Discrete-Time Fourier Transform For a real sequence xn] X(e/o)land Xre(e Jo are even functions of @, whereas 0(o) and Xm(ejo )are odd functions of o Note: X(ejo)=X(eo)lee(o+27k =X(e yo 6() for any integer k The phase function 0(o) cannot be uniquely specified for any DTFT Copyright C 2001, S K MitraCopyright © 2001, S. K. Mitra 5 Discrete-Time Fourier Transform • For a real sequence x[n], and are even functions of w, whereas and are odd functions of w • Note: for any integer k • The phase function (w) cannot be uniquely specified for any DTFT | ( ) | j X e w (w) ( ) jw re X e ( ) jw Xim e ( 2 ) ( ) | ( ) | j j j k X e X e e w w  w  + = ( ) | ( ) | j j X e e w  w =