Discrete Fourier Transform Chapter 5 Part A ◆Definition ●●● ●●● ●●● 色● ◆Matrix Relations ●●0●6 Finite Length The Discrete Fourier DFT Computation Using MATLAB Discrete Transforms Transform(DFT) Relation between DTFT and DFT and their inverses 1.Definition 1.Definition 1.Definition Definition .From the definition of the DTFT we thus have Using the notation W=pr,the DFT is The simplest relation between a length-N X)=X(e。-2aN sequence x(n),defined for OsnN-I and its usually expressed as: DTFT X(e)is obtained by uniformly =∑x(n)e2saN, 0≤k≤N-1 X(k)=∑x(n)W,0≤k≤N-1 sampling X(ei)on the @-axis between Osos m=0 2πato4=2πkW,0s≤N-1 .X(k)is also a length-N sequence in the The inverse discrete Fourier transform (IDFT) frequency domain is given by .DFT is obtained by sampling the DTFT over The sequence X(k)is called the discrete 1 one principal value interval in the frequency x(n)= ∑Xk)w,0≤n≤N-1 Fourier transform(DFT)of the sequence x(n) domain 4 6Chapter 5 Chapter 5 Finite Length Discrete Transforms Part A The Discrete Fourier Transform (DFT) Discrete Fourier Transform Definition Matrix Relations DFT Computation Using MATLAB R l ti b t DTFT d DFT d Relation between DTFT and DFT and their inverses 3 1. Definition Definition Th i l li b l h he simplest relation between a length-N sequence x(n), defined for 0nNˉ1 and its DTFT X j (e ) is obtained by uniformly sampling X(ej) on the -axis between 0 2 at k= 2 k/N, 0k Nˉ1 DFT is obtained by sampling the DTFT over DFT is obtained obtained by sampling sampling the DTFT over one principal value interval in the frequency domain 4 1. Definition From the definition of the DTFT we thus have 2 / 1 2 / () ( ) () 0 1 j k N N j kn N X k Xe k N     X(k) is also a length N sequence in the 2 / 0 ( ) , 0 1 j kn N n x n e k N       X(k) is also a length-N sequence in the frequency domain The sequence X(k) is called the discrete Fourier transform (DFT) of the sequence x(n) 5 1. Definition Using the notation WN=e-j2 /N g , the DFT is N , usually expressed as: 1 () () 0 1 N kn Xk W k N     The inverse discrete Fourier transform (IDFT) 0 () () , 0 1 kn N n X k x n W kN       The inverse inverse discrete discrete Fourier Fourier transform transform (IDFT) is given by 1 1 N 0 () () , 0 1 N kn N k xn X kW n N N       6