# Signal parameter estimation of complex exponentials using fourth order statistics: additive Gaussian noise environment

- Pradip Sircar
^{1}Email author, - Mukesh K Dutta
^{1}and - Sudipta Mukhopadhyay
^{2}

**Received: **30 December 2014

**Accepted: **2 July 2015

**Published: **17 July 2015

## Abstract

A novel approach based on fourth order statistics is presented for estimating the parameters of the complex exponential signal model in additive colored Gaussian noise whose autocorrelation function is not known. Monte Carlo simulations demonstrate that the proposed method performs better than an existing method which also utilizes fourth order statistics under the similar noise condition. To deal with the non-stationarity of the modeled signal, various concepts are introduced while extending the estimation technique based on linear prediction to the higher order statistics domain. It is illustrated that the accuracy of parameter estimation in this case improves due to better handling of signal non-stationarity. While forming the fourth order moment/ cumulant of a signal, the choice of the lag-parameters is crucial. It has been demonstrated that the symmetric fourth order moment/ cumulant as defined in this paper will have many desirable properties.

## Keywords

## Introduction

The estimation of the model parameters of a non-Gaussian stationary signal in the presence of additive Gaussian noise is conveniently carried out by a higher-order cumulant-based technique (Swami and Mendel 1991). Note that the non-Gaussian signals are frequently encountered in active sonar (Wegman et al. 1989). It is known that the cumulants of order greater than two of Gaussian processes are zero, whereas the cumulants of non-Gaussian processes carry higher order statistical information. As a consequence, the cumulant-based techniques are particularly suitable when the noise is colored Gaussian with unknown power spectral density (PSD).

A transient signal is traditionally represented by the parametric model consisting of a sum of complex exponentials, and the model parameters are estimated by Prony’s method (Henderson 1981; Kumaresan and Tufts 1982). Parameter estimation of the sum of complex exponential signals is closely related to the problem of system identification, because the impulse response of a linear time-invariant (LTI) system with distinct poles has the similar parametric form. Moreover, the sum of complex exponential signals provides a more general model than the sum of complex sinusoidal signals. In this way, the parametric representation can be extended from a stationary signal to a non-stationary signal, as explained in the sequel. When the signal derived from a multicomponent complex exponential random process is corrupted with white noise, an extension of Prony’s method to the second-order statistics domain can provide better accuracy of estimation of model parameters (Cadzow and Wu 1987; Sircar and Mukhopadhyay 1995). It should be mentioned, however, that special care is to be taken in the extension of the estimation technique to the higher order statistics domain because the signal being modeled is non-stationary in nature (Sircar and Mukhopadhyay 1995), and the cumulants of a non-stationary process are, in general, time-dependent.

In the presence of additive noise which is colored Gaussian with unknown autocorrelation sequence, an estimation technique utilizing the fourth order moment/ cumulant sequence of the modeled signal will be desirable. In this paper, several new concepts related to higher order statistics, which can be implemented for estimating the parameters of a transient signal in noise are presented. It is demonstrated how the concept of accumulated moment can be introduced in the context of the non-stationarity of the modeled signal (Sircar and Mukhopadhyay 1995). The case where a single data record is available is studied separately because a replacement of the ensemble average by a temporal average will alter the result for this type of signal.

While forming the higher order moments and cumulants, the selection of the definition to be used and the choice of the lag-parameters in the definition are crucial (Swami and Mendel 1991). It is demonstrated that the moments and cumulants defined and used in the proposed methods will have many desirable properties. Moreover, it is shown by Monte Carlo simulation that the method presented here will perform better than another method (Papadopoulos and Nikias 1990), even though both of the methods utilize the fourth order statistics (FOS) of the sampled signal.

In the present work, we consider fourth order moment/ cumulant of a non-stationary signal for parameter estimation. The FOS has been used for parameter estimation with multiplicative noise model (Swami 1994), for detection of transient acoustic signals (Ravier and Amblard 1998), and for blind identification of non-minimum phase finite impulse response (FIR) system (Belouchrani and Derras 2000). In recent time, the FOS has been applied for blind source identification where number of sources exceeds number of sensors (Ferréol et al. 2005), for underdetermined independent component analysis (ICA) (Zarzoso et al. 2006; Lathauwer et al. 2007), and for ICA with multiplicative noise (Blanco et al. 2007). Lately, it has been shown that the FOS-based direction-of-arrival (DOA) estimation of non-Gaussian source has many advantages compared to the subspace based methods (Zeng et al. 2009).

In the continuation work, the method based on fourth-order moment/ cumulant, as developed in this paper, has been applied for parameter estimation of non-stationary signals in multiplicative and additive noise (Gaikwad et al. 2015). We assume that the additive noise is Gaussian and the signal component comprising of the multiplicative noise is non-Gaussian. It has been demonstrated in the accompanying paper that the concept of accumulated moment, as presented here, can be extended to the concept of accumulated cumulant while estimating parameters of some non-stationary signals in multiplicative noise (Gaikwad et al. 2015).

The paper is organized in the following way: The accumulated fourth order moment for a non-stationary signal is defined in "Accumulated fourth order moments". The case of signal corrupted with noise is considered in "Noise-corrupted signal case", where the fourth order cumulant of the signal is derived. In "Deterministic signal case", the case of deterministic signal is considered, and the fourth order time cumulant of the signal is defined. The simulation study is presented in "Simulation study", and the conclusion is drawn in "Conclusion".

## Accumulated fourth order moments

*X*[

*n*] consisting of

*M*complex exponential signals with the circular frequencies \({\omega }_i\), damping factors \({\alpha }_i\) (negative), amplitudes \(A_i\) and phase \({\phi }_i\) is expressed as

*m*it becomes,

*n*between properly selected \(n_1\) and \(n_2\) used in (10) and (15), we compute the time-varying autocorrelation functions (ACFs) of

*X*[

*n*] considered to be a random sequence due to its random phase \({\phi }_i\) (1),

*n*as shown below,

It should be noted that the accumulated ACFs are functions of lag *k* in a similar way as the discrete-time random process *X*[*n*] is function of time *n*. Consequently, the accumulated ACFs will satisfy the linear prediction equations in lag \(k (= m-\ell )\) (10, 15). Note here that for a wide-sense stationary (WSS) process the ACF is time-invariant, and it satisfies the prediction equation in lag *k*. Apparently, the role of accumulated second-order moment for the non-stationary process is quite similar to the role of second-order moment for a WSS process (Sircar and Mukhopadhyay 1995).

*X*[

*n*] as follows,

*n*between properly selected \(n_1\) and \(n_2\) as shown below,

It is apparent from (25) that the accumulated fourth order moment (AFOM) is time-invariant as desired. Remember that the underlying process is non-stationary in nature. Moreover, it is to be noted that the AFOM sequence will satisfy the linear prediction equations in lag *k* (Sircar and Mukhopadhyay 1995). However, the modes of the AFOM sequence \(Q_{4X} [k]\) are not the same modes as those of the discrete-time signal *X*[*n*]. The modes of AFOM sequence are, indeed, the square and product modes. For every mode \(z = |z|\angle {\theta }\) in *X*[*n*], the square mode \(z^2 = |z|^2\angle {2\theta }\) is included in \(Q_{4X} [k]\). Again, for every pair of modes \(z_1 = |z_1|\angle {\theta _1}\) and \(z_2 = |z_2|\angle {\theta _2}\) in the signal, the product mode \(z_1 z_2 = |z_1| |z_2| \angle {(\theta _1 + \theta _2)}\) is also included in the AFOM sequence. Clearly, if there are *M* modes in the original signal, the AFOM sequence will contain a total \(L = M + {}^M C_2 = M(M+1)/2\) modes. Accordingly, the sequence will satisfy the linear prediction equations of order L or higher (Deprettere 1988); and once the modes of the AFOM sequence are extracted, there will be redundancy of information for computing the modes of the underlying signal.

## Noise-corrupted signal case

*Y*[

*n*] is represented as

*W*[

*n*] is the random noise whose statistics may not be completely known. In general, the moments of

*X*[

*n*] and

*Y*[

*n*] will be different. In all cases, it is assumed that the noise sequence is zero-mean, and the signal and noise sequences are uncorrelated.

It is easy to show that the symmetric fourth order moments of the noisy and noiseless signals are same, that is, \(R_{4Y} [n,k] = R_{4X} [n,k]\), with an additional condition that the noise sequence is white. This case together with the complementary case where the noise correlation is known, however, can be adequately dealt with in the second-order statistics domain (Cadzow and Wu 1987; Sircar and Mukhopadhyay 1995). Therefore, these situations are not considered here any further.

*Z*[

*n*] which may be either

*X*[

*n*] or

*Y*[

*n*], does not change when the noiseless signal becomes noisy, that is, \(C_{4X} [n,k] = C_{4Y} [n,k]\), where we define the symmetric fourth-order cumulant as

*Z*[

*n*] (Swami and Mendel 1991).

Note that the fourth order cumulant will satisfy the *M*-order linear prediction equations in lag exactly in the same way as the signal does that in time (Sircar and Mukhopadhyay 1995; Deprettere 1988). Moreover, the modes of the cumulant are the square modes of the signal, and the product modes as defined earlier are missing.

## Deterministic signal case

In last two sections we considered the case of random signal in some detail. In this section, a treatise is presented for the non-random signal. Although the observed sequence can be thought of as a sample of some discrete-time random process, any replacement of ensemble average by temporal average will likely produce a different set of results because the underlying signal is non-stationary in nature.

*n*(and

*m*), indices

*i*and

*l*[see (28)] by taking summation over respective variables. Similarly, each of \(t_{\ell 2}\) is independent of all variables except

*u*, and every \(t_{\ell 3}\) is made independent of all six variables by summation. It is to be pointed out that if the mean \(X_0 = 0\), the coefficients \(t_{\ell 2}\) and \(t_{\ell 3}\) will be identically zero. In this case, each of \(t_{\ell 1}\) will again be a non-zero factor.

Comparing (25) and (37), it can be observed that the \(\tilde{C}\)-sequence consists of the square and product modes of the signal, together with the original signal modes. If there are *M* modes in the sampled signal, the number of modes in the \(\tilde{C}\)-sequence will be \(L = M + M(M+1)/2 = M(M+3)/2 \,\,\). Consequently, the sequence will satisfy the linear prediction equations of order more than *L*. Remember that the unity mode may also be present.

When the signal is corrupted with noise, the \(\tilde{C}\)-sequence which may be termed as the fourth order time cumulant (FOTC), will deviate. However, it is likely that this deviation will be small when the superimposed noise is zero-mean Gaussian and uncorrelated with the signal. Remember that we are computing the time-average here.

## Simulation study

The discrete-time signal \(X[\tilde{n}]\) of (1) is sampled at \(\tilde{n} = 0,1,\ldots ,2N+1\) with \(N = 50\) utilising the following parameters, \(M = 2\), \(A_1 = 1.0\), \(\alpha _1 = - 0.02\), \(\omega _1 = 0.3\), \(\phi _1 = 0\), \(A_2 = 1.0\), \(\alpha _2 = - 0.01\), \(\omega _2 = 0.5\), \(\phi _2 = \pi /4\). The time-origin is then shifted by making \(n = \tilde{n}-N\) such that the sequence \(\{X[n]; n = -N,\ldots ,0,\ldots ,N\}\) is now available for processing. The shifting does not change the modes of the signal, only the complex amplitudes of the components change to new values [refer to (1)].

Next, the zero-mean complex colored Gaussian noise is added to the signal setting the signal-to-noise ratio (SNR) at various designated levels. The colored noise is generated by passing a complex white Gaussian process through a coloring filter as mentioned in (Papadopoulos and Nikias 1990). The fourth order time cumulants \(\tilde{C}[k]\) are then computed by (34) for the lags \(k = -K,\ldots ,0,\ldots ,K\) with \(K = 25\), that is a total of 51 points. The time-frame is chosen with \(n_1 = 8\) and \(n_2 = 25\). The corresponding frame-length \((n_2-n_1+1)\) lies between *N* / 3 and 2*N* / 5 ensuring optimal performance (Hua and Sarkar 1990).

There are \(L = 5\) modes in the \(\tilde{C}\)-sequence. We use backward linear prediction model with the extended model order \(J=17\) and the singular value decomposition based total least squares (SVD-TLS) technique for extraction of the *L* signal \((1/z_i^{\star })\) and \((J-L)\) noise modes (Kumaresan and Tufts 1982; Cadzow and Wu 1987; Deprettere 1988). Remember that the modes of the \(\tilde{C}\)-sequence are the square, product and original modes of the sampled sequence *X*[*n*].

For the purpose of comparison, the same noise-corrupted data record is processed employing the fourth order cumulant (FOC) method as proposed in (Papadopoulos and Nikias 1990). The extended model order used is same in each of the cases, which ensures that the computational efforts for extracting modes are roughly equal for the methods.

It is seen in Figure 1 that for the \(\tilde{C}\)-sequence, the square and product modes are accurately determined, whereas the original signal modes are often undetected at this SNR level. Moreover, it is found that the square modes are estimated with better accuracy than the product modes. Note that it is easy to identify and utilize the square modes of the signal for estimating the modes of the underlying signal. First the square-roots of all the estimated modes of the \(\tilde{C}\)-sequence are to be computed, then the complex amplitudes are to be estimated for all these computed modes utilizing the original signal sequence (Cadzow and Wu 1987). Selecting the original signal modes for *M* larger amplitudes will then be relatively straightforward. Note that in this way, the signal modes will be determined utilizing the square modes of the \(\tilde{C}\)-sequence. Thus, it should be the strategy that the modes of the underlying signal are to be determined utilizing the accurate estimates of the square modes of the \(\tilde{C}\)-sequence, because the estimates of the signal modes of the \(\tilde{C}\)-sequence may be inaccurate when the signal samples are corrupted with noise.

## Conclusion

In this paper, a new method based on utilizing the computed fourth order time cumulants of a transient signal is presented for estimating the parameters of the modeled signal. The proposed method performs better than the FOC method developed in (Papadopoulos and Nikias 1990), even though the latter is another method based on the fourth order statistics.

It is known that the fourth order moments and cumulants can be defined and evaluated in a variety of ways (Swami and Mendel 1991). It is note-worthy that when the fourth order moments/ cumulants are utilized in the problem of model parameter estimation of a transient signal in noise, the methods which employ the cumulants defined and evaluated in different ways do produce results with different accuracies.

Our investigation shows that the proposed method based on a new definition of the fourth order cumulants does satisfy some optimality criteria. In particular, it is demonstrated that the symmetric fourth order cumulant sequence as defined here is independent of time, even though the underlying signal is non-stationary in nature.

In the accompanying paper, it is demonstrated that the method based on symmetric fourth-order moment/ cumulant can be applied for parameter estimation of signals in multiplicative and additive noise (Gaikwad et al. 2015). The concept of accumulated fourth-order moment has been extended to the concept of accumulated fourth-order cumulant for parameter estimation of non-stationary signals.

## Declarations

### Authors’ contributions

All authors have made contributions to conception and design, analysis and interpretation of data, and they have been involved in drafting the manuscript. Each author has read and approved the manuscript in the submission. All authors read and approved the final manuscript.

### Compliance with ethical guidelines

**Competing interests** The authors declare that they have no competing interest.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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