电子科技大学：《数字信号处理》课程教学资源（PPT课件讲稿，英文版）Chapter 3 TransformDomain-Representation of Discrete-Time Signals

83.1 Discrete-Time Fourier Transform · The dTFtX(eio) of a sequence x叫lisa continuous function of o It is also a periodic function of a with a period 2T X(e (0o+2k -j(Oo+2 tk)n =∑ xinle yoon2mhn=∑xmle-10n=Y(eo

§3.1 Discrete-Time Fourier Transform • The DTFT X(ej) of a sequence x[n] is a continuous function of  • It is also a periodic function of  with a period 2: =   =−  +  −  +  n j o k j o k n X e x n e ( 2 ) ( 2 ) ( ) [ ] j kn n j n x n e e o −   =− −  =  2 [ ] [ ] ( ) o o j n j n x n e X e   =− −  =  =

83.1 Discrete-Time Fourier Transform · Therefore X(e0)=∑ xInle@ 1=-00 represents the Fourier series representation of the periodic function As a result, the Fourier coefficients xn can be computed from X(eJo) using the Fourier integral xn]=∫X(e)e Jon 2丌

§3.1 Discrete-Time Fourier Transform • Therefore =   =−  −  n j j n X (e ) x[n]e    =  −   x n X e e d j j n ( ) 2 1 [ ] As a result, the Fourier coefficients x[n] can be computed from X(ej) using the Fourier integral represents the Fourier series representation of the periodic function

83.1 Discrete-Time Fourier Transform Inverse discrete-time fourier transform xn]=∫X(e)eoao 2几一 Proof: XIn ∑x[(]k le Joe.jon 2兀

§3.1 Discrete-Time Fourier Transform • Inverse discrete-time Fourier transform:    =  −   x n X e e d j j n ( ) 2 1 [ ]           =  −   =− −  x n x e e d j j n   [] 2 1 [ ] Proof:

83.1 Discrete-Time Fourier Transform 2. The order of integration and summation can be interchanged if the summation inside the brackets converges uniformly, i.e. X(ej@o) exists ·Then ∫∑xle-o|eoo 2 ∑xq5inz(n=O 丌(n

§3.1 Discrete-Time Fourier Transform • The order of integration and summation can be interchanged if the summation inside the brackets converges uniformly, i.e. X(ej) exists • Then            −   =− −  x e e d j j n   [] 2 1 ( ) sin ( ) [ ] 2 1 [ ] ( )        − − =         =     − =− −  =− n n x e d x j n       

83.1 Discrete-Time Fourier Transform OW SIn gt(n 丌(n-()0,n≠C S|n-0 Hence sing(n ∑x(] l) ∑x(]δ[n-]=x[r π(n-C)c

§3.1 Discrete-Time Fourier Transform • Now [ ] [ ] [ ] ( ) sin ( ) [ ] x n x n n n x =  d − =  −  −   =−  =−       = d[n − ]      = =  −  − n n n n 0, 1, ( ) sin ( ) Hence