The Use of Windows for f Harris

7/29/2019 The Use of Windows for f Harris

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On then Use of Windows for Harmonic Analysiswith the Discrete Fourier Transform

FREDRIC J. HARRIS, MEXBER, IEEE

HERE IS MU CH signal processing devoted to detect ionand estimation. Detection is the task of detetmitdng ifa specific signal set is pteaettt in an obs&tion, whflcestimation is the task of obtaining the va.iues of the parameters

derriblng the signal. Often the s@tal is complicated or iscorrupted by interfeting signals or noise To facil i tate thedetection and estimation of signal sets . the obsenation isdecomposed by a basis set which spans the s ignal space [ 1)For ma ny problems of engineering interest, the class of aigttl lsbeing sought are periodic which leads quite natuallv to adecomp osition by a basis consistittg of simple petiodic fun=-tions, the sines and cosines. The classic Fourier tran.,f ot,,, hthe mechanism by which we M able to perform this decom-posttmn.

BY necessi ty , every observed s ignal we pm- must be offinite extent. The extent may be adjustable and Axtable.but it mu st be fire. Proces%ng a fi i te-duration observation~POSCS mteresting and interacting considentior,s on the ha-momc analysic rhese consldentions include detectabil i tyof tones in the Presence of nearby strong tones, rcoo habil ityof similarstrength nearby tones, tesolvabil i ty of Gxifting tona,and biases in estimating the parameten of my of the alon-menhoned signals.

For pract icali ty , the data we p- are N uni fomdy spacedsamples o f the obsetvcd s ignal. For convenience. N is highJycomposi te, and we wi l l zwtme N is evett . The harmott icestm~afes we obtain Utmug Jt the discrae Fowie~ tmnsfotm(DFT) arc N mifcwm ly spaced samples of the asaciatedperiodic spectra . This approach in elegant and attnctivewhen the proce~ scheme is cast as a spectral decomposi t ionin an N-dimensional orthogonal vector space 121. U nfottu-nately, in mm Y practical situations, to obtain me aningfulresul ts this elegance must be compmm ised. One such

t=O,l;..,N- l.N.N+l.

We cherve that by defining a buds set ovet an ordered int, we am defining the rpcc tmm over a l ine (called the quen=Y Uu) from W hich we dnW the concepts of bandindmd of frwuencia c lose to and fat from a given frequen(rhich is related to l-e%htion).For nmpled s@als. the basix set spatming the interval ofscxxm dr h identid with the scqucn ces obtained by unifosatnpka of the wmspon ding wntinuous spanning Set uthe index N /2.

k=O,,.-.-.N/2

n=O.,;--.N-

We note hen that the trigonometric functions are unique that uniformly spxed samples (ovet an integer number periods) fotm ortlt~ottal sequen ces. Arbitrary onhogonfunctions , zimiluly sampied , do not form orthogonal quences. We also note that aa interval of length NT seco1s not the same as the inteml coveted by N samples separaby intervals of T secon ds. This is easily understood when

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real i.ze that the interval oveq which the samples are t*cn isclosed on the left and is open on the right (i.e., I-)). Fig. 1demon stntes this by sampl ing a function which k even abouti ts midpoint and of duration hT sccondr.Since the DFT e ssentiaUy co nsiders squences to be per iodic,we can consider the mirnng end point to be the beming ofthe next per iod of the per iodic extension of this sequence. Infact, under the per iodic extensiott, the next sam ple (at 16 sinFi i . 1.) is indM&gGhable from the sample at zero seconds.This apparent lack of symm etry due to the missing (butimpl ied) end point is P +ource of umfusion in sampled windowde&n. Thk can be traced to the ear ly work related to con-vergence factors for the part ial sum s of the Four ier ser ies. Thepart ial sum s (or the Gnite Four ier transform) always includean odd number of paints and exhibi t even sym metry aboutthe or igin. Hence much of the l i tekrc and many softwarel ibr.uia incnrponte windows de-signed with true even sym-metry rather than the impl ied symm etry with the missing endpoint!We must remember for DFT prwe&ng of sampled data thateven aym mctry means that the projection upon the sampledsine sequences is identically zero; i t doa not mean a matchingleft and r ight data point about the midpoint. To dist inguishthi3 symm etry from cawentiond ~emten we wiU refer to i tas DFT-even ( i .e., a cmwen tio~I even sequence with the r ight-end point removed). Another example of DFT-even sym-metry is presented in Fi i. 2 aa samples of a per iodicaIlyextended tr iangle wave.

If WC evaluate a DFT-even sequence via a f ini te Four iertransform (by treating the +N/Z point as a zero-value point) ,the resul tant continuous per iodic function exhibi ts a non zeroimaginary compon ent. The DFT o f the same wquenre is a setof samples of the t i i te Four ier transform, yet thex samplesexhibi t an imaginary component equal to ICTO . Wh y the dis-par i ty? We mlut remember that the mixing end point underthe DFT symm etry contr ibutes an imaginary smusoidalcomponen t of per iod Zn/(N/Z) to the f ini te transform(corresponding to the odd componen t at sequence posi t ionN/2). The sampl ing posi t ions of the DFT are at the mult iplesof 277/N. which, of course. correspond to the zeros of theim&m , sinuoidal compone nt. An example of this for-tu~tou sampl ing is show in Fig. 3. Notice the sequencei(

scqcnee.is decomposed into i ts even and odd parts. wi th the odd partsupplying the imaginary sine component in the ft i tetransform.

I I I . SPECTRAL LEAKAGEThe selection of a ft i te- t ime interval of NT seconds and ofthe orthogonal tr igonometr ic basis (continuous or sampled)

over this interval leads to an interesting pectiti y of thespectral expansion. From the continuum of possible fre-quencies. only those which coincide with the basis wi l l protectonto a single basis vecto r; all other frequen cies will exhibttnon zero projections on the enti re basis set. This 1s oftenreferred to as spectral leakage and is the resul t of processingft i te-duration records. Al though the amount of leakage 1sinfluenced by the sampl ing per iod, leakage Is not caused bythe sampl ing.

An intui tive approach to leakage is the undentandmg thatsimals with frequencies other than those of the basis set arenot per iodic in the observation window. The petiodic exfen-sion of a signal not comme nsurate with i ts natural per iodexhibi ts discontiwi t ies at the boundaCes of the observatmn.The discontinu ities are responsiblr for spectra l contribution s(61 leakage) over the entire basis xl The fo~rn) ni ihis ducontmu ity are demonstrated in Fig. 1.

Windows are weighting functions appl ied to data to reducethe spectral leakage associated with fmite observation inter-vals From one viewpoint, the window is appl ied to data(a.5 a multiplicative weighting) to reduce the order of the di.-contlnurty at the boundary of the per iodic extension. This isaccompl ished by matching as many orders of der ivative (ofthe we&red data) as possible at the boundary. The easiestway to achieve this match mg is by sett ing the value of thesedenvarwes 10 zero or near to zero. Thus windowed data aresmoothly brought to zero at the boundar ies so that theper iodic extension of the data is continuous in many ordersof der ivative.


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HARRIS: SE OF WlNDOWS FOR HARMONIC ANALY SIS

where+=Zk.Nl nndk=O,l ; .~,N- I

From another view point, the window is mukiplioativtdyapplied to the basis set so that a signal of arbitrary frequencywill exhibit a ugnificanf projection only oo those basis rectorshaving a frequency close to the signal frequen cy. Of causeboth viewpo ints lead to identical results. We can w insightinto window design by occasionaUy swi tching between theseVi.Wp0inti.

IV . WINDOWS ANND FIGURFS OF MERITWindo ws are wed in harmonic pnalysk to reduce the unde-sirable effec ts related to spectnl le. Windo ws implct on

man y attributes of a harmonic prooeso r; these indude detec-tabil i ty, resolution, dynam ic ran ge. confidence, and ease ofimplemen tation. We would l ike to identify the major parun-eten that wil l al low perfcrma nce compa risons between dil-fennt windows. We can best ident i fy the= pawneten byexamining the effec ts on harmonic analysis of a window .

An essentially ba ndlimited signal f(f) with Fourier transformF(w) can be described by the uniformly sampled data set/(nT). This data set defme s the periodically extended rpec-trum FT( m) by its Fourier series expansion as iden- as

+-F(w) = I_ _ f(r) exp (-jut) df

+mF(w) = 1 f(nn exp (-iwIn=-_

+-ITf(f) = I FT(w) exp (+;a) dwl2n-?f,T

(3s)

(3b)

(3.9

and where IF( = 0, Iwllfl2n/nFT(W) = F(U), IWI c + IZnlTl .

For (real-world) machine proces s&, the data mo st be offinite e xtent, and the rumm rtion of (3b) can only by per-formed as a fmte approximation as indicated a;i

+N 2F,(w)= i /(nneexP(-iwnn , Neven (41)=-N/2

(NP-1Fb(W)= z ((n77ecxp(-jwn77 , Neven (4b)n = -N/2(NI2l-L

F,(w,) = x /(nT) exp (-;w~n77, Never, (4~1=-N,l.v 1Fdwc) = z J(nT) exp (-;w*nT). N even (4d)n=o

We recognize (4s) ~1 the finite Fourin tnnafonn, , sumticm addrcratd for the corwenience of i ts even symm eE.qution (4b) is the finite Fourier transform with the xend Point d ckted. and (4~) is the DF T ymplin# of (Of c&l- for sctui proaaing, PC desire (for count@ paa in a4withms) that the index start . t zero. We accplkb this by sbifk the start& point of the data N/2 tions. ,zhaw iw (4~) to(dd). Equation (4d) is the forward Dl ie N/2 s hift will affect on ly the phue an&s of the tnform. a for th? amvw.dence of symm etry we wi l l addresswindow 11 being centered St the origin. We alao identify convenkaa u rmjor source of window misappliuhon. shi f t of N/2 pc4nia and i ts dtmt phase shi f t is often olooked or h impm pCa y ham&d in the definition of wiodor when usd wi th the DFT . T his is put intkrly so wthe window@ is performed as. s~echal convolution. Seedismion on the Fbmin# window under the cm= windows.The question nor Poled is. to what sxte nt is the fisumrm tion of (4b) a manin@ approxinutim of the in6nsumm atim of (3b)? In facts we addrcr the quest ion fomore encnl case of m arbitnry window applird to the tfunction (or &ried) a.5 plxxntcd in

F,(u) = i7 ww 9mn exp (-iwnna=--where

an dw(nT)=O. N even

w(n7-J = w(-nn,L.et us *ov e umine the effects of the window on

Jpcctnl estima te. Equlioa (5) shhoan that the transfomF,(w) t i the transform of . product. As indicated Ur fouowing equation, ttds is eqtivaknt to the con*olution the two comxponding tnnsforms (see Appendix):

F,(w) = /- F(x) W(w x)dx/2n_m

F,(w) = F(w) l W(w)

Equation (6) is the key to the effec ts of procoed& rtitextent data. The equation an he interpreted in two cquilent waya , ahic,, wi be more c,,i,y vinulired with the of an cxampk. The example we choose U the samprectanzle wiodowa; ~(7 ) = 1.0. We know W(w) is h&let kernel (4 I presec.ted as

NU T sinW(w) =exp ( 1

u.TUT1

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excep t for the l inear phase shift term (which wil l change dueto the N,2 point shift for realizabili ty), a single period of thetransform has the form indicated in Fig. 5. The observat ionmm m,~ (6) is that the value of F,(w) at a part icular W,say w - weI is the sum of al i of the spectral contribut ions ateach w wcigbted by the window centered at we and measuredat w (see Fig. 6).A. Equivalent Noise Bandwidth

From Fi i . 6, we observe @at the ampl i tude of the harmonicest imate at P given frzqoency is biased by the accumulatedbroad-band noise included in the bandwidth of the windo w.In this ywe, the wiodoa behaves a9 a Nter, gathering conm-butiona for i ts est imate over i ts bandwidth. For the harmonicdetect ion problem, we de&e to minimize this accumulatednoise &oat, md we accompl ish this wi th smal l -bandwidthwindow. A cawenient measure of this baodwidth is theequivalent noise bandwidth (ENBW) of the window. This isthe width of a ncta,&e f t i ter wi th the same pcok power sainthat would accumti tc the same noise power (se-e Fi i . 7).

The xcumolated noise power of the window is dcfmed as

I+-IT

Noise Power = No IW(w)l dw/?n (8)-,Twhere No Is the noise power pet uni t bandwidth. Pamevastheorem al lows (8) to be computed by

Noixe Power = F 2 w(nT).nThe peak power gain of the window occurs at w = 0. the zerofrequency power gain. and is defined by

Peak Signal Gain * W(0) - x w(nT ) (lOa)nPcakPow erGaio- W(O)= c w(nT ) ~ (lob)[ 1

Tbur the ENBW (normal ized by N,/T , the noise power perbin) is given in the following equation and is tabulated for thewindo ws of this report in Table Iz w(nT)

2 (I I)

A coocept c losely t ied to ENBW is procesiog gaiz IPG)and processing loss (PL) of a windowed transform We canthink of the DFT as a bank of matched fdten, where each

Nter is matched to one of the complex s inusoidal sequencer ofthe basis set [ 31. From this perspect ive, ,ve can examme thePC (sometimes cal led the coherent gain) of the f i lter. and wecan examic the PL due to the window having reduced thedata to zero vplues near the boundaries. Let the i,,put sampledsequence be defmed by (12):

/(nT) = A CXP (+;wrnT) +&IT) I IZJwhere q(nT) is a whi te-noise sequence wi th variance 0:. Thenthe s ign.4 component of the windowed spectrum (the matche


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__-

-

-q-T-K-p7I I


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reduction in proportiona lity factor is important as It repre-nents a known bias on spectral amplitudes. Coherent powergain. the square of coherent gain. is occas ion& the parameterlisted in the literature. Coherent gain (the summation of (13))normalized by its maximum value N is listed in Table I.,-be incoherent component of the windowed transfon is,$ven by7%) jnc.iac =x w(nT)q(nT) exp (-ianT) (14a)n

and the incoherent power (the meansquarr value of this co,-panent where E { } is the expectation operator) is given byE{IF(wr)/ . , , I l = 2 x w(nT)w(mT)EIq(nT)q*(mr))n m

exp (-jw*nT) exp (+jwrmT)= 0; x w(nr). (I%)n

Notice the incoherent power gain is the sum of the squares ofthe window tertes, and the coherent power gain is the squareof the sum of the window terms.Finally, PS, which is dcfqxd as the ratio of output s&nal-tcmoise ratio to input signal-tcmoise ratio, is given.by

pG~S,,N, -A [? w(nr)]p & w(nT)Wi A/O:,.[ 1wcnn;:w(nr)n

Notice PG is the reciprocal of the normalized ENBW. Thuslarge ENBW suggesta a reduced processing gain. This is reason-able, tice an increased noise bandwidth permits additionalnoise to contribute to a spectral estimate.c. Over17pConelationWhen the fast Fourier transform (FFT) is used to processlong-time sequences a partition length N is fust selected toestabbsb the required spectral re-sc~lution of the enolysis.Spectral resolution of the FFT is defied in (16) where Af isthe resoltion, f, l s the sample frequency selected to latifythe Nyqut criterion . end 0 is the cafficient rcflecfing thebandwidth increuc due to the perticulu window selected.Note that [f,/N] is the minimum te.wlution of the FFf whichwe denote as the FFf bin width. The unfficicn t ,4 s usua llyselected to be the ENB W in bina u listed in Table IAf=fl ;0 (16)

If the window and the FFT are applied to nonovcr,appingpartitions of the sequence. as shhown in Fig . 8. a slgrur icantpart of the reties is ignored due to the windows exhibitingsmall values near the boundaks. Far instance. if the transformIS being used to detect rho,+duration tone-like signals . the nonoverlapped analysrs could rmss the event if it occurred neuthe boudancs. To avoid this loss of data. the transfons axusually applied to the overlapped panition sequences as shownin Fig. 8. The overlap LS lmost always 50 or 75 percent. Thxoverlap processing of course mcreases the work load to coverthe total sequence length, but the rewards warrant the extraeffort.

rY-: e-1Fig. 9. Relationship between indices on Overhppcd intervals.

A,, imp,,rtant question related to overlapped processing iswhat is the degree of comlation of the random componentsin successie transforms? This cnm1ation, as a function offractional overlap r. is deftned for a relatively flat noise spec-trum OYCThe window bandwidth by (17). Fig. 9 identifieshow the indices of (I 7) relate to the overlap of the intervals.The correlation coefficient

is computed and tabulated in Table 1. for each of the windowslisted for SO-and 75.percent overlap.Often in a spectral analysis. the squared magnitude of succes-sive ttansfotms are averaged to reduce the va,iance of the mea-su~ements (S] We know of cotse that when we average Kidentically disttibutcd independent measurements. the vati-awe of the average is related to the indiv idual vatiansr of themeasurements bykg. -__-diea K t 181

Now we can ask what i s the reduction in the variance when weavenge measuemenfs which are correlated as they are forovcrfappcd transforms? Welch 151 has supplied an answer tothis question which we present here, for the spec ial case of SO-and 75.percent overlapdn,. 1. =Kl%ea + ?c(O.S)] iz; lc(OS)],

50 percent overlap=;ll+2c (0.75)+ 2c(o.s)+ Zc(O.2S )l

$ lc(O.75)+ 2~(0.5)+3~(0.25j1.75 percent overlap. (19)

The negative terms in (19) are the edge effects of the averageand can be ignoted if the number of temu K is larger thanten. For good windows. ~ (0.25) Is small campared ,a 1.0.


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and can also be omitte d from (19) with negligible error. Forthis reason, ~(0.25) was not l isted in Table I. Note, that forgood w indows (see last paragraph ot Section IV-F), transformstaken with SO-percent over lap are essential ly independent.D. Scal loping Loss

An important consideration related to minimum detectablesignal is cal led scal lopiag lass or picket- fence effect. We haveconsidered the windowed DFT as a bank of matched ff i tenand have examined the procexing gain and the reduction ofthis gain ascr ibable to thb window for tones ma tched to thebasis vectols. The b&s vectors are tones with frequenciesequal to mult iples of f,/N (with f, being the sample fre-quency). These frequencies are sample points from thespectrum. and are nomm lly referred to as DFT wtpt pointsor as DFT bins. We now address the question, what is theadditional loss in procasing gain for a tone of frequen cy mid-way between two bin frequencies ( that is, at frequencies(k+ ll2)f,lWReturning to (13), w ith w* replaced by w(~+~,~), v/e deter-mie the processing gain for this hall-bin frequen cy shift asdefined inFkJ( I, I) ) Is igN! = A x w(n73 exp (-iw( ,, , ,nT),n

We ah de!ine the scal loping loss as the ratio of coherent gainfor a tone located hal f a bin from a DFT sample point to thecoherent gain for a tone located at a DFT sample point, asindicated in

scalloping Las =IT w(nn=w(-jin)( w(+F)!

1 w(n?- l = WJ)n(2Ob)

Scal loping loss represents the maximun reduction in PC dueto signal frequency. This loss has been computed for the win-dows of this report and has been included in Table I.

We now make an mterert ing observation. We define worstcase PL as the sun, of maximum sca,,oping loss of a windowand of PL due to that wmdow (both in decibel). This numberis the reduction of output s&m&to-noise ratio as a resul t ofwindowing and of wont case frequency location. This ofcourse Is related to the minimum detectable tone in broad-band n oise. It IS interesting to note that the wont case loss isahays between 3.0 and 4.3 dB. Windows with wont casePL exceeding 3.8 dB are very poor windows and should not

be used. Addit ional comm ents on poor windwvx wfound in Section N -C. We can wncludr from tL< comloss f i of Table I and from Fig. 1: that for the deteof single tones in broad-band noise. nearly any window than the rectangle) is as gcal LS any other. The di fferbetween the var ious windows is less than 1.0 dB and forwindows is Icar than 0.7 dB. The detection of t&s presence of other tones is, however, qui te another proHere the w indow dccs have P marked affect, as wi l l be destrated shcdy.F. Specml Leakwe Rrvidted

Returning to (6) and to Fig. 6, we observe the spmeaslnme nt is affected not only by the broadhand spectrum, but also by the narrow-band spatrun: rhic!~ within the bandwidth of the window. In fact, a given spcompanent say .t w - wg wil l contr ibute output (or wobserved) at mother frequency. say .t w = w. accordinthe gain of the window centered .t wo and measured aThis is the effect normrl ly referred to u spectral leakageIs demonstrated in Fi i . 10 with the transform of a f ini te t ion tone of frequency wgThis leakage cases P bias in the ampl i tude and the poof P harmonic estimate. Even for the case of a singleharmonic l ine (not at a DFT sa mple point) , the leakage the kernel on the negative-frequency axis biases the kernthe posi t ive-frequency br ie. This bias is mos t severe and bothersome for the detection of smal l s ignals in the preof nearby huge signals. To reduce the effects of this biawindow should exhibi t low-ampl i tude sidelobes far fromcentral main Lob e, and the transi tion to the low sideshould be very rapid. One indicator of how wel l a wisupp resses leakage h the peak sidelobe level (relative tmain lobe): another is the asym ptotic rate of fal loff of sidelobes . TIew indicators are listed in Table 1.

Fig. I1 suggests another cr i ter ion with which w%. shouconcerned in the window selection process. Since the wiimposes an effective bandwidth on the spectral l ine, we be interested in the minimum separation between two strength l ines such that for arbi trary spectral locations respective main lobes can be resolved. The classic cr i ter ionthis resolution is the width of the window at the hal f-ppoints ( the 3.0.dB bandwidth). This cr i ter ion reflects ththat two equalstrength main lobes separated in frequencyless than their 3.0.dB bandwidths wi l l exhibi t a single sppeak and wi l l not be resolved as two dist inct l ines. problem with this cr iter ion is that i t does not work fcoherent addi t ion we f ind in the DFT . The DF? opoints are the coherent addi t ion of the spectral co mponweighted through the window at a given frequency.


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I f two kernels M contr ibuting to the coherent summ ation,the sum at the crc6soer point bvxnidly hal f-way betweenthem) must be smal ler than the individual pea!~s i f the twopeaks arc to be resolved. Thus at the crossover points 01 fhckernels, the gain from each kernel mw t be less than 0 .5,01 thecrcssovcr points must occur beyond the 6.0-dB points of thewindoan. Table I l ists the 6.C-dB bandwidths of the var iOUwindow e xamined in this report. From the table, we see thatthe 6.0sdB bandwidth var ies from 1.2 bins to 2.6 bins, where abin is the fundamental frequency resolution w,/h . The3.0-dB bandwidth does have uti l ity as P pafomm ncc lni icatoras shown in the next paragraph. Remem ber however. i t is the6.0-dB bandwidth which defme s the resolution of the win.dowed DFT.Fro,,, Table I, we see that the noise handwidth alwaysexceeda the 3.0-dB bandwidth. The di fference between thetwo, mfercnced to the 3.0-dB bandwidth, appears to be asensi t ive indicator of overall window perfotmnnce. We haveobserved that for al l the wad windows on the table, thisindicator was found to be in the range of 4.0 to 5.5 percent.Tbae window8 for which this ratio is outside that rangeeither have L wide m ain lobe or a bigb sidelobe structure and,hena. arc chancter ized by hi& processing las.s or by poor

i;..two-tone de tection capabi l it ies. Those windows for whichthis mtio ia inside the 4.0 to 5.5perant range are found inthe lower lcfturraa of the ~erformvla comp arison ch art1; (Fe. 12). which is deaaibed next,.< Whi le Table I dces l ist the comm on performance param-

;; cten of the windova e xamined in this report, the mas s of: numben U not s~tenins. We do nti that the aidelobe :, kvd ( to mducs hhs) and the wont case procrtsiag Ices ( toxauindn dstsctabi&y) am probably the meat importantpaameten on the table. Fig. 12 shows the relative posi t ionof the t idorr u i hmtion of tbwc parameters. Windowsrcaidiug in the lower left comer of the fwe are the good-

~~perforudng windgaa . They exhibit low-sidelobe levels andlow aomt am proasdne loss. We urge. the reader to read

~seetioru VI and VU; Fii. 12 praents a lot of infonm tioo,but not the Ml Itmy .v . c -c WrNwws

We wiII nc.w utdw ame reU-kaown (and some not ~weU-know window% For each window we wi l l comm ent on theL jutifiatim for ita ua and identify Ita significant p-eten._ AU the t idowa ui iJ be wkd aa even (about the or i@n)aeqtrsncu with m odd muhbex of Q&&S. To convert the win-dew to DFFeve n. the t,&cnd point wi l l be discarded and

the seqcqucna dl be shIfted so that the left cod point coin-cidea with the origin. We uiUus s normalized coordinates with

defined asw(n)= 1.0, =O.I:~~,N- I C:lhI

The spectral window for the DFT window sequence 1s @*en in

Nln-(S) = exp -,((2 1. 1e,g -___ [ 1 I:,


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,in

that the f ini te sum is s imply the part ial sum of the series. and the spectral window corresponding to the DF T sequeoFrom this v iewpoint we can cast the quest ion in terms of the given inconvergen ce properties of the partial sum s of Fouier series.From this work we know the part ial sum is the least mean-square error approximation to the infmite sum

We observe that mean square convergence IS a convementanalyt ic concept, but i t is not attract ive for f inite est imates otior numencd approximations. Meansquare est imates tend to

w@~=~exp [- i (N;- 1) e]

orcdlate about thei r means. and do not exhibit umtorm con-vergenc e. (The approxima tion in a neighborhood of a point ofcont lnul ty may get worse i f more terms are added to thepamal rum.l We normally observe this behavior near pomtr ofdlrcantm uity as the figin& we call Gibbs phenomenon . It isth,s orc ibtory behavior we UC try ing to control by the se ofother wndows.

W(n) = 1 o Jr?! NN/2 n=--,.2 -l.o,l.--.~ (23a)and ~1 shown in Fig 14. The same wmdow for a DF T 1sdefined as

The transform of this window is seen to be the sqDirichlet kernel. Its main-lobe width (between zero crossix twice that of the nctan%er and the fu-st ndelobe leapproximately 26 dB down from the main-lobe peak. twxe that of the rectandcs. The s idelobes fal l off at per octw e, reflecting the discontinuity of the nndow resin the first derivative (rather tha n in the funcuon Itulf, triangle is the simple st window which exhihtts a nonneeatranrftx w 3%~ property can be retired by ronvo lvi,,.: wmda w (of hal f-extent) wi th i tx l f . The resul tant wi ,dotransform is the square of the o, i@nal windows tnnsfonn.

A wndo w sequence derived by self-conrolm a parent dew contains approximately twice the number of samplthe parent window, hence curesponds to . tnwwxne tnpolynomiaJ (i ts Z-transform) of approximately twi t-eorder. (Convol~ two recta,@es each of N/2 points result in a ttianglc of N + 1 points when the zero end pare counted.) The transform of the window wi l l now extwce ar many zeros as the parent transform (to accountthe increased order of the associate d trigonome tric nomial , . But how has the transform appl ied these extra availabie from the increased order polynomial? The


10/33

of the cosine function . These propeties are particularlyattractive under the DFT. The wmdo w for a f inite Four iertransform is defied as

convolved window simply places repented zeros at each IOCa-tion for which the parent transform had a zero. This, Ofcame, nor only seta the trash-m to zero zt those points, butalso ,ets the t int der ivative to zero at those points. I f theintent of the i,,creaed order of polynomial is to hold downthe sidelobe Ierslr, then doubling up on the zeros is a waste fultwtic . The additionnl zerca might better be placed betweenthe existing zeroa (near the Id pelts of the sidelobes) tohold down the sidelobea rather than at locations for whichthe tnnaform i, already equal to zero. In fact we will observein subsequent winders that very few good windows exhibi treprted roota.hckin.g up for a mom ent. i t is interesting to examine thetrim& window in term s of putul+un conrer~ncc ofFour ier win. Fejcz observed that the pamnl sums of Four ierseries were poor nume ricd approxima tions I8 I. F0lUiercoeff ic ients were ury to gwzrate however, and he questionedif some simple madit iut ion of coeff ic ients might lead to anew set wi th mom desirable conrerynce properdes Theoscillation of the partial sum , and the contraction of thoseos.ziUations i ts the order of the partiaJ su m increased. suggestedthat an .veng of the part iaJ surm would be a smootherfunction. Fi i . IS prernts an expulsion of twc. part ial sumsnear P diswntmuity. Notice the ..- of the two expansionsis smoo ther thm either. Contintxiru in this line of reasoning.an averwe expansion FN(B) mi#d be defined by

F~(e)=~IF~.,(e)+F~-*(e)+~+F,(e)l (24)where FM(O) is the M


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(b)(b) o*-magDi*de Of tr .rnfmln.


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ne ymp,ed Haming window can be wr i tten as the sum Ofthe sequences indicated in

-

w(n) = 0.5 + 0.3 cos [2n 1N J )

_.N ..,,2 - l ,O,l ; . . , ;- 1. (Z&a)

Exh sequence hu the easi ly recognized DFT indicated iniv(e)=osD(e)+0.2 5 D e-y +fl e+;( ) ( )I t2W

where

D(e)=exp(+it) f .sin Te[ 1We recognize the Dir ichIet kernel at the or&in as the transformof the constant 0.5 samples and the pair of translated kernelsas the tr tnsfonn of the single cycle of cosine samples. Notethat the translated kernels are located on the fuxt zeros of thecenter kernel, and arc half the size of the center kernel. Alsothe sideobcs of the translated kernel are about hal f the sizzand are of opposi te phase of the sidelobes of the Centralkernel . The summa tion of the three kernels sidelobes bcti inphse opposi t ion. tends to cancel the sidclobc structure. mClnceII ing summ ation is demonstrated in Fig. 20 which d epictsthe summ ation of the Dir ichlet kernels (wi thout the phase-shi ft term).The pattial cancel ling of the sidelobe structure su%g ssts 1cm,s tr ,,ct isc technique to defme new windows. The matwe&horn of these are the Hamm ing and the BLzckma nwindows which are presented in the next two sections.For the special case of the DFT . the HamU g window issampled at mult ipks of 2*/N, which of course am the loca-t ions of the tcrca o f the central Dir ichict kernel . Thus onlythree nonzero sampka are taken in the SampIhlg pPxa.- Theposi t ions of these samples UC ,t -2x/N, 0, and +2x/N. Thedue of the sam ples obtained from (28b) (including the phasefactor exp (- j(N/2)8) to wautt for the N/2 shi ft) M - $,++, - ; , rcrpecti reIy. Note the minus s&us. l -hue resul tsfrom the shift in the origin for the window. Without the shi ft .the pha term h missing and the urfticknts are all positivef , +, ; . These am in- for DFT proardng. but theyfind their w ay into much of the Htentruo and practice.Rather than apply the window aa a product in the t imedomain, we always have the option to apply i t IS a conwl: it ion in the frequency domain. The attraction of the Harmingwindow for this appl ication is twofold; t int. the windowspectra is nonzero at only thee data pain , and second. thesample values M binary fnctioaa, xbich can be implementedas t ight shi fts. Thus the Haming-windowed spectral pointsobtained from the rectangle-windowed spectral points areobtained as indicated in the following equation as two realadds and two binary shi fts ( to mult iply by f) :f(k)/Hdw = f [f(k) f (F(k 1)

or as 2N reai adds and 2N binary shifts on the spectral Jar;l~One other mi ldly important consideration, i f the window !s iobe appl ied to the t ime data. is that the samples of the u,r ,~.Io~~mus t be stored somewh ere, which normal ly means additionalmem ory or hardware. It so happens that the samples of thecosine for t ix Hatming window are already stored in themachine as the tr ig-table for the FFT ; thus the Hamungwindow requires no additional storage .

D. Hamming Window [7/The Hamming window can be thought ofas a modif iedHatming w indow. (Note the potential source of confuion m

the simi lar it ies of the two name,.) Referr ing back to Fig.. I7and 20, we note the inexact cancel lat ion of the sidelobes fromthe summ ation of the three kernels. We can construct d wiwdew by adjusting the relative size of the kernels as indicated inthe following to achieve a more desirable form of cancellation

27 7w(n)=u+(I -a)cos N[ 1w(e)=ao(e)+o.s(l-a) D e-2 +D e+z

[ ( N) ( N)](3Oa)

Perfect cancel lation of the fmt sidelobe (at 0 = 2.5 [?n/Nl)occm when ( I = 25146 (a = 0.543 478 261). I f a is selected a0.54 (an approtimation to 25146). the new zero occurs af0 A 2.612n/Nl and a marked improvement in sidclobe level 1sreal ized. For this value of (1. the window is cal led the Ham-ming .window and is identif ied by

0.54 + 0.46 cos [ 1 nN 0.54 0.46 CM ;n ,[ 1

+.=(k+l)ll /R-. (29)

I n=O,l .2;.. .N- I. (30b)The coeff ic ients of the Hamm ing window are near ly the setwbxh a&eve mm mm sidelobe levels. I f a is selected to be0,53856 the sidelobe level is -43 dB and the resultant windowLS P rpccial case of the Blackman-Harr is windows presented inSccaon V-E. The Hamm ing window is shown in Fig. 2 I.Vouce the deep attenuation at the missing sidelobe posi t ion.Note also that the smal l discontinui ty at the boundary of thewmdo w has resul ted m a I/w (6.0 dB per octave) rate ofThus P Harming window appl ied to a real transform of length fal loff. The better sidelobe canceUation does resul t in a muchN can be wrformed as N real mult iplier on the tune sequence lower initial sidelobe level of 42 dB. Table I l ists the param-


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eters of this window. Also note the loss of binary weighting;hence the need to perform mult ipl ication to apply theweighting factors of the spectral convolution.E. Blackman Window [7/

The Hamm ing and Harming windows ale examples of wir-dows constmcred as the summ ation of shi fted Dlr ichlet ker-nels. This data window is defined for the f ini te Four ier trans-form in 131a) and ior the DF T 1n(3lb):equation(3lc)is theresul tant spectral window for the DFT given as a summ ationof the Dti ichiet kerneis D(8) defined by W(9) I, , (2,c,;

of this form with a0 and 0, being oonzero. We see thspectral windows are summ ations of three-shi fted kerneWe can construct windows with any K nonzero coefand achieve a (2K- I) summ ation of kernels. We rehowever. that one way to achieve windows with a narro

lobe is to restrict K to a smal l integer. Blackman exthis window for K = 3 and found the values of the coeff ic ients which place zeros at 0 = 3.5 (?n/N) and a(2nlN). the posi t ion of the third and the fourth sirespectively, of the central Dir ichlet kernel . Thesvalues and their two place approximations are

7936N a0 = ---- 0.426 590 71 = 0,42I86089240(3l.a) a = --& 0.496 560 62 = 0.5018608.=O.I:. . . .V~ 1 ~ 1430a2 = 18608 A 0.076 848 67 = 0~08m=o

(3lb) The window which uses these two place approximatio

W(O)= x ,- I)2 -+(+)+D(O+;m)]~known as the Blackman window. When we descr iwindow with the exact coeff ic ients we wi l l refer

m=o the exact Blackman window. The Blackman window ,31Cl fined for the finite transform in the following equatithe window is shown in Fig. 22:Sublect to constraint

N /2, z m = 1.0~

Wini=O~42+0.5Ocnr[~n] +0.08+2,].

im=o N

We ian see that the Harming and the Hamm ing windows are =---..


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The exact Blackman window is sbom in Fig. 23. The sidelobelevel is 5 I dR down for the exact Blackman window anal is 58dB down for the Blackman window. As an observation. notethat the coeff ic ients of the Blackman window sum to zero(0.42 -0.50 +0.08) at the boundaries while the exac t coei-t ic icntn do not. Thus the Blackman window is continuouswith a continuou s fr% derivative at the boundary and falls offl ike I/w or I8 dB per octave. The exact terms ( l ike theHamming wmdow) have a discontinuny at the boundary andfalls off l ike l/w or 6 dB per octav e. Table I Lists the param-

sidelobe level. We have also constru cted families of 3~ an,, 1~Lerm window in which we trade ~nain-lobe width Ior SIIIC/L)II~ ~level . WC caU this fami ly the Blackman-Harr is wmduu U,,have found tha t the minimum )- term window can achlrvr I Isidelobe level of -67 dB and that the minim um d-term urnsdow can acbwe a sidelobe level of -92 dB. These wmdowiare defined for the DFT by

eten of these two windows. Note that for this class of w,,-down. the ao coeff icxnt is the coherent gain of rhc wmdo w n=O .I.::--..v- 1~ ,3?lUsing a gradient search technique 191, we have found the Thr l~tcd coei i ic ients correspond to the mmm ~um 3.termwindows which for 3. and 4.nonzero terms achieve a mintmum wmdo w uhlch is presented in Fig. 24, another 3-rerm wm dou


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3.Term 3-T- 4-TCrm GTW(47 dB) (41 dB) (-92 **, (-74 dB,

00 0.42323 0.44959 0.35875 0.4021701 0.49755 0.49364 0.48829 0.49703a* 0.07922 0.05677 0.14128 0.09392a3 0.01168 0.00183

( to establ ish another data point in Fi i 12). the minimum 4-term window ( to also eatabl isb a data point in Fi i . 12). mdanother &term window which is pxsented in Fig. 25. Thepart icular d- term window shown is one which performs reflin a detection example descr ibed in Sxtion VI (see F& 69).The p-eters of these windows ue l ined i , , Table I. Note inpart icular where the Blackman and the Blackman-Hanis win-dows reside in Fig. 12. They are surpr isingly good windowsfor the smal l number of terms in their tr igonometr ic ser ies.Note, i f we were lo extend the Line corme&ng the Blackmae-Harr is fami ly i t would intereat the Hamm ing window which.in Section V-D . we noted is neviy the minimum sidelobe level2-tern Blackman-Harr is window.We also mentioo that agood approximation to the Blackman-Harr is 3- and 4-term windows ca,, be obtained as xa,edsampler of the Kaise-Eeszzl windows tmns(one (see SectionV-H). We have used this approximation to construct b- termwindows for adjustable bandwidth convolutional fdten asreponed in [ 101. This approximation is defmed as

ho :!! !? .,=2%- m=:1,2,() ) .c c3 (341The 4 coeff ic ients for this approximation when a = 3.0 are*o = 0.40243, o, = 0.49804. ,,I = 0.0983 I , and a, = 0.00122.Notice how close thus terms are to the selected 4.termBlackman-Harr is ( - 74 dB) window. The window dcfmcd hyOme coeff ic ients is shown in Fig. 26. Like the prototypefrom which i t came ( the Kaiser-Bessel wi th a = 3.0). thu sWm dow exbihi ts sidelobes just shy of -70 dB from the reamlobe. On the rcale sh ow,,. the two are indist inguisbahlrThe parameters of this window are also Listed in Table I andthe wmd ow is entered in Fig. 12 as the 4sample Kaiser-ksc. I f was these 3. and 4Smple Kaiser-Bessel prototype

windows (pmmetcr ized on a) which were the star t ing ctions for the gradient minimizatioo which lads to the n~~-Hur in windows. The optimization star t ing with mefft ients baa vir tual ly no effect on the main- lobe chanMica but does dr ive down the aidelobes ap pmtintely 5

Numennu bmst+ton have co-ted windows as ucts, as sum s, a.3 satiom, or a.3 convolutions of simple t ions and of other ximple t idoas. xx window have constructed for certain desirable fcshn s, no, the lawwhich is the attnctioo of simple functions for genera& window terms. In gmm nf, the constructed windows tendto be 8c.d windows, and ocosionnl ly arc very bad windWe hare l l rudy eramincd some ximplc window coostmc tioThe Fejer @arUett) window. for t ianoe, is the convolutof two rectangle windows; the Hamm ing window is the sua rcct+nde and a Harming window; and the cos(X, winis the product of two t&am& windows We wi l l now enmother mnstructed windows that have appeared m the Itwe.~ We wiU present them so they me avai lable for comso. later we wi l l examine windows constructed ,n awith som e cr i ter ia of optima,i ty.(sce Sectrons VG . H. 1I) . Each window is identif ied only for the foute Four ier form A simple shi ft of N/2 paints and r ight end-pomt t ion ml l supply the DFf version. The slgruficant f igure~~rformancc for these windows are also found in Table I~

1) Rirsz (Bochner. Punen ] Window fll/~~ The Raeu dew. identif ied as

lb+)= 1.0. n =Ii NJ2 i i thr cinlplest ~ontinuou polynom inl window II exhlhidi~:ontmuoos fnt der ivative at the boundar ies; hrnl ::tramfo fas Off bke I/w . The window is rhowvn in27. The fnt ddclobe is -22 dB from the main lohe~ window is simi lar to the cosine lobe (26) IS can he demstrated by e xamm mg i ts Taylor ser ies expulsion.21 Riemmn Window 1121: The Riemann window, defby r .

w(n)= NoGGi

is the central lobe of the SING kernel . This wmdow is -


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t im,ous, wi th o di ,cootiouous fnt der ivative at the boundary.It is simi lar to the Ricsz md cosine lobe windows. TheRicmm n window U shown in Figs 20.31 de in Val id-Powsin (Jackson. Punen) Window [II ] . Thede la Val lC-PouSan window is P piecewise cubic curve ob-tained by self-convolving two triangles of half exten t or fourrectangles o f one-fourth extent. I t is defined as

The window is continuou s up to itr third derivative so that itssidelpbes fal l off l ike I/w . The window is shown I I : IF 20~Notice the trade off of main-lobe width for sidelobe level.Compare this wth the rectangle and the triangle. I t is a non-naativ~ window by vir tue of i ts sel f-convolution const~~tioa~.41 Tukcy Window [13l: The Tukey window, often cal ledthe cosine-tapered window. is best imagined as a wsine lobe ofwidth ia/Z)N convolved with a rectangle window of widthI 1 .O nil)A Of course the rrsultdrrl transfu rm AS he productof the two corresponding transforms. The window rspieszntsan attempt to smoothly ret the data to zero at the bouaddtizswhi le not signi f icantly reducin& the processing gam of thrwmdow ed transform The window evolves from the rectangleto the Harming windou as the parameter o var ies from zero lo

(37: umty . The fami ly of windows exhibi ts a confusing anay 31


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tidelobe levels arising from the product of the two componentInnsfons. The window is dcti.ed by

i

I .o. N06116UZ

u(n) =0.5

,(38)

fbhe tindow is shown in Figri. 3&32 for values of (I equal to0.25. 0.50. and 0.75. respectively.

51 Bohrnon Window 1141: The Ekhman window is o&

tained by the convolution of two half-duration cosine lobes(261). thus its transform is the square of the cosine lobzstmm.fohn (see Fii. 16). in the time domain tbc wndow I 11be described as a product of a trim@ window with a singlecycle of a Mine with the same period and. the, a comectiveterm added to %et the first derivative to zero at the boundary.Thu the second derivative is cmtinuous , and the disconti-nutty resides in the third derivative. 7%~ transform fatIs off likeI/w. The window is defmcd in the folloti~ and is showo inFii 33:


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6) Poisson Window 1121: The Poison window is P two- observed in Table I as a large equivalent rroise bandw Idth andsided exponential defmed by as a large worst case processing 109s.In Iwin)=exp -a-( )/2 O


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derivative at the origin and is l/w. Notice as a increaser,forc ing more of the exponential into the Harming window.the zeros of the nidelobc structure disappear and the lobesmrrge into the asymptote. This window is shown in Figs,37-39 for va lues o f (I equal to 0.5, 1.0, and 2 ~0. respe ctively.Awn note the very large main-lobe width

8: Cauchg (Abel, Poisson) Window (I5]: Tt~he Cauchy wti -dow is a family param etefized on Q and defined byIw(n) =

In 1 O.O+ GJThe wmdo w is shown in Figs 40-42 ior valws of Q ~quzl to3.0, 4 0. and 5~0. respectively. Note the transform of the

Cauchy window is a two-s ided exponent,, , (see Podews), wh ich w hen presented on a log-mumtude essentially an isoreles triangle. This C~LIYI the wezhibif a very wide m ain lobe and to have I large EEX

Windows ax smooth posi t ive funct ions wlh tal l concer.trated) Fourier transforms. From the guncertainty principle, we know W C cannot dmul taconcen trate both a signal and its Fourier transform measu re of concentra tion is the mean-squa re time dand the mean-square bandwidth W. we know allsatisf y the inequality of nv>;


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with equ ality being achiclcd only for the Gau5ian ~~1161.Thus the Gauszian pulse. character ized by minimum time-bandwidth product, is a rcuonablc candidate for a window.when WC se the GaU pulse as a window we havr to tnu-ute or discard the tai ls. By rest&t& the puke to be f ini teIcwth, the window no longer is minimum time-bandwidth.If the trvnution point is beyond the thrwAgmr point, thecrro would be smal l . and the window should be a good*PProumation to minimum time-hndwidth.

The Catin window is dcf,ed by

dn) = exp n=Clr 11T-/2 (44) (Mb)Thu window is paameter ized 0x1 a. the reciprocal of thestandard deviation. a measure o f the width of i ts Four iertranrfomr. lnclease d a wi l l decrease with the width of the-doe, and reduce the sever i ty of the discontinui ty at thebounda ries. ThrS will result ti an increased width tra nsform


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mam lobe and decreased sidelohe levcls~ The wmdow ispresented ir. F&s 43, 44. and 42 for values of a equli !o I,?,3.0. and 3.5, respectively. Nate the rapid drop-off rate ofsldelcbe lrvel in the exchange of ddelobc level for mam-lobewidth. The fiurer cf merit for this wlndcw arc lured inTable I.

.apenure to achieve a narrow main-lobe beam partwn simultaneously restricting sidelobe rc~ponx. (The adrsignrr calls his weighting procedure Ihrrdinx.) Theform solution to the minimum main-lobe width for asidelobe level is the Dolph-C hebyshcv window (shaThe continuous solution to the problem exhibits imp ulthe bvundtics which restricts continnuous realizationsaFprOX~atlOns (the Taylor approximrfion). The discrsampled window is not so redtxictsd . and the solution unpiemented e,.actly.

The relatvx~ T,C.U! = cos (n9) describes a mapping bethe nth-order Cbebys hev (algebraic) polynomial and thwdcr trxganom e:tic polynomiai. Tix Dolph4Yhebyx


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window is defied with this mapping in the following equa-tion, i,, terms of u,,ifom, ly spaced runpla of the window sFourier tnnsform.

whereO


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presented in Figs. 46-49 for values of Q equal t 2.5, 3 .0,3.5, and 4.0, respectively. Note the uniformity of the sidelobestructure; almost sinusoidal! It is this uniform oscil lationwhich is responsible for the impulses in the window.J. Kaiser-Bessel Window /I81

Let us examine for a mm ent the optimali ly criteria of thelast two sections. In Section V-G we sought the functionwi th minimum t ime-bandwidth product. We know this to bethe Gsus%iul. In Section V-H we sought the function withrestricted tim e duration. which minimized the main-lobewidth for P given sidelobe level. We now consider a simllarproblem. For a restricted energy, determine the function ofrestricted time duration T which mnximhcs the energy in theband of frequentica W. Slepian, PoUa k, and Landau ( I9 1,1201 have determined this function as a family parametetiedOva the time-bandwidth product. the prolate-spheroidal wavefunctions of order zem . Kaiser hu discovered a rimplc ap-Proximatian to these functions in terms of the zero-ordermodified Besel function of the fti kind. Tbe Kaise-Besselwindow is defined by

Jow(n)=-

where

The p-eta 110 is half of the time-bandwidth product. Thetransform is approximately that of

This window h presented in Fii. SO-53 for nls of acqdto 2.0. 2.5. 3.0, and 3.5, -tidy. Note the trade ofbetween sidelobe level and main&be width.1. Borcilon-Temes Window 1211

We now examine the lut criterion of 0ptimaLity for a window. We have already described the Slcpian, Poti, andLandau criterioo. Subject to the wnstnin ts of tixcd en-and fued duration. determine the function which maxim& athe energy in the band of ffequcn tics W. A related criterion,subject to the constraints of fued arca md fried duration. to determine the function which ll lsmds the tl lerp? (nthe we&b&d energy) outside the band of frequeocies W ~#iis a reasonable criterion since we rxa@ze that the tnnsfonof a good window should minimix the en- i t gathm fmmfrequencies removed from i ts cater frcqwy. Ti l l now. whave bee responding t thi, .g,al by m.xim iz@ the onlcca-tration of the transform at its main Lobe.A closed-form solution of the unwei&tcd minimum-en crgrcriterion has not been found. A rolutia dcfme d as an cxpm -sion of prolate~pheroidal ware functions dots exist sod i i of the form shown in


24/33

i


25/33

Here the X,, is the e&envplue corresponding to the associatedprolate-spheroidal wave function I $tLln(x. y) I . and the na isthe selected hal f t ime-bandwidth product. The summ ationconverges qui te rapidly, and is often approximated by the i rstterm or by the tint two term s. The fust term happens to bethe solution al the Slepian. Poll&, and Landau pro blem.which we have already examined as the Kaiser-Bessel window.

A closed-form solution of a weighted minimum-energycriterion. presented in the following equation has been foundby Barci lon and Tema :

Minimtie IHWl&dw. (481Thiscr i tet ion isone which is a comprom ise between the Dolph-Chebyshev and the Kaiser-Bessel window cr i teria.

L.ike the DolphXbcbysbw window, the Four ier truuform more easi ly defmed, and the window timej lmpla an otamed by an inverse DFT and an appropriate scale factor. Ttransform samples are defmed by

A cos Iv(k)1 + BW)=(-IP

y? sin [y(L)] 1[C+ABI [p] + LO]

whereA =sinh(O=mB=cosh(C)= IO

(4


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C=cosh- (IOU)1= cash ;C[ 1y(k)= N cos

(gee alsO (45).) this window is presented in Figs. 54-56 fatvduer of a equal to 3.0, 3.5, and 4.0. respectively. The ainlobe stmc tre is practically indistinguishable fro the Kaiser-&s] m&+&e. me fuures of meri t l is ted on Table 1 suggesteat f,,r the same sidelobe level, this window does indeedt ide between the Kaiser-Bessel and the Dolph-Cbeby*evh&w , it is interesting to examine Fig. 12 and note wherew ,+jndow is located w i th respect to the Kaiser-Besseledow; rt&ing s imi larity in PerfOtmanCe!

we now describe a s imple experiment which dIaat ic~Ydemo,,strstes the inf luence a window exerts on the detect ionof a we& spectrl l l ine in the presence of a strong nearby l ine., f two ~pccval , i , ,es reside in Dm bins, the rectangle windowavows each to be identi f ied wi th no interaction. ,To demon-strate this, consider the s ignal composed of two frequencieS,O f i ,N and 16 f , /N (corresponding to the tenth and thedteenth DF T bins) and of ampl i tudes 1.0 and 0.01 (40.0 dBscpustion ), respe ctively. The power Epectru Of this Sigllalobtained by a DF T is shown in Fig. 57 as a l incar interpola-t ion between the DF T wtput points.

We now mo dify the signal sl ightly so that the larger signalresides midway between two D FT bins; in part iculnr, at 10.5f,/h'. The smaller signal sti l l resides in the sixteenth bin. Thepower spectrum of this s ignal is show in Fig. 58. We notethat the sidelobe structure of the larger signal has comp letely

-amped the main lobe of the smal ler s igznl In fact, we know(see Fig. 13) that the sidelobe amplitude of the rectangle win-dow af 5.5 bins from the center is only 25 dB down from thepeak. Thus the second s ignal (5.5 bins away) could not bedetected because i t was ore than 26 dB down. and hence,hidden by the sidelobe. (The 26 dB corns from the 25d8sidelobe level minus the 3.9dB processing Ions of the windowplus 3.0 dB for a high confidence detection.) We also notethe obvious asymm etry around the main lobe centered at 10.5bm s. Thrs is due to the coherent addition of the sidelobestruucturcs of the par of kernels located at the plus and minus10.5 bin positions. We are obsetig the self-leakage betweenthe positive and the negative frequencies. Fig. 59 is the powerspectru m of the signal pair, oditied so that the largwamp litudesignal resides at the 10.25.bin position. Note the change Lnasym met ry of the main-lobe and thr rcdncti~!n in i l ls ridclobelevel . We st i ll can not observe the second , . ,givd 1ocatc. l a!bin position 16.0.

We now apply di f ferent windows to the twxtone s ignal todemo nstrate the difference in second-tone detecta bil i ty. ForSOme of the windows, the pwrer resolut ion OCC UIS when thelarge signal is at 10.0 bins rather than a t 10.5 bins. We wil lalways Present the window with the large signal at the loca-tion corresponding to wars t-case resolution.

The f i rs t windo* we apply is the tr iangle window (see Fig.60) The s ide;obzs have fak,, by a factor of two over therec*=WJc w mdowr lobes (e.g.. the -35dB ieve, has fal len to-70 dBI. The ridrlabes of the larger signal have fallen toaPPro*lnUtelY -45 dB a( the second signal so that it LI barel)

dc tectable. If there were any noise in the signal. the secofCme would probably not have been detected.thca

The next windows we apply are the axa fami ly. m ecosine lobe, a = I .O, shown in Fig. 6 I we observe a pha!ncellation in the sidelobe of the large signal located at nall signal position. T&is annnot be considered a detec te also see the sp ectral leakage of the main lobe o et equency axis. Signals below this leakage level would not : tectcd. With a = 2.0 we hzve the Harming wmdo w, which

InWfrd


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Fi t?. 63. ca f. ,K) widow.

presented in Fig. 62. We detect the second s ignal and observe1 3.O-dB null betw een the two lobes. This is sti l l a marginaldetect ion. For the cos (x) window presented in Fig. 63, wedetect the second signal and obsm e a 9.O-dB null betw eenthe lobes. We al.w see the improved sidelobe response . Finallyfor tbe co%(x) window presented in Fig. 64, we detect thesecond signal and ob~rve a 7.O-dB null between the lobes.Here WC wi tnes the reduced return for the trade betweensidelobe level and main-lobe width. In obtaining furtherreduction in sidelobe level we have caused tbc increased mati-

~8Obe width to encroach upon the second SignaL.

I!i!1;

t We next apply the Hamm ing window and present the resul th.. FS 65 Here we observe the second s ignal some 35 dB-e a ph== am. approximately 3.0 dB over the s idelobe response of

a det-Xtl - e be s iamI. Here. too, we observe the phase cancel lat ion the s idelobe stmcture of tJut kernel . Note the rapid be over the (I

&the leakage between the positive and the negative fre- falloff of the sidelobe Leakage has confined the artifacts

&d not be eCY comp onen ts. Signals more than 50 dB down wo uld small portion of the spectral line.o w ich 1s t be detected in the ; ;resenze of the larger s ignal . We next apply the exact Blackman coeff ic ients and wThp BLaclwaa window is applied next and we szn the revolts the results :;n Fig. 67. Again the second Ugnal is well d

2 Fig. 66. Tbe presence of the smaller amplitude kernel is with a 24-ziB null between the two kernels. The sinow WV apparent. There ir a 17-dB null between the two strctre of the larger kernel now extends over the dCna k IX: artifact at the base of the lqe-signal kernel is spectral range. This leafrage is not terribly severe as it is


28/33

P-t&70. .amp tc---ri la. a. 73. de I# vstt.5-Polli0 windowto the phuc can-tion of a sidelobe in the law &Ikernel.The result of a Riemam window is presented in FU. 72.Here. too, we bare no detuztion of the second s&ml. WC dohave a muU aul l due to piuse rm wlfation at the second swad. We also have a Loge ddelobe rapome.The ,,ext window. the de la Vail&Poussin or the rlf-convolved triangle. is shown io Fig 7 3. T be second n .qnnll ua.dy found lad the power spec trum exhibits a 16.O-dB null.An artifact of the window (itz lower sidelobe) shorn up.hoaew r, at the fi ith DF T bin as a signal approxima tely 53.0dB down See Fy 29.The rcdt of applyins the Tukcy fam i ly of window ixpraeoted in Fm . 74-76. In Fig. 74 (the 25percent taper)WC l ice the hck of ,ecmd+wal detection due to the b&b side-lobe smxturc of the dominant rectangle window. In Fi&. 75(the 50pe rccnt taper) WC observe a lack of second-ngmldetection . with the second s%?m al actually fi l ling in one of theoulk of the furt s&m l.% kernel. In Fig. 76 (the 76-percenttaper) we witnes s a marginal detection in the stil l high side--.. .


29/33

The B&mm construct ion window is appl ied and presentedin Fia. 77. The second signal has been detected and :hv nullbetween thr two 1&es is approximately 6.0 dB. Thk is notbad. but we can st i ll do better. Note rbere the Bohman wi ,~dew resides m Fig. 12.

The resul t of apply ing the Poisson-window famdy 1s pre-sm:ed i : . Fig.. 7.?-50~ Ti-x second :Unal is nor detccrsd f@rany of the selected parameter values due to the higkidrlobe

levels of the larger signal. We anticipated tha poor mance in Table I by the large dif fermce betrem the and the ENBW.

The result of applying the Hanning-Po isson family dows is presented in Figs. 81-83. Here. too. the secondis either not de tected in the presence of UIC hi&Jidestructure or the detection is bewildered by the utifnc ts.Tb.e Cauchy-fami ly windows have been apphcd aresults are presented in Figr. 84-86. Here too w e havrof sat is factory detect ion of the second s&nl l and tks idelobe mpo rse. Tbis was predicted by the lqc di f fbetween the 3.0 dB and the cquivaknt noise bandwidlisted in Table I.

We now apply the Gausian fami ly of windows and the resul ts in F@. 87-89. The second si .mnl L detectedthree fwen. we note a we further depress the sstrulic:,ue to enhance secon dsi~a, detection, the null dto approximately 16.0 dB and then becomes poorer main-lobe width increases and start s to overlap the the smdler signal.

The DolphXhebyzhtv fami ly of windows is presenFigs. 90-94. We observe strong detect ion of the second


30/33

Note the difference in phase cancellat ion near the base of thLarge s&nab Fig 93. the 7MBsidelobe window. exhibi ts a1 MB nul l between the two m ain lobes but the sidelobes havadded constructively (along with the scal loping loss) to th-62.O-dB level . In Fig. 94, we see the 8WB udelobe -0oU

lobe being sampled off of the peak and being refcrenwd as exhibited sidelobes below th e 7048 level and stil l manag ed tMD dB. Fi i . 90 and 91 dcmoo stnte t ic sensi t iv i ty of the bold the nul l between the two lobes to approximatley 18.sidelobe coherent addi t ion to main&& posi t ion. In Fig. 90 dB.the larger sigh is at bin L0.5; in Fig 91 i t is at bin 10.0. The KG.ser-Bessel fami ly is presented in Figs 95-98. Here


31/33

too. we have strong second-signal detection. Again. w e see theeffec t of trading increased main-lobe width for decreasedsldelobe level. The null between the two lobes reaches a max i-mu m of 22.0 dB as the sidelobe stru~~ture fa lls and then be-com es poorer with further sidelobe level improve ment Notethat this window can maintab, a 20.OdB nul l between the twoWna l lobes and sti hold the leakage to more than 70 dBdown over thr enti re spectrum.Figs. 99-101 present the performanc e of the Barcilon-Tews window. Note the strong detection of the second s,gnal .

There are slight sidelobe artifac ts. The window can maa 20,OdB nul l between the two signal lobes. The performaof this window is sl ightly shy of that of the Kaiser-Bewmdo w. but the two are remarkably simi lar.

VI I . CoNCLUsloNsWe have examined some classic windows and some winwhich satisfy some criteria of optim ality. In particular.have desctihed their effe cts on the problem of general


32/33

0.*- 1- i\

.m - j;!_ :

man ic analysis of tones in broadband noise and of tones inthe presence of other tones. We have observed that when theDFT Is used as a hvmonic energy detector, the worst caseprocesing 1os1 due to the windows sppearr to be lowerbounded by 3.0 dB and (far good window s) upper boundednear 3.75 dB. This suggesta that the choice of part icularwindows has very l i tt le effect on worst case performance inDFl energy detection. We have concluded that a good perfor-mance indicator for the window 15 the di fference between theequivalent noise bandwidth and the 3.048 bandwidth nor-mal ized by the 3.o-dB bandwidth. The windows which per-form weU (as indicated in Fig. 12) exhibit vallres for thisratio between 4.0 and 5.5 percen t. The range of this ratiofor the window s listed in Table I is 3.2 to 22.9 percent.For mult iple- tone detection via the DFT , the windowemployed does have a considerable effect. Maximum dynamicrange of multitone detection requires the transform of thewindow to exhibit a highly conc entrated central lobe withvery-low sidelobe structure. We have demonstrated thatmany classic windam satisfy this miter ion with varymg

degrees of suxess and some not at al l . W e have demonsr; . , , the optimal windows (Kaiser-Bessel. Dolph-Chebyshcv, aidBarci lon-Temes) and the Blackman-Hanis windows performbest in detection of nearby tones of dgnificantly differmrampl i tudes. Also for the same dynamic range, the three opti -mal windows and the Blackman-Harr is window are roughlyequivalent wi th the Kaiser-Bessel and the Blackman-Ham s.demonstrating minor perfornmnce advantages over the othersWe note that whi le the Dolphxhebyshev window appears tobe the best window by vir tue of i ts relative posi t ion in Fig. 12.the coherent addition of its constan t-level sidelobes d etractsfrom i ts perfomwtce in mult i tone detection. Also the PIIIC-lobe st,, ,ctwe of fhe Dolph-Chebyshev window rxhi lvt\extreme sensi t iv i ty to coeff ic ient errors This would affectits performa ce in mach ines operating with fixed-point arith-metic. This suggests that the Kaiser-Bessel or the Blackman-Harris window be declared the top performer. My preferenceis the Kaiser-Bessel window. Among other reasons. the cur i -ticienfs are easy to generate and the tradeall of sidelob?level as a function of time-band tidth product is fairly amp le,For many appl ications, the author would recommend thr 4~samp le Blackm an-Harris (or the 4samp le Kaiser-Brssrl)window. These have the distinctiw~ czf being defini+ by :+ :.:easily generatcd coefficie nts and of being able to ix apl., l iar a spectral convolution after the DFT .

We have called attention to a petisten t error in the applica-tion of windows when performing convolution in the fre-quency domain. i .e., the omission of the al ternating signs onthe window sample spectrum to account for the shi fted t imeOgln. We have also identified and clarified a source ofconfusion concerning the e~ennesz of windows under the DFT .Final ly. we comm ent that all of the conclusions presentedabout window performan ce in spectral analysis are also ap-plicable to shading for array processing of spatial sampleddata, including FFT beamforming.


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HARRIS: SE OF WINDOWS FOR HARMONlC ANAY SIS

an d

Then

*Pf(l) = L F(w) exp (-jwr) dwl2n

+Nl,Yw)= x w(nT) exp (+jwnT)..-V,*

F,(w) = x w(nT)jC(nT) exp (+jwnT)n--

becomesIS

F,(w)= 1 F(x) exp (-jxnT)dxjZnn= -exp (+jwnT)

/l - fW

= F(x) x w(nT) exp I+j(w x)nTl drj2n-- =a

/*a *TV,*= F(x) x w(nT) exp Ifj (w - x)nTl dx/2n_m =-IV,*

=I-.m

F(x) W(w x)dx/2nor

F,(W) = F(W) * W(W).

.

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The Use of Windows for f Harris

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