Statistical frequency measuring error of the quadrature demodulation technique for noisy single-tone pulse signals

Abstract

The standard deviation of the centre frequency of a signal is investigated for the quadrature demodulation technique (QDT). Signal frequencies can be measured by QDT unaffected by the amount of available signal periods. It can be used for the measurement of nonstationary signals generated by laser Doppler velocimeters. The dependence of the frequency measuring error on the averaging time of noisy single-tone Gaussian pulse signals is analysed. Assuming a quantum noise process, it is shown that the minimum measuring error results for an averaging time of approximately 1/ e 2 of the pulse duration of the signal. Alternatively, defined weighting of the measured values leads to a monotonically decreasing measuring error with increasing averaging time until the Cramer-Rao lower bound is reached. Therefore, the weighted QDT provides the lowest measuring error of all linear unbiased frequency estimators. It allows the evaluation of small frequency changes of laser Doppler signals, e.g. from micro-turbulent flows. The theory presented here is verified by Monte Carlo simulations and experiments.

Measurement of velocity gradients in boundary layers by a spatially resolving laser Doppler sensor

“…Besides the used FFT signal processing technique, the quadrature demodulation signal processing technique (QDT) can be applied [20,21,22], see section 5. The advantages of the QDT are its independence of the frequency measurement accuracy on the signal period number as well as against unsymmetrical signal shapes, which can occur at the boundaries of the measurement volume.…”

“…The quadrature demodulation technique (QDT) comprises the generation of a sine-cosine signal pair, the measurement of the signal phase and its evaluation in order to determine the signal frequency [20][21][22]. Figure 10 shows an exemplary scheme for the generation of quadrature signals, employing a heterodyne LDA system.…”

Measurement time dependency of asymptotic Cramér–Rao bound for an unknown constant in stationary Gaussian noise

“…For instance, the Heisenberg uncertainty principle in quantum physics from 1927 links the uncertainties of position and momentum of a quantum particle [1], which can be also formulated as an uncertainty relation between energy and time [2]. Note that this uncertainty relation describes a natural behavior of both physical quantities and does not describe insufficiencies of a measurement system.…”

“…[2,[9][10][11]. Especially parameter estimation for discrete time Gaussian processes was investigated by many researchers, e.g.…”

Optic simulation and optimization of a laser Doppler velocity profile sensor for microfluidic applications

“…Consequently, the spatial resolution is enhanced significantly with uncertainties in the micrometer range, opening new perspectives for the measurement of flows with high-velocity gradients. An optimized signal processing technique has already been developed that works close to the theoretical limit, the Cramér-Rao lower bound, 6,15,17 hence minimizing the uncertainty of the signal processing. However, an investigation of the uncertainty arising from distortions of the fringe system has not been performed yet.…”