Chapter 8 the discrete fourier transform 8. 1 representation of periodic sequences the discrete fourier series 8.2 the fourier transform of periodic signals 8.3 properties of the discrete fourier series 8. 4 fourier representation of finite-duration sequences Definition of the discrete fourier transform 8.5 sampling the fourier transform(point of sampling) 8.6 properties of the fourier transform 8.7 linear convolution using the discrete fourier transform 8.8 the discrete cosine transform(DCT
8.1 representation of periodic sequences:the discrete fourier series 8.2 the fourier transform of periodic signals 8.3 properties of the discrete fourier series 8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform 8.5 sampling the fourier transform(point of sampling) 8.6 properties of the fourier transform 8.7 linear convolution using the discrete fourier transform 8.8 the discrete cosine transform(DCT) Chapter 8 the discrete fourier transform
8. 1 representation of periodic sequences: the discrete fourier series A小=x>Xkx(.n=0,N-1 X[k]=∑xnW(2),k=0,N-1 O J -erin/N X[k+rN]=∑mW (k +rN)n XTi n=0
8.1 representation of periodic sequences: the discrete fourier series kn j kn N N N n kn N N k kn N W e X k x n W k N X k W n N N x n 2 / 1 0 ~ ~ 1 0 ~ ~ [ ] [ ] (2), 0,.. 1 [ ] (1), 0,.. 1 1 [ ] − − = − = − = = = − = = − [ ] ( ) ~ [ ] ~ [ ] ~ 1 0 X k N k rN n X k rN x n W N n − = = + + =
EXAMPLE xn 10 012345678910 Figure 8.1 N=10 XIk Wio se-j4nk/10 sin k/2 k in (ck/10)
sin( /10) sin( / 2) [ ] 4 0 4 /1 0 1 0 ~ k k X k W e n kn j k = − = = Figure 8.1 EXAMPLE. N=10
XIkll 1012345678910 20 k ≮X[k k x denotes indeterminate s (magnitude=o) phase x denotes: magnitude=0, phase is indeterminate Figure 8.2
Figure 8.2 phase X denotes：magnitude=0，phase is indeterminate
8.2 the fourier transform of periodic signals 2丌 丌 2丌 X(e)=∑ [k]6(O 2k k N N N 0 other dispersion in time domain results in periodicity in frequency domain; Periodicity in time domain results in dispersion in frequency domain D FS is a method to calculate frequency spectrum of periodic signals
8.2 the fourier transform of periodic signals − = = = − = 1 0 0 2 2 [ ] ~ ) 2 [ ] ( 2 ~ ( ) ~ N k j other k N N X k N k X k N X e dispersion in time domain results in periodicity in frequency domain; Periodicity in time domain results in dispersion in frequency domain. DFS is a method to calculate frequency spectrum of periodic signals
IX(ejo)l. IXIk 4丌 20 a ≮X(e),≮X|k T 10 20 (b) FIgure 8.5
FIGURE 8.5
8.3 properties of the discrete fourier series DES DES DES [n]X[k],x[]X1[k,x2[m]x2[k DES 1.linearity: ax,[n]+bx2In axi[k]+bx2k N=4, 12 points DFS two periodic sequences with different period N=6, 12 points DFS both period=12 compositive sequence N=12, 12 points dFs
[ ] ~ [ ] ~ [ ], ~ [ ] ~ [ ], ~ [ ] ~ 1 1 2 2 X k DFS X k x n DFS X k x n DFS x n 8.3 properties of the discrete fourier series [ ] ~ [ ] ~ [ ] ~ [ ] ~ 1. : 1 2 1 2 aX k bX k DFS linearity ax n bx n + + N=4，12 points DFS N=6，12 points DFS compositive sequence N=12，12 points DFS two periodic sequences with different period both period=12
des km 2. shift of a sequence: xn-m w Xk l DES w xi Xk-a DES 3.duality: XIn Nx[k]
[ ] ~ [ ] ~ 2. : X k N k m W DFS shift of a sequence x n m − [ ] ~ [ ] ~ X k l DFS x n N nl W − − [ ] ~ [ ] ~ 3. : Nx k DFS duality X n −
4.sy mmetry properties DFS DFS x*[n]X*[一k],x*[一1 ⅹ*[k] DFS 1 Re{X[m}=(x山+X*[m)(X[]+*[-k])=Xk] <)2 DFS 1 jImin=-(x[n]-x *nD) Xk]—X*[k])=X。[k )2 DFS 1 x=(x]+x*[一n)(X队]+X*k])=Re{[k]} DES x[]=(x[n]-X*[-m) 付2(X-Xk)=jm(k}
X*[k] DFS~ x *[ n] ~ X*[ k], DFS~ x *[n] ~ 4.symmetry properties: − − X [k] ~ X*[ k]) ~ X[k] ~ ( 2 DFS 1 x *[n]) ~ x[n] ~( 2 1 x[n]} ~ Re{ = + + − = e X [k] ~ X*[ k]) ~ X[k] ~ ( 2 DFS 1 x *[n]) ~ x[n] ~( 2 1 x[n]} ~ jIm{ = − − − = o X[ ]} ~ X*[k]) Re{ ~ X[k] ~ ( 2 DFS 1 x *[ n]) ~ x[n] ~( 2 1 x [n] ~ e + = k = + − [ ]} ~ *[ ]) Im{ ~ [ ] ~ ( 2 1 *[ ]) ~ [ ] ~( 2 1 [ ] ~ X k X k j X k DFS xo n x n x n − = = − −
For a real sequence: [n]=xIn Xk]=X*[k Rexik=refxl-k Im(Xk=-ImXl-k X[]=X[-k] 4X[k]=-4X[-k]
[ ] *[ ] ~ ~ x n = x n [ ] ~ [ ] ~ [ ]| ~ [ ]| | ~ | [ ]} ~ [ ]} Im{ ~ Im{ [ ]} ~ [ ]} Re{ ~ Re{ X k X k X k X k X k X k X k X k = − − = − = − − = − *[ ] ~ [ ] ~ X k = X −k For a real sequence: