Signal-to-noise ratio estimation using higher-order moments

doi:10.1016/j.sigpro.2005.06.003

Signal Processing

Volume 86, Issue 4, April 2006, Pages 716-732

https://doi.org/10.1016/j.sigpro.2005.06.003 Get rights and content

Abstract

We consider the problem of estimation of the signal-to-noise ratio (SNR) of an unknown deterministic complex phase signal in additive complex white Gaussian noise. The phase of the signal is arbitrary and is not assumed to be known a priori unlike many SNR estimation methods that assume phase synchronization. We show that the moments of the complex sequences exhibit useful mean-ergodicity properties enabling a “method-of-moments” (MoM)-SNR estimator. The Cramer–Rao bounds (CRBs) on the signal power, noise variance and logarithmic-SNR are derived. We conduct experiments to study the efficiency of the SNR estimator. We show that the estimator exhibits finite sample super-efficiency/inefficiency and asymptotic efficiency, depending on the choice of the parameters. At $0 dB$ SNR, the mean square error in log-SNR estimation is approximately $2 {dB}^{2}$ . The main feature of the MoM estimator is that it does not require the instantaneous phase/frequency of the signal, a priori. Infact, the SNR estimator can be used to track the instantaneous frequency (IF) of the phase signal. Using the adaptive pseudo-Wigner–Ville distribution technique, the IF estimation accuracy is the same as that obtained with perfect SNR knowledge and 8–10 dB better compared to the median-based SNR estimator.

Introduction

We address the problem of estimation of the signal-to-noise ratio (SNR) of a constant amplitude complex phase signal [1], [2], [3] $s_{n}$ ,¹ in additive complex white stationary Gaussian noise,² $W_{n}$ . The noise is assumed to have zero-mean and unknown variance $σ_{w}^{2}$ . The phase signal is deterministic with unknown amplitude $A$ , phase $φ_{n}$ , and is of the following form: $s_{n} = A e^{j φ_{n}} .$ We assume that $A$ is real. If A is not real, the complex part of $A$ can be absorbed by $e^{j φ_{n}}$ . This is the typical scenario of a time-varying signal in stationary noise. The noisy observations denoted by $X_{n}$ , are given by $X_{n} = s_{n} + W_{n}, 0 ⩽ n ⩽ N - 1,$ where $N$ is the length of the random observation sequence. It is desired to estimate the signal-to-noise ratio, denoted by $ξ$ and defined as $ξ = A^{2} / σ_{w}^{2}$ .

The basic difference in the above model from those commonly used [4] is that $s_{n}$ , the signal of interest is deterministic but with a time-varying spectrum. Also, many SNR estimation algorithms [4], [5] assume that the phase is known/estimated (phase-synchronous SNR estimation). In the new technique, we do not assume a priori knowledge/estimate of the phase. Also, unlike many communication system applications, the phase signal is not constrained to have a constant frequency. We allow for arbitrary frequency variation which is not known a priori. This is a generalized signal definition and, is useful in many practical problems in digital communication systems [6], [7], RADAR [8], SONAR, helicopter return signal analysis [9], aircraft flight parameter estimation [10], etc.

The organization of the paper is as follows: In Section 2, we state and prove the properties of the random sequence $X_{n}$ that enable the moments-based SNR estimation. We derive the estimators for the signal power, noise variance and logarithmic SNR (Section 3.3) and show their statistical efficiency with respect to the CRB. We demonstrate, experimentally that the new estimator exhibits finite sample inefficiency/super-efficiency, depending on the choice of the parameters, $A^{2}$ and $σ_{w}^{2}$ . However, it is asymptotically efficient. In Section 4, we compare the new estimator to the median-based SNR estimator. We consider the application to instantaneous frequency estimation (Section 4.2), and show that, in the presence of noise, the moments-SNR estimator improves the accuracy of the adaptive window pseudo-Wigner–Ville distribution based instantaneous frequency (IF) estimation technique significantly.

Section snippets

Properties of $| X_{n} |^{2}$ and $| X_{n} |^{4}$

We have $X_{n} = A e^{j φ_{n}} + W_{n}$ , where $W_{n}$ is independent and identically distributed (i.i.d) complex Gaussian distributed random noise of zero-mean and variance $σ_{w}^{2}$ . The ensemble averages of the sequences, $X_{n}$ and $| X_{n} |^{k}$ ( $k$ odd), $E {X_{n}}$ and $E {| X_{n} |^{k}}$ ( $k$ odd) are functions of time, i.e., they are non-stationary. However, the sequences $| X_{n} |^{k}$ ( $k$ even) are wide-sense stationary. Of particular interest are the sequences $| X_{n} |^{2}$ and $| X_{n} |^{4}$ which exhibit interesting properties as shown below:

(1) The sequence $| X_{n} |^{2}$ is

Estimators of $A^{2}$ and $σ_{w}^{2}$

The mean-ergodic and wide-sense stationarity properties of the sequences, $| X_{n} |^{2}$ and $| X_{n} |^{4}$ can be used to derive the estimators of $A^{2}$ and $σ_{w}^{2}$ .

We note that $E {| X_{n} |^{2}} = A^{2} + σ_{w}^{2} and$ $E {| X_{n} |^{4}} = A^{4} + 2 σ_{w}^{4} + 4 A^{2} σ_{w}^{2} .$ These equations are non-linear in the unknowns, $A^{2}$ and $σ_{w}^{2}$ . At first, it appears that it is difficult to solve explicitly for $A^{2}$ and $σ_{w}^{2}$ . However, noting the kind of non-linearity of the unknowns involved, we can solve for $A^{2}$ and $σ_{w}^{2}$ as follows: $A^{2} = \sqrt{2 (E {| X_{n} |^{2}})^{2} - E {| X_{n} |^{4}}} and$ $σ_{w}^{2} = E {| X_{n} |^{2}} - \sqrt{2 (E {| X_{n} |^{2}})^{2} - E {| X_{n} |^{4}}$

Performance comparison to the median-based-SNR estimator

The median-based SNR estimator [16], denoted by $ξ_{Med}$ , also does not require phase a priori. It is defined as $ξ_{Med} = \frac{\hat{A^{2}}}{{\hat{σ}}_{w}^{2}} where$ $\hat{A^{2}} + {\hat{σ}}_{w}^{2} = \frac{1}{N} \sum_{n = 0}^{N - 1} | X_{n} |^{2} and$ ${\hat{σ}}_{w}^{2} = \frac{{\hat{σ}}_{wr}^{2} + {\hat{σ}}_{wi}^{2}}{2},$ where ${\hat{σ}}_{wr}$ and ${\hat{σ}}_{wi}$ are obtained by median filtering of $X_{n}$ as below ${\hat{σ}}_{wr} = \frac{median (| R {X_{n} - X_{n - 1}} |, n = 1, 2, 3, \dots, N - 1)}{0.6745}$ and ${\hat{σ}}_{wi} = \frac{median (| I {X_{n} - X_{n - 1}} |, n = 1, 2, 3, \dots, N - 1)}{0.6745},$ where $R$ and $I$ indicate the real and imaginary parts respectively. A similar estimator for the noise variance has been suggested in [17, p. 459]. The median estimator

Conclusion

We have derived a higher-order moments-based SNR estimator for deterministic phase signals with arbitrary time-varying phase, in i.i.d complex Gaussian noise. The estimator does not require a priori knowledge/estimate of the phase. The statistical properties of the new estimator have been obtained experimentally and compared to the theoretical bound. The experiments have shown the finite sample super-efficiency/inefficiency and asymptotic efficiency property of the SNR estimator. The utility of

Acknowledgements

We thank the reviewers for careful scrutiny, useful comments and references to “super-efficiency” [15] that helped significantly in improving the quality of the paper.

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Signal-to-noise ratio estimation using higher-order moments

Abstract

Introduction

Section snippets

Properties of |Xn|2 and |Xn|4

Estimators of A2 and σw2

Performance comparison to the median-based-SNR estimator

Conclusion

Acknowledgements

Signal Processing

Time–Frequency Analysis

On instantaneous amplitude and phase of signals

IEEE Trans. Signal Process.

A comparison of SNR estimation techniques for the AWGN channel

IEEE Trans. Comm.

Cramer–Rao bounds of SNR estimates for BPSK and QPSK modulated signals

IEEE Comm. Lett.

Performance bounds for polynomial phase parameter estimation with non-uniform and random sampling schemes

IEEE Trans. Signal Process.

Hybrid FM-polynomial phase signal modeling: parameter estimation and Cramer–Rao bounds

IEEE Trans. Signal Process.

Aircraft flight parameter estimation based on passive acoustic techniques using the polynomial Wigner–Ville distribution

J. Acoust. Soc. Amer.

Probability, Random Variables and Stochastic Processes

Properties of $| X_{n} |^{2}$ and $| X_{n} |^{4}$

Estimators of $A^{2}$ and $σ_{w}^{2}$