![]() The second stage was the quadratic interpolation. Consider the zero-padding to be the first stage of processing. This gave a factor of 4 reduction in bin width. Rows 9 and 10 showed results of zero padding (adding zero’s to the end of the 512 sample record until it was 2048 number long). Rows 7 and 8 show effect of not using window….as expected, we get poorer results when we don’t use a window. Since I’m sure Entek is at least as smart as me, I’m sure they can do the same thing and get that precise (5% of bin width or better precision for clean sinusoid). The error in estimating frequency ranges from about 1% to 5% of bin width when window is used. Rows 2 thru 6 are the results of simple quadratic peak estimation (only) and windowing. Results are shown in tab labeled summary. Quadratic interpolation applied as the final step of peak estimation in all trials Reconstruction of the DTFT applied in some but not all of the trials Zero-padding applied in some but not all of the trials Hanning window applied in all of the trials, except where noted The input signal is a pure sinusoid in all trials, except where noted. ![]() ![]() Sample rate = 2000 hz (0.0005 seconds between samples).ĥ12 samples were used for 512 point FFT, 256 positive frequencies, going from 0 …1000hz In the attached spreadsheet, I have done some experiments using the following FFT parameters: ![]() Then the peak is found where d/dF = 0… i.e where B+2*C*F = 0…. From our 3 input magnitudes at 3 known frequencies, we have 3 equations in 3 unknowns and can solve A, B, C. We fit those points to a quadratic form: Magnitude = A + B*F + C*F^2. We find the highest spectral magnitude, and also look at the two neighboring points of the FFT. I don’t know what algorithm Entek uses, but the simplest method is quadratic interpolation. The smoothness arises from the sinc function that would be used to build the continuous DTFT from the samples. There are a variety of means to estimate these peaks in absence of the full data and I would say as a rough description they rely on the smoothness of the continuous curve that would pass through the discrete points located at bin centers. So it does not have the unlimited precision, but I tend to think it will estimate most well-behaved peaks (far above noise floor and nothing else around) to a precision of around 5% of bin width or better, based on my own numerical exercises where quadratic interpolation was used with windowing (below)/ In the Entek E-monitor software, we have peak label feature available to us even though the time samples may not be saved and neither is the phase of the FFT results. In this way we can determine the frequency of the peak to any desired level precision within our floating point capabilities (**notice I said “precision”, not “accuracy”… claiming accuracy is a much tougher burden to prove). So if the FFT spectrum tells us we have a peak in some frequency range, we could compute (reconstruct) the DTFT for any value of frequency we want within that frequency range and use a numerical algorithm (bisection algorithm) to find the maximum value within that frequency range. This continuous valued-spectrum is called the DTFT (as opposed to DFT or FFT). Interpolation between bin centers to estimate the frequency of the peak (using peak label feature) is an interesting subject to me, and I think it is not discussed very much in vibration training materials or books.Īn interesting fact is that if you have the time samples that were used as input to an FFT (or the FFT results including phase angle), then you can compute a continuous-valued spectrum, giving a spectrum magnitude value for any frequency that we choose below Fmax (not just the bin centers). I started a new thread because the other one is focusing on the pump case study.
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