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An Examination Of Signal Resolution In An FFT Analyzer

by Jack Frarey

### ABSTRACT

This paper examines the effect that the Hanning window and other windows have on signal resolution. The WNF (Window Noise Factor) equation is shown to be inadequate when attempting to resolve two closely spaced signals. The paper demonstrates this through a series of tests with in-close signals using three different windows. Further testing is done with the in-close signals set at different areas of the sample window. The relative signal amplitudes were also shown to influence required signal spacing. All test results are clearly shown in multiple graphs. The paper concludes with the comment never try to skimp on resolution.

### PREVIEW

“INTRODUCTION

Anyone who has attended an Annual Meeting session on vibration nomenclature will readily understand the difficulties one can encounter when proposing a term definition. The definition suggested in the abstract deals with the number of lines of spectral resolution that one must employ to successfully separate two closely spaced signals.

In addition to simple separation, it should be required to give accurate information of the amplitude and frequency (within the window and line spacing constraints) of the signals.

There has been some confusion between the term resolution and the term noise bandwidth of a spectral line for various weighting windows. This paper resulted from an attempt to clarify the difference in terms. In setting up several test cases, some surprising results were obtained. These results showed that one could miss seeing a signal or give erroneous values for its amplitude and frequency. This paper will address three topics:

Large signal resolution
False resolution
High dynamic range resolution

LARGE SIGNAL RESOLUTION

If the two signals to be separated are approximately the same magnitude, then the rules for large signal resolution apply. The magnitudes may vary from 0.2 times the magnitude of the larger to being equal in magnitude. Later in this paper we will address the case where the magnitude of the two signals differ by more than this factor. The equation that has been used for resolution has been:

Resolution= WNF=(FMAX/nl)*wf

Where:
Resolution = lowest resolvable frequency difference
WNF = Window Noise Factor
FMAX = frequency span of the spectrum
nl = number of lines in the spectrum
wf = Window factor = 1, 1.5, and 3.8 for rectangular, Hanning and flattop windows, respectively

This equation is really for the window noise factor or how much wider than one bin the effective window is for the banning and flat top window. This section of this paper will show that the required resolution should be twice that shown in the above equation.

In order to demonstrate this, two signals have been mixed and fed to a 400 line, 400 Hz spectrum. This is convenient since every line is equivalent to 1 Hz. These signals are synthesized in software using Data Physics Signal Cale software and their DP420 hardware system. Figure 1 shows two signals at 120 and 121 Hz. The time trace in the upper left shows one cycle of the beat while the three spectra for the rectangular, Hanning and flat top window have expanded frequency scales for clarity.”