optimal algorithm for modulation classification, IEEE Commun. Lett., vol. 2, no. 5, pp , May [4] A. K. Nandi and E. E. Azzouz, Algorithms for automatic modulation recognition of communication signals, IEEE Trans. Commun., vol. 46, no. 4, pp , Apr [5] K. Umebayashi, R. Morelos-Zaragoza, and R. Kohno, Evaluations of the multimode PLL using the modulation identification techniques in ISDB-S, in Proc. Software Defined Radio Technical Conference and Product Exposition (SDR), Phoenix, Arizona, Nov. 2004, pp [6] A. Swami and B. M. Sadler, Hierarchical digital modulation classification using cumulants, IEEE Trans. Commun., vol. 48, no. 3, pp , Mar [7] R. Beran, Minimum Hellinger distance estimates for parametric models, The Annals of Statistics, vol. 5, no. 3, pp , May [8] D. G. Simpson, Minimum Hellinger distance estimation for the analysis of count data, Journal of the American Statistical Association, vol. 82, no. 399, pp , Sept [9] D. Donoho and X. Huo, Large-sample modulation classification using Hellinger representation, in Proc. Signal Processing Advances on Wireless Communication (SPAWC), Paris, France, Apr. 1997, pp [10] X. Huo and D. Donoho, A simple robust modulation classification method via counting, in Proc. International Conference on Acoustic Speech and Signal Processing (ICASSP), Seattle, WA, May 1998, pp [11] W. G. Cowley, Phase and frequency estimation for psk packets: bounds and algorithms, IEEE Trans. Commun., vol. 44, no. 6, pp , Jan [12] S. Stein and J. J. Jones, Modern communication principles: with application to digital signaling. NY: McGraw Hill, [13] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington: Dover Publications,