▁it . ▁ ▁But ▁as ▁Harris ▁points ▁out , ▁the ▁D FT ▁requires ▁an ▁imp lied ▁shift ▁of ▁the ▁symmetric ▁sequence ▁from ▁a ▁domain ▁ ▁known ▁as ▁a ▁finite ▁Fourier ▁transform ▁data ▁window , ▁into ▁ ▁a ▁D FT ▁data ▁window . ▁ ▁Effect ively , ▁the ▁real - valu ed ▁function ▁being ▁sample d ▁is ▁multip lied ▁by ▁a ▁complex - valu ed ▁phase ▁function : ▁ ▁  ▁ ▁Therefore , ▁a ▁straightforward ▁D FT ▁of ▁length ▁ 2 M + 1 ▁would ▁sample ▁at ▁frequency ▁intervals ▁of ▁ 1 /( 2 M + 1 ), ▁and ▁the ▁samples ▁would ▁be ▁complex - valu ed , ▁whereas ▁samples ▁at ▁intervals ▁of ▁ 1 /( 2 M ) ▁are ▁all ▁real , ▁although ▁the ▁odd - number ed ▁ones ▁are ▁neg ated ▁relative ▁to ▁the ▁original ▁D T FT . ▁ ▁To ▁compute ▁samples ▁at ▁intervals ▁of ▁ 1 /( 2 M ), ▁using ▁a ▁D FT , ▁its ▁length ▁must ▁be ▁ 2 M . ▁ ▁Two ▁methods ▁available ▁are ▁simple ▁trunc ation ▁( de let ing ▁the ▁last ▁sample ) ▁and ▁periodic ▁summ ation ▁( com bin ing ▁the ▁first ▁and ▁last ▁samples ▁by ▁addition ). ▁ ▁Both ▁result ▁in ▁a ▁sequence ▁length ▁of ▁ 2 M , ▁to ▁which ▁a ▁D FT - even ▁window ▁can ▁be ▁applied . ▁ ▁The ▁difference ▁is ▁that ▁trunc ation ▁affect s ▁the ▁D T FT ▁function ▁being ▁sample d , ▁and ▁periodic ▁summ ation ▁affect s ▁only ▁the ▁sampling ▁intervals . ▁ ▁In