8.25 Periodic Convolution For two periodic sequence, both with period N DES DES x[<>X[l],[可<2[小]x小=X1]x2 then[=∑xm[n-m]=∑[m年[n-m Proof: m=0 N-IM X[]=∑[川W=∑∑无m元[n-m知 n=0 n=0m=0 N N N ∑mx[m一m=∑m(W知[ n=0 ∑利如]=]x m=0 29
29 8.2.5 Periodic Convolution 1 1 1 3 3 1 2 0 0 0 N N N kn kn N N n n m X k x n W x m x n m W − − − = = = = = − 1 2 0 1 1 0 N m N kn N n x m x n m W − = − = = − 0 2 1 1 km N N m x m W X k − = = then Proof: ◆For two periodic sequence, both with period N: ( ) 1 0 1 2 N km N m x m W X k − = = = X k X k 1 2 X k X k X k 3 1 2 = 1 1 , DFS x n X k 2 2 . DFS x n X k If 3 1 2 1 0 N m x n x m x n m − = = − 2 1 1 0 N m x m x n m − = = −
8.25 Periodic Convolution ∑[m]n-m]←DS、文[][ major differences between periodic convolutions and aperiodic convolutions The sum is over the finite interval0≤m≤N-1 The value of 2[n-m in the interval 0<m<N-1 repeat periodically for m outside of the interval duality x3-n21。DFS、,[k|=
30 8.2.5 Periodic Convolution ➢ The sum is over the finite interval . 0 1 − m N ➢ The value of in the interval repeat periodically for m outside of the interval x n m 2 − 0 1 − m N 1 2 X k X k 1 2 1 0 N m x m x n m − = − ⎯⎯→ DFS x n x n x n 3 1 2 = 1 3 1 2 0 1 N l X k X l X k l N − = ⎯⎯→ DFS = − ◆ major differences between periodic convolutions and aperiodic convolutions: duality
Example 8.4 Periodic Convolution X lm alm] i,mI [-m] 0=∑[m[m x2[1-m]=x2[(m-1) [=∑m[ x2D2-m=x2[-(mn-2) x[2]=∑[m2[2-m] m=0
31 Example 8.4 Periodic Convolution 2 1 3 0 1 1 1 N m x x m x m − = = − x m1 x m2 x m 2 − x m 2 1− x m 2 2 − 2 1 3 0 1 2 2 N m x x m x m − = = − 2 1 3 0 1 0 N m x x m x m − = = −
8.3 The Fourier Transform (DTFT) of discrete-time Periodic signal Periodic sequences are neither absolutely summable nor square summable, hence they don 't have a strict Fourier Transform (DTFD) x小=10an,x(e")=∑276(m+27r) fT xIn=e e)=∑26(m-1+2x) x=∑aerr,x(e")=∑∑ 2za16(1-12+2xr 32
32 8.3 The Fourier Transform (DTFT) of discrete-time Periodic Signal ◆Periodic sequences are neither absolutely summable nor square summable, x n for all n =1 ( ) ( ) 0 2 2 jw r X e w w r =− = − + ⎯⎯→ FT k k k jw n x n a = e hence they don’t have a strict Fourier Transform (DTFT) : ( ) 2 2 ( ) jw r X e w r =− = + ⎯⎯→ FT 0 jw n x n e = ⎯⎯→ FT ( ) 2 2 ( ) jw k k r k X e a w w r =− = − +
8.3 The Fourier Transform of Periodic signal We can represent periodic sequences as sums of complex exponentials: DFS DFS can be incorporated within framework of dtft ODTFT of periodic sequences: Periodic impulse train with values proportional to dFs coefficients 2e%图p2 = 凶利∑Xl (2/N)kn Periodic k=0 IDTET ()∑2+()Dr proof next ∑∑x410N+2zr)=∑o 2丌k -00k=0 N 33
33 8.3 The Fourier Transform of Periodic Signal ◆We can represent Periodic sequences as sums of complex exponentials: DFS ◆DFS can be incorporated within framework of DTFT. ◆DTFT of periodic sequences: Periodic impulse train with values proportional to DFS coefficients ( ) j X e = x n = - - 2 2 k k N X k N = = ( ) 1 0 1 2 N k j n N k X k N e − = x n = ( ) j X e = 2 2 ( ) r k k w r k a =− − + k a , X k N = k= 2 k N 1 0 2 - 2 2 N r k k N X k r N − =− = = + k k k j n a e FT DTFT Periodic = x n IDTFT proof next