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An Examination Of Signal Resolution In An FFT Analyzer

by Jack Frarey

### ABSTRACT

This paper provides examples of how windowing and the three primary windows used in vibration analysis (rectangular, hanning & flat-top) affect the real frequency resolution of signals. The author makes a strong case that the standard formula for frequency resolution should be modified by a factor of 2 to provide the true frequency resolution possible for all signal combinations. Reading this short paper provides a better feel for how each window type affects the frequency resolution of our machine vibration signals.

### PREVIEW

“Signal resolution for the FFT analyzer may be defined as the minimum number of lines of spectral resolution necessary to successfully separate two closely-spaced signals. Signal resolution should provide accurate information about the amplitude and frequency (within the window and line-spacing constraints) of the signals.
There is some confusion between resolution and noise bandwidth of a spectral line for various weighting windows. This article attempts to clarify the differences. Several test cases provided some surprising results. They showed that a signal could be missed or erroneous values obtained for its amplitude and frequency. This article addresses the topics of large signal resolution, false resolution, and high dynamic range resolution.

Large Signal Resolution
If two signals to be separated are approximately the same magnitude, the rules for large signal resolution apply. The magnitudes may vary from 0.2 times the magnitude of the larger to being equal. The equation that has been used for resolution is

resolution = WNF = (Fmax/nl )*wf

resolution = lowest resolvable frequency difference
WNF = window noise factor
Fmax = frequency span of the spectrum
nl = number of lines
wf = window factor = 1; 1.5; 3.8 for rectangular, Hanning, and flat top windows

This equation is for the window noise factor; that is, how much wider than one bin is the effective window for the Hanning and flat top windows. It is shown in the following examples that the required resolution should be twice that given in the equation.
Two signals were mixed and fed to a 400 line, 400 Hz spectrum, convenient because every line is equivalent to 1 Hz. The signals were synthesized using Data Physics Signal Calc soft-ware and their DP420 hardware system. Figure 1 shows two signals, at 120 Hz and 121 Hz. The time waveform in the upper
left is one cycle of the beat. The frequency scales for the three spectra for the rectangular, Harming, and flat top windows have been expanded for clarity. None of the windows, including the rectangular one, can resolve the two signals, thus confirming that the equation for resolution is not correct.
Figure 2 contains two signals, at 120 Hz and 122 Hz. The rectangular window separates the two signals; the Hanning and flat top windows do not resolve them. Thus, if the resolution is two times that given by the equation, the rectangular window resolves the two signals.”