Digital modulation classification using constellation shape Bijan G. Mobasseri Department of Electrical and Computer Engineering, Villanova University, Villanova, PA 19085
[email protected]
Abstract This work reports on a new digital modulation recognition algorithm. We have proposed and implemented a technique that casts modulation recognition into shape recognition. This is possible by treating constellation shape as the key modulation signature. Traditional techniques have been based on various statistics extracted from the time domain samples of a signal. Here, transmitted "information" is taken as the geometric shape of the constellation. Received information is the recovered constellation, deformed by noise and other channel effects. The challenge is to make correct inferences about the original shape based on a single deformed observation. We first demonstrate that fuzzy c-means is capable of reconstructing the original constellation with a high degree of fidelity in spite of random noise, fading and other factors. The vertex set of the reconstructed constellation then defines a single observation based on which a decision has to be made as to its origin. For recognition purpose, we have used Bayesian inference engine based on the maximum a posteriori probability of the observed shape. To implement this inference, the reconstructed constellation is modeled by a discrete spatial random field. The statistics of this random field is an extension of the binomial process. For candidate modulations, their corresponding random field models are computed off-line and the a posteriori probability of an unknown constellation is maximized. Experimental results are shown for various modulation standards including V.29, V.29_fallback, QPSK, 8-PSK and 16QAM. For most cases, we show consistent performance above 90% for Eb/No~0dB and above. Keywords: Modulation classification; Constellation; Digital modulation; Shape recognition ______________________________________________________________________________ 1. Introduction Recognition of the modulation type of an unknown signal provides valuable insight into its structure, origin and properties. Automatic modulation classification is used for spectrum surveillance and management, interference identification, military threat evaluation, electronic counter measures, source identification and many others. For example, if the modulation type of an intercepted signal is extracted, jamming can be carried out more efficiently by focusing all resources into vital signal parameters. Other applications may include signal source identification. This is particularly applicable to wireless communications where different services follow well known modulation standards. It is important to spell out what modulation recognition is, and as importantly, what it is not. Picking up on a point first articulated by Liedtke[8], modulation recognition is an intermediate step on the path to full message recovery. As such, it lies somewhere between low level energy
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detection and a full fledged demodulation. Therefore, correct recovery of the message per se is not an objective, or even a requirement. In fact, it is possible to experience very high BERs and high correct modulation classification at the same time. Case in point is modulation types with L-fold ambiguity for which constellation maps onto itself for a rotation of ±2πK L radians. In a nondifferentially encoded system, such rotations can cause very large BER's. What is stable, however, is constellation shape; a signature that will be exploited throughout this paper. This example also illustrates that modulation classification and symbol detection are potentially two different problems. Modulation classification algorithms have generally followed two main approaches; statistical pattern recognition and decision theoretic. Pattern recognition approach attempts to extract a feature vector for later use in a statistical classifier. Ensuring optimality has always been a problem with this approach. More recent efforts have favored the decision theoretic viewpoint. A likelihood function is formed from the observed signal followed by statistical hypothesis testing. Such formulations, however, have been narrowly defined for specific modulation classes. Frequently, the optimality of likelihood functional has been compromised for practical reasons. The approach presented in this work differs from the existing modulation recognition approaches in several fundamental ways. The premise of the approach is the following: if a digitally modulated signal can be uniquely characterized by its constellation it should also be identifiable by the recovered constellation at the receiver. In effect, constellation shape is now defined as the key modulation signature. The recovered constellation is of course distorted in a variety of ways depending on the specific receiver structure as well as channel. Random noise disturbs constellation vertices. Loss of phase lock in a coherent receiver causes a fixed rotation or slowly spinning constellation; errors in carrier frequency tracking causes a local spin of individual constellation points. We demonstrate that constellation, as a global feature, provides a robust, stable and broad means of modulation classification The rest of the paper is organized as follows: Section 2 is an overview of previous work. Section 3 outlines a constellation recovery algorithm,. Section 4 models the reconstructed constellation as a discrete binomial random field. Section 5 formulates constellation matching as a Bayesian inference. Section 6 contains experimental results. Section 7 reviews existing results and compares them with those reported in this work. Section 8 is conclusions and closing remarks. 2. Previous Work The existing methods for modulation classification span 4 main approaches. Statistical pattern recognition, decision theoretic, M-th law non-linearity and filtering and ad hoc. The following is a review of the existing literature to date. Early on it was recognized that modulation classification is, first and foremost, a classification problem well suited to pattern recognition algorithms. A successful statistical classification requires
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the right set of features extracted from the unknown signal. There have been many attempts to extract such optimal feature. Histograms derived from functions like amplitude, instantaneous phase, frequency or combinations of the above have been used as feature vectors for classification, Jondral[7], Dominguez et al.[5], Liedtke[8]. Also of interest is the work of Aisbett[1] which considers cases with very poor SNR. The current state of the art in modulation classification is the decision theoretic approach using appropriate likelihood functional or approximations thereof. Polydoros and Kim[12] derive a quasi-log-likelihood functional for classification between BPSK and QPSK modulations. In a later publication, Huang and Polydoros[6] introduce a more general likelihood functional to classify among arbitrary MPSK signals. They point out that the S-classifier of Liedtke, based on an ad hoc phase-difference histogram, can be realized as a noncoherent, synchronous version of their qLLR. Statistical Moment-Based Classifier(SMBC) of Solimon and Hsue[15] are also identified as special coherent version of qLLR. Wei and Mendel[17] formulate another likelihood-based approach to modulation classification that is not limited to any particular modulation class. Their approach is the closest to a constellation-based modulation classification advocated here although they have not made it the central thesis of their work. Carrier phase and clock recovery issues are also not addressed. Chugg et al [4] use an approximation of log-ALF to handle more than two modulations and apply it to classification between OQPSK/BPSK/QPSK. Lin and Kuo[9] propose a sequential probability ratio test in the context of hypothesis testing to classify among several QAM signals. Their approach is novel in the sense that new data continuously updates the evidence. Nonlinear operations on signals are probably the most indigenous to communications theory but are also ad hoc in many respects. For example, a square-law classifier produces bulges in the spectrum of a BPSK signal at twice the carrier frequency. No such component appears for a QPSK signal. Many have suggested this property to distinguish between the two. Square-law classifier is a special case of delay-and-multiply operation; r( t) r( t − ∆) . The spectrum of this product can be used as a harmonics detector. If energy is detected at dc for =0, but not at symbol rate for =half symbol rate, then the unknown modulation is an OQPSK signal rather than a BPSK or QPSK[12]. This approach is extended as the Mth law nonlinearity and filtering where the Mth power of an MPSK signal produces a spectral bulge at Mf c . Reichert[13] compiles these approaches under nonlinear algorithms in modulation recognition. There have been other approaches to modulation classification. A new method has been proposed by Ta[16] which uses the energy vectors derived from wavelet packet decomposition as feature vectors to distinguish between ASK, PSK and FSK modulation types. Apparently, only binary modulations are tested but more importantly no noise or other disturbances are involved. Marinovich et al.[10] use singular value decomposition to identify that part of the signal subspace
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that separates the classes the most. They use this approach to discriminate between MSK and OQPSK signals. Beidas and Weber[3] use higher order statistics. The latter approach is applied to MFSK signals only. More recently, Nandi and Azzouz[11] have used neural nets for modulation classification. Their database is limited to 2/4 ASK/PSK/FSK. 3. Reconstructing Received Signal's Constellation The focus in a constellation-based modulation recognition is on the reconstructed constellation. Let the received signal be modeled by arbitrary combinations of {amplitude, phase}. There are no fundamental limitations to add other dimensions such as frequency, however. N jθ j 2 πf t + θ r( t) = Re{ s( t) + n( t) } = Re ∑ Rk e k p( t − ( k − 1) Ts ) e ( c c ) + n( t) 0 < t < T = N Ts k = 1 (1) amplitude
where N is the number of observed symbols, R k kth symbol 2π θk ∈ i,i = 0,...,M − 1 , M number of phases, Ts symbol duration, T observation interval, fc M carrier frequency, θ c carrier phase tracking error and p(t) the basic baseband pulse. Noise is white with a two-sided spectral density of No/2. Working with the complex envelope portion r˜( t,θ c ) =
N
∑R e k
j (θ k + θ c )
p( t − ( k − 1) Ts ) + Rn e jθn
(2)
k=1
Following a serial to parallel conversion, N symbols are simultaneously fed to N parallel bank of correlators. This arrangement allows for a parallel constellation recovery. The output of the filter bank is an Nx2 vector with the kth row defined by r˜k (θ c ,ε ) = r˜i, k (θ c ,ε) + jr˜q, k (θ c ,ε ) =
∫
(k + ε )T s (k − 1+ ε )T s
r˜( t,θ c ) dt
(2)
k = 1, . . .N , Integration time is offset by the symbol timing error ε. Expanding (3) r˜k (θ c ,ε ) = r˜i, k (θ c ,ε) + jr˜q, k (θ c ,ε ) = Rk e j (θ k + θ c )
∫
(k + ε )T s (k − 1+ ε )T s
[ p( t − ( k − 1) T ) + R e ]dt jθ n
s
n
k = 1, . . .N , (4) Likelihood functions based on r˜k constitute the current thinking in modulation recognition. Due to the underlying complexities, however, only approximations to likelihood functions are N
{
M used. For example, in Huang and Polydoros[6] the sign of ∑ Re ( r˜k ) k=1
} is used for modulation
recognition under a coherent and synchronous environment (θ c = 0,ε = 0) .
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3.1 Impact of fading, clock recovery and symbol timing error Modulation classification most often is a non-cooperative process. Little is known about carrier frequency, phase, baud rate, symbol timing and propagation characteristics. In addition, channel fading can have a significant impact on the reconstructed constellation. There are standard techniques for carrier frequency estimation (CPS), baud rate recovery (Mth law nonlinearity) and clock recovery(early-late gate synchronizer). All such techniques are accompanied by errors which affect constellation shape. We will examine a few common cases below. In a carrier-coherent situation, θ c is assumed known. As such, it only introduces a fixed, known rotation of the entire constellation. Under a more realistic condition of carrier-noncoherent receiver, θ c is random but is assumed constant during the observation interval T. This will also cause a fixed constellation rotation but the angle of rotation will change form one reconstruction to another. Under a symbol-noncoherent scheme,θ c = 2 π fcτ ( t) is random but remains fixed for the duration of a symbol only. For this assumption to hold, the rms delay spread στ
