ESTIMATION ERRORS IN 1/fγ NOISE SPECTRA WHEN EMPLOYING DFT SPECTRUM ANALYZERS

Abstract

Spectra estimation in the field of low frequency noise measurements (LFNMs) is almost always performed by resorting to Discrete Fourier Transform (DFT) based spectrum analyzers. In this approach, the input signal is sampled at a proper frequency fs and the power spectrum of sequences of N samples at a time are calculated and averaged in order to obtain an estimate of the spectrum at discrete frequency values fk = kΔf, where the integer k is the frequency index and Δf = fs/N is the frequency resolution. As the number of average increases, the statistical error, which is inversely proportional to the resolution bandwidth, can be made very small. However, if the spectrum of the signal is not a slowly changing function of the frequency, as in the case of 1/fγ processes, spectra estimation by means of the DFT also results in systematic errors. In this paper we discuss the dependence of these errors on spectral parameters (the spectrum amplitude, the frequency f, the spectral exponent γ and the DC power) and on measurement parameters (the spectral window, the resolution bandwidth Δf and the instrumentation AC cutoff frequency). Quantitative expressions for the systematic errors are obtained that, besides helping in the interpretation of the results of actual LFNMs, can be used as a guideline for the optimization of the measurement parameters and/or for the estimation of the maximum accuracy that can be obtained in given experimental conditions. This quantitative analysis is particularly important since while we find that, in general, the systematic error at a given frequency fk = kΔf can be made small if k is made large, which implies that Δf must be much smaller than fk, possibly in contrast with the need for a Δf as large as possible in order to reduce the measurement time, the magnitude of the error depends on the selected spectral window. The role of the instrumentation AC cutoff frequency f AC on the systematic error is also investigated and quantified and it is demonstrated that the error increases as f AC reduces. This last result is very important since, often, f AC is chosen much lower than the frequencies of interest and this choice may result in an increase of the systematic error.