▁of ▁a ▁signal ▁and ▁a ▁system ; ▁typically ▁a ▁= ▁- ∞ ▁and ▁b ▁= ▁+ ∞ . ▁ ▁Consider ▁two ▁wave forms ▁f ▁and ▁g . ▁By ▁calculating ▁the ▁convolution , ▁we ▁determine ▁how ▁much ▁a ▁revers ed ▁function ▁g ▁must ▁be ▁shift ed ▁along ▁the ▁x - axis ▁to ▁become ▁identical ▁to ▁function ▁f . ▁The ▁convolution ▁function ▁essentially ▁revers es ▁and ▁sl ides ▁function ▁g ▁along ▁the ▁axis , ▁and ▁calcul ates ▁the ▁integral ▁of ▁their ▁( f ▁and ▁the ▁revers ed ▁and ▁shift ed ▁g ) ▁product ▁for ▁each ▁possible ▁amount ▁of ▁sl iding . ▁When ▁the ▁functions ▁match , ▁the ▁value ▁of ▁( f * g ) ▁is ▁maxim ized . ▁This ▁occurs ▁because ▁when ▁positive ▁areas ▁( pe aks ) ▁or ▁negative ▁areas ▁( tr ough s ) ▁are ▁multip lied , ▁they ▁contribute ▁to ▁the ▁integral . ▁ ▁Fourier ▁transform ▁The ▁Fourier ▁transform ▁is ▁a ▁function ▁that ▁transform s ▁a ▁signal ▁or ▁system ▁in ▁the ▁time ▁domain ▁into ▁the ▁frequency ▁domain , ▁but ▁it ▁only ▁works ▁for ▁certain ▁functions . ▁ ▁The ▁constraint ▁on ▁which ▁systems ▁or ▁signals ▁can ▁be ▁transformed ▁by ▁the ▁Fourier ▁Trans form ▁is ▁that : ▁▁▁ ▁This ▁is ▁the ▁Fourier ▁transform ▁integral : ▁ ▁Usually ▁the ▁Fourier ▁transform ▁integral ▁isn ' t ▁used ▁to ▁determine ▁the ▁transform ; ▁instead , ▁a ▁table ▁of ▁transform ▁pairs ▁is ▁used ▁to ▁find ▁the ▁Fourier ▁transform ▁of ▁a ▁signal ▁or ▁system . ▁The ▁inverse ▁Fourier ▁transform ▁is ▁used ▁to ▁go ▁from ▁frequency ▁domain ▁to ▁time ▁domain