1.Definition 1.Definition 1.Definition To verify the validity of IFDT,we multiply .W=e which is usually called twiddle both sides of the expression by Ww and sum Making use of the identity factor has many useful features the result from n=0 to n=N-1,resulting in [N,for k-1=rN,r isan integer It is the first root of the NN-th roots of unity 分mw-】 1 -0 0,otherwise The modulus is I (on the unit circle) 岁w=0 N -0 ·Hence ·联=WgP=Wg 1 ∑∑X)-h omg=XO =0k-0 =0 W8-1 W2=-1 X∑- N- =0 1.Definition 1.Definition 1.Definition Mapping Relations between time-domain and Type 1:Continuous-Time Fourier Transform Type 2:Continuous-Time Fourier Series frequency-domain transforms (CTFT) (CTFS) (Time-domain) (Frequency-domain) Continuous Aperiodical 普x(0 X.(2) P x() >X,U0)6s LDiscrete Periodical x.().d 1, X.())e-ao d Periodical Discrete -Aperiodical◆ Continuous 0-上x.U x)=∑X.Uk2,)ea 121. Definition WN=e-j2 /N which is usually called twiddle N factor has many useful features It i th fi t t f th It is the first root of the N N-th t f it th roots of unity The modulus is 1 (on the unit circle) 1 k kN W W N N   kN k / 2 W W N N    1 1 0 N k N k W     k 1 0 1 WN  / 2 1 N WN   7 1. Definition To verify the validity of IFDT, we multiply both sides of the expression by and both sides of the expression by W ln and sum the result from n = 0 to n=Nˉ1, resulting in WN N NN 1 11 0 00 1 () () N NN ln kn ln N NN n nk xnW X kW W N              0 00 1 1 1 ( ) n nk N N k ln N N X kW N            0 0 1 1 1 ( ) n k N N k ln N N Xk W           0 0 8 ( ) N k n Xk W N     1. Definition Making y use of the identity   1 , for , isan integer N k ln N N k l rN r W          Hence 0 0, otherwise N n W   Hence 1 0 ( ) () N ln N x nW X l    n0 9 1. Definition Mapping Relations between time-domain and frequency-di f oma n transforms (Time-domain) (q y Fre uency-domain) Continuous Aperiodical Discrete Periodical Periodical Discrete Periodical Discrete Aperiodical Continuous 10 1. Definition Type 1: Continuous-Time Fourier Transform (CTFT) Continuous Aperiodical ( ) a x t ( ) X j a  Continuous Aperiodical Aperiodical Continuous ( ) () j t X j x t e dt a a        1 () ( ) 2 j t a a xt X j e d       11 2 1. Definition Type 2: Continuous-Time Fourier Series (CTFS) Continuous Aperiodical ( ) a x t 0 ( ) X jk a  Continuous Periodical Aperiodical Discrete 1 T / 2 0 0 / 2 1 ( ) () p p T jk t a a T p X jk x t e dt T       p 0 0 () ( ) jk t a a k x t X jk e      12 k