▁and , ▁more ▁specifically , ▁that ▁the ▁function ▁goes ▁sufficiently ▁rapidly ▁toward ▁zero . ▁ ▁App lications ▁▁ ▁Window ▁functions ▁are ▁used ▁in ▁spectral ▁analysis / mod ification / res yn thesis , ▁the ▁design ▁of ▁finite ▁imp ulse ▁response ▁filters , ▁as ▁well ▁as ▁beam form ing ▁and ▁anten na ▁design . ▁ ▁Spect ral ▁analysis ▁▁ ▁The ▁Fourier ▁transform ▁of ▁the ▁function ▁cos   ω t ▁is ▁zero , ▁except ▁at ▁frequency   ± ω . ▁ ▁However , ▁many ▁other ▁functions ▁and ▁wave forms ▁do ▁not ▁have ▁convenient ▁closed - form ▁transform s . ▁Alternatively , ▁one ▁might ▁be ▁interested ▁in ▁their ▁spectral ▁content ▁only ▁during ▁a ▁certain ▁time ▁period . ▁ ▁In ▁either ▁case , ▁the ▁Fourier ▁transform ▁( or ▁a ▁similar ▁transform ) ▁can ▁be ▁applied ▁on ▁one ▁or ▁more ▁finite ▁intervals ▁of ▁the ▁wave form . ▁ ▁In ▁general , ▁the ▁transform ▁is ▁applied ▁to ▁the ▁product ▁of ▁the ▁wave form ▁and ▁a ▁window ▁function . ▁ ▁Any ▁window ▁( including ▁rect angular ) ▁affect s ▁the ▁spectral ▁estimate ▁computed ▁by ▁this ▁method . ▁ ▁Cho ice ▁of ▁window ▁function ▁▁ ▁Window ing ▁of ▁a ▁simple ▁wave form ▁like ▁cos   ω t ▁causes ▁its ▁Fourier ▁transform ▁to ▁develop ▁non - zero ▁values ▁( comm only ▁called ▁spectral ▁leak age ) ▁at ▁frequencies ▁other ▁than ▁ ω . ▁The ▁leak age ▁tends ▁to ▁be ▁worst ▁( high est ) ▁near ▁ ω ▁and ▁least ▁at ▁frequencies ▁far th est ▁from   ω . ▁ ▁If ▁the ▁wave