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# Signal parameter estimation using fourth order statistics: multiplicative and additive noise environment

- Chandrakant J Gaikwad
^{1}, - Hemant K Samdani
^{1}and - Pradip Sircar
^{1}Email author

**Received:**20 November 2014**Accepted:**5 June 2015**Published:**24 June 2015

## Abstract

Parameter estimation of various multi-component stationary and non-stationary signals in multiplicative and additive noise is considered in this paper. It is demonstrated that the parameters of complex sinusoidal signal, complex frequency modulated (FM) sinusoidal signal and complex linear chirp signal in presence of additive and multiplicative noise can be estimated using a new definition of the fourth order cumulant (FOC), and the computed accumulated FOC (AFOC). Analytical expressions for the FOC/AFOC of the above signals are derived. The concept of accumulated cumulant is introduced to handle the case of a non-stationary signal, for which the fourth order cumulant may be a function of both time and lag. Simulation study is carried out for all the three signals. In case of complex sinusoidal signals, the resul ts of parameter estimation show that the proposed method based on the new definition of fourth order cumulant performs better than an existing method based on fourth order statistics. The proposed method can be employed for parameter estimation of non-stationary signals also as mentioned above. For comparison purpose, the Cramer-Rao (CR) bound expressions are derived for all the signals considered for parameter estimation. The simulation results for non-stationary signals are compared with the CR bounds.

## Keywords

- Parameter estimation
- Multiplicative noise
- Fourth-order cumulant
- Higher-order statistics

## Background

In many applications, such as Doppler radar signal processing (Besson and Castanie 1993), synthetic aperture radar image processing (Frost et al. 1982; Lee and Jurkevich 1994), optical imaging under speckle or scintillation condition (Frankot and Chellappa 1987; Jain 2002), transmission of signals over fading channels (Makrakis and Mathiopoulos 1990a, b; Proakis 2001), speech processing in signal-dependent noise (Kajita and Itakura 1995; Quatieri 2002), and more, we need to consider the noise component to be both multiplicative and additive to the signal component.

In literature, signal parameter estimation in multiplicative and additive noise has been reported employing the non-linear least squares (NLLS) techniques (Besson and Stoica 1995; Besson and Stoica 1998; Ghogho et al. 2001; Besson et al. 1999), the cyclostationary approaches (Shamsunder et al. 1995; Zhou and Giannakis 1995; Giannakis and Zhou 1995; Ghogho et al. 1999a, 1999b), and the methods based on higher order statistics (Dwyer 1991; Swami 1994; Zhou and Giannakis 1994). In the NLLS techniques, a random amplitude observed signal is matched with a constant amplitude modelled signal in the least squares sense. When the random amplitude process is zero mean, we match the squared observed signal with the squared modelled signal. The NLLS estimators lead to an optimization problem which needs to be solved by an iterative technique. For a linear chirp signal, we need to perform a two-dimensional search where the initial guess, global convergence, convergence rate, and more are crucial issues (Besson et al. 1999). In the approaches based on cyclic statistics, we utilize the properties of the underlying signal. For a random amplitude polynomial phase signal, if the polynomial order is \((p+1)\), then the process will be \(2^p\)-order cyclostationary, i.e., the signal moments and cumulants of order \(2^p\) will be (almost) periodic. Using the cyclic moments/cumulants of order \(2^p\), the \((p+1)\)th order coefficient in the phase polynomial can be estimated. Having estimated the highest order polynomial coefficient, the signal can be demodulated to reduce the polynomial order, and the process can be repeated to estimate the next highest order polynomial coefficient. For the cyclic estimator to work, it is necessary that the random amplitude process be bandlimited, and higher the polynomial order, the more stringent the requirement on the bandlimitedness of the amplitude process. Some other issues are: (1) When finite data samples are used, the peaks in the cyclic moments/cumulants may be difficult to discern; (2) Due to the sequential procedure, there is cumulative effect that significantly degrades the accuracy of lower order polynomial coefficients (Shamsunder et al. 1995).

In the present work, our focus is on higher order statistics. We do not consider any other approaches for comparison or otherwise. In the methods based on higher order statistics, our concern is to develop a way to reduce the higher dimensionality of higher order moments and cumulants. Another issue is to tackle the non-stationarity of the observed signal, which makes the moments and cumulants time-varying in nature. In the paper, we address these issues and find some solutions.

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. Therefore, when the additive noise process is Gaussian and the signal process modulated by the multiplicative noise is non-Gaussian, one may use the methods based on third or fourth order cumulants of the signal for estimating signal parameters (Swami and Mendel 1991; Swami 1994).

Different slices of higher order cumulants are utilized for parameter estimation of various harmonic and modulated signals. Higher dimensionality of higher order cumulants are conventionally tackled by taking appropriate slices of cumulants such that the slices retain the pertinent information about the signal (Swami and Mendel 1991; Swami 1994). However, the selection of appropriate slices for various signals of interest may be a complicated task. Moreover, when the signal is non-stationary in nature, the moments and cumulants of the signal may depend on both time and lag (Sircar and Mukhopadhyay 1995; Sircar and Syali 1996; Sircar and Sharma 1997; Sircar and Saini 2007). Therefore, the utilization of such time-varying moments and cumulants for parameter estimation of signals may be quite challenging.

In the accompanying paper, a new definition for calculating the symmetric fourth order moment and cumulant of a transient signal has been proposed (Sircar et al. 2015). It has been demonstrated that with the choice of the lag-parameters in the definition, the computed moment and cumulant of the non-stationary signal will have some desirable properties. In the present work, we use the same definition for computing the symmetric fourth order moments and cumulants of some stationary and non-stationary signals in multiplicative and additive noise.

The multi-component signals considered in this paper for parameter estimation are complex sinusoidal signal, complex frequency modulated (FM) sinusoidal signal, and complex linear chirp signal. The complex amplitude modulated (AM) sinusoidal signal case can be treated as an extension of the complex sinusoidal signal case with main and side lobes. Thus, this case is not considered separately. The concept of accumulated fourth order moment, as developed in the accompanying paper (Sircar et al. 2015), has been extended to the concept of accumulated fourth order cumulant while estimating parameters of the complex FM sinusoidal signal in multiplicative noise.

The paper is organized as follows: In "Symmetric fourth order cumulant", we give the definition of fourth order moment and cumulant used in this work, and derive the analytical expressions for the symmetric fourth order cumulant or accumulated cumulant of the above multi-component signals in multiplicative and additive noise. We analyze the "Deterministic signal case" and discuss the effects of replacing the ensemble average by the time average. In the next section "Simulation study" is presented, and the "Conclusion" is given in last section. The Cramer-Rao (CR) bound expressions for the simulated examples are derived in Appendices A–C.

## Symmetric fourth order cumulant

*Y*[

*n*] comprising of the sum of

*M*signals in presence of multiplicative and additive noise,

*i*th multiplicative noise process, \(S_{i}[n]\) is the

*i*th signal process,

*W*[

*n*] is the additive noise process, and

*X*[

*n*] is the composite signal component comprising of multi-component signal and multiplicative noise.

It is assumed that *W*[*n*] is the zero-mean complex Gaussian noise process independent of the multiplicative noise processes. Since the fourth order moment and cumulant of the Gaussian process are zero, we need to study the fourth order statistics of *X*[*n*], which will be same as that of *Y*[*n*].

*X*[

*n*] as follows (Sircar et al. 2015),

*X*[

*n*] is defined as

*n*and lag

*k*, we will use the concept of accumulated fourth order cumulant (AFOC) (Sircar and Mukhopadhyay 1995; Sircar et al. 2015). The resulting AFOC sequence will be a function of lag only.

### Complex sinusoidal signals

*X*[

*n*] consisting of

*M*complex sinusoids of angular frequencies \(\omega _i\)’s in multiplicative noise can be expressed as

*X*[

*n*] as given by (2), we compute

*X*[

*n*] as defined by (3),

*X*[

*n*] of (4) is a stationary signal. Once the FOC sequence is computed, it is easy to extract its frequencies which are set at twice the frequencies of the signal.

### Complex FM sinusoidal signals

*X*[

*n*] consisting of

*M*complex frequency modulated (FM) sinusoids of carrier angular frequencies \(\omega _i\)’s, modulating angular frequencies \(\xi _i\)’s and modulation indices \(\beta _i\)’s in multiplicative noise can be expressed as

*X*[

*n*] as given by (2), we calculate

*X*[

*n*] as given by (3),

*X*[

*n*] comprises of narrow-band FM sinusoids with small values of \(\beta _u\)’s.

*n*and lag

*k*. This is not unexpected because the signal

*X*[

*n*] of (14) is a non-stationary signal (Sircar and Sharma 1997; Sircar and Saini 2007). We compute the accumulated FOC (AFOC) \(Q_{4X}\) by summing \(C_{4X}\) over an appropriately selected time frame \([n_1,n_2]\) (Sircar and Mukhopadhyay 1995; Sircar et al. 2015),

Once the AFOC sequence is computed, we extract its frequencies which are set at twice the carrier frequencies of the signal *X*[*n*], together with the side-frequencies at 2 times carrier plus/minus modulating frequencies.

### Complex linear chirp signals

*X*[

*n*] consisting of

*M*complex linear chirps of on-set angular frequencies \(\omega _i\)’s and rates of increase of angular frequencies or chirp rates \(\gamma _i\)’s in multiplicative noise can be expressed as

*X*[

*n*] is computed by (2) as follows

*X*[

*n*] as given by (3), is computed as

## Deterministic signal case

In this section, we discuss the non-random signal case. 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 not likely produce the same result when the underlying signal may not necessarily be stationary and ergodic.

*X*[

*n*], we compute the \(\tilde{C}\)-sequence as follows (Sircar et al. (2015))

*n*(and

*m*), indices

*i*and

*l*(see 8) 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. Note 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 (13) and (33), it can be observed that the \(\tilde{C}\)-sequence consists of the square and product modes of the signal, together with the low amplitude 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.

*X*[

*n*] comprises of narrow-band FM sinusoids with small values of \(\beta _u\)’s. Note that \(T_6\), \(T_7\), \(T_8\) are non-zero only when \(X_0 \ne 0\).

In the presence of additive noise, the \(\tilde{C}\)-sequence may deviate, but 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 doing time averaging here.

## Simulation study

Simulation study is carried out for the complex sinusoidal signals, complex FM sinusoidal signals, and complex linear chirp signals. The common simulation parameters used for all the signals are the number of realizations equal to 500, the multiplicative noise amplitude \(\alpha _i\) to be i.i.d. and Rician distributed, and its phase \(\phi _i\) to be i.i.d. and \(U[0,2\pi )\), and the additive noise *W*[*n*] to be complex zero-mean white circular Gaussian process.

### Complex sinusoidal signals

*Y*[

*n*] taken for simulation consists of

*M*complex sinusoidal signals in multiplicative and additive noise.

*M*= 2, the angular frequencies \(\omega _{i}\) = \(2\pi \left( {f_{i}}/{f_s}\right)\) with \(f_{1}=70\) Hz and \(f_{2}=150\) Hz, the sampling rate \(f_s=800\) Hz, and the number of data points \(N=513\). The amplitude \(\alpha _{i}\) and the phase \(\phi _{i}\) of the multiplicative noise and the additive noise

*W*[

*n*] are as stated above.

The sequence \(\bar{Y}[n]\) is computed by subtracting the mean of *Y*[*n*] from each value of the data sequence. The new sequence \(\bar{Y}[n]\) is used to compute the FOTC as given by (30).

*L*complex sinusoids, satisfies the

*L*th order prediction equation. The order

*L*becomes \(L=M(M+3)/2=5\). We use the extended order modelling for noise immunity and form forward prediction error filter (PEF) \(\mathcal{{D}}_J(z)\) as

\(\mathbf{d} = [1\,\,\,d_1\,\,\,d_2\, \ldots \,d_J]^T\).

### Complex FM sinusoidal signals

*Y*[

*n*] taken for simulation is

*W*[

*n*] are same as stated earlier.

The sequence \(\bar{Y}[n]\) is computed by subtracting the mean of *Y*[*n*] from each value of the data sequence. The new sequence \(\bar{Y}[n]\) is used to compute the FOTC as given by (30). The FM signal will contain modes corresponding to the carrier frequency \(f_c\), and two side bands \(f_c+f_m\) and \(f_c-f_m\), and consequently, the resulting signal will have 6 modes. Thus, the FOTC will contain \(L=M(M+3)/2=27\) modes.

### Complex linear chirp signals

*W*[

*n*] are same as stated earlier.

*Y*[

*n*] from each value of the data sequence. The new sequence \(\bar{Y}[n]\) is used to compute the FOTC as given by (30). The magnitude spectrum of the computed FOTC is shown in Figure 8.

\(\omega _{d,1}=(2(\omega _1-\omega _2)+2\gamma _1\ell )\) with \(\gamma _{d,1}=((\gamma _1-\gamma _2)\ell )\),

\(\omega _{d,2}=((\omega _1-\omega _2)+(\gamma _1+\gamma _2)\ell )\) with \(\gamma _{d,2}=((\gamma _1-\gamma _2)\ell /2)\),

\(\omega _{d,3}=((\omega _1-\omega _2)+2\gamma _1\ell )\) with \(\gamma _{d,3}=((\gamma _1-\gamma _2)\ell /2)\),

\(\omega _{d,4}=((\omega _2-\omega _1)+(\gamma _1+\gamma _2)\ell )\) with \(\gamma _{d,4}=((\gamma _1-\gamma _2)\ell /2)\),

\(\omega _{d,5}=((\omega _2-\omega _1)+2\gamma _1\ell )\) with \(\gamma _{d,5}=((\gamma _2-\gamma _1)\ell /2)\), and

\(\omega _{d,6}=(2(\omega _2-\omega _1)+2\gamma _2\ell )\) with \(\gamma _{d,6}=((\gamma _2-\gamma _1)\ell )\).

The plots show that the estimates of chirp rates are quite accurate for the SNR level above 12 dB. The variance of estimate is 3–5 dB higher than the CR bound in each case. The bias for \(f_{r,1}\) or \(f_{r,2}\) is very small. Thus, the parameters of the chirp signals in presence of additive and multiplicative noise can be estimated accurately by using the FOTC values of the signal and the method described in (Peleg and Porat 1991; Barbarossa et al. 1998).

## Conclusion

In this paper, the parameter estimation approach based on the symmetric fourth-order cumulant (FOC) or accumulated FOC (AFOC) is proposed for some stationary or non-stationary signals in multiplicative and additive noise. The derivations of the symmetric FOC are carried out for the multi-component complex sinusoidal, complex FM sinusoidal and complex linear chirp signals.

In case of parameter estimation of complex sinusoidal signal, the proposed method performs better than the method presented in (Swami 1994) at all SNR levels, even though the latter is also another method based on the fourth order statistics.

The simulation results show that using the method based on the new definition of the FOC or AFOC as developed in this paper, the parameters of various stationary and non-stationary signals can be estimated accurately in multiplicative and additive noise environment. The CR bounds are computed in each case for comparison of the variances of estimated parameters.

The new definition of symmetric fourth-order moment and cumulant, as proposed in (Sircar et al. (2015)) and in this paper, reduces the dimension of fourth-order moment/cumulant drastically from three lag-variables to one lag-variable. Moreover, the symmetric FOC is found to be time-independent for some non-stationary signals like complex exponentials and linear chirps. In our future research, we like to explore the full potential of symmetric FOC by applying the proposed method for analysis of various other stationary and non-stationary signals in multiplicative and additive noise. As further research, we need to present results for comparison of performance of our method and that of the methods based on the NLLS and cyclic statistics.

## Notes

## 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. 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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