14SOUND AND VIBRATION/MARCH 2006lyzed, which almost never happens without planning. We candevelop a formula for testing if a sine wave signal componentis bin centered as follows. A sine wave is given by Equation 1:where f is the frequency in cycles/second (Hz). The period ofa sine wave P (seconds per cycle) is the reciprocal of the fre-quency (Equation 2):Since we have to work with digitized data, define:N = number of samples in data segment being analyzedfs = sample rate in samples/secondThe time interval between samples h is the reciprocal of thesampling rate fs (Equation 3):The signal duration T is given by Equation 4:A Comprehensive Windows TutorialHoward A. Gaberson, Oxnard, CaliforniaThis article is a tutorial attempt to provide an easier analy-sis of how windows work. I begin by looking at individualspectrum bins as affected by off-bin-centered tones with sixdifferent windows. I define and use the convolution theoremto explain why the windows do what they do. I review the so-phisticated misunderstood fat-center-peaked, picket-fence-looking FFTs of various windows that come from adding ze-ros to increase resolution. I then use these drawings in astep-by-step graphical convolution to show how this or thatwindow will affect the FFT of your data. This is a procedureto demonstrate how any oddly shaped, even graphically de-fined, window affects data. These techniques are tools to al-low you to invent a graphical window to affect data in newways. To demonstrate how to use graphically defined win-dows, I use digitized drawings of a Hawaiian mountain andthe popular cartoon beagle as windows. To use these windows,I demonstrate FFT-IFFT interpolation and resampling, whichin itself is interesting. Finally, I compare how eight windowsaffect the spectrum of typical vibration data.We need a window for vibration spectrum analysis becausethe beginning does not match the end of the data segment weare analyzing and because we virtually never have an integernumber of periods of any cyclic information in the signal seg-ment/chunk. The discrete Fourier transform (DFT) analysis isreally a Fourier series expansion of whatever segment we in-put. It is considered periodic. Mismatched ends and nonintegernumbers of periods garbles the result. Windows somewhat al-leviate that garbling but in a complicated way.Windows are shapes – like a hill, a hump in the center, andthey go to zero at the ends. When you multiply these windowshapes term by term with a segment of data, you force the endsof the data segment to zero, so at least the ends match the be-ginning. You cannot really fix the noninteger number of peri-ods. Figure 1 shows a window application. In Figure 1a we havea 1,024 value plotted list of acceleration data; in Figure 1b, a1024 value Hanning window plotted for the same time values.Figure 1c shows the product of the two. The beginning and endscertainly match.I am sure there are more than 100 windows described in theliterature. Figure 2, shows six of them that I happen to use andwill illustrate here: boxcar (or uniform), Hanning, Hamming,Gaussian, Kaiser Bessel, and flat-top (this is very different fromthe boxcar or uniform window). Signal analyzers frequentlyoffer five or six to choose from, with variably helpful discus-sions of when to use each. Hanning is the most widely recom-mended. Reference 1 gives 44 and Reference 2 gives eight more;these might be the most widely referenced windows papers, butthese authors hadn’t yet heard of the flat-top. Ron Potter3 givesabout five windows and is famous for an unpublished HP pub-lication. Potter did have a flat-top in there. Finally, the vener-able Brüel & Kjær Technical Review published the first fullydisclosed flat-top window and a nice version of the KaiserBessel window.4,5 These two publications are downloadablefrom the B&K web site. I read a paper6 from the 2004 Interna-tional Modal Analysis Conference where the investigator wasworking on keeping the side lobes extremely low because ofall the new high-resolution, 24-bit, data-collecting hardware.If you really want to dig into things, Harris7 wrote a difficult64-page article with excellent drawings that I recommend.Defining the Bin-Centered ConceptNon-bin centered or not, an integer number of periods orcycles in the analyzed data segment is the topic to start with.A sine wave component of our signal is bin centered if it hasan integer number of periods in the data segment being ana-Figure 1. This shows a 1,024-point Hanning window applied to 1,024-point chunk of air handler acceleration.Figure 2. Six common window shapes.(1)yft= sin2p(3)hfs=1(2)Pf=1(4)TNhNfs==15INSTRUMENTATION REFERENCE ISSUEThe number of periods Np has to be given by the duration overthe period (Equation 5):By substituting Equations 2 and 4 into Equation 5, we find:Now, as long as Np is a whole number, the sine wave in thedata segment is bin centered or has an integer number of peri-ods. Such test segments of a sine wave signal are needed forcalibrating a window coefficient in our spectrum calculationprograms, so this is a handy formula to keep in mind. In thecase of a sampling rate of 1,024/second and if our number ofsamples is also 1,024, the number of periods will be equal tothe frequency. I will set things up that way here. To illustratehow the end and beginning not matching affects things, I willdraw an expanded view of the end of a 16-Hz, 1,024-point sinewave chunk connected to the beginning of the identical 16-Hz,1,024-point sine wave chunk. This is shown for 16 Hz and sev-eral frequencies close to 16 Hz in Figure 3. I have expandedthe region where the end of the first sine meets the beginningof the second sine to make it clear that a discontinuity occurswhen the frequency is not a whole number. Notice that the twocurves match perfectly (end to beginning) only for the case of16 Hz, which makes the signal bin centered.Non-Bin-Centered EffectsNow we are in an excellent position to consider the effectsthat those six windows I mentioned help us alleviate (but can-not really cure) the picket fence and leakage effects of a non-bin-centered signal on the spectra we compute. In case you arerusty, I want to review some spectrum fundamentals so you willknow what to expect. I use the acronyms DFT and FFT syn-onymously. DFT means the discrete Fourier transform, and FFTmeans the fast Fourier transform, the calculation procedureused to calculate the DFT. All signal analyzers and data col-lectors use an FFT to initially compute