A novel approach to dictionary construction for automatic modulation classification

doi:10.1016/j.jfranklin.2014.02.002

Journal of the Franklin Institute

Volume 351, Issue 5, May 2014, Pages 2991-3012

https://doi.org/10.1016/j.jfranklin.2014.02.002 Get rights and content

Abstract

In almost all the work carried out in the area of automatic modulation classification (AMC), the dictionary of all possible modulations that can occur is assumed to be fixed and given. In this paper, we consider the problem of discovering the unknown digital amplitude-phase modulations when the dictionary is not given. A deconvolution based framework is proposed to estimate the distribution of the transmitted symbols, which completely characterizes the underlying signal constellation. The method involves computation of the empirical characteristic function (ECF) from the received signal samples, and employing constrained least squares (CLS) filtering in the frequency domain to reveal the unknown symbol set. The decoding of the received signals can then be carried out based on the estimate of the signal constellation. The proposed method can be implemented efficiently using fast Fourier transform (FFT) algorithms. In addition, we show that the distribution estimate of the transmitted symbols can be refined if the signal constellation is known to satisfy certain symmetry and independence properties.

Introduction

The purpose of modulation identification is to determine the modulation format of an unknown signal in order to demodulate and decode it correctly. Identification of digital modulation formats from noisy measurements has attracted much interest in diverse areas ranging from military applications such as electronic warfare to civilian applications such as cognitive radio (CR) and adaptive modulation [1], [2], [3]. The current strategies to tackle the modulation identification problem can be grouped under two categories: likelihood-based (LB) methods and feature-based (FB) methods (See [1] and references therein). LB methods attempt to solve this problem using classical detection theory by minimizing the probability of incorrect classification under different levels of prior statistical information about the received signal. The likelihood of the received signal samples is computed for each candidate modulation. Depending on whether the unknown quantities are treated as random variables or deterministic unknowns, there are three techniques: average likelihood [4], generalized likelihood [5], and hybrid likelihood [6]. With multiple candidates for the modulation format, the classification decision is made in favor of the modulation with the largest likelihood. Average likelihood based approach presupposes proper modeling for the prior probability density functions (PDFs) of the unknown channel parameters and data symbols. As a result, it is susceptible to model mismatch errors. It is also computationally demanding due to multidimensional integration needed to compute the average. On the other hand, generalized likelihood technique employs multidimensional maximization over the unknown parameters but it cannot satisfactorily discriminate among nested constellations due to its tendency to favor the denser constellation. In the hybrid likelihood technique, this problem is alleviated by averaging over the unknown data symbols, and then the resulting average likelihood function is maximized over the remaining unknown parameters. Recently, suboptimal noncoherent ML schemes have been proposed for identifying quadrature amplitude modulation (QAM) constellations under frequency mismatch and phase shift [7]. They offer reduced computational complexity with reasonably small performance loss when compared to the optimal noncoherent ML scheme.

While LB methods are optimal in the Bayesian sense, FB methods are suboptimal but their appeal derives from lower computational complexity and resistance to various model mismatches. FB methods rely on the extraction of discriminative features from signal characteristics such as amplitude, phase, frequency, zero-crossing interval, wavelet transform, Radon transform, characteristic function, signal cyclostationarity, moments, cumulants and cyclic cumulants [1]. A decision is made by comparing the sampled and theoretical values of the reference features. Among these, cumulant is the most widely used feature. The main reason is that they are not affected by additive white Gaussian noise (AWGN) since higher-order cumulants of Gaussian distribution are all zero [2]. Additionally, they are robust to carrier phase and frequency offsets, as well as colored Gaussian noise and impulsive non-Gaussian noise. However, higher-order cumulants are needed to differentiate higher-order constellations of the same modulation class, and a large number of samples is required to accurately estimate a higher-order cumulant. Recently, higher classification performance with lower number of samples has been reported using metrics based on Kolmogorov–Smirnov, Kuiper and variational distances between the cumulative distribution functions (CDFs) corresponding to different modulation orders [8], [9], [10].

Both LB and FB approaches aim to decide on the correct modulation format of the received noisy signals via hypotheses testing. In the case of digital amplitude-phase modulations (e.g., M-ary pulse amplitude modulation (M-PAM), M-QAM, M-ary phase shift keying (M-PSK) and M-ary amplitude-phase shift keying (M-APSK)), the problem simplifies to the identification of the signal constellation using the demodulated complex signal values assuming that the signal characteristics needed for downconversion and sampling are known or can be estimated reliably. Although efficient LB and FB algorithms are proposed in the literature for modulation classification, their performance relies on the implicit assumption that the modulation dictionary contains the true modulation type. Furthermore, the classification accuracy will drop as the number of candidate modulations in the dictionary increases. Since modulation identification typically needs to be performed in a noncooperative environment, it is beneficial to construct a complete dictionary of the candidate modulations employed by the target communications system. The discovery of the unknown digital amplitude-phase modulations is addressed in this paper by formulating it as a deconvolution problem and estimating the distribution of the transmitted symbols, which completely characterizes the underlying signal constellation. The method involves computation of the empirical characteristic function (ECF) from the received signal samples, and employing constrained least squares (CLS) filtering in the frequency domain to reveal the unknown symbol set. Upon estimation of the distribution of the transmitted symbols, the discovered constellation can be added to the dictionary. The decoding of the received signals can also be carried out based on the estimate of the signal constellation. If a dictionary is already available, the proposed method can be used to dynamically discover the unknown modulations that are not already considered in the dictionary and add them to the dictionary for hypothesis testing while pruning any modulation that is not observed from the dictionary. Such an approach can significantly help reduce the false detection rate. A particular advantage of the proposed method is that it is formulated entirely in the characteristic function (CF) domain, and can be implemented efficiently using fast Fourier transform (FFT) algorithms. In addition, we show that the distribution estimate of the transmitted symbols can be refined if the signal constellation is known to satisfy certain symmetry and independence properties.

The remainder of this paper is organized as follows. In Section 2, we introduce the AWGN channel model, discuss how flat BF channel conditions can be incorporated, and define the notation. In Section 3, we formulate the estimation of the distribution of the transmitted symbols as a deconvolution problem and explain the motivation behind this formulation. In Section 4, we provide a CF domain representation of the problem. We also discuss the effects of AWGN and finite sample size. In Section 5, we introduce the CLS filtering based approach to obtain the distribution estimate of the transmitted symbols. Also in this section, we present an efficient numerical method that is based on the FFT algorithm. In Section 6, some simplifications and refinements while obtaining the PDF/CF estimates are investigated under certain symmetry and independence conditions. In Section 7, numerical examples are presented to corroborate the applicability of the method developed. Section 8 concludes the paper.

Section snippets

System model

We consider a coherent and synchronous reception scenario where the symbols are transmitted through an intersymbol interference (ISI) free, AWGN channel and demodulated into the complex signal values without any phase or frequency distortion. After preprocessing, we obtain a discrete-time baseband system model [8], [9], [10], [11] $y_{n} = s_{n} + η_{n}, n = 1, \dots, N,$ where $s_{n}, y_{n}$ and η_n are respectively the complex-valued transmitted modulation symbol, the received signal, and the noise at time n. The transmitted

Motivation and problem statement

Since the transmitted modulation symbol $s$ and the noise $η$ are independent, the PDF of the received signal vector $y$ can be written as a convolution between the PDFs of $s$ and $η$ . We have $s ~ p_{s} (t) = \sum_{m = 1}^{M} π_{m} δ (t - μ_{m})$ and $η ~ N (0, σ^{2} I)$ , where $δ (\cdot)$ is the Dirac delta function and $I$ is the identity matrix. Hence, the PDF of $y$ can be expressed as a bivariate Gaussian mixture density with M homoscedastic components [1], [3], i.e., $p_{y} (t) = (p_{s} ⁎ p_{η}) (t) = \sum_{m = 1}^{M} π_{m} f (t; μ_{m}, σ^{2} I),$ where $⁎$ denotes convolution, and $f (\cdot; μ_{m}, σ^{2} I)$ is

Representation in the characteristic function domain

The convolution given in Eq. (5) can be equivalently expressed as multiplication in the CF domain. To see this, we recall that for a 2D random vector $x$ with PDF $p_{x} (t)$ , the joint CF is defined as [16] $Φ_{x} (ω) ≜ E {e^{j ω^{T} x}} = \iint_{R^{2}} e^{j ω^{T} t} p_{x} (t) d t,$ where $E {\cdot}$ denotes the expectation with respect to $p_{x} (\cdot)$ . It is noted that the CF is the complex conjugate of the continuous-time Fourier transform (CTFT) of $p_{x} (t)$ . Since $p_{x} (t)$ is real, the CF has conjugate symmetry, i.e., $Φ_{x} (ω) = Φ_{x}^{⋆} (- ω)$ , where $⋆$ denotes complex

Constrained least squares filtering

The naive approach for the estimation of $Φ_{s} (ω_{1}, ω_{2})$ is direct inverse filtering, where an estimate is computed by dividing ${\hat{Φ}}_{y} (ω_{1}, ω_{2})$ by the noise CF $Φ_{η} (ω_{1}, ω_{2})$ , i.e., ${\hat{Φ}}_{s} (ω_{1}, ω_{2}) = \frac{{\hat{Φ}}_{y} (ω_{1}, ω_{2})}{Φ_{η} (ω_{1}, ω_{2})} = Φ_{s} (ω_{1}, ω_{2}) + \frac{Φ_{Ξ} (ω_{1}, ω_{2})}{Φ_{η} (ω_{1}, ω_{2})} .$ Theoretically, with infinite data, $Φ_{s} (ω_{1}, ω_{2})$ can be perfectly recovered from ${\hat{Φ}}_{y} (ω_{1}, ω_{2})$ regardless of the noise power σ² since $Φ_{η} (ω_{1}, ω_{2}) \neq 0$ for all ω₁ and ω₂. However, in practice, the result of direct inverse filtering is of little value. First of all, the error $Φ_{Ξ} (ω_{1}, ω_{2})$

Further remarks

When it is known that the signal constellation possesses certain independence and symmetry properties, this information can be exploited to improve the performance of the proposed approaches as discussed below.

(1) When the real and imaginary components of the transmitted modulation symbol $s$ are independent, the components of $y$ are independent under the circularly symmetric AWGN channel model given in Eq. (1).⁵

Numerical results

In this section, we provide simulation results to evaluate the performance of the proposed method in estimating the underlying signal constellation. We also investigate its sensitivity to variations in the parameters. The 2D PDF estimates of the transmitted symbols are shown using contours. The assessment is based on the graphical examination of the output. We assume the AWGN channel model given in Eq. (1) with $η_{n} ~ CN (0, 2 σ^{2})$ . Without loss of generality, we restrict to unit power constellations.

Conclusion

In this paper, a deconvolution problem was formulated for the estimation of the distribution of the transmitted symbols, which completely characterizes the signal constellation. The method of constrained least squares filtering was employed to obtain the solution. The only requirement for the proposed procedure is the knowledge of the probability density function or the characteristic function of the noise at the receiver. As a result, our method can be applied in the presence of non-Gaussian

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^☆: This material is in part based upon work supported by the Government under Contract No. W15P7T-12-C-A040.

¹: B. Dulek was a postdoctoral researcher at the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, USA, when this work was performed.

View full text

A novel approach to dictionary construction for automatic modulation classification☆

Abstract

Introduction

Section snippets

System model

Motivation and problem statement

Representation in the characteristic function domain

Constrained least squares filtering

Further remarks

Numerical results

Conclusion

Survey of automatic modulation classification techniquesclassical approaches and new trends

IET Commun.

Hierarchical digital modulation classification using cumulants

IEEE Trans. Commun.

Likelihood-ratio approaches to automatic modulation classification

IEEE Trans. Syst. Man. Cybern. C: Appl. Rev.

Maximum-likelihood classification for digital amplitude-phase modulations

IEEE Trans. Commun.

Noncoherent maximum likelihood classification of quadrature amplitude modulation constellationssimplification, analysis, and extension

IEEE Trans. Wirel. Commun.

Fast and robust modulation classification via Kolmogorov–Smirnov test

IEEE Trans. Commun.

Computationally efficient modulation level classification based on probability distribution distance functions

IEEE Commun. Lett.