Quantification of noisy MRS signals with the Pseudo-Wigner Distribution

This paper investigates the estimation errors induced by noise in the quantification of damped sinusoids with the Pseudo-Wigner Distribution (PWD). A constant amplitude single frequency noise is first considered. This simple model shows how underestimation or overestimation errors depend on the relative phase between signal and noise for a fixed signal-to-noise ratio (SNR). In a second step, cross-terms and noise amplitude fluctuations are identified as the main sources of discrepancy between the theoretical model and a practical situation where wideband noise is linearly added to a synthetic free induction decay (FID) signal. Cross-terms can be attenuated by band-pass filtering noise or by using a weighting (or smoothing) window in the PWD. An original procedure is then derived to make an on-line and noise-specific estimation of the statistical error in the quantification step. This property of the Wigner distribution is a unique feature in quantitative MRS. Confidence intervals are evaluated for a single damped sinusoid corrupted by eight random noise sequences with three different SNR's, 20, 10 and 5 dB, respectively. They are shown to match the statistical ranges of quantification results obtained with linear regression, until the SNR drops below 10 dB. Estimation accuracy of amplitude and damping constant is finally evaluated from the comparison of the Cramér-Rao (CR) lower bounds with the variance of estimation errors. CR-bounds are shown to be nearly achieved at each SNR.