Abstract:
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The minimum number of samples that must be taken from a
sinusoidal signal affected by white Gaussian noise, in order to find its
frequency with a predetermined maximum error, is derived. This analysis
is of interest in evaluating the performance of velocity-measurement systems
based on the Doppler effect. Specifically, in laser Doppler anemometry
(LDA) it is usual to receive bursts with a poor signal-to-noise
ratio, yet high accuracy is required for the measurement. In recent years
special attention has been paid to the problem of monitoring the temporal
evolution of turbulent flows. In this kind of situation averaging or filtering
the data sequences cannot be allowed: in a rapidly changing environment
each one of the measurements should rather be performed
within a maximum permissible error and the bursts strongly affected by
noise removed. The method for velocity extraction that will be considered
here is the spectral analysis through the squared discrete Fourier transform,
or periodogram, of the received bursts. This paper has two parts. In
the first an approximate expression for the error committed in LDA is
derived and discussed. In the second a mathematical formalism for the
exact calculation of the error as a function of the signal-to-noise ratio is
obtained, and some universal curves for the expected error are provided.
The results presented here appear to represent a fundamental limitation
on the accuracy of LDA measurements, yet, to our knowledge, they have
not been reported in the literature so far. |