Signal-to-noise ratio estimation using higher-order moments
Introduction
We address the problem of estimation of the signal-to-noise ratio (SNR) of a constant amplitude complex phase signal [1], [2], [3] ,1 in additive complex white stationary Gaussian noise,2 . The noise is assumed to have zero-mean and unknown variance . The phase signal is deterministic with unknown amplitude , phase , and is of the following form:We assume that is real. If A is not real, the complex part of can be absorbed by . This is the typical scenario of a time-varying signal in stationary noise. The noisy observations denoted by , are given bywhere is the length of the random observation sequence. It is desired to estimate the signal-to-noise ratio, denoted by and defined as .
The basic difference in the above model from those commonly used [4] is that , 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 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, and . 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 and
We have , where is independent and identically distributed (i.i.d) complex Gaussian distributed random noise of zero-mean and variance . The ensemble averages of the sequences, and ( odd), and ( odd) are functions of time, i.e., they are non-stationary. However, the sequences ( even) are wide-sense stationary. Of particular interest are the sequences and which exhibit interesting properties as shown below:
(1) The sequence is
Estimators of and
The mean-ergodic and wide-sense stationarity properties of the sequences, and can be used to derive the estimators of and .
We note thatThese equations are non-linear in the unknowns, and . At first, it appears that it is difficult to solve explicitly for and . However, noting the kind of non-linearity of the unknowns involved, we can solve for and as follows:
Performance comparison to the median-based-SNR estimator
The median-based SNR estimator [16], denoted by , also does not require phase a priori. It is defined aswhere and are obtained by median filtering of as belowandwhere and 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.
References (24)
- et al.
The evil of super-efficiency
Signal Processing
(November 1996) - B. Boashash (Ed.), Time Frequency Signal Processing—A Comprehensive Reference, Elsevier, Amsterdam,...
Time–Frequency Analysis
(1995)On instantaneous amplitude and phase of signals
IEEE Trans. Signal Process.
(March 1997)- et al.
A comparison of SNR estimation techniques for the AWGN channel
IEEE Trans. Comm.
(October 2000) - R. Matzner, F. Englberger, An SNR estimation algorithm using fourth-order moments, Proceedings of IEEE International...
Cramer–Rao bounds of SNR estimates for BPSK and QPSK modulated signals
IEEE Comm. Lett.
(January 2001)- B. Li, R.A. DiFazio, A. Zeira, P.J. Pietraski, New results on SNR estimation of MPSK modulated signals, Proceedings of...
- et al.
Performance bounds for polynomial phase parameter estimation with non-uniform and random sampling schemes
IEEE Trans. Signal Process.
(February 2000) - et al.
Hybrid FM-polynomial phase signal modeling: parameter estimation and Cramer–Rao bounds
IEEE Trans. Signal Process.
(February 1999)
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
Cited by (30)
Non data-aided estimation of modulation index of USB TT&C waveform with sinusoidal tone
2023, Digital Signal Processing: A Review JournalInstantaneous frequency estimation and localization
2016, Time-Frequency Signal Analysis and Processing: A Comprehensive ReferenceTime-Frequency Signal Analysis and Processing: A Comprehensive Reference
2015, Time-Frequency Signal Analysis and Processing: A Comprehensive ReferenceInstantaneous frequency in time-frequency analysis: Enhanced concepts and performance of estimation algorithms
2014, Digital Signal Processing: A Review JournalXWD-algorithm for the instantaneous frequency estimation revisited: Statistical analysis
2014, Signal Processing