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').

2 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.

3 For example frequencypure sine wave. 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'.

4 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.

5 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.

6 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.

7 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: 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 .

8 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.'Coherent Power Gain' measures thecontributes to all other FFT in signal power due to theThe contribution is weighted by theleakage model.

9 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.

10 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. 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.)