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SignalProcessing

 CauchyWindow
 multiply an array of samples by a Cauchy windowing function

 Calling Sequence CauchyWindow( A, alpha )

Parameters

 A - Array of real or complex numeric values; the signal alpha - real numeric constant

Options

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

Description

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

$w\left(k\right)=\frac{1}{1+\mathrm{\alpha }{\left(\frac{2k}{N}-1\right)}^{2}}$

 • 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[CauchyWindow] 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)
 > $\mathrm{CauchyWindow}\left(a,1.23\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{-}{0.352875467614135}& {0.265002466090137}& {-}{0.448254737799080}& {0.416842233869133}& {-}{0.0175786586344172}& {0.00618068433753250}& {-}{0.0959470323276664}& {0.341992308195404}& {0.0635379962356947}& {0.0972704186641977}& {-}{0.333933326209121}& {0.280380160842164}& {-}{0.344277911891418}& {-}{0.314801285948255}& {-}{0.377805807232448}& {0.427317119472900}& {-}{0.166411974553774}& {-}{0.147247278253721}& {0.0564529332270645}& {0.0110310061045436}& {-}{0.248085389562273}& {-}{0.236422416158014}& {0.337500757666865}& {0.183206111127723}& {0.439142111011870}& {-}{0.254052053526393}& {-}{0.412317779444077}& {0.170053859310775}& {-}{0.326512663156941}& {0.155124351224409}& {-}{0.0257055240375264}& {-}{0.225761690815533}& {0.361838539539466}& {-}{0.233743896638183}& {0.326364331999999}& {0.453864680505483}& {-}{0.344738056042437}& {-}{0.153525510251193}& {0.383668723562791}& {0.428822658962375}& {-}{0.461023885476222}& {-}{0.320080784067603}& {-}{0.229422122147427}& {0.359086641884572}& {-}{0.342845938639314}& {0.221260211642880}& {0.199654106297647}& {-}{0.231353965675476}& {-}{0.0747618325408860}& {-}{0.446272087704214}& {-}{0.267307603660255}& {0.394092143916853}& {0.399341689943202}& {-}{0.257936590598686}& {-}{0.353436005172344}& {0.197481482785267}& {-}{0.155627817035103}& {0.193198586595645}& {0.127479263013725}& {-}{0.0573810737247943}& {0.109350879092252}& {-}{0.236985740278002}& {-}{0.384756827129465}& {-}{0.0780680099112651}& {-}{0.0558135540425543}& {-}{0.0870223602475498}& {-}{0.272245206421024}& {0.249719217357452}& {-}{0.465139822371910}& {0.368304261994571}& {-}{0.482297575835479}& {-}{0.0790757120606860}& {-}{0.290363789170504}& {-}{0.331565621439322}& {0.251209758000197}& {-}{0.274896614335636}& {0.203050889676314}& {0.453260866652530}& {-}{0.298813882368121}& {-}{0.209467300078748}& {0.430489957537353}& {-}{0.444683731085242}& {0.514271960817434}& {0.143631538126241}& {-}{0.503612732909089}& {0.324483080728237}& {0.270502197110994}& {0.359719821207258}& {0.261513971673840}& {-}{0.412123422253680}& {0.437012386773880}& {-}{0.279463516431662}& {0.160193064490377}& {0.0370435305620965}& {-}{0.168388494109647}& {-}{0.258851762417949}& {0.359476178507860}& {0.359260173094526}& {0.483758162988665}& {-}{0.201007726435844}& {\mathrm{...}}& {"... 924 row vector entries not shown"}\end{array}\right]$ (2)
 > $c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'={'\mathrm{float}'}_{8},'\mathrm{order}'='\mathrm{C_order}'\right):$
 > $\mathrm{CauchyWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'={'\mathrm{float}'}_{8},'\mathrm{order}'='\mathrm{C_order}'\right),0.72,'\mathrm{container}'=c\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.582346657273673}& {0.583299214981220}& {0.584253018928849}& {0.585208066041645}& {0.586164353200515}& {0.587121877241932}& {0.588080634957668}& {0.589040623094534}& {0.590001838354117}& {0.590964277392514}& {0.591927936820075}& {0.592892813201129}& {0.593858903053730}& {0.594826202849386}& {0.595794709012795}& {0.596764417921581}& {0.597735325906029}& {0.598707429248817}& {0.599680724184749}& {0.600655206900496}& {0.601630873534319}& {0.602607720175810}& {0.603585742865621}& {0.604564937595201}& {0.605545300306524}& {0.606526826891824}& {0.607509513193328}& {0.608493355002986}& {0.609478348062207}& {0.610464488061589}& {0.611451770640649}& {0.612440191387560}& {0.613429745838878}& {0.614420429479280}& {0.615412237741290}& {0.616405166005014}& {0.617399209597875}& {0.618394363794339}& {0.619390623815653}& {0.620387984829575}& {0.621386441950108}& {0.622385990237232}& {0.623386624696635}& {0.624388340279450}& {0.625391131881989}& {0.626394994345470}& {0.627399922455759}& {0.628405910943100}& {0.629412954481849}& {0.630421047690213}& {0.631430185129981}& {0.632440361306261}& {0.633451570667218}& {0.634463807603807}& {0.635477066449514}& {0.636491341480091}& {0.637506626913293}& {0.638522916908620}& {0.639540205567052}& {0.640558486930793}& {0.641577754983006}& {0.642598003647559}& {0.643619226788765}& {0.644641418211120}& {0.645664571659053}& {0.646688680816665}& {0.647713739307475}& {0.648739740694164}& {0.649766678478323}& {0.650794546100198}& {0.651823336938439}& {0.652853044309850}& {0.653883661469135}& {0.654915181608651}& {0.655947597858161}& {0.656980903284584}& {0.658015090891748}& {0.659050153620150}& {0.660086084346706}& {0.661122875884510}& {0.662160520982595}& {0.663199012325690}& {0.664238342533978}& {0.665278504162863}& {0.666319489702733}& {0.667361291578718}& {0.668403902150466}& {0.669447313711902}& {0.670491518491001}& {0.671536508649557}& {0.672582276282957}& {0.673628813419949}& {0.674676112022424}& {0.675724163985189}& {0.676772961135743}& {0.677822495234061}& {0.678872757972373}& {0.679923740974950}& {0.680975435797886}& {0.682027833928887}& {\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[CauchyWindow] command was introduced in Maple 18.