Analytic Nakagami fading parameter estimation in dependent noise channel using copula
© Gholizadeh et al.; licensee Springer. 2013
Received: 17 November 2012
Accepted: 4 July 2013
Published: 16 July 2013
In this paper, the probability density function (PDF) estimation is introduced in the framework of estimating the Nakagami fading parameter. This approach provides an analytic procedure for finding the fading parameter. Using the copula theory, an accurate PDF estimate is obtained even when the desired signal is corrupted in a noisy environment. In the real world, the noise samples could be highly dependent on the main signal. Copula-based models are a general set of statistical models defined for any multivariate random variable. Thus, they depict the statistical behavior of a received signal including two dependent terms, representative of the desired signal and noise. Previous works in the Nakagami parameter determination have mainly examined estimation based on either a noiseless sample model or an independent trivial noisy one. In this paper, we consider a more comprehensive situation about the noise destruction and our investigation is done in low signal-to-noise ratios. The parametric bootstrap method approves the accuracy of the analytically estimated PDF, and simulation results show that the new estimator has superior performance over conventional estimators.
The Nakagami-m distribution is considered as one of the most important models among all the statistical ones that have been proposed to characterize the fading envelope due to multipath fading in wireless communications . With a simple exponential family form, the Nakagami-m distribution often leads to closed-form analytical results. The Nakagami fading exploits Nakagami probability density function (PDF) for the envelope of received signal which possesses two parameters: scale and shape parameters. The latter is more important and called the fading parameter or m-parameter. Determining m is a problem in Nakagami PDF estimation.
The most prominent conventional procedures used for the estimation of the Nakagami fading parameter, m, are based on either maximum likelihood estimation or moment-based estimators [2, 3]. Among maximum likelihood (ML)-based conventional methods, the Greenwood-Durand estimator is well known in m-parameter estimation . There are also analytic and bootstrap bias-corrected ML estimators for estimating m that improve conventional ML estimators . On the other hand, the inverse normalized variance and generalized method of moments (GMM) are the moment-based procedures, in which the latter presents the Nakagami parameter estimation in noisy environment with acceptable performance .
However, all aforementioned approaches either do not take into account noisy cases or consider trivial noises. In this paper, we intend to estimate the Nakagami parameter in a dependent noise environment based on the PDF of received signal and the copula concept. We present a comprehensive noise model that is not confined to restrictive assumptions, such as uncorrelatedness or independence. Signal-dependent noise is used in the paper which is a more realistic assumption in practical applications such as image transmission , radar , and wireless communications . The copula theory is one of the best methods used for modeling the dependency in conventional works . The copula theory is an elegant concept introduced by Sklar in order to find the link between a joint PDF and its marginal PDFs . Then, copula functions have been extensively studied and a comprehensive discussion of their mathematical properties has been presented [11, 12].
Now, a new copula-based method is presented to estimate the Nakagami fading parameter in the faded received signal contaminated by dependent noise. The novelty of our approach lies on determining the analytic PDF of the received data under the assumption of dependent noise by using the copula theory in order to estimate the parameter of the Nakagami-m fading model. To measure the accuracy of the estimation approach, the result is examined by using a goodness of fit test called parametric bootstrap algorithm .
This paper is organized as follows: In Section 2, we recall basic facts and definitions on copulas and the dependency problem. Section 3 includes the main signal and noise model used in the paper. The introduced models are such comprehensive ones which precisely correspond to actual signals. Using copula, the PDF of the received signal is estimated in Section 4, and the fading parameter is determined. Finally, simulation results are given in Section 5, and some conclusions are drawn in Section 6.
One popular method of modeling the dependencies is the copula approach. The word copula is a Latin noun which means ‘a link, tie, or bond’ and was first employed by Sklar in mathematical and statistical problems . Mathematically, copula is a function that combines univariate PDFs to obtain a joint PDF with a particular dependency structure. In this paper, the fading parameter is estimated based on the PDF of the received signal, given that the received signal has been corrupted by a signal-dependent noise. Due to the signal-dependent nature of noise, we are required to determine the PDF of a signal that is composed of two dependent components. Thus, the copula concept is a tool that is compatible with our problem, and it facilitates the PDF estimation procedure. The foundation theorem for copula was introduced by Sklar which states that for a given joint multivariate PDF and the relevant marginal PDFs, there exists a copula function that relates them. In a bivariate case, Sklar’s theorem is as follows:
If F x and F y are continuous, then C is unique; otherwise, C is uniquely determined on the (range of F x ) ×(range of F y).
Conversely, if C is a copula and F x and F y are CDFs, then the function F xy defined by (1) is a joint CDF with margins F x and F y.
The proof of the theorem can be found in . Since C is a rather particular type of function, it possesses some inherent properties. A thorough description of these properties is found in . The mentioned properties state that a copula is itself a CDF, defined on [0,1]2, with uniform margins.
The copulas have two main families. One of them is the family of elliptical copulas. The most common elliptical copulas are normal and Student’s t. The key advantage of an elliptical copula is that one can specify different levels of dependency between the margins. Another important class of copulas is known as the Archimedean copulas. Archimedean copulas are popular because they are constructed easily and allow modeling the dependence in arbitrarily high dimensions with only one parameter, governing the strength of dependence .
where ρ is the normal copula parameter and Φ −1 is the inverse of the univariate standard normal CDF. It is called the normal copula because, similar to bivariate normal distribution, it also enforces dependency by using pairwise correlations among the variables. However, in the normal copula, the marginal distributions are arbitrary.
where α is the Clayton copula parameter.
3 Signal model
In this section, a new model is introduced for the received signal in which noise and fading are statistically embedded. By estimating the PDF of the received signal, the nonlinear minimum mean square error (NMMSE) estimator principle is invoked to estimate the fading parameter. To transmit information over a communication channel, there are different methods to modulate the information. Quadrature amplitude modulation, phase shift keying, frequency shift keying, and continuous phase modulation are some prominent modulation methods. All these methods use a sinusoidal function as the carrier signal. Hence, we base signaling assumption on a sinusoidal transmission entering a multipath environment infested by generally a non-Gaussian noise. Thus, the procedure is extensible to all of the above types of modulation schemes.
where a i , , and θ i are the attenuation factor, Doppler shift, and the phase on the i th path, respectively, and ℜ(·) denotes the real part of its argument.
where I(t) and Q(t) are the inphase and quadrature components of signal s(t)/A. When L is large, the inphase and quadrature components will be normally distributed. Therefore, the phase Θ is uniformly distributed over [0,2π).
where P(·,·) is the incomplete gamma function. Nakagami is a general fading distribution that reduces to the Rayleigh for m=1 and to the one-sided normal distribution for m=0.5. It also approximates the Rician and lognormal distributions.
The scale parameter Ω is defined as the second-order moment of the Nakagami-m fading envelope and is calculated straightforward. Thus, we focus on estimating the primary parameter, i.e., the shape parameter m.
On the other hand, when the envelope R(t) has Nakagami distribution, we can assume that the phase Θ(t) is uniformly distributed over [0,2π) .
In order to start the formulation of our problem, firstly, let us suppose that s(t) in (8) is received without any additive noise. So, the received signal is s(t), and the PDF of s(t) is needed. Based on (10), we find the PDF of the product of two random variables (RVs) at time t, i.e., R(t) and A cos(ω c t+Θ(t)).
In (19) and (20), we obtained an expression for the PDF of the received signal s(t) in a noiseless scenario. Next, we intend to derive an expression if a noise term is also present along with s(t).
where the parameter γ is between 0 and 1. It is well known that the model defined in (22) mimics many types of random variates [18, 19]. This second sub-channel signifies the statistical importance of signal characteristics which could be modified by a random medium denoted by
The dependency between the faded signal and noise helps to have a more actual model for the received signal.
The samples of signal ν(t) can be at hand, for example, by using an out-of-band measurement of the noise. In this way, the data transmission is not interrupted and the samples of ν(t) can be measured as accurately as needed. In other words, in (23), during the times that out-of-band measurements are performed, the presence of the signal is contemplated as s(t)=0; therefore, this allows us to estimate the parameters of ν(t). Alternatively, the samples can also be obtained by sending a ‘zero’ signal and sampling the output of the receiver.
The new presented model helps to ensure that final results are reliable even in an actual non-Gaussian environment.
where f s(s) and f ν (ν) are obtained from (19) and (25), respectively, and η is an auxiliary variable.
Up to now, both PDF of the faded signal and the dependent noise were estimated analytically. In the next section, we determine the total PDF of the received signal.
4 Parameter estimation
where F s(s) and F n(n) are the marginal CDFs of the faded signal and noise, and θ c is the dependence parameter in the copula density function. Therefore, the joint PDF is equal to the product of the marginal PDFs and the copula density function, i.e., the term c(F s(s),F n(n);θ c ).
Until now, we have determined the PDF of the received signal analytically. However, it is necessary to validate the estimated PDF. The possible discrepancy between the hypothesized PDF model and the observed data is measured by the so-called goodness of fit statistics. In order to decide whether the observed discrepancy is substantial, performing a statistical test is required. Bootstrapping is an ideal procedure to estimate the accuracy of a parameter estimator . This method calculates confidence intervals for parameters. To assess a parameter estimator using the bootstrap method, we examine whether the estimated parameter is in a corresponding confidence interval or not. The bootstrap method, and also the confidence interval that the bootstrap method presents for the fading parameter in our work, is described in Appendix Appendix 3: bootstrapping. The results in Appendix Appendix 3: bootstrapping about using the bootstrap method for calculating the 95% confidence interval of the fading parameter are satisfactory.
In the next section, some simulations approve the obtained results.
5 Simulation and results
The efficiency of the proposed method for fading parameter estimation based on the estimated analytic PDF is evaluated using some simulations. It is essential that the proposed method be compared with other prominent conventional methods. The moment-based estimation , enhanced moment-based method , ML-based estimation , and GMM procedure  are known as the conventional methods. The mentioned methods are compared with the two proposed copula-based estimators. The difference between the two copula-based methods is due to the type of copula being used.
Note that since the results of the two moment-based  and enhanced moment-based methods , in Figures 2 and 3, are not reliable for SNR values from −10 to 0 dB, we depict them only for SNR values from 0 to +10 dB.
In addition to simulation, the statistical accuracy of the proposed copula-based estimator is examined by the bootstrap method which is introduced in Appendix Appendix 3: bootstrapping.
In this work, we propose a new estimation method based on the PDF and copula function to estimate the Nakagami-m fading parameter in wireless communications. The copula concept, which was originally proposed in econometrics literature, provides an analytic approach for finding the PDF of the received signal. It models the dependency between the faded signal and noise, and facilitates the separation of dependent noise from the desired part of the received signal. Therefore, in this paper, a novel approach is employed to estimate the Nakagami-m parameter that analyzes the noise behavior much better than other conventional procedures. Moreover, a comprehensive model is also suggested for the noise behavior that is a suitable representation for the actual noisy environment. The presented copula-based method has precise results in low SNRs, while no other conventional methods can be reliable in such SNRs even in independent noise background. The parametric bootstrapping method is used to test the accuracy of the estimations. In addition to goodness of fit tests, simulation results also show the validity of the estimations.
Appendix 1: the proof for (18) and (19)
Firstly, we show how (18) is derived. Then, the proof of (19) is presented.
The proof for (18)
where Θ i s are the real roots of the equation s 2=g(Θ), and g ′(Θ) is the derivative of g(Θ).
In this paper, Θ is uniformly distributed over [0,2π). Thus, only two solutions, which are in the interval [0,2π), are acceptable. The function f Θ (Θ) equals for these two values, and it equals 0 for any Θ n outside the interval [0,2π). Therefore, the proof of (18) is concluded.
The proof for (19)
Thus, the first equality of (19) is proven. The latter equation of (19) is obtained by setting , and the proof of (19) is concluded.
Appendix 2: K-distribution
In this section, it is proven that if the variance of the normally distributed noise ν(t) becomes a random variate, the noise ν(t) will have the K-distribution instead of normal distribution.
As mentioned, the variance of noise ν(t) is estimated given samples of ν(t). The samples can be broken up into small segments of length N 1 such that the samples in each segment have almost identical variance.
which has chi-squared behavior with N 1 degrees of freedom.
where τ has chi-squared distribution with N 1 degrees of freedom, and is normally distributed with mean zero and standard deviation b.
where Γ(·) is the gamma function.
Since the function f ν (ν) is even, the calculation of f ν (ν) for negative values of ν is a straightforward matter. Therefore, the proof is concluded.
Appendix 3: bootstrapping
Let us consider r=(r 1,r 2,…,r N ) as a random sample of the received signal r(t) with the PDF f r(r) in (30). The sample is used to estimate the certain fading parameter, m, associated with the PDF f r(r). A statistic, T=T(r), might be used to estimate m from the data. In this paper, the mentioned statistic is introduced in (36). The bootstrap method determines a measure of the statistical accuracy of the estimator T(r) in estimating the parameter m. The method can statistically quantify the error between m and the statistic T.
95% Confidence interval for fading parameter m in normal copula-based estimator for ρ p =0.9 ( B=100 )
95% Confidence interval
95% Confidence interval for fading parameter m in Clayton copula-based estimator for ρ p =0.9 ( B=100 )
95% Confidence interval
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