上海交通大学：《数字信号处理 Digital Signal Processing（B）》教学资源_Handouts_ch9 DFT

FS Digital Signal Processing--DFT/FFT Algorithms x(t)=ao+a cosst+az cos22t+...+b sint+b2 sin2t+.. d+(d cosn+b,sinnt) a. x(t)dt a,= x(t)cosn tdt bn= Clo+T x(t)sinntdt 上游充通大

Digital Signal Processing—— DFT/FFT Algorithms FS 0 1 1 2 1 1 1 2 1 x t a a t a t b t b t ( ) cos cos 2 ... sin sin 2 ...            0 1 1 1 ( cos sin ) n n n a a n t b n t         0 0 0 1 ( ) t T t a x t dt T    0 0 1 2 ( )cos t T n t a x t n tdt T     0 0 1 2 ( )sin t T n t b x t n tdt T    

FS RS Digital Signal Processing--DFT/FFT Algorithms x(t)=d+(a cosnt+bsin n2t) n= b=sinm2 n= =d+c(cosp cosnt-sinp sinn!) n= =C+∑ccos(n2t+p,) n- ao Co cn=va,"+b2 -b an =Cn cos pr =-arctan b cos br =-Cn sin pn 上游充通大粤

Digital Signal Processing—— DFT/FFT Algorithms FS 0 1 1 1 ( ) ( cos sin ) n n n x t a a n t b n t         2 2 0 1 1 2 2 2 2 1 ( cos sin ) n n n n n n n n n a b a a b n t n t a b a b             0 1 1 1 (cos cos sin sin ) n n n n a c n t n t           0 1 1 cos( ) n n n c c n t         0 0 a c  2 2 n n n c a b   arctan n n n b a    2 2 sin n n n n b a b     2 2 cos n n n n a a b    cos n n n a c   sin n n n b c   

FS Digital Signal Processing--DFT/FFT Algorithms ∑c,cos2f+n,)=∑c.(eao+eao) n=1 c em) n=l 00 n=12 00 n=1 n= 00 00 =∑X(n2)e+∑X(-n2)ear n=l =∑X(n2)emar+∑Xn2,)ear n=l X(n2)=K(n2em=号c,em 1X2=X(-n2=c, X(-n2 )=X(-n)e=c,e 上游充通大￥

FS Digital Signal Processing—— DFT/FFT Algorithms 1 1 ()() 1 1 1 1 cos( ) ( ) 2 n n j n t j n t n n n n n c n t c e e                   1 1 1 1 1 1 ( ) ( ) jn t jn t n n X n e X n e               1 1 1 1 ( ) 2 n n jn t jn t j j n n c e e e e            1 1 1 1 1 1 2 2 n n j j jn t jn t n n n n c e e c e e               1 1 1 1 1 1 ( ) ( ) jn t jn t n n X n e X n e             1 1 1 ( ) ( ) 2 n n j j X n X n e c en       1 1 1 ( ) ( ) 2 n n j j X n X n e c en           1 1 1 ( ) ( ) 2 X n X n c      n

FT Digital Signal Processing--DFT/FFT Algorithms x(t)=∑Xen=∑Xn22mr 1=-00 X,=2)=%0mh •T/2 imX。=m(e T→00 X TX= X.=2n 天 21 X(j)="x(t)edi 上游充通大

Digital Signal Processing—— DFT/FFT Algorithms FT 1 1 1 ( ) ( ) jn t jn t T n n n x t X e X n e            1 / 2 1 / 2 1 ( ) ( ) T jn t n T T X X n x t e dt T        1 / 2 / 2 lim lim ( ) T jn t n T T T T TX x t e dt        1 1 2 n n n X X TX f     ( ) ( ) j t X j x t e dt       

FT Digital Signal Processing--DFT/FFT Algorithms x0=立0emdk T/2 T- 21 0-立2@e x(t)→x(t)n21→2， 21>d2 e)-小2元 )=2元 上游充〔大学

Digital Signal Processing—— DFT/FFT Algorithms FT 1 1 / 2 / 2 1 ( ) [ ( ) ] T jn t jn t T T T n x t x t e dt e T          1 2 T    1 1 / 2 1 / 2 ( ) [ ( ) ] 2 T jn t jn t T T T n x t x t e dt e            ( ) ( ) T x t x t  1 n   1 ， ， ，   d 1 ( ) [ ( ) ] 2 j t j t x t x t e dt e d             1 ( ) ( ) 2 j t x t X j e d        