10SOUND AND VIBRATION/JANUARY 2003Effects of Windowing on the Spectral Content of a SignalPierre Wickramarachi, Data Physics Corporation, San Jose, CaliforniaFigure 1. Spectrum of a sine wave with theRectangular window: (a) f = f0; (b) f = f0 – ∆f/8;(c) f = f0 – ∆f/2.Figure 2. Comparison of spectra for the Rect-angular window (a) and the Hanning window(b).Figure 3. Spectrum of a sine wave with theHanning window: (a) f = f0; (b) f = f0 – ∆f/8; (c)f = f0 – ∆f/2.Fourier analysis is commonly used toestimate the spectral content of a mea-sured signal. When choosing the appro-priate window, one needs to be aware ofits advantages and pitfalls in order to fitthe measurement situation. The follow-ing deals with some practical consider-ations on the effects of windowing.1The Fourier series assumes periodicityof the signal in the time domain. An FFTis actually a Fourier Series performedupon an interval, Tspan = n∆t where n isthe number of samples observed and ∆tis the constant time between samples.Since the sampled time signal may notexactly contain an integer number of pe-riods, this assumption may not be trulysatisfied.In effect, the truncation of the originalsignal corresponds to its multiplicationwith a Rectangular window of lengthTspan. The Fourier series then assumesthat the signal is the succession of ver-sions of this truncated signal in the timedomain leading to a spectrum with har-monic components at frequencies equalto multiples of ∆f = 1/TSpan.Let us examine the situation with a sinewave of frequency f0. In theory the cor-responding spectrum is a peak at f0.When a noninteger number of periods isacquired, this results in signal leakage,characterized by the smearing of thespectrum. Figure 1 illustrates this phe-nomenon by comparing three cases,where f is the sine wave frequency and ∆fthe frequency resolution.Case 1(a) allows us to determine theideal situation where an integer numberof periods (200) is set for the signal gen-erator. The corresponding frequency is f0= 508.626 Hz. In practice this case is notlikely to occur, because the frequencythat is being measured rarely falls on afrequency line. On the other hand, case1(b) represents a typical situation wherethe leakage is clearly visible. Here f hasbeen slightly decreased, which results ina non-integer number of periods withinTspan. The maximum leakage is obtainedin case 1(c). Why? The answer lies in thespectrum of the window as shown in Fig-ure 2(a).The FFT emulates a bank of parallelbandpass filters with the center frequen-cies exactly centered on integer multiplesof ∆f. The width and shape of each filteris identical and are given by the spectrumof the observation window shown in Fig-ure 2. Note that the filter shape is char-acterized by multiple lobes separated byzero values at multiples of ∆f and that allfilters in the bank ‘overlap.’When f of an applied sine correspondsexactly to a filter center-frequency [case1(a)], only that filter will respond becausef corresponds to an amplitude notch of allother filters in the bank. Conversely, if fis not exactly on a frequency line [case1(b)], the energy at f is smeared over ad-jacent frequencies because the secondarylobes of all other filters overlap f withnonzero gain and these filters respond inproportion to this gain. This perverse ef-fect is maximum when f = f0 – ∆f/2 [case1(c)], since the frequency f coincides withthe peak of each side lobe.If side lobes could be reduced in am-plitude, this error would decrease aswell. This is why people have used anumber of windows to weight the trun-cated signal such that the starting valueand the ending value are zero. This pro-duces a signal that appears periodic inTspan, meeting the basic assumption of theFourier Series. Weighting avoids thesharp discontinuities induced by theRectangular window and yields reduced-amplitude side lobes as desired. Figure2(b) shows the spectrum of the wellknown Hanning2 window. Observe thatthe amplitude of the first side-lobe is re-duced from –13.2 dB to –32.2 dB. Moreimportantly, notice that the amplitudes ofsubsequent side-lobes fall off at 60 dB/decade as opposed to 20 dB/decade forthe Rectangular window.These improvements come at a cost.The width of the primary lobe essentiallydoubles, eliminating the first set of zero-amplitude points. The primary lobe of theRectangular window has a –3 dB band-width of 0.85 ∆f. That of the Hanningwindow is increased to 1.4 ∆f. However,the benefits far outweigh the cost asshown in Figure 3. Here the Hanning win-dow is applied to the three cases previ-ously examined. Indeed results with theHanning window are close to case 1(a)done with the Rectangular window (idealcase).However, the Hanning window exhib-its a deficiency. Like the Rectangular win-dow, its primary lobe has significant cur-vature or ‘ripple’ across the ±∆f band.When a sine falls “exactly between cells,”its amplitude is reported 15% (–1.42 dB)lower than it would be at the filter cen-ter-frequency. The Rectangular windowexhibits this same fault, but more pro-nounced at 36% (–3.92 dB).When the application requires an accu-rate measure of peak amplitude (e.g. ro-tating machinery), the Flat-Top windowis usually selected. Its spectrum is char-acterized by a nearly flat main lobe acrossfi ±∆f, which reduces maximum ampli-tude error to 0.1%! As for the side lobes,their amplitudes remain at –70 dB belowthat of the main lobe, which strongly re-duces leakage. However this windowmust be used with care, particularly if theperiodic signal of interest is ‘buried’ inbroadband noise. The Flat-Top windowshould only be applied to clean periodicwaveforms. It is indeed a poor choice forrandom-signal or mixed-signal analysisbecause it lacks selectivity. A HanningPitfallsLeakageSome amplitude errorLarge noise bandwidthNo roll-offAdds artificial dampingApplicationsTransient(Impact testing)Periodic, randomPeriodicImpact testing(Lightly damped structures)AdvantagesRaw dataIdentification of closely- spaced frequenciesLittle leakageFrequency accuracyLittle leakageAmplitude accuracyShorter Tspan (quick measurements for modal analysis)Table 1. Comparison of windows for FFT analysis.WindowRectangular . . . . . . .Hanning . . . . . . . . . .Flat-Top . . . . . . . . . .Exponential . . . . . . . . Nb1 ∆f1.50 ∆f3.43 ∆fVariableFrequency, HzV, dB8005002500–20–100–80–60–40(a)(c)(b)20 dB/decadefifi−2fi−1fi+2fi+1V, dB0∆f(a)(b)–8060 dB/decade31 dB13