able ▁windows ▁ ▁Gaussian ▁window ▁▁ ▁The ▁Fourier ▁transform ▁of ▁a ▁Gaussian ▁is ▁also ▁a ▁Gaussian ▁( it ▁is ▁an ▁eigen function ▁of ▁the ▁Fourier ▁transform ). ▁ ▁Since ▁the ▁Gaussian ▁function ▁extends ▁to ▁infinity , ▁it ▁must ▁either ▁be ▁trunc ated ▁at ▁the ▁ends ▁of ▁the ▁window , ▁or ▁itself ▁window ed ▁with ▁another ▁zero - ended ▁window . ▁ ▁Since ▁the ▁log ▁of ▁a ▁Gaussian ▁produces ▁a ▁par ab ola , ▁this ▁can ▁be ▁used ▁for ▁nearly ▁exact ▁quadratic ▁interpolation ▁in ▁frequency ▁estimation . ▁ ▁The ▁standard ▁deviation ▁of ▁the ▁Gaussian ▁function ▁is ▁ σ   ·   N / 2 ▁sampling ▁periods . ▁ ▁Conf ined ▁Gaussian ▁window ▁ ▁The ▁conf ined ▁Gaussian ▁window ▁yields ▁the ▁smallest ▁possible ▁root ▁mean ▁square ▁frequency ▁width ▁ ▁for ▁a ▁given ▁temporal ▁width ▁  . ▁These ▁windows ▁optimize ▁the ▁R MS ▁time - fre quency ▁band width ▁products . ▁They ▁are ▁computed ▁as ▁the ▁minimum ▁eigen ve ctors ▁of ▁a ▁parameter - dependent ▁matrix . ▁The ▁conf ined ▁Gaussian ▁window ▁family ▁contains ▁the ▁S ine ▁window ▁and ▁the ▁Gaussian ▁window ▁in ▁the ▁limit ing ▁cases ▁of ▁large ▁and ▁small ▁, ▁respectively . ▁ ▁Appro xim ate ▁conf ined ▁Gaussian ▁window ▁▁ ▁Def ining ▁  ,   ▁a ▁Conf ined ▁Gaussian ▁window ▁of ▁temporal ▁width ▁    ▁is ▁well ▁approxim ated ▁by : ▁ ▁where ▁ ▁is ▁a ▁Gaussian ▁function : ▁ ▁The ▁standard ▁deviation ▁of ▁the ▁approximate ▁window ▁is ▁asympt ot ically ▁equal ▁( i . e . ▁large ▁values ▁of ▁) ▁to ▁