UnderstandingFFT Windows

APPLICATION NOTE

AN014

IntroductionFFT based measurements are subject to errors froman effect known as leakage. This effect occurs whenthe FFT is computed from of a block of data which isnot periodic. To correct this problem appropriate win-dowing functions must be applied. The user mustchoose the appropriate window function for the specif-ic application. When windowing is not applied correct-ly, then errors may be introduced in the FFT ampli-tude, frequency or overall shape of the spectrum. Thisapplication note describes the phenomenon of leak-age, the various windowing functions and theirstrengths and weaknesses, and examples are givenfor various applications.

FFT Background Most dynamic signal analyzers (Figure 1) computetime and frequency measurements. Time measure-ments include capturing time traces of measured sig-

nals, including filtering and statistical measures.Frequency measurements that are computed by mostDSAs include Fast Fourier Transform, Power SpectralDensity, Frequency Response Functions, Coherenceand many more. These signals are computed in theDSP from the digitized time data. Time data is digi-tized and sampled into the DSP block by block. A blockis a fixed number of data points in the digital timerecord. Most frequency functions are computed fromone block of data at a time. A block of data is alsocalled a time record or time window.

The Fast Fourier Transform (FFT) is the FourierTransform of a block of time data points. It repre-sents the frequency composition of the time signal.Figure 2 shows a 10 Hz sine waveform (top) and theFFT of the sine waveform (bottom). A sine wave iscomposed of one pure tone indicated by the single dis-crete peak in the FFT with height of 1.0 at 10 Hz.

LeakageThe FFT computation assumes that a signal is periodic ineach data block, that is, it repeats over and over againand it is identical every time. Note this was the case inFigure 2 because there are an interger number of cyclesof the sine wave in the data record. Another type of sig-nal that satisfies the periodic requirement is a transientsignal that starts at zero at the beginning of the time win-dow and then rises to some maximum and decays againto zero before the end of the time window.

Figure 1. Dactron FocusTM (left) and the Dactron PhotonTM (right)Dynamic Signal Analyzers

Figure 2. Time waveform of sine function (top) and FFT (bottom).

When the FFT of a non-periodic signal is computedthen the resulting frequency spectrum suffers fromleakage. Leakage results in the signal energy smear-ing out over a wide frequency range in the FFT when itshould be in a narrow frequency range. Figure 3 illus-trates the effect of leakage. The left-top graph showsa 10 Hz sine wave with amplitude 1.0 that is periodicin the time frame. The resulting FFT (bottom-left)shows a narrow peak at 10 Hz in the frequency axiswith a height of 1.0 as expected. Note the dB scaleis used to highlight the shape of the FFT at low levels.The right-top graph shows a sine wave that is not peri-odic in the time frame resulting in leakage in the FFT(bottom-right). The amplitude is less than the expect-ed 1.0 value and the signal energy is more dispersed.The dispersed shape of the FFT makes it more difficultto identify the frequency content of the measured sig-nal.

Windowing Reduces Leakage In a signal analyzer the time record length is adjustablebut it must be selected from a set of predefined val-ues. Since most signals are not periodic in the prede-fined data block time periods, a window must beapplied to correct for leakage. A window is shaped sothat it is exactly zero at the beginning and end of thedata block and has some special shape in between.This function is then multiplied with the time data blockforcing the signal to be periodic. A special weightingfactor must also be applied so that the correct FFTsignal amplitude level is recovered after the window-ing. Figure 4 shows the effect of applying a Hanningwindow to a pure sine tone. The left-top plot shows asine tone that is not periodic in the time window with-out the windowing function resulting in leakage in theFFT (left-bottom).

When a Hanning window is applied (top-right), then theleakage is reduced in the FFT (bottom-right). The result-ing spectrum is a sharp narrow peak with amplitude of1.0. Notice that it does not have exactly the same shapeas the FFT of the original periodic sine wave in Figure 3,but the amplitude and frequency errors resulting fromleakage are corrected. A Windowing function minimizesthe effect of leakage to better represent the frequencyspectrum of the data.

Windowing functions are most easily understood in thetime domain; however, they are often implemented in thefrequency domain instead. Mathematically there is nodifference when the windowing is implemented in the fre-quency or time domains, though the mathematical proce-dure is somewhat different. When the window is imple-mented in the frequency domain, the FFT of the windowfunction is computed one time and saved in memory andthen it is applied to every FFT frequency value correctingthe leakage in the FFT. This gives rise to one measureof the window's characteristics, known as the side lobe.The FFT of a window has a peak at the applied frequencyand other peaks, called side lobes, on either side of theapplied frequency. The height of the side lobes indicateswhat affect the windowing function will have on frequen-cies around the applied frequency. In general, lower sidelobes reduce the leakage in the measured FFT butincrease the bandwidth of the major lobe.

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Figure 3. Comparison of periodic sine wave (left) and FFT to nonperiodic (right) with leakage in the FFT.

Figure 4. Comparison of non periodic sine wave and FFT withleakage (left) to windowed sine wave and FFT showing no leakage(right).

Figure 5 shows the Hanning windowing function andits FFT. The highest side lobe is -32 dB. Comparethis with the Flat Top windowing function in Figure 6.The highest side lobe is much lower (-74), but themain lobe bandwidth is significantly wider.

A comparison of an FFT of a non-periodic sine wavewith Hanning and Flat Top windows is shown in Figure7.

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Figure 8 shows a frequency response function of a beammeasured with an impact hammer and accelerometerwith and without a window. In this case, leakage drasti-cally affects the overall shape of the spectrum. Theunwindowed spectrum totally obscures the first anti-res-onance and it also caused some aplitude errors in thespectrum peaks that correspond to the structure’s res-onances.

Choosing a Windowing FunctionFFT windows reduce the effects of leakage but can noteliminate leakage entirely. In effect, they only change theshape of the leakage. In addition, each type of windowaffects the spectrum in a slightly different way. Many dif-ferent windows have been proposed over time, each withits own advantage and disadvantage relative to the oth-ers. Some are more effective for specific types of signaltypes such as random or sinusoidal. Some improve thefrequency resolution, that is, they make it easier todetect the exact frequency of a peak in the spectrum.Some improve the amplitude accuracy, that is, theymost accurately indicate the level of the peak. The besttype of window should be chosen for each specific appli-cation.

Figure 5. Hanning window (left) and FFT (right).

Figure 6. Flat Top window (left) and FFT (right).

Figure 7. Comparison of Hanning and Flat Top windowing func-tions with a non-periodic sine wave.

Figure 8. Frequency response function of a beam with and with-out window.

Bartlett Highest Side lobe: -26dB

Blackman Highest side lobe: -74 dB

Flat Top Highest side lobe: -93dB

Hamming Highest side lobe-42dB

Hanning Highest side lobe: -32 dB

Kaiser-Bessel Highest side lobe:-70 dB

Tukey Highest side lobe: -13 dB

Welch-Highest side lobe: -21dB

Window Best for theseSignal Types

FrequencyResolution

SpectralLeakage

AmplitudeAccuracy

Barlett Random Good Fair Fair

Blackman Random ormixed

Poor Best Good

Flat top Sinusoids Poor Good Best

Hanning Random Good Good Fair

Hamming Random Good Fair Fair

Kaiser-Bessel Random Fair Good Good

None (boxcar) Transient &SynchronousSampling

Best Poor Poor

Tukey Random Good Poor Poor

Welch Random Good Good Fair

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The most common windows and their features are given below. This table can be used tochoose the best windowing function for each application.

Time Domain Window Shapes

Overlap ProcessingOne of the disadvantages of windowing functions is thatthe beginning and end of the signal is attenuated in thecalculation of the spectrum. This means that more aver-ages must be taken to get a good statistical representa-tion of the spectrum, increasing the time to complete themeasurement. Overlap processing is a feature that isavailable in most signal analyzers that can recover thelost data and reduce the measurement time. This pro-cessing reduces the total measurement time by recover-ing a portion of each previous frame that otherwise is lostdue to the effect of the windowing function as shown inFigure 9. The top pane shows the original continuousinput signal. Below the Input Signal are shown the over-lapping windowed frames. Next are the unaveraged FFTsfrom each frame, and finally at the bottom is the averageof the FFTs. Overlap processing is particularly effective atreducing the measurement time for low frequency tests(generally under 50 Hz) for which the frame acquisitiontimes are very long.

Windowing for Impact MeasurementsAnother type of windowing, developed especially for modalanalysis using an impact hammer, is the exponential win-dow. This window function, shown in Figure 10, has twoparts, the pre-window at the beginning of the time frame,and the exponential window. The pre-window includes ahold-off period that eliminates any noise before theimpact. The length of this hold-off period can be specifiedby the user to coincide with the pre-trigger time reducingthe effects of noise. The exponential window applies anexponential decay that forces the response data to zeroby the end of the frame resulting in a guaranteed period-ic signal. It should be noted that this will result in an overestimate of the damping of the structure because thewindowing function artificially damps the signal in a short-er time.

Older signal analyzer hardware was limited in memoryand computational resources, limiting the data record toas few as 256 points. This limitation made exponentialwindowing necessary. Modern signal analyzers containmore memory and computational resources. When pos-sible, it is always preferable to use no window by increas-ing the record length to capture the entire waveform inone time record.

Figure 9. Overlap processing shortens the acquisition time byrecovering a portion of each previous frame that otherwise is lostdue to the effect of the FFT window

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Figure 10. Force/exponential widow function is used for modalanalysis with impact hammer excitation.

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Examples The following examples describe three applications andthe outline the typical decision making process on select-ing the appropriate windowing function.

Case 1. A structure is excited with a mechanical shakerusing broadband random noise and an accelerometer isplaced on the structure to identify the resonant frequen-cies as accurately as possible. According to Table 1,Hanning, Hamming, Tukey and Welch produce good fre-quency resolution. Hanning is the most commonly usedwindow function for random signals because it providesgood frequency resolution and leakage protection withfair amplitude accuracy. Figure 11 shows a comparisonof the spectrum with and without a window. The windowreduces the leakage and provides more accurate ampli-tude measurements for the resonant frequencies.

Case 2. The same structure in Case 1 is next subjectedto a pure sine tone at the first resonance frequency toaccurately measure the amplitude ratio between the exci-tation and the structural response level. According toTable 1, a Flat Top window function gives the best ampli-tude accuracy for sinusoidal signals. Figure 12 showsthe frequency response function with the Flat Top win-dow. Note that the Flat Top window changes the shapeof the peak from a sharp peak to a flat peak, as suggest-ed by the name.

Figure 11. Spectrum with and without window.

Figure 12. Frequency response function using Flat Top window.

Case 3. A measurement is performed on a lightlydamped structure using an impact hammer. A few testimpacts show that the structure continues to vibrateafter the impact for a time of approximately 1.1 seconds.The time record on the signal analyzer is set to 800 mil-liseconds and therefore an exponential window is used.Figure 13 shows the time response of the structure with-out the window in the top frame. Note that the vibrationhas not died out at the end of the time record. The bot-tom frame shows the results with the window applied.The vibrations are forced to zero at the end of the timerecord by the exponential window.

The plots in figure 13 show the disadvantages of usingan exponential window. Estimates of damping extractedfrom the resulting frequency response measurementswill be affected. This effect results from the exponentialwindow adds artificial damping to the measurements.

Figure 13. Time response of lightly damped structure withoutexponential window (top) and with window (bottom).

Copyright © 2003 LDS AN014 1203Focus and Photon are trademarks of LDS.

ConclusionsAll FFT based measurements assume that the signal isperiodic in the time frame. When the measured signal isnot periodic then leakage occurs. Leakage results in mis-leading information about the spectral amplitude and fre-quency. An FFT window can be applied to reduce theeffects of leakage. There are many windows to choosefrom, each with advantages for specific applications. Youmust understand the effects of leakage and know thetradeoffs and advantages of the various windowing func-tions to accurately interpret frequency domain measure-ments.

ReferencesHartmann, William M., "Signals, Sound, and Sensation,"American Institute of Physics, New York, 1997.

Ramirez, Robert W., "The FFT, Fundamentals andConcepts," Prentice-Hall, New Jersey, 1985.

Ziemer, Rodger E., et. al., "Signals & Systems," PrenticeHall, New Jersey, 1998.

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