The-FFT-Fundamentals-and-Concepts.pdf

7/27/2019 The-FFT-Fundamentals-and-Concepts.pdf

1/199


2/199


3/199


4/199


5/199


6/199


7/199


8/199


9/199


10/199


11/199


12/199


13/199


14/199


15/199


16/199


17/199


18/199


19/199


20/199


21/199


22/199


23/199


24/199


25/199


26/199


27/199


28/199


29/199


30/199


31/199


32/199


33/199


34/199


35/199


36/199


37/199


38/199


39/199


40/199


41/199


42/199


43/199


44/199


45/199


46/199


47/199


48/199


49/199


50/199


51/199


52/199


53/199


54/199


55/199


56/199


57/199


58/199


59/199


60/199


61/199


62/199


63/199


64/199


65/199


66/199


67/199


68/199


69/199


70/199


71/199


72/199


73/199


74/199


75/199


76/199


77/199


78/199


79/199


80/199


81/199


82/199


83/199


84/199


85/199


86/199


87/199


88/199


89/199


90/199


91/199


92/199


93/199


94/199


95/199


96/199


97/199


98/199


99/199


100/199


101/199


102/199


103/199


104/199


105/199


106/199


107/199


108/199


109/199


110/199


111/199


112/199


113/199


114/199


115/199


116/199


117/199


118/199


119/199


120/199


121/199


122/199


123/199


124/199


125/199


126/199


127/199


128/199


129/199


130/199


131/199


132/199


133/199


134/199


135/199


136/199


137/199


138/199


139/199


140/199


141/199


142/199


143/199


144/199


145/199


146/199


147/199


148/199


149/199


150/199


151/199


152/199


153/199


154/199


155/199


156/199


157/199


158/199


159/199


160/199


161/199


162/199


163/199


164/199


165/199


166/199

PROCEEDINGS O F THE IEEE, VOL. 66 , NO. 1 , JANUARY 1978

On the Use o f Windows fo r Harmonic Analyswith the Discrete Fourier Transform

FREDRIC J . HARRIS, M E M B E R , IEEE

A h m w - T h i s Pw!r mak= available a concise review of data win- compromise consists of applying windows to the daws pad the^ affect On the Of in the data set, or equivalently, smoothing the spectral sampl'7f aoise9 m the ptesence of sdroag bar- The twooperations to which we subject thedamomc mterference. We dm call attention to a number of common-= in berp~crhfwindows densed with the fd ~ - sampling and windowing. These operations can be petransform. This paper includes a comprehensive ca t dog of data win- in either order. Sampling is well understood, windowi

related to sampled windows for DFT's.I . I N T R O D U C T I O N

HERE IS MUCH signal processing devoted to detectionand estimation. Detection is the task of determiningifa specific signal set is present in an observation, while

estimation is the task of obtaining thevalues of the parametersdescribing the signal. Often the signal is complicated o r iscorrupted by interfering signals or noise. To facilitate thedetection and estimation of ignal sets, theobservation isdecomposed by a basis set which spans the signal space [ 11.For many problems of engineering interest, the class of signalsbeing sought are periodic which eads quite naturally to adecomposition by a basis consisting of simple periodic func-tions, the sines and cosines. The classic Fourier transform isthe mechanism by which we are able to perform this decom-position.By necessity, every observed signal we process must be offinite extent. The extent maybe adjustable and selectable,but it must be finite. Processing a finite-duration observationimposes interesting and interacting considerations on the har-monicnalysis.These considerationsnclude detectabilityof tones in the presence of nearby strong tones, resolvabilityof similarstrength nearby ones, resolvability of shifting tones,and biases in estimating the parameters of any of the afore-mentioned signals.

For practicality, the data we process areN uniformly spacedsamples of the observed signal. For convenience, N is highlycomposite, and we will assume N is even. Theharmonicestimates we obtain through the discrete Fourier ransform(DFT) are N uniformly spaced amples of the associatedperiodic spectra. This approach is elegantnd attractivewhen the processing scheme s cast as a spectral decompositionin an N-dimensional orthogonal vector space [21. Unfortu-nately, in many practical situations, t o obtain meaningfulresults this elegancemuste compromised. One such

Manuscript received September 10, 197 6; revised April 11, 1977 andSeptember 1, 197 7. This work was supported by Naval UnderseaCenter (now Naval Ocean Systems Center) Independent ExploratoryDevelopment Funds.and the Department of Electrical Engineering, School of Engineering,The author is with the Naval Ocean Systems Center, San Diego, CA,San Diego State University, San Diego, CA 921 82.

11. HARMONIC ANALYSIS OF FINITE-EXTEDATA AND THE DFTHarmonic analysis of finite-extent data entails the p

of the observed signal on a basis set spanning the obinterval [ 1 , [3 I . Anticipating the next paragraph, wT seconds as a convenient time nterval and NT seconobservation interval. The sines and cosines with perioto an integer submultiple of N T seconds form an orbasis set for continuous signals extending over N TThese are defined as

sin [ % k t ] O < t < N T .We observe that by defining a basis set over an orderk, we aredefining the spectrum over a line(called quency axis) from which we draw the concepts of baand of frequencies close to and f a r from a given f(which is related to resolution).

For sampled s i g n a l s , the basis set spanning the intervseconds is identical with the sequences obtained bysamples of the corresponding continuous spanning sthe index N / 2 ,

sin [ 3 T ] sin [ 5 ] J n = O , l , * . . ,We note here that the trigonometric functions are uthat uniformly spacedsamples overan integer nuperiods) form rthogonal sequences. Arbitrary rthfunctions, similarly sampled, donot formorthogonquences. We also note that an interval of length N Tis not the same as the interval covered by N samples sby intervals of T seconds. This is easily understood


167/199

52 PROCEEDINGS OF THE IEEE, VOL. 66 , NO. 1 , JANUARY 1

Nt h T - ~ e c ample 'Fig. 1 . N samples of an even function taken over anNT econd interval.

- 1 4 - 5 4 - 3 - 2 - 1 0 1 2 4 5 0P*"odlC e x t m w n Of

mpMYWe-P w o d ~x tmion of

- 9 4 . 7 4 - 5 4 - 3 - 2 . 1 0 1 2 4 5 6 7 8 9Fig. 2. Even sequence under DFT and periodic extension of sequenceunder DFT.

ze that the interval oveq which the samples are taken isis open on the right (i.e., [-)). Fig. 1this by sampling a function which is even about

of duration N T seconds.Since the DFT essentially considers sequences to be periodic,

ng end point to be the beginning ofxt period of the periodic extension of this sequence. In

16 s in1.)is indistinguishable from the sample at zero seconds.This apparent lack of symmetrydue to the missing (butis a sourceof confusion in sampled window

This can be traced to the early work related to con-e factors for the partial sums of the Fourier series. The

finite Fourier transform) always includenumber of points and exhibit even symmetry about

origin. Hence much of the literature and many softwaredesigned with true evensym-

!We must remember for DFTprocessing of sampled data that

n symmetry means that the projection upon the sampledis identically zero; it does not mean a matching

right data point about the midpoint. To distinguishsymmetry from conventional evenness we will refer to it

DFT-even (i.e., a conventionaleven sequence with the ight-point removed). Anotherexample of DFT-even ym-

presented in Fig. 2 as samplesof a periodicallyIfwe evaluate a DFT-even sequence via a finite Fourier

(by treating the + N / 2 point as a zero-value point),continuous periodic function exhibits a non zero

nary component. The DFT of the same sequence is a setsamples of the finite Fourier transform, yet these samples

an imaginary component equal to zero. Why the dis-We must remember that the missing end point underymmetryontributesn imaginary sinusoidal

eriod 2 n / ( N / 2 ) tohe finite transforming to he odd component at sequence position

The sampling positions of the DFT are at the multiples21r/N, which, of course, correspond to he zerosof the

4 - 3 - 2 . 1 0 1 2 3 '

Fig. 3. DFT sampling of finite Fourier transform of a DFT esequence.

is decomposed into its even and odd parts, wi th.the odd psupplying the imaginary sine component in the fintransform.

111. SPECTRAL EAKAGEThe selection of a fite-time interval of N T seconds and

the orthogonal rigonometric basis (continuous or samplover this interval leads to an interesting peculiarity of spectral expansion.Fromhe ontinuum of ossible fquencies, only those which coincide with thebasiswprojonto a singlebasis vector; al l other frequencies will exhinon zero projections on he entire basis set. This is oftreferred to as spectral leakage and is the result of processifinite-duration records. Although the amount of eakageinfluenced by the sampling period, leakage is not caused bthe sampling.An intuitive approach to leakage is the understanding th

signals with frequencies other than those of the basis set not periodic in the observation window. The periodic extsion of a signal not commensuratewith ts natural periexhibits discontinuities at the boundaries of the observatiThe discontinuities are responsible for spectral contributio(or leakage) over the entire basis set. The forms of this dcontinuity are demonstrated in Fig. 4.

Windows are weighting functions applied to data to reduthe spectral leakageassociated with finite observation intvals. Fromoneviewpoint, the window is applied todat(as a multiplicative weighting) to reduce the order of the dcontinuity at the boundary of the periodic extension. Thiaccomplishedby matching as many orders ofderivative the weighted data) aspossible at he boundary. The easiway to achieve this matching is by setting the value of thederivatives to zero or near to zero. Thus windowed data smoothlybrought to zero atheboundaries so that he


168/199

HARRIS: USE OF WINDOWSFORHARMONIC ANALYSIS

Fig. 4. Periodic extension of sinusoid not periodic in observationinterval.Fromanother viewpoint, the window is multiplicatively

applied t o the basis set so that a signal of arbitrary frequencywill exhibit a significant projection only on those basis vectorshaving a frequency close to the signal frequency. Of courseboth viewpoints lead t o identical results. We can gain insightinto window design by occasionally switching between theseviewpoints.

IV . W I N D O W S AN D FIGURES OF M E R I TWindows are used in harmonic analysis to reduce the unde-

sirable effects related to spectral leakage. Windows impact onmany attributes of a harmonic processor; these include detec-tability, resolution, dynamic range, confidence, andeaseofimplementation. We would like to identify the major param-eters that willallow performance comparisons between dif-ferent windows. We can best identify these parameters byexamining the effects on harmonic analysis of a window.An essentially bandlimited signal f ( t )with Fourier transformF ( u ) can bedescribed by he uniformly sampled datasetf ( n T ) . This data set de fi es th e periodically extended spec-trum F T ( u )by its Fourier series expansion as identified as

F(o)=[--( t ) xp-jut)t ( 34

We recognize (4a) as the f i t e Fourier transform, a tion addressed for he convenienceof its even symEquation (4b) is the f i t e Fourier transform with thendpointdeleted, and (4c) is theDFT samplingof Of course for actual processing, we desire (for countposes in algorithms) that the index start at zero. Weplish this by shifting the starting point of the data Ntions, changing (4c) to 4d). Equation (4d) is the forwaThe N/2 shift will affect only the phase angles of thform, so for the convenience of symmetry we will addwindows as being centered at the origin. We also idenconvenience as a major source of window misapplicatioshift of N/2 points and its resultant phase shift is oftelooked or is improperly handled in he definitionwindow when used with the DFT. This s particularly sthe windowing is performed as a spectral convolution. discussion onhe Hanningwindow underhe cowindows.

The question nowposed is, to what extent is thsummation of (4b) a meaningful approximation of thesummation of (3b)? In fact, we address the questiomore general case of an arbitrary window applied to function (or series) as presented in

F,(u) = w ( n T ) f ( n ~ )xp (-junT)+-n = - mwhere

Nw ( n T )= 0, In1 >5, N evenand

*IT Let usnowexamine the effects of the windowoF T ( ~ )xp (+jut) du /2= (3c) spectral estimates. Equation (5) shows thathe trF,(u) is the transform of a product. As indicatedfollowing equation, this is equivalent to the convolthe two corresponding transforms (see Appendix):

= J - r / T

IF(u)l= 0, I u I2 3 [27r/TIand whereFor (real-world)machineprocessing, thedata mustbe off i t e extent, and thesummation of (3b) can only beper-formed as a finite approximation as indicated as

+ N l zFa(u)= f ( n r ) exp-junT) , N even (4a)n =- N / 2

Fb(u)= f(nT)exp(-junT) , Neven4b)( N l Z 1 - ln= - N / 2

orF,(u) = F ( u ) W ( 0 ) .

Equation ( 6 ) is the key to the effects ofprocessingextent data. The equation can be interpreted in two lent ways, which will be more easily visualized withof an example. The example we choose is the rectanglewindow; w ( n T ) = 1.0. We know W ( uDirichlet kernel 141 presented as

N - 1


169/199

5 4 PROCEEDINGS O F THE IEEE, VOL. 66, NO. 1, JANUARY 1

kFig. 5 . Dirichlet kernel for N point sequence.

Except for the linear phase shift term (whichwill change dueto the N / 2 point shift for realizability), a single period of thetransform has the form indicated in Fig. 5 . The observationconcerning ( 6 ) is that the value of F,(w) at a particular w ,say o = 00, is the sum of all of the spectral contributions ateach w weighted by the window centered at wo and measuredat w (see Fig. 6 ) .A . Equivalent Noise Bandw idth

From Fig. 6 , we observe that the amplitude of the harmonicestimate at a given frequency is biased by the accumulatedbroad-band noise included in the bandwidth of the window.In this sense, the window behaves as a filter, gathering contri-butions for its estimate over its bandwidth. For the harmonicdetection problem, we desire to minimize this accumulatednoise ignal,and we accomplish this with small-bandwidthwindows. A convenient measure of this bandwidth is theequivalent noise bandwidth (ENBW)of the window. This isthe width of a rectangle filter with the same peak power gainthat would accumulate the same noise power (see Fig.7).

The accumulated noise power of the window is def ied as+nlT

Noiseower = N o I W ( 4 1 2 d w / 2 n (8)where N o is the noise power per unit bandwidth. ParseVal'stheorem allows (8) to be computed by

Noise Power =- w 2 ( nT).OT nThe peak power gain of the window occurs at w = 0, the zerofrequency power g a i n , and is def ie d by

Peak Signal Gain W ( 0 )= w ( n T ) ( loa)n

Peak Power Gain = W 2 ( 0 ) = [F ( n T ) ] . (lob)2Thus the ENBW (normalized by N O I T , he noise power perbin) is given in the following equation and is tabulated for thewindows of this report in Table I

GainA concept closely allied to ENBW is processing gain (PG)

processing oss (PL) of a windowed transform. We can

0 D W-0Fig. 6. Graphical interpretation of equation (6). Window visualizea spectral Nter.

- -AI L - - + wFig. 7 . Equivalent noise bandwidth of window.fi te r is matched to one of the complex sinusoidal sequencethe basis set [ 3] . From this perspective, we can examine PG (sometimes called the coherent gain) of the fiter, and can examine the PL due to the window having reduced data to zero values near the boundaries. Let the input sampsequence be defined by (12 ) :

f ( n T )= A exp ( + j o k n T )+ 4 ( n T ) (where q ( n T ) is a white-noise sequence with variance 0: . Ththe signal component of the windowed spectrum (the matchfilter output) is presented in

F ( Q ) lsignal= w ( n T )A exp ( + j o k n T ) exp ( - j w k n T )n

= A w (n T) . (1We see that the noiseless measurement (the expected valuethe noisy measurement) is proportional to the input amplituA . The proportionality factor is the s u m of the window termwhich is in act hedc signalgain of the window. Forectangle window this factor is N , the number of terms in window. For any other window, the gain is reduced due

n


170/199

ARRIS: USE O F WINDOWS FOR HARMONIC ANALYSISTABLE IWINDOWS ND FIGURESF MERI T

H I G H E S T O V E R L A PO R S TI D E -S I D E - C O R R E L A T I O. W BA SEC A L L O P.WBQ U I V . C O H E R E N TO B EW I N D O W ( P C N T IWROCESSOS SWO I S E G A I NA L L .O B EL E V E LId61

O F F( d B I O C T ) B W W %5%0Ld B )

( B I N S )OS SdB) (B INS1(B I N S )

R E C T A N G L E 505.0.21.92.92.89.oo.oo613T R I A N G L E 251.9.78.07.82.2 8.33.501227

D E L A V A L L E -POUSSIN

- 53 59.3.55.72.9082.92.3824

T U K E Y U=0.25250.580.07.73.31.26.63 -18 -19= 0.75 362.7.57.1 1.24.15.22 0.7518 -15=0.50444.1.38.39.96.01.10.881814

E O H M A N -46 -24 0.41 74.5.38.54.02.71.79POISSON a = .0

154.8.08.64.46.45. 6 5.326243.0 279.9.w.23.09.21.30.44619U=4.0 70.4.58.21.03.75; 08.25631

H A N N I N G - U = .5 -3544.6.65.94.87 1.87.02.2918O N E=2.096 O.30.50.11.64.73.38 -1839OISSON U = 1.0

121.3.14.33.x.5 46 1.4318

CAUCHY a = x o98.3.53.28 1.13. 68.06.28630= 5.0

138.8.20.83.36.50.76.33635= 4.0201.6 1.90 .71.34 1.48.42631

G A U S S I A N a = 2 . 547.2.52.73.94.79. 9 0.37669= 3.5

57.5.18.40.25.55 1.64.43655= 3.0 207.7.8 6.14.w.33.39.51642 10DOLPH- a = 2 . 5 -50 0 0.53 1.390.48 1 . 3 3

85.9.31.48.10.65.73 0.4280= 4 . 016110.2.1 7.35.25.55.62 64.7

.010.4570= .5

3.23.44.44.51229.6.85.1 2.70C H E E Y S H E V a = 3.0 -60

K A I S E R - a = .0 1.w.20.46.4350.49646BESSEL U=2.5 -57 -6748.8.57.74.89.83 1.93

53.9.390.37682=3.5

111.80

59.50.40

2.203.56.02.71669= .0

165.73.38.m.57.65.44

B A R C I L O N - a = .0 -53 -6 0.47 1.56 1.49 1.34 3.27 2.07 63.0 14T E M E S u = .5 -58 -674.4.36.52. 0 5 1 .6 9.77.41668= 4 . 0 10

8.6.23.40.18.59.67.43

E X A C T B L A C K M A N 142.7.13.29.33.52.57.46651

B L A C K M A N

B L A C K M A N - H A R R I S 97.281.45.13.66.71.42667I N I M U M 3 -S A M P L E

96.7.35.47.10.68.73.42 -1858

' M I N I M U M 4 -S A MP L E 36.0.72.85.83.90.00.36692B L A C K M A N - H A R R I S'61 dE 3 -SAMPLE 121 .O.19.34.27.56.61.45661B L A C K M A N - H A R R I S

74 d B 4 -S A M P L E

7KAISER-BESSEL

53.9.44.56.02.74BO.40669- S A M P L E a = 3.0B L A C K M A N - H A R R I S 7

3.9.44.56.03.74.79.40674


171/199

56 PROCEEDINGS OF THEEEE, VOL. 66 , NO. 1 , JANUARY reduction in proportionality factor is important as it repre-sents a known bias on spectral amplitudes. Coherent powergain, the squk of coherent gain, is occasionally the parameterlisted in the literature. Coherent gain (the summation of (13))normalized by its maximum valueN is listed in Table1.

The ncoherent component of the windowed transform isgiven by

F ( w k ) nois = w ( n T )q ( n T ) exp ( - j w k n T ) (144and the incoherent power (the meansquare value of this com-ponent where E { } is the expectation operator)is given by

n

E { IF(Wk) Inois12} = w ( n T ) w ( m T ) E ( q ( n T )q * ( m T ) }n m.exp ( - j w k n T ) exp (+ jwkrnT)

= ui w 2 ( n T ) . (14b)Notice the incoherent power gain is the sum of the squares ofthe window terms, and the coherent power gain is the squareof the sum of the window terms.

Finally, PG, which is def ied as the ratio of output signal-to-noise ratio to input ignal-to-noise ratio, is given by

n

c ...

nNotice PG is the reciprocal of the normalized ENBW. Thuslarge ENBW suggests a reduced processing gain. This is reason-able, since an increased noisebandwidthpermits additionalnoise to contribute t o a spectral estimate.C . OverlapCorrelation

When the fast Fourier transform (FFT) is used to processlong-time sequences a partition length N is first selected toestablish theequired spectral resolution of the analysis.Spectral resolution of the FFT is defined in (1 6) where A f isthe resolution, f , is the sample frequency selected to satisfythe Nyquist criterion, and f l is the coefficient reflecting thebandwidth increase due to he particular window selected.Note that [ f J N ] is the minimum resolution of the FFTwhichwe denote as the FFT bin width. The coefficient f l is usuallyselected to be the ENBW in bins as listed in Table I

A f = f l (5). (16)If the window and the FFT are applied to nonoverlapping

partitions of the sequence, as shown in Fig. 8, a significantpart of the series s gnored due to the window's exhibitingsmall values near the boundaries. For nstance, if the transformis being used to detect short-duration tone-like signals, the nonoverlapped analysis could miss the event if i t occurred nearthe boundaries. To avoid this loss of data, the transforms areusually applied to the overlapped partition sequences as shownin Fig. 8. The overlap is almost always 50 or 75 percent. This

! r _ _ - _ _ :: - . , ' j +Original SequenTe"., c---- . , !


172/199

HARRIS: USE OF WINDOWS FOR HARMONIC ANALYSIS

Nan RerolvaLde PC&% Resolvable Peaks

Fig. 10. Spectraleakage effect of window. Fig. 1 1 . Spectral resolution of nearbyernels.

and can also be omitted from (19) with negligible error. Forthis reason, c(0.25) was not listed in Table I. Note, that .forgood windows (see last paragraph of Section IV-F), transformstaken with 50-percentoverlap are essentially independent.D . Scalloping LossAn important consideration related to minimum detectable

signal is called scalloping loss or picket-fence effect. We haveconsidered the windowed DFT as a bankof matched fdtersand have examined the processing gain and the reduction ofthis gain ascribable to thk window for tones matched to thebasisvzctors. The basis vectors are toneswith requenciesequal to multiples, of ,/N (with f, being the sample fre-quency). These frequencies areample pointsromhespectrum, and are normally referred to as DFT output pointsor as DFT bins. We now address thequestion, what is theadditional loss in processing gain for a tone of frequency mid-way betweenwo bin frequencies (that is, at frequencies(k+ 1/2)f,/N)?Returning to (1 3),with wlc replaced by w ( k + / 21, we deter-mine the processing gain for this half-bin frequency shift asdefined in~ ( ~ ( 1 1 2 ) )signal = A w(nT) ~ X P- i o ( l / z ) n ~ ) ,n

where q l I 2 ) =-- -ws 7r2 N N T' (204We al so define the scalloping loss as the ratio of coherent gainfor a tone located half a bin from a DFT sample point to thecoherent gain for a one ocated at a DFT sample point, asindicated in

n(20b)

Scalloping loss represents the maximum reduction in PG dueto signal frequency. This loss has been computed for the win-dows of th s report and has been included in Table I.E . Wors t Case Processing Loss

We now make an interesting observation. We define worstcase PL as the sum of maximum scalloping loss of a windowand of PL due to tha t window (both in decibel). This numberis the reduction of output signal-to-noise ratio as a result ofwindowing ndofworst ase frequency location. This ofcourse is related to the minimum detectable one n broad-

beused. Additional ommentsonpoor windows wifound in Section IV-G. We can conclude from the combloss f i e s of Table I and from Fig. 12 that for the deteof single tones in broad-band noise, nearly any window (than the rectangle) is as goodas any other . The diffebetween the various windows is less than 1.0 dB and for windows is less than 0.7 dB. The detection of tones inpresence of other tones is, however, quite another problHere the window does have a marked affect, as will be destrated shortly.F. Spectral Leakage Revisited

Returning to ( 6 ) and to Fig. 6 , we observe the spemeasurement is affected notonlyby hebroadband spectrum, but also by the narrow-band spectrum whichwithin the bandwidth of the window. In fact, a given specomponent say at w = wo wi contribute output (or wobserved) at another frequency, say at w = w, accordinthe gain of the window centered at 00 and measured aThis is the effect normally referred to as spectral leakageis demonstrated in Fig. 10 with the transform of a finite tion tone of frequencywo

This leakage causes a bias in the amplitude and the posof aharmonic estimate. Even for hecise of a singleharmonic line (not at a DFT sample point) , the leakage the kernel on the negative-frequency axis biases the kernthe positive-frequency line. This bias is most severe and mbothersome for the detection of small signals in the preof nearby large signals. To reduce the effects of this biawindow should exhibit low-amplitude sidelobes far fromcentral main lobe, and the transition to he low sideshould bevery rapid. One indicator of howwell a winsuppresses eakage is the peaksidelobe evel(relative tomain lobe): another is the asymptotic rate of falloff of sidelobes. These indicators are listed in Table I.G . Minimum Resolution BandwidthFig. 11 suggests another criterion with which we shouconcerned in the window selection process. Since the wiimposes an effective bandwidth on the spectral line, we wbe interested in the minimum separation between two estrength lines such that for arbitrary spectral locations respective main lobes can be resolved. The classic criterioth s resolution is the width of the window at the half-ppoints (the 3.0-dB bandwidth). This criterion reflects thethat two equalstrength main lobes separated in frequencless than their 3.0-dB bandwidths will exhibit a single sppeak nd wiU not be esolved s two distinct lines. problem with this criterion is that it does not work fo


173/199

58 PROCEEDINGS O F THEEEE, VOL. 66, NO. 1 , JANUARYIf two kernels are contributing to the coherent summation,

the sum at the crossover point (nominally half-way betweenthem) must be smaller than the individual peaks if the twopeaks are to be resolved. Thus at the crossover points of thekernels, the gain from each kemelmust be less han 0.5, or thecrossover points must occur beyond the 6.0-dB points of thewindows. Table I lists the 6.0-dB bandwidths of the variouswindows examined in thi s report. From the table, we see thatthe 6.0-dB bandwidth varies from 1.2 bins to 2.6 bins, where abin is theundamentalrequency resolution wJN. The3 .O-dB bandwidth does have utility as a performance indicatoras shown in the next paragraph. Remember however, it is the6.0-dB bandwidth which defies the resolution of the win-dowed DFT.

From Table I, we see that he noise bandwidth alwaysexceeds the 3.0-dB bandwidth. The difference between thetwo, referenced to he 3.0-dB bandwidth,appears to be asensitive indicator of overallwindow performance. We haveobserved that or all the goodwindows on the table, thisindicator was found t o be in the range of 4.0 to 5.5 percent.Thosewindows for which this ratio is outside that rangeeither have a wide main lobe or a high sidelobe structure and,hence, are characterized by high processing oss or by poortwo-tonedetection capabilities. Those windows for whichthis ratio is inside the 4.0 to 5.5-percent range are found inthe lower left comer of theperformance comparison chart(Fig. 121, which is described next.

While Table I does list thecommonperformance param-eters of the windows examined in this report, he massofnumbers is not enlightening. We do realize that the sidelobelevel (to reduce bias) and the worst case processing oss (tomaximize detectability) are probably the most importantparameters on the table. Fig. 12 shows the relative positionof the windows as a function of these parameters. Windowsresiding in the lower left comer of the figure are the good-performingwindows. They exhibit lowsidelobe levels andlowworst case processing oss. We urge the reader to readSections VI and VII; Fig. 12 presents a lot of information,but not the full story.

V. CLASSIC WINDOWSWe will now catalog some well-known (and some not well-

known windows. For each window we will comment on thejustification for its use and identify its significant parameters.All the windows will be presented as even (about the origin)sequences with an odd number of points. To convert the win-dow to DFTeven, the right end point will be discarded andthe sequence will be shifted so that the left end point coin-cides with the origin. We will use normalized coordinates withsample period T = 1 O, so that w is periodic in 2n and, hence,will be identified as 8 . A DFT bin wbeconsidered toextend between DFT sample points (multiples of 2nlN) andhave a width of 2nlN.A . Rectangle (Dirichlet) Window 161

The rectangle window is unity over the observation interval,and can be thought of as a gating sequence applied o the dataso that they areof finite extent .The window fora finiteFourier transform s defined as

N Nw ( n )= 1.0, n = - - ; - . , - l , O , l ; - - , ~ (2la)L L

WORSTW E ROCESSNG L O S S . dBFig. 12. Comparisonof windows: sidelobe levelsand worst case ping loss.

defined asw ( n ) = 1.0, n = 0 , l ; - - , N -.

The spectral window for the DFTwindow sequence is giv

The ransform of thi s window is seen to be the Dirkernel, which exhibits a DFT main-lobe width (betweencrossings) of 2 bins and a first sidelobe level approximatdB down from the main-lobe peak. The sidelobes fal l6.0 dB per octave, which is of course the expected ratefunctionwith a discontinuity. The parameters of thewindow are listed in Table I.With the rectangle window now defmed, we can answquestion posed earlier: in what sense does the finite su(22a) approximate he infinite s u m of (22b)?

+N 2n=-N/2

F ( @ = f ( n )exp ( - i n 8 )

F ( 8 ) = f ( n ) exp ( - i d ) .+-n=--

We observe the finite s u m s the rectangle-windowed versthe M i t e sum. We recognize that he infinite s u mFourier series expansion of some periodic function for


174/199


Fig. 13 . (a) Rectangle window. (b) Log-magnitude of transform.

T 1.25 4 Od B1 1 , -20(a) (b)

Fig. 14 . (a) Triangle window. (b ) Log-magnitude of transform.

that he frnite sum is simply the partial sum of the series.From this viewpoint we can cast the question in terms of theconvergence properties of the partial s u m s of Fourier series.From this workwe know the partial sum is the least mean-square error approximation to the infinite s u m .

We observe that mean square convergence is a convenientanalytic concept, but it is not attractive for finite estimates orfor numerical approximations. Mean-square estimates tend tooscillate about their means, and do not exhibit uniform con-vergence. (The approximation ina neighborhood of a point ofcontinuity maygetworse if more terms areadded to thepartial sum.) We normally observe this behavior near points ofdiscontinuity as the ringing we call Gibbs phenomenon. It isthis oscillatory behavior we are trying to control by the use ofother windows.B. Triangle (Fejer, Bartlet) Win dow [7]

The triangle window for a finite Fourier transform is definedas

In NW(n)=l.O-- n = - -N / 2 '' . * * , - l , O , l ; - - , - (23a)2and is shown nFig. 14. The amewindow for a DFT isdefmed as

f Nn = 0 , ; . . -W ( n )= (23b)

and the spectral window corresponding to the DFT sequegiven in

The transform of this windows een to be the sqDirichlet kernel. Its main-lobe width (between zero crosis twice that of the rectangle's and the first sidelobe evapproximately 26 dBdownfrom the main-lobe peak, twice that of the rectangle's. The sidelobes fall off at -per octave, reflecting the discontinuityof the window rein the first derivative (rather than in the function itself)triangle is the simplest window which exhibits a nonnegtransform. This property canberealized by convolvinwindow (of half-extent) with itself. The resultant wintransform is the square of the original window's transformA window sequence derived by self-convolving a parendow contains approximately twice the number of sampthe parent window, hence corresponds to a trigonompolynomial (its Z-transform) of approximately twicorder. (Convolving two rectangles each of N / 2 pointresult in a triangle of N + 1 points when the zero end pare counted.) The transform of the window will now etwice as many zeros as the parent transform ( to accouthe increased order of the associated trigonometric


175/199

60 PROCEEDINGS O F THE IEEE, VOL. 66 , NO. 1 , JANUARY

Fig. 1 5 . Two partial sums and their average.

zf+3

convolved window simply places repeated zeros at each loca-tion for which theparent ransform had a zero. This, ofcourse, not only sets the transform to zero at those points, butalso sets the firstderivative to zero at those points. If theintent of the increased order of polynomial is to hold downthe sidelobe levels, then doubling up on thezeros is a wastefultactic. The additional zerosmight better be placed betweenthe existing zeros (near the local peaks of the sidelobes) tohold down the sidelobes rather than at locations for whichthe transform is already equal to zero. In fact we will observein subsequent windows that very few good windows exhibitrepeated roots.

Backing up for a moment, it is interesting to examine thetrianglewindow inerms ofartial-sumonvergence ofFourier series. Fejer observed that the partial s u m s of Fourierserieswere poor numerical approximations [ 8 ] . Fouriercoefficients were easy to generate however, and he questionedif somesimple modification of coefficients might ead to anewet with moredesirable onvergence properties. Theoscillation of the partial sum, and the contraction of thoseoscillations as the order of the partial sum increased, suggestedthat an averageof the partial sums .would be asmootherfunction. Fig. 15 presents an expansion of two partial sumsnear a discontinuity. Notice the average of the two expansionsis smoother than .either. Continuing in thi s line of reasoning,an average expansion F N ( e ) might be defined by

of the cosine function. These properties are particattractive under the DFT. The window for a finite Ftransform is defined as

w(n)=cosQ [ i n ] , n = - -N2 * * * , - 1 , 0 , 1 ; . * and for aDFT as

w(n)=sinQ [ i n ] , n = 0 , 1 , 2 ; * . , N - 1.25Notice the effect due to the change of the or ig in. Thecommon values of a are the integers 1 through4, with 2the most well known (as he Hanning window). This wis identified for values of a equal to 1 and 2 in (26a), (27a), and (27b), (the a for the fi nt e transforms, thfor the DFT):

a = 1O (cosine lobe)w(n)=cos [ i n ] , n = - - * * . , - l , O , l ; * . , - (N2 2

a = 1 O (sine lobe)w(n)=sin [in], n = 0 , 1 , 2 ; * . , N - 1

a = 2.0 (cosine squared, raised cosine, Hanning)

a = 2.0 (sine squared, raised cosine, Hanning)

= O S 1.0-cos -1 , n = 0 , 1 , 2 , . . - , N -[ E l 1The windows are shown for a nteger values of 1 througFigs. 16 through 19. Notice as a becomes larger, the winbecome smoother and the ransform reflects this incrsmoothness in decreased sidelobe level andfaster falloff owhere F M ( 6 ) is the M-term partial sum of the series. This is sidelobes, but with an increased width of the main lobe.

easilyisualized in Table 11, which lists the nonzero coeffi- Of interest in thi s family, is the Harm wicients Of the first four sums and their (afterheustrianmeteorologist, Julius Von Hann) [71.tion. We see that the Fejer convergence factors applied to th e only is thi s window continuous, but so is its first derivaFourier series coefficients is, in fact, a triangle window. The since he discontinuity of this window resides in the sesummability. octave. Let us closely examine the transform of this win

of Partial sums is known as the method Of cesko derivative, thebansform falls offat or at - 18 dBC. CosQ(X)Windowse wi gain some iteresting insightnd learn of a c

This is actually a family of windows dependent upon the application of the window under the DFT.parameter a,with a normally being an integer. Attractions of The correct name of this window is Hann. The term Han


176/199


T1.25

0 B

(a)Fig.

'Tti

-.16 . (a) CO S (nn/n? window. (b) Log-magnitude of transform.

0 B

1.251.w

(a)Fig. 1 7 . (a) Cos2 ( n n / N ) win1

1.25T

(b )dow. (b) Log-magnitude of transform.

(a) (b )Fig. 18. (a) COS' (nn/N)window. (b) Log-magnitude of transform.

I 1.25II 1.w


177/199

62 PROCEEDINGS OF TH E IEEE, VOL. 66, NO . 1 , JANUARYThe sampled Hanning window can be written as the s u m of

the sequences ndicated inw ( n ) = 0.5 + 0.5 COS - ,[, I

N2 2n = - - * * - , - l , O , 1, *. . , -- 1. 28a)

Each sequence has he easily recognized DFT indicated in

where

Fii. 20 . Transform of Hanning window as a sum of three Dkernels.

We recognize the Dirichlet kernel at the or ig in as the transformof the constant 0.5 samples and the pair of translated kernelsas the transform of the single cycle of cosine samples. Notethat the translated kernels are located on the f i t zeros of thecenter kernel, and are half the size of the center kernel. Alsothe sidelobes of the translated kernel are about half the sizeandareof opposite phaseof the sidelobes of the centralkernel. The summation of the three kernels sidelobes being inphase opposition, tends to cancel the sidelobe structure. Thiscancelling summation is demonstrated inF i g . 20 which depictsthe summation of the Dirichlet kernels (without he phase-shift terms).

The partial cancellingof the sidelobe structure suggests aconstructive technique to define newwindows. Themostwell-knownof these are the Hamming and the Blackmanwindows whichare presented in he next two sections.

For he specialcaseof the DFT, he Hanningwindow issampled at multiples of 2n/N, which of course are the loca-tions of the zeros of the central Dirichlet kernel. Thus onlythree nonzero samples are taken in the sampling process. Thepositions of these samples are at -2n/N, 0, and +2n/N. Thevalue of the samples obtained from (28b) (including the phasefactor exp (-j(N/2)0) to account for the N/2 shift) are - $,+ - $, respectively. Note he minus s i g n s . These resultsfrom the shift in the or ig in for the window. Without the shift,the phase term is missing and the coefficients are all positive$, 3, $. Theseare incorrect for DFT processing, but heyfind their way into much of the literature and practice.domain, we always have the option to apply it as a convolu-tion in the frequency domain. The attraction of the Hanningwindow for ths application is twofold; f i t , the windowspectra is nonzero at only three data points, and second, thesample values are binary fractions, which can be implementedas ight shifts. Thus he Hanning-windowed spectral pointsobtained romhe rectangle-windowed spectral points areobtained as indicated n he following equation as two realadds and two binary shifts (to multiply by 3):

Rather hanapply he window as a product in the time w ( n ) =

F ( k ) H d n g = 3 [ F ( k ) 3 [ F ( k - 1)+ F ( k + 1111 I R ~ .29)

N2 * , - 1 , 0 , 1 , * . . - 2n = - -

or as 2N real adds and 2N binary shifts on the spectraOne other mildly important consideration, if the windowbe applied to the time data,is that the samples of the wmust be stored somewhere, which normally means addmemory or hardware. It so happens that the samples ocosine for he Hanning windoware already stored machine as the trig-table for he FFT; thus the Hwindow requires no additional storage.D. Hamming Window /7]The Hamming windowcan be thought of as amodHanning window. (Note the potential source of confusthe similarities of the two names.) Referring back to Fand 20, we note the nexact cancellation of the sidelobethe summation of the three kernels. We can construct dow by adjusting the relative size of he kernels as indicathe following to achieve a more desirable form of cancell

w ( n ) = a + ( l - a ) c o s [$4Perfect cancellation of the f i t sidelobe (at 0 = 2.5 [2occurs when a = 25/46 (a 0.543 478 261). If a s sele0.54 anapproximation to 25/46), he new zerooccu6 G 2.6 [2n/Nl and a marked improvement in sidelobe lrealized. For ths value of a, he window is called theming window and s identified byI.54 + 0.46 COS [ n ] ,

0.54 - 0.46 cos [ $ n ] ,I n = 0 , , 2; *. ,N - 1.

The coefficients of the Hamming window are nearly twhich achieve minimum sidelobe levels. If a s selected0.53856 the sidelobe level is -4 3 dB and the resultant wis a special case of the Blackman-Harris windows presenSection V-E. The Hammingwindow is shown in FiNotice the deep attenuation at the missing sidelobe poNote also that the smal l discontinuity at the boundary


178/199

HARRIS: USE OF WINDOWS FO R HARMONIC ANALYSIS

T 1.2sT

(a) (b)Fig. 21. (a) Hamming window. (b ) Log-magnitude o f Fourier transform.

1.25

(a) (b)Fig. 22. (a) Blackman window. (b ) Log-magnitude of transform.

eters of this window. Also note the loss of binary weighting;hence the need to perform multiplication to apply theweighting factors of the spectral convolution.E. Blackman Window 171

The Hamming and Hanning windows are examples of win-dows constructed as the summation of shifted Dirichlet ker-nels. This data window is defined for the finite Fourier trans-form in (31a) and for the DFT in (31b); equation (3 IC)s theresultant spectral window for the DFT given as a summationof the Dirichlet kernels D ( 8 ) defined by W ( 8 ) n (21c);

(3 la)

of this form with u o and u 1 being nonzero. We see thaspectral windows are summations of three-shifted kernels

We can construct windows with any K nonzero coeffiandachieve a (2K- 1) summation of kernels. We recohowever, that one way to achieve windows with a narrowlobe is to restrict K to a small nteger.Blackmanexaminethis window for K = 3 and found the values of the nocoefficients which place zeros at 8 = 3.5 (2n/N) and at 8(2n/N), he position of the third and thefourth siderespectively, of the centralDirichleternel.Thesexactvalues and their two place approximations are

7938Q O =--18608 10.4 26 590 71 N- 0.42

Q l = - -186089240 L 0.496 560 62N 0.50

Q z =--186081430 I 0.076 848 67N 0.08.(31b) The windowwhichuses these wo place approximati

Nl 2 known as the Blackmanwindow.Whenwe escribe window with the "exact"coefficients wewill refer tothe exact Blackmanwindow. The Blackmanwindow sdfined for the finite transform in the following equatiom=O

(31c) the window s shown in Fig. 22:Subject to constraint

Nf 2 W ( n )= 0.42 + 0.50 cosQ = 1.0.


179/199

64

T 1.25T

PROCEEDINGSOF THE IEEE, OL. 66, NO. , JANUARY

I f \(a) 0)

Fig. 23. (a) Exact Blackman window. (b) Log-magnitude of transform.

(a) (b)Fig. 24. (a) Minimum 3-term Blackman-Harris window. @) Log-magnitude of transform.

(a) 0)Fig. 25. (a) 44- Blackman-Harris window. (b) Log-magnitude of transform.

The exact Blackman window is shown in Fig. 23. The sidelobelevel is 51 dB down for the exact Blackman window and is 58dB down for the Blackman window. As an observation, notethat he coefficients of the Blackmanwindow sum to zero(0.42 -0.50 M.08) at he boundaries while the exact coef-ficients donot. Thus the Blackmanwindow is continuouswith a continuous first derivative at the boundary and f a l l s offlike l/w3 or18 dB peroctave. The exact terms (Like theHamming window) have a discontinuity at the boundary andfalls off like l /w or 6 dB per octave. Table I lists the param-eters of these two windows. Note that for this class of win-dows, the a. coefficient is the coherent gain of the window.

Usingagradientsearch technique [9] , wehave found the

sidelobe level. We have also constructed families of 3- aterm windows in which we trade main-lobe width for sidlevel. We call this family the Blackman-Harriswindow. have found that the minimum 3-term window can achieve asidelobe level of -67 dB and that the minimum 4-termdow can achieve a sidelobe evel of -92 dB. These windoare defiied for theDFT byw ( n ) = a o - a1 cos --n +a2 cos -2n - a 3 cos -3n() ( ) (

n = 0 , 1 , 2 ; * * , N - 1.The listedcoefficients correspond to the minimum3-te


180/199


(-67 dB)3-TellIl ( -61 dB)3-Term (-92 dB)4 -Term (-74 dB)4-TUIQ

a0 0.42323.44959.35875a1 0.49755.49364.48829a2 0.07922 0.05677.14128.09392a3 _ _ _ --- 0.01168.00183(to establish another data point in Fig. 12), the minimum 4-term window (to also establish a data point in F'ii. 12), andanother 4-term window which is presented in Fig. 25. Theparticular 4-term window shown is one which performs wellin a detection example described in Section VI (see Fig. 6 9 ) .The parameters of these windows are listed in Table I. Note inparticular where the Blackman and the Blackman-Harris win-dows reside in Fig. 12. They are surprisingly good windowsfor the small number of terms in their trigonometric series.Note, if we were to extend the line connecting he Blackman-Harris family it would intersect the Hamming window which,in Section V-D ,we noted is nearly the minimum sidelobe level2-term Blackman-Harris window.

We also mention that aood approximation to th e lackman-Harris3- and 4-term windows can be obtained as scaledsamples of the Kaiser-Bessel window's transform (see SectionV-H). We have used this approximation to construct 4-termwindows foradjustablebandwidthconvolutional filters asreported in [ 101. This approximation is defined as

uo =- a , = 2 -, m = 1 ,2 , 3 ) .o bmC C (34)The 4 coefficients for this approximation when Q = 3.0 area. = 0.40243, u1 = 0.49804, a2 = 0 . 0 9 8 3 1 , and a3 = 0.00122.Notice how close these erms are to he selected 4-termBlackman-Harris (-74 dB) window. The window defined bythesecoefficients is shown in Fig. 26. Like theprototypefrom which it came (the Kaiser-Bessel with CY = 3 .0 ) , thiswindow exhibits sidelobes just shy of -70 dB from the mainlobe. On the scale shown, the wo are indistinguishable.The parameters of th is window are also listed in Table I and

windows (parameterized on a) which were the startingtions for the gradient minimbation which leads to the--Harris windows. The opthization.-starhgwitcoefficients hasvirtually no effect on the main4obe chaistics but does drive down the sidelobes approximately 5F. ConstructedWindows

Numerous investigators have constructed windows aucts, as sums, as sections, or as convolutions of simpltions and of other simple windows. These windows havconstructed or certain desirable features,not the lwhich is the attraction of simple functions for generatwindow terms. In general, the constructed windows teto be good windows, and occasionally are very bad winWe have already examined some simple window construThe Fejer (Bartlett) window, for instance, is the convof two rectangle windows; the Hamming windowis the a rectangle and a Hanning window; and the cos4(X) wis the product of two Hanning windows. We will now eother constructed windows that have appeared in theture. Wewpresent them so they are available for coson. Later we will examine windows constructed in with some criteria of optimality,(see Sections VG, HJ) . Each windowis identified only for he f i t e Fourieform. A simple shift of N/2 points and right end-poition will supply the DIT version. The significant figperformance for thesewindows are also found in TableI ) Riesz (Bochner, Panen) Window [ I 1 : The Riedow, identified as

is the simplest continuous polynomial window. It exhdiscontinuous first derivative at heboundaries;hence tstransform falls off like l / d . The window is shown27. The first sidelobe is -22 dB from he main lobewindow is similar to the cosine lobe (26) as can be strated by examinhg ts Taylor series expansion.2) Riemunn Window (121: The Riemann window, bY


181/199

66 PROCEEDINGS OF THE IEEE, VOL. 66 , NO. 1. JANUARY

in 0 d8

-15 -20 -15 -10

-25420

1i- 5 !i1 1.15

I

Fa. 7. (a) Riesz window. (b ) Log-magnitudeof transform.

12 5

Fa. 8. (a) Riemann window. (b ) Log-magnitudeof transform.

I 12 5

Fig. 29. (a) The de la Vall6-Pouspin window. (b) Log-magnitudeof transform.

tinuous, with a discontinuous first derivative at tile boundary.It is similar to theKesz and cosine lobe windows. TheRiemann window is shownia Fig. 2.8.3) d e la Vall&Poussin (Jackson, Parzen) Window I I ] The

de la VallB-Poussin window is apiecewisecubiccurve ob-tained by self-convolving two triangles of half extent or fourrectangles of one-fourth extent. It is defined as

1 1 . 0 - 6[d21.0-%], O< I n l < - N2 [LO - $$, N- < I n I < - .4 2

The window is continuous up to its third derivative so thsidelobes f a l l off like l /w4. The window is shown in FiNotice the trade of.ofmain-lobe wid&-fer-sidelobeCompare this with the rectangle and the triangle. It is anegative window by virtue of its self-convolution constru4 ) TukeyWindow [13/: The Tukey window, often cthe cosine-tapered window, is best imagined as a cosine lowidth ( a / 2 ) N convolved with a ectanglewindow of(1 O - a / 2 ) N . f course the resultant transform is the prof the two corresponding transforms. The window reprean attempt to smoothly set the data to zero at the bounwhile not significantly educing the processinggain owindowed transform. The window evolves from the rect


182/199


&/-25 - I5

-25B20 -i:tt5 i o i s

sidelobe levels arising from the productof the tw o componenttransforms. The window is defined by

(38)The window is shown in Figs. 30-32 for values of Q equal to

tained by the convolution of two half-duration cosin(26a), thus its transform is the square of the cosinetransform (see Fig. 16). In the time domain the windbe described as a product of a triangle window with acycle of a cosine with the same period and, then, a coterm added to set the frrst derivative to zero at the bouThus the second derivative is continuous, and he dnuity resides in the third derivative. The transform fallsl/w4. The window is defined in the following and is shFig. 33:


183/199

68 PROCEEDINGS O F THE EEE, VOL. 66 , NO. 1 , JANUARY

(a) @)Fig. 33. (a) Bohman window. (b ) Log-magnitudeof transform.

[ 1 2 5

(a) @)Fig. 34. (a) Poisson window. (b) Log-magnitudeof transform a = 2.0).

T 1 2 5T

(a) (b)Fig. 35. (a) Poisson window. (b) Log-magnitudeof ransform (a = 3.0).

6 ) Poisson Window [ 1 2 ] : The Poissonwindow is a two- observed in Table I as a large equivalent noise bandwidth7) Hanning-Poisson Window: The Hanning-Poissonwinsided exponential de f ie d by as a largeorstaserocessingoss.N, o g l n I < . (40) dows constructed as the product of the Hanning and2 Poissonindows. The family is def ied by

This is actually a familyofwindows parameterized on hethe transform can fal l off no faster than l /w . The window isshown in Figs. 34-36 for valuesf a equal to 2.0,3.0, and 4.0,41)respectively. Notice as the discontinuity at he boundaries

variable a. Since it exhibits a discontinuity at the boundaries, w ( n )= 0.5


184/199


I 1.25

(4 @)Fig. 36. (a ) Poismewindow. (b) Logmagnitude of transform (u = 4.0).

(4 (b)Fig. 37. (a) Hllming-Pobon window. (b) Log-magnitude of transform (u = 0.5).

(8) (b)Fig.38. (a) Hanning-Poisson window. (b) Log-magnitude of transform (a = 1.0).

derivative at he on& and is I/&. Noticeas a increases,forcing more of the exponential into he Hanningwindow,the zeros of the sidelobe structure disappear and he lobesmerge into heasymptote. This window isshown in Figs.37-39 for values of a equal to 0.5, 1.O, and 2.0, respectively.Again note thevery large main-lobe width.8) Cauchy (Abel , Poisson) Window ( 1 S J : The Cauchy win-

dow is a family parameterized on a and defined by1w ( n ) = N2 , O < l n l < - .2 (42)1.0+ [$

Cauchy window is a two-sided exponential (see Poissodows),whichwhenpresented on a og-magnitude caessentially an isosceles riangle. This causes the winexhibit a very wide main obe and to have a large ENBWG. Gaussian or Weiersfrass Window ( I S ]

Windows are smooth positive functions with tall thconcentrated) Fourier transforms. From the genuncertainty principle,we know we cannot simultaconcentrateboth a signal and its Fourier transform. measure of concentration is the mean-square time duand the mean-square bandwidth W , we know all fusatisfy the inequality of


185/199

70 PROCEEDINGS OF THE IEEE, VOL. 66 , NO. 1, JANUARY

T1 1.m 'fi O-I I ' "(4 (b)Fi.40. a) Chuchy window. (b) Log-magnitude of ransform (a = 3.0).

( 8 ) 9)Fii.41. (a) Cauchy window. (b) Log-magnitude of tr8nafom (a = 4.0).

with equality being achieved only for the Gaussian pulse [ 161. a Dirichlet kernel as indicated inThus the Gaussian pulse, characterized by minimum time-bandwidth product, is a reasonable candidate for a window.When we use the Gaussian pulse asa window we have to trun-a t e or discard the tails. By restricting the pulse to be f i t elength, the window no longer is minimum time-bandwidth.If the truncation point is beyond the threesigma point, theerror should be small, andhe window should be a good 2 Qapproximation to minimum time-bandwidth.

2 Q

Th e Gaussianwindow is defmed by (4This window is parameterized on a, he reciprocal of standard deviation, a measure of the width of itsFourtransform. Increased a will decreasewith the width of


186/199


1 . 2 5

1 l.m

I

c

8-1 0 7

( 4 (b)Fig. 42 . (a) Cauchy window. (b) Log-magnitude of transform (a = 5.0) .

TT 1.25

0d l

-m

(a) (b)Fig. 43 . (a) Gaussian window. (b ) Log-magnitude of transform (a = 2.5).

T1 1. 25

90(a) (b)

Fig. 44 . (a) Gaussianwiudow. (b) Log-magnitude of transform (a = 3.0).

main lobe and decreased idelobe evels. ~ The window _i spresented in Figs. 43 , 44, and 45 for values of a equal to 2 .5 ,3.0, and 3 .5 , respectively. Note the rapid drop-off rate ofsidelobe level in the exchange of sidelobe level for main-lobewidth. The figures of merit for his window are listed nTable I .H. Dolph-Chebyshev Window [ I 7J

Following the reasoning of the previoussection, we seek awindowwhich, for a known finiteduration, n some senseexhibits a narrow bandwidth. We now take a ead from the

aperture to achievea n m a w main-lobe-beam pattersimultaneouslyestrictingidelobeesponse. (The adesignercallshisweighting procedure shading.) Theform solution to the minimummain-lobe width for sidelobeevel is the Dolph-ChebyshevwindowshadinThe continuous solution to the problem exhibits impthe boundarieswhich restr icts ontinuous realizatiapproximations (the Taylor approximation). The dissampled window is not so restricted, and the soht ionimplemented exactly.

The relation T,(X) cos (ne) describes a mapping b


187/199

72 PROCEEDINGS OF THE IEEE, VOL. 66 , NO. 1 , JANUARY

II 1 1 51.m

Fe. 5. (a) Gaussisnwindow. (b)Log-rmgnitudeofM m a = 3.5).

I 125

i 5

m 0-

-.(4 @)Fw. 6. (a) Dolph-Chebyslm window. (b) Lmg-magrdude of transform (a = 2.5).

T -20-40

0-* 0 .(a) @)

Fi. 7. (a) Dolph-Chebylev window. (b)Log-magnitudeof transform (a = 3.0).

window is defined with this mapping in the followingequa- andtion, in terms of uniformly spaced samples of the window'sFourier transform, 1" - tan-' [ X / ~ E K F I , IX G 1 Ocos-1(X)

co s [N os-1 [o cos (*;)]IW(k) (- 1)k cosh [N osh-I @)I ' To obtain the corresponding window time samples w ( n )simply perform a DFT on the samples W ( k ) and then s

for unity peak amplitude. The parameter Q represents the(4s) of the ratio of main-lobe level to sidelobe level. Thus a vwhere a equal to 3.0 represents sidelobes 3.0 decadesownromthe main lobe, or sidelobes 60.0 dB below the main lobe.

I'IV-


188/199


(a) (b)Fig. 48. (a) Dolph-Chebyshevwindow. (b)Log-mn.gnitude of ransform (u = 3.5).

Fig. 9. (a) Dolph-Chebyshevwindow. (b) Log-magnitadeof M o m u = 4.0).

presented in Figs. 46-49 for valuesof a equal to 2.5, 3.0,3.5, and 4.0, respectively. Note the uniformity of the sidelobestructure; almost sinusoidal! It is this uniform oscillationwhich is responsible for he impulses in the window.I . Kaiser-Bessel Window [ l 8 ]

Let us examine for a moment the optimality criteria of thelast two sections. In Section V-Gwe sought he unctionwith minimum time-bandwidth product. We know this to bethe Gaussian. In Section V-Hwe sought the unctionwithrestricted time uration, whichminimized the main-lobewidth for a given sidelobe level. We now consider a similarproblem. For a restricted energy, determine the function ofrestricted time duration T which maximizes the energy in thebandof frequencies W. Slepian, Pollak,andLandau [ 191 ,[ 2 0 ] have determined this function as a family parameterizedover the time-bandwidth product, the prolate-spheroidal wavefunctions of order zero. Kaiser has discovered a simpleap-proximation to these functions in terms of the zero-ordermodified Bessel function of the fiist kind. The Kaiser-Besselwindow is defined by

The parameter nu is half of the timebandwidth produtransform is approximately t h a t ofN IdSP - ( N 6 / i l 21.w ( e )G-o(m) Ja2n2 - ( ~ e / 2 ) 2This window is presented in Figs. 50-53 for values ofto 2.0, 2.5, 3.0, and 3.5, respectively. Note the rabetween sidelobe level and main-lobe width.

J. hrc i lon-Temes Window [21]We now examine the last criterion of optimality for

dow. We have already described the Slepian, PollLandau criterion. Subject to the constraints of fmedand f ved duration, determine the function which mathe energy in the band of frequencies W. A related crsubject to the constraints of f iied area and fmed durato determine the function whichminimizes the enethe weighted energy) outside the band of frequenciesis a reasonable criterion since we recognize that the traof a good window should minimize the energy it gathefrequencies removed from its center frequency. Till nhave been responding to ths goal by maximizing the tration of the transform at its main lobe.A closed-form solution of the unweighted minimumcriterion has not been found. A solution defined as ansion of prolate-spheroidal wave functions does exist aof the form shown in


189/199

14 PROCEEDINGS OF THE IEEE, VOL. 66 , NO. 1, JANUARY

Tt

-1 20 -1 5 -10 -5 0

1 3 5

1.00

Fig.50. (a) KPirer-Jhsd window. (b) Logmagnitude of transform (u = 2.0).

.+. 1.m

(a) @)Fig. 1 . (a) Kaiser-Bessel window. (b) Logmagnitude of ransform (a = 2.5).

T

(a)Fig. 52.

Tt

II! !l

L!

(b1(a ) Kaiser-Bessel window. (b ) Log-magnitude of transform (a = 3.0).

1.25

1.00


190/199


T*

11 2 5

1.m

Fa. 4. (a) B.r&n-Tma window. (b) Log-magnhde of transform (u = 3.0).

II 1.25lm

(a) (b)Fa. 5. (a) Barcilon-Temes window. (b) Log-magnitude of tr8asform (u = 3.5).

Ti . 56. (a) Barcilon-Temes window. (b) Lag-nugnhdeof transform (u = 4.0).

Here the A,,, is the eigenvalue corresponding to th e associated Like the Dolph4%ebyshev window, the Fourier transprolate-spheroidal wave function I$42Jx, y ) , and the tra is more easily defined, and the window timesamplesthe selected half time-bandwidth product. The summation tained by an inverse DFT and an appropriate scale factconverges quite rapidly, and is often approximated by the first transform samplesare defined byterm or by the first two terms. The first term happens to bethe olution of the Slepian, Pollak, and Landau problem,which we have already examined as the Kaiser-Bessel window. A cos y(k)l +B r$ sin ~ ( k )criterion,resented in the following equation has been found [ C + A B l [p?]' + 1-01by Barcilon and Temes:

A closed-form solution of a weighted minimum-energy W(k) (- 1)k

P


191/199

16 PROCEEDINGS OF THE IEEE, VOL. 6, NO . 1 , JANUARY

C = cosh-' (1OQ)@ =osh [ i c ]

y(k)=Ncos-' [ B cos ( 4 1(See also (45).) This window is presented in Figs. 54-56 forvalues of equal to 3.0, 3.5, and 4.0, respectively. The main-lobe structure is practically indistinguishable from the Kaiser-Bessel main-lobe. The f i e s f merit listed on Table I uggestthat for the same idelobe evel, this window does ndeedreside between the Kaiser-Bessel and he Dolph-Chebyshevwindows. It is interesting to examine Fig. 12 and note whereths window is locatedwith respect to he Kaiier-Besselwindow; striking similarity in performance!

VI. HARMONIC ANALYSISWe now describe a simple experiment which dramatically

demonstrates the influence a window exerts on the detectionof a weak spectral line in the presence of a strong nearby ine.If two spectral lines reside in DFT bins, the rectangle windowallows each to be identified with no interaction. ,To demon-strate this, consider the signal composed of two frequencies10 f , / N and 16 f , / N (corresponding to the enth and thesixteenth DFT bins) and of amplitudes 1.0 and 0.01 (40.0 dBseparation), respectively. The power spectrum of this signalobtained by a DFT is shown in Fig. 57 as a linear interpola-tion between the DFT outpu t points.

We now modify the signal slightly so that the larger signalresides midway between two DFT bins; in particular, at 10.5f s / N . The smaller signal s t i l l resides in the sixteenth bin. Thepower spectrum of this signal is shown in Fig. 58. We notethat the sidelobe structure of the larger signal has completelyswamped the main lobe of the smaller signal. In fact, we know(see Fig. 13) that the sidelobe amplitude of the rectangle win-dow at 5.5 bins from the center is only 25 dB down from thepeak. Thus the secondsignal ( 5 . 5 bins away) could not bedetected because it was more than 26 dB down, and hence,hidden by the sidelobe. (The 26 dB comes from the 25dBsidelobe level minus the 3.9dB processing loss of the windowplus3.0dB for a high confidence detection.) We also notethe obvious asymmetry around the main lobe centered at 10.5bins. This is due to he coherentaddition of the sidelobestructures of the pair of kernels located at the plus and minus10.5 bin positions. We are observing the self-leakage betweenthe positive and the negative frequencies. Fig. 59 is the powerspectrum of the signal pair, modified so that thearge-amplitudesignal esides at the 10.25-bin position. Note the change inasymmetry of the main-lobe and the reduction in the sidelobelevel. We still can not observe the second signal located atbin position 16.0.

We now apply different windows to the two-tone signal todemonstrate the difference in second-tone detectability. Forsome of the windows, the poorer resolution occurs when thelarge signal is at 10.0 bins rather than at 10.5 bins. We willalways present the window with the large signal at the loca-tion corresponding to worst-case resolution.

The first window we apply is the triangle window (see Fig.60). The sidelobeshave allenby a factor of two over therectangle windows' lobes (e.g., the -35d B level has fallen to

+-Q 1

T-e a c

!-e a +I1

o ~ o z u ~ e a ~ e o m mI 1 I kFii. 56 . Rectangle window.

I-60 *Fig. 5 9 . Rectangle window.

O? il- m 1 I \

Fii. 60 . Triangle window.detectable. If there were any noise in the signal, the sectone would probably not have been detected.

Thenext windows we apply are the cosa(x) family. the cosine lobe, a = 1 O, shown in Fig. 6 1 we observe a pcancellation in the sidelobe of the large signal located atsmall signal position. This cannot be considered a detectWe alsosee the spectral leakage of the main lobe over


192/199

HARRIS: USE OF WINDOWS FOR HARMONIC ANALYSISAmd.1.mnot

Fig. 61. Cos (nn/N)window.

- m

-40

-60

presented in

Fig. 63 . Cos ( n f f / N ) indow.

Fig. 62. We detect the second signal and observea 3.0-dB null between the two lobes. This is still a marginaldetection. For the cos3(x) window presented in Fig. 63, wedetect the second signal and observe a 9.0dB null betweenthe lobes. We also see the improved sidelobe esponse. Finallyfor the cos4(x) window presented in Fig. 64, we detect thesecondsignalandobserve a 7.0dB null between the lobes.Here we witness the reducedreturn for he rade betweensidelobeevel andmain-lobewidth. In obtainingurtherreduction in sidelobe level we have caused the increased main-lobe width o encroach upon he secondsignal.

We next apply the Hamming window and present the resultin Fig. 65. Here we observe the second signal some 35 dBdown,approximately 3.0dBover the sidelobe responseofthe large signal. Here, too, we observe the phase cancellationand the leakage between the positive and the negative fre-quencycomponents. Signals more han 50 dB down wouldnot be detected in the presence of the larger signal.

The Blackman window is applied next andwe see the resultsin Fig. 66. The presenceof the smaller amplitude kernel is

I n

Fig. 64. Cos (nn/N)window.

jot l i* I/ I

Fig. 66. Blockman window.

Fig. 67. Exact Blackmanwindow.

the sidelobe structure of that kernel. Note the rapid falloff of the sidelobe leakage has confined the artifacsmal l portion of the spectral line.We next apply the exact Blackman coefficients and w

the results in Fig. 67. Again the second signal is well dwith a 24 dB null between the two kernels. The si


193/199

PROCEEDINGS OF THE IEEE, VOL. 66 , NO.1, JANUARY

= - I 1

6OdB down relative to he peak. There is another. smallartifact at 5OdB down on the low frequency side of the largekernel. This s definitely a single sidelobe of the l argekernel.This artifact is essentially removed by the minimum 3-termBlackman-Harris window which we see in F i i 68. The nullbetween the two signal main lobes is slightly smaller, at ap-proximately 20 dB.

Next the 4-term Blackman-Harris window is applied to thesignal and we see the results in Fig. 69. The sidelobe struc-tures ar e more than 7OdB down and as such are not obserrredon this scale. The two signal lobes are well defined withapproximately a 19 dB null between them. Now we apply the4-sample Kaiser-Bessel window to the signal and see the re-sults in Fig. 70. We have essentially the same performance aswith the 4-term Blaclanan-Harris window. The only obser-vable difference on this scale is the small sidelobeartifact68 dB down on the low frequency side of the l nrge kernel.This group of Blackmanderived windows perform admiraMywen for their simplicity.

Fs. 1. Riesz window.

Fa. 2. Riemmn window.

- B

4

9

to the phasekernel.

Fa. 3. de t V&Poussin window.cancellation of a sidelobe in the large s i g

The result of a Riemann window is presented in FigHere, too, we have no detection of th e second signal. Whave a small null due to phase cancellation at the,secondnal. We also have a large sidelobe response.Thenext window, thede la VaIli-Poussin or heconvolved triangle, is shown in Fig. 73. The second signeasily found and the power spectrum exhibits a 16.0-dBAn artifact of the window (its lower sidelobe) showshowever, at the BlhDFT bin as a signal approximatelydB down. See Fig. 29.

The result of applying the Tukey family of windowpresented in F i . 74-76. In Fig. 74 (the 25-percent tawe see the lack of second-signal detection due o thehighlobe structure of the dominant rectangle window. In Fig(the 50-percent taper) we obseme a lack of secondddetection, with the second signal actually filling in one onulls of the hfft signals' kernel. In Fig. 76 (the 76-pe


194/199


I \i \ AO I h i A & l ~ W i i 9 b ~ ~Fig. 74. Tukey (25-percent cosine taper) window.

Fig.75. Tukey (50-percent cosine taper) window.

-20

-a

-e3

Fig. 76. Tukey (75-percent cosine taper) window.

The Bohman construction window is applied and presentedin Fig. 77. The second signal has been detected and the nullbetween the two lobes is approximately 6.0 dB. This is no tbad, but we can s t i l l do better. Note where the Bohman win-dow resides in Fig. 12.

The result of applying the Poisson-window family is pre-

o i m ~ a i i m i i, ,Fig. 78. Poisson window (a = 2.0).

o ~ o ~ m a s ~ m, , , * , , . -

Fig. 79. Poisson window (a = 3.0).

-20

-40

- M

O l b ~ ~ & i W L i QFa. 0. Poisson window (a = 4.0).

levelsof the largersignal. We anticipated hispoor mance in Table I by the large difference between the and the ENBW.

The result of applying the Hanning-Poisson family dows is presented in Figs. 81-83. Here, too, the secondis either not detected in the presenceof thehighsidstructure or thedetection is bewildered by the artifacts.The Cauchy-familywindowshavebeen applied aresults are presented in Figs. 84-86. Here to o we haveof satisfactory detection of the second signaland thsidelobe response. This was predicted by the large difbetween the 3.0dB and the equivalent noise bandwilisted in Table I.

We now apply the Gaussian family of windows and the results in Figs. 87-89. The second signal is detectethree figures. We note as we further depress the sstructure to enhance second-signal detection, the null dto approximately 16.0 dB and hen becomes poorermain-lobe width increasesand starts to overlap the lthe smaller signal.


195/199

PROCEEDINGS OF THE EEE. VOL. 66, NO. 1. JANUARY

-m -

b l & 2 b 2 & Q X X ? b h i GF= 85. Caucby window (u = 4.0).

Ib l & a b ; P ; D ; D d 7 A b S i r EFii.86. Csuchy window (a = 5.0) .- . 10 5 1.00S l l r Z lU.0 QO1F F T M m d .

-60 -

-zD n-a

-a

ng. 4. Cauchy window (a = 3.0). Fii. 88. Gaussian window (a = 3.0).in a l l cases, but it is distressing to see the uniformly high side-lobe structure. Here, we again see the coherent addition ofthe sidelobes from the positive and negative frequency kernels.Notice that the smaller signal is not 4OdB down now. Whatwe are seeing is the scalloping loss of the large signals' main-lobe being sampled off of the peak and beiug refereaced aszero dB. Figs. 90 and 91 demonstrate the sensitivity of the

Note the difference in phase cancellation near the base oflarge signal. Fig. 93, the 7MB-sidelobe window, exhibits18-dB null between the two main lobes but the sidelobes hadded constructivdy (along with the scalloping loss) to-62.O-dB level. In Fig. 94, we see the 80-dB sidelobe winexhibited sidelobes below the 70-dB level and stillmanagehold the null between the two lobes t o approximatley 1


196/199

HARRIS: USE O F WINDOWS FOR HARMONIC ANALYSIS

Fig. 89 . Gaussian window (a = 3.5). Fig. 93 . Dolph-Chebyshev window (a = 3.5).

Fig. 90. Dolph-Chebyshev window (a = 2.5). Fig. 94. Dolph-Chebyshev window (a = 4.0).

s,pn* I 10.5 I mS # p n * 2. 16.0 0.01FFT Sin Am w

Fig. 91. Dolphzhebyshevwindow (a = 2.5). Fig. 95 . Kaiser-Bessel window (a = 2.0).

-eo i

Siqd 1. 10.5 1 mF F T B m Am#,Siqd 2 16.0 0.01

Fig. 92. Dolph-Chebyshev window (a = 3.0). Fig. 96. Kaiser-Bessel window (a = 2.5).

too, we have strong second-signal detection. Again, we see theeffectof trading increasedmain-lobe width for decreasedsidelobe level. The null between the two obes reaches a maxi-mum of 22.0 dB as the sidelobe structure falls and then be-comes poorer with further sidelobe level improvement. Notethat this window can maintain a 20.0-dB null between the twos i g n a l lobes and still hold the leakage to more than 70 dBdown over the entire spectrum.

There are slight sidelobe artifacts. The window can maa 20.0dB null between the two signal lobes. The perfoof this window is slightlyshy of that of the Kaisewindow, but the two are emarkably similar.

VII. CONCLUSIONSWe have examined some classic windows and some w


197/199

82 PROCEEDINGS OF TH E IEEE, VOL. 66, NO. , J ANUARY

-m

-40

-8u

s+m 1. 1 0 5 1.mjilru 2. 16.0 0.01F F I B i n W .

1o l b ~ i a D & & S b o & a mFig. 97. Kaiser-We1 window (a = 3.0).

, + k

Fig. 98 . Kaiser-Bessel window (a = 3.5).

' ;I , I l ' !O : O M X 1 ~ 5 0 6 0 7 O % O 9 0 l W, k

Fig. 99. Barcilon-Temes window (a = 3.0).

monic analysis of tones in broadband noise and of tones inthe presence of other tones. We have observed that when theDFT is usedasa harmonic energy detector, he worstcaseprocessingoss due to he windows ppears to be lowerbounded by 3.0 dB and (for good windows) upper boundednear 3.75dB.This uggests that he choice of particularwindows has very little effect on worst case performance inDFT energy detection. We have concluded that a good perfor-mance indicator for the window is the difference between theequivalentnoise bandwidth and the3.0dB bandwidth nor-malizedby the 3.0-dB bandwidth. The windowswhichper-form well (as indicated in Fig. 12) exhibit values for hisratio between4.0 and 5.5percent. The range of this atiofor the windows listed in Table I is 3.2 to 22.9 percent.For multiple-tone detection via the DFT, the windowemployed does have a considerable effect. Maximum dynamicrange of multitone detection requires the transform of thewindow to exhibit ahighly concentrated central obe withvery-lowidelobe structure. We have demonstrated thatmany classicwindowsatisfy thisriterion with varying

- m

-a

-60 -1 , , , , , I , , , , tO ~ O ~ ~ ~ ~ ~ ~ O

Fig. 100. Bardon-Temes window (a = 3.5).

- I t

I, 10 Ib in i o lo k & J 70 i A l%

Fig. 101 . Barcilon-Temes window (a = 4.0).

degrees of success and some not at all. We have demonstrtheoptimal windows Kaiser-Bessel,Dolph-Chebyshev, Barcilon-Temes) and the Blackman-Hamswindows perfbest indetection of nearby tones of significantly diffeamplitudes. Also for the same dynamic range, the three omalwindows and the Blackman-Harriswindow are rouequivalent with the Kaiser-Bessel and the Blackman-Hademonstrating minor performance advantages over the othWe note that while the Dolph-Chebyshev window appearsbe the best window by virtue of its relative position in Figthe coherent addition of its constant-levelsidelobes detrfrom its performance in multi tone detection. Also the slobe structure of the Dolph-Chebyshevwindow exhextreme sensitivity to coefficienterrors. This would aits performance in machines operating with fixed-point ametic. This suggests that the Kaiser-Bessel or the BlackmHarris window be declared the top performer. My preferis the Kaiser-Bessel window. Among other reasons, the cficients are easy to generate and the trade-off of sidelevel as a function of time-bandwidth product is fairly simFor many applications, the author would recommend thsamplelackman-Hams (orhe 4-sampleKaiser-Bessel)window. These have the distinction of being defined by aeasily generated coefficients and of being able to be applas a spectral convolution after the DFT.

We have called attention to a persistent 'error in the apption of windowswhenperforming convolution in he quency domain, i.e., the omission of the alternating signthe window sample spectrum to account for the shifted torigin. We havelso identified and clarified sourceconfusion concerning the evenness of windows under the DFinally, we comment that all of the conclu

Date post:	14-Apr-2018
Category:	Documents
Upload:	bubo28
View:	214 times
Download:	0 times

The-FFT-Fundamentals-and-Concepts.pdf

Documents