FFT window functions - Chris Bore

1 Fordwater, Pond Road, Woking, Surrey GU22 0 JZTelephone: (01483) 740138 Fax: (0148) 740136 email: Web: window functionsFFT window functions Limits on FFT analysisLimits on FFT analysisWhen using FFT anaysis to study thefrequency spectrum of signals, thereare limits on resolution betweendifferent frequencies, and ondetectability of a small signal in thepresence of a large are two basic problems: thefact that we can only measure thesignal for a limited time; and thefact that the FFT only calculatesresults for certain discrete frequencyvalues (the 'FFT bins'). The limit onmeasurement time is fundamentalto any frequency analysis technqiue:the frequency sampling is peculiarto numerical methods like the measurement timeLimited measurement timeThe first problem arises because thesignal can only be measured for alimited time. Nothing can be knownabout the signal's behaviour outsidethe measured interval. We have toassume something about the signaloutside the measured interval, andthe Fourier Transform makes animplicit assumption that the signal isrepetitive: that is, the signal withinthe measured time repeats for real signals will havediscontinuities at the ends of themeasured time, and when the FFTassumes the signal repeats it willassume discontinuities that are notreally sharp discontinuities havebroad frequency spectra, these willcause the signal's frequencyspectrum to be spread leakageSpectral leakageIt is easy to gain an insight bybut they can bethinking about the special case of aarranged: for example frequencypure sine wave.

2 This has aanalyses are often made by tuning afrequency spectrum which is a singlestimulus freuqency precisely so thatspectral line: but the frequencyits frequency exactly fits thespectrum calculated by the FFT willmeasurement a spread out line. Eachspectral line will be spread out inSpectral leakage is not an artifact ofthe same FFT, but is due to the fact thatThe spreading means that signalfinite measurement which should beconcentrated only at one frequencySpectral leakage causes at least twoinstead leaks into all the otherdistinct This spreading ofenergy is called 'spectral leakage'.First, any given spectral componentSince spectral leakage is related toenergy, but also noise from thediscontinuities at the ends of thewhole of the rest of the time, it will be worseThis will degrade the signal to noisefor signals that happen to fall there are large signals may, by coincidencelarge signal component may beor by design, fall in such a way thatsevere enough to mask otherthere happens to be no discontinuitysmaller signals at differentat the ends of the : for these signals the effect ofspectral leakage may be example a pure sine waveIn effect, the process of measuring asampled for an exact number ofsignal for a finite time is equivalentcycyles would match up quiteto multiplying the signal by acorrectly when made repetitive: therectangular function of unitrepetitive signal would be exactly theamplitude.

3 The rectangular functionsame as the 'real' signal and so nolasting for the duration of thespectral leakage would example would be a signalThe signal is measured during awhich fell smoothly to zero at eachfinite measurement time or ' window '.end of the measurement interval:This idea leads to the rectangularsuch a signal would have nofunction being called a 'rectangulardiscontinuities when madewindow'.repetitive, and so would not sufferfrom spectral special cases are infrequent,signal measurement signal was measured only for afinite time. For a sine wave to havea single line spectrum it must existfor all time. Any practical method ofcalculating the frequency spectrumof a signal suffers from spectralleakage due to the finitemeasurement leakage is not related inany way to the fact of havingsampled the signal, but only to thewill contain not just the signalSecond, the spectral leakage from aThe effects of spectral leakage canbe reduced by reducing thediscontinuities at the ends of theFordwater, Pond Road, Woking, Surrey GU22 0 JZTelephone: (01483) 740138 Fax: (0148) 740136 email: Web: leads to the idea of signal within the measurementtime by some function that smoothlyTo help in choosing a suitablereduces the signal to zero at the endwindow function some quantitativepoints.

4 Hence avoidingmeasures are process of multiplying the signalhave to judge from context whetherdata by a function that smoothlywe are talking about the frequencyapproaches zero at both ends, isor the time domain function.)called 'windowing': and themultiplying function is called aThe FFT as a series of filtersThe FFT as a series of filters' window ' is easy to analyse the effect of awindow function: the frequencyspectrum of the signal is convolvedwith the frequency spectrum of thewindow Equivalent Noise BandwidthSpectral leakageSpectral leakagenoise performance of the window . ItOne way to visualise spectralwhich would accumulate the sameleakage is as spreading of theAnother productive way to visualisenoise power with the same peakfrequency leakage is to regard the FFTpower gain. This is a fruitfulEach frequency should contributecentred on each spectral to one FFT bin but spectralCoherent power gainCoherent power gainleakage causes the energy to beThe filter's frequency response is thespread by the window function so itshape of the window function.

5 'Coherent Power Gain' measures thecontributes to all other FFT in signal power due to theThe contribution is weighted by theleakage model. Each FFT bincoherent signal at the ends of thewindow function centred at theincludes contributions from all othermeasurement component and evaluatedfrequencies in the bandwidth of theat the FFT , weighted by the windowAgain, this measurement relates toIn the special case of thecontributions from broad bandfilters matched to each frequencyrectangular window (that is, nonoise as well as narrow at all), the window functionsignals at other 1 in the interval and 0 outside theinterval. Its Fourier transform isDetection or resolution?Detection or resolution?For an ideal discrete line frequencyknown as the 'sinc' function, orcomponent, the 'noiseless' signalmore formally the 'Dirichlet kernel'.There are two common situations:contribution to the FFT bin isThe shape of the Fourier transformincluding accumulated broadbandof a window function is sometimesnoise.

6 To detect a narrow bandcalled the 'kernel'.signal in the presence of noise, we(Confusingly, the Fourier Transformbe done by using a narrowof a window function is also oftenbandwidth window the ' window function': weas equivalent to a series of filters,concept, and easy to is the inverse of the spectralwindow function suppressing afunction. This will includethe model of the FFT as a bank ofdetection of a spectral component inproportional to the signalthe presence of broadband noise; oramplitude. The proportionality factordistinguishing between narrow bandis the sum of the window terms,spectral components. The choice ofwhich is just the DC gain of thewindow function may be different inwindow. The 'Coherent Gain' is thethe two cases of detection orsquare of this, or in other words theEquivalent noise bandwidthEquivalent noise bandwidthA given FFT bin includescontributions from other frequencieswant to minimise the noise.)

7 This can(ENB) of the window measures theis the width of a rectangle filterFordwater, Pond Road, Woking, Surrey GU22 0 JZTelephone: (01483) 740138 Fax: (0148) 740136 email: Web: power gain is the square ofsignal frequency lies exactly halfa spectral sum of the window between FFT a rectangular window the DC'Scalloping Loss' is defined as theworst when detecting small signalsgain is N, the number of terms inratio of coherent gain for a signalin the presence of nearby largethe window : but for any otherfrequency component located the DC gain will be reducedway between FFT bins, to thebecause the window goes smoothlycoherent gain for a signal frequencyTo minimise the effects of spectralto zero at the ends of thecomponent located exactly at an FFTleakage, a window function's FFTmeasurement have low amplitudeThe reduction in DC gain isThis is just the ratio of the windowthe fall off to the low sidelobesimportant because it represents afunction's value one half a frequencyshould be scaling of the amplitudes ofsample off centre.

8 To the its value atthe frequency spectrum whichthe centre peak sidelobe level is a usefulrequires correction for any absoluteindicator of how well a windowmeasurements to be case processing lossWorst case processing lossfunction suppresses spectralCoherent gain is usually normalisedWorst case processing loss isthe dividing by N, so that thedefined as the sum of Scallopingnormalised coherent gain of aLoss and Processing resolution bandwidthMinimum resolution bandwidthrectangular window is LossLossoutput signal to noise ratio resultingminimum separation needed'Processing Loss' is the ratio of inputfunction and the worst caseof equal amplitude, so that they cansignal to noise to output signal tofrequency location. It is of coursebe , which is the Coherent Powerrelated to the minimum tone thatGain divided by the noise be detected in broadbandBy resolved, is meant that thereFor a signal made up of an idealthe two line frequency componentSpectral leakageSpectral leakagepolluted by white noise, theThe 'rule of thumb' for resolvability isProcessing Loss is, not unexpectedly,The calculated spectrum is not onlythe width of the window at the halfthe reciprocal of the Equivalentaffected by broadband noise, butpower points (the 3 dB bandwidth):Noise Bandwidth (ENB).

9 Also by the narrow band frequencybecause two frequency componentsScalloping lossScalloping if separated by less than theirThe analysis of Equivalent and Processing Loss sofar related only to signalBut this assumes incoherentcomponents whose frequenciesaddition. There is an importantexactly match the FFT with this criterion whenThe model of the FFT as a series ofaddition which is involved with thefilters centred on the FFT binsFFT is coherent, not incoherentsuggests that frequencies that donot coincide with the FFT bins willThe FFT output is the coherentbe attenuated as the filter's responseA frequency component ataddition of frequency components,falls off away from the centrefrequency T will contribute to theweighted by the window function at another frequency Teach frequency. Because of theThis effect will vary as the signalwindow's Fourier transform, centreddefines resolution rather than the 3frequency ranges between theat T and measured at T.

10 This isdB FFT bins, and is called 'theusually referred to as 'spectralpicket fence effect' or 'scalloping'.leakage'.If two frequency components areIt is a reasonable assumption thatSpectral leakage can change thefunctions at the crossover pointthe worst case occurs when thecalculated amplitude and position of(halfway between the peaks) mustThis is a measure of the reduction ofAn interesting measure is thefrom the combination of the windowbetween two frequency be a local minimum betweencomponents either of noise or of theof equal strength show a single0aproportionally to the gain of thecoherence, the 6 dB bandwidth0 aThe effects of spectral leakage aresidelobes away from the centre, andleakage: so is the rate of fall off to3 dB bandwidth, and so cannot beapplied to the FFT because theinvolved the sum of the windowFordwater, Pond Road, Woking, Surrey GU22 0 JZTelephone: (01483) 740138 Fax: (0148) 740136 email: Web.