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SignalProcessing

 ExponentialWindow
 multiply an array of samples by an exponential windowing function

 Calling Sequence ExponentialWindow( A, alpha )

Parameters

 A - Array of real or complex numeric values; the signal alpha - numeric value strictly between $0$ and $1$

Options

 • container : Array, predefined Array for holding results
 • inplace : truefalse, specifies that output should overwrite input

Description

 • The ExponentialWindow( A, alpha ) command multiplies the Array A by the exponential windowing function, with parameter $\mathrm{\alpha }$, and returns the result in an Array having the same length.
 • The exponential windowing function $w\left(k\right)$ with parameter $\mathrm{\alpha }$ is defined as follows for a sample with $N$ points.

$w\left(k\right)={\mathrm{\alpha }}^{\frac{k}{N}}$

 • The parameter $\mathrm{\alpha }$ must lie in the open interval $0,1$.
 • Before the code performing the computation runs, A is converted to datatype float[8] or complex[8] if it does not have one of those datatypes already. For this reason, it is most efficient if A has one of these datatypes beforehand. This does not apply if inplace is true.
 • If the container=C option is provided, then the results are put into C and C is returned. With this option, no additional memory is allocated to store the result. The container must be an Array of the same size and datatype as A.
 • If the inplace or inplace=true option is provided, then A is overwritten with the results. In this case, the container option is ignored.

 • The SignalProcessing[ExponentialWindow] command is thread-safe as of Maple 18.

Examples

 > $\mathrm{with}\left(\mathrm{SignalProcessing}\right):$
 > $N≔1024:$
 > $a≔\mathrm{GenerateUniform}\left(N,-1,1\right)$
 ${a}{≔}\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{-}{0.785218492150308}& {0.588413964957000}& {-}{0.993165822699668}& {0.921578288543971}& {-}{0.0387801709584892}& {0.0136057925410569}& {-}{0.210756972897798}& {0.749600215815009}& {0.138966357801110}& {0.212285134010017}& {-}{0.727212007157506}& {0.609271531458945}& {-}{0.746508821379394}& {-}{0.681121068540962}& {-}{0.815677223727108}& {0.920580454170705}& {-}{0.357731881551445}& {-}{0.315850691869855}& {0.120832127984613}& {0.0235598362050951}& {-}{0.528712330386043}& {-}{0.502768306992949}& {0.716167932841928}& {0.387918812688441}& {0.927826197817923}& {-}{0.535605234093965}& {-}{0.867390423081817}& {0.356968106236309}& {-}{0.683916721958668}& {0.324222652241588}& {-}{0.0536105097271503}& {-}{0.469822424929590}& {0.751377623062582}& {-}{0.484332469291986}& {0.674785583745689}& {0.936373751610519}& {-}{0.709695004858078}& {-}{0.315371678676457}& {0.786426438484342}& {0.877079485449941}& {-}{0.940901432652028}& {-}{0.651838099118323}& {-}{0.466202749870718}& {0.728111944627018}& {-}{0.693676937371493}& {0.446705075912178}& {0.402212079148740}& {-}{0.465064398013056}& {-}{0.149959974456579}& {-}{0.893211717717351}& {-}{0.533857398666442}& {0.785364017821850}& {0.794103573076428}& {-}{0.511805256363005}& {-}{0.699780572205783}& {0.390154657885433}& {-}{0.306801157072187}& {0.380043311044574}& {0.250223507639021}& {-}{0.112387157976628}& {0.213712436612696}& {-}{0.462156727444381}& {-}{0.748708907514812}& {-}{0.151586118619889}& {-}{0.108139840420336}& {-}{0.168242880143225}& {-}{0.525201478973032}& {0.480703854002059}& {-}{0.893447801005097}& {0.705915172118695}& {-}{0.922403736039998}& {-}{0.150907000061125}& {-}{0.552928699180485}& {-}{0.630023401696236}& {0.476304094772787}& {-}{0.520089327357710}& {0.383331325836480}& {0.853844197466971}& {-}{0.561684322543443}& {-}{0.392888241447509}& {0.805707171559335}& {-}{0.830475841183217}& {0.958363623823972}& {0.267084791325033}& {-}{0.934454344213010}& {0.600780255626888}& {0.499754573684187}& {0.663151745684446}& {0.481067702174187}& {-}{0.756487140897663}& {0.800444356631489}& {-}{0.510770577006043}& {0.292151435278357}& {0.0674125049263240}& {-}{0.305776782333851}& {-}{0.469037371221931}& {0.649966387543828}& {0.648178403731437}& {0.870920942630620}& {-}{0.361100737471134}& {\mathrm{...}}& {"... 924 Array entries not shown"}\end{array}\right]$ (1)
 > $\mathrm{ExponentialWindow}\left(a,0.23\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{-}{0.784092331012393}& {0.586727367503991}& {-}{0.988898750179820}& {0.916302737447939}& {-}{0.0385028746995195}& {0.0134891308854116}& {-}{0.208650179925450}& {0.741042645791122}& {0.137182862647516}& {0.209260113085191}& {-}{0.715821274645983}& {0.598868036519053}& {-}{0.732709596245725}& {-}{0.667571729699757}& {-}{0.798304625609140}& {0.899681410664950}& {-}{0.349109230330512}& {-}{0.307795456452361}& {0.117581631898536}& {0.0228931742335138}& {-}{0.513014774947955}& {-}{0.487141371753411}& {0.692912953050209}& {0.374784255420278}& {0.895125263253460}& {-}{0.515986908861099}& {-}{0.834420914363845}& {0.342907244778551}& {-}{0.656035219616044}& {0.310558896100976}& {-}{0.0512775473166679}& {-}{0.448732719036138}& {0.716620024230060}& {-}{0.461265480096529}& {0.641726320866645}& {0.889221542144119}& {-}{0.672990881729076}& {-}{0.298632320347489}& {0.743616318420761}& {0.828145119427394}& {-}{0.887132134098562}& {-}{0.613706338940877}& {-}{0.438300924261334}& {0.683553322574328}& {-}{0.650291662791864}& {0.418165794402080}& {0.375975386165980}& {-}{0.434104298890600}& {-}{0.139776138204210}& {-}{0.831359337130374}& {-}{0.496176647357574}& {0.728884544587725}& {0.735938594694381}& {-}{0.473637258021889}& {-}{0.646665489774594}& {0.360023862718691}& {-}{0.282701538219005}& {0.349688184919882}& {0.229907246863564}& {-}{0.103114070054204}& {0.195797762081797}& {-}{0.422808679835180}& {-}{0.683981416767635}& {-}{0.138282563735227}& {-}{0.0985077504356053}& {-}{0.153037569187316}& {-}{0.477050101038402}& {0.436005876931923}& {-}{0.809208825557004}& {0.638440795425121}& {-}{0.833039998222847}& {-}{0.136091460283828}& {-}{0.497928870524380}& {-}{0.566541260407573}& {0.427696666784032}& {-}{0.466343770364326}& {0.343225255050994}& {0.763414217685141}& {-}{0.501476518462397}& {-}{0.350270867866055}& {0.717280301089372}& {-}{0.738270244924879}& {0.850737095642760}& {0.236750494691078}& {-}{0.827135215194690}& {0.531019858390454}& {0.441091385372297}& {0.584468894766690}& {0.423381008964853}& {-}{0.664818981530406}& {0.702440745254990}& {-}{0.447590754397073}& {0.255646556264562}& {0.0589045815243775}& {-}{0.266802444024015}& {-}{0.408666870193618}& {0.565496042584256}& {0.563131623191929}& {0.755563170976589}& {-}{0.312821871099248}& {\mathrm{...}}& {"... 924 row vector entries not shown"}\end{array}\right]$ (2)
 > $c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'={'\mathrm{float}'}_{8},'\mathrm{order}'='\mathrm{C_order}'\right):$
 > $\mathrm{ExponentialWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'={'\mathrm{float}'}_{8},'\mathrm{order}'='\mathrm{C_order}'\right),0.72,'\mathrm{container}'=c\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.999679246699440}& {0.999358596281560}& {0.999038048713360}& {0.998717603961850}& {0.998397261994052}& {0.998077022776998}& {0.997756886277729}& {0.997436852463299}& {0.997116921300772}& {0.996797092757220}& {0.996477366799730}& {0.996157743395396}& {0.995838222511324}& {0.995518804114629}& {0.995199488172440}& {0.994880274651894}& {0.994561163520137}& {0.994242154744329}& {0.993923248291639}& {0.993604444129247}& {0.993285742224341}& {0.992967142544124}& {0.992648645055805}& {0.992330249726607}& {0.992011956523762}& {0.991693765414512}& {0.991375676366111}& {0.991057689345822}& {0.990739804320919}& {0.990422021258687}& {0.990104340126421}& {0.989786760891427}& {0.989469283521021}& {0.989151907982529}& {0.988834634243288}& {0.988517462270647}& {0.988200392031963}& {0.987883423494604}& {0.987566556625950}& {0.987249791393389}& {0.986933127764323}& {0.986616565706161}& {0.986300105186323}& {0.985983746172242}& {0.985667488631359}& {0.985351332531126}& {0.985035277839006}& {0.984719324522471}& {0.984403472549005}& {0.984087721886103}& {0.983772072501267}& {0.983456524362014}& {0.983141077435868}& {0.982825731690364}& {0.982510487093049}& {0.982195343611480}& {0.981880301213222}& {0.981565359865853}& {0.981250519536961}& {0.980935780194143}& {0.980621141805009}& {0.980306604337176}& {0.979992167758274}& {0.979677832035943}& {0.979363597137832}& {0.979049463031602}& {0.978735429684923}& {0.978421497065477}& {0.978107665140955}& {0.977793933879058}& {0.977480303247499}& {0.977166773214000}& {0.976853343746294}& {0.976540014812124}& {0.976226786379245}& {0.975913658415419}& {0.975600630888421}& {0.975287703766035}& {0.974974877016056}& {0.974662150606291}& {0.974349524504553}& {0.974036998678669}& {0.973724573096476}& {0.973412247725819}& {0.973100022534555}& {0.972787897490552}& {0.972475872561688}& {0.972163947715849}& {0.971852122920934}& {0.971540398144851}& {0.971228773355518}& {0.970917248520866}& {0.970605823608832}& {0.970294498587367}& {0.969983273424430}& {0.969672148087992}& {0.969361122546031}& {0.969050196766540}& {0.968739370717519}& {0.968428644366980}& {\mathrm{...}}& {"... 924 row vector entries not shown"}\end{array}\right]$ (3)
 > $u≔{\mathrm{~}}_{\mathrm{log}}\left(\mathrm{FFT}\left(c\right)\right):$
 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}$

 > 

Compatibility

 • The SignalProcessing[ExponentialWindow] command was introduced in Maple 18.
 • For more information on Maple 18 changes, see Updates in Maple 18.