8.25 Periodic Convolution ∑[m]n-m<DS、[k[k ◆ The sum is over the finite interva|0≤m≤N-1 ◆ The value of x, n-m] in the interval0≤msN-1 repeat periodically for m outside of that interval =[2 2|DFS、X 3k=N2XIYalk
28 8.2.5 Periodic Convolution ◆The sum is over the finite interval 0 m N −1 ◆The value of in the interval repeat periodically for m outside of that interval x n−m 2 ~ 0 m N −1 X k X k 1 2 1 1 2 0 N m x m x n m − = − ⎯⎯→ DFS x n x nx n 3 1 2 ~ ~ ~ = − = = − 1 0 3 1 2 ~ 1 ~ ~ N l X l X k l N ⎯⎯→ DFS X k
Example 8.4 Periodic Convolution ,m 0=∑年[m[-m x2[1-m]=x2[(m-1) 元2[1-m [!]=∑m]元[1-m] x2D2-m=x2[(m-2) [2-m [2]=∑m2[2-m] 29
29 Example 8.4 Periodic Convolution 1 3 1 2 0 1 1 N m x x m x m − = = − x m1 x m2 x m 2 − x m 2 1− x m 2 2− 1 3 1 2 0 2 2 N m x x m x m − = = − 1 3 1 2 0 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 DTFT) x]= 1,rmnn∠FT>x%B 2o(v+2丌 x[n]=e Mom FI x(e"=22r8(w-Wwo+2r r) r三-00 =mr,x(“)=∑∑2z ra6(-+2丌 00 30
30 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 (DTFT) : xn=1 for all n X (e ) (w r) r j w = 2 + 2 =− ⎯⎯→ FT jw n x n e 0 = ( ) ( ) =− = − + r j w X e 2 w w 2 r 0 ⎯⎯→ FT ⎯⎯→ FT k k k jw n x n a = e ( ) ( ) =− = − + r k k k j w X e 2 a w w 2 r
8.3 The Fourier Transform of Periodic signal =∑e"”",x(e")=∑∑2ra6(m-+2x) k r=-00 We can combine dfs and fourier transform Fourier transform of periodic sequences Periodic impulse train with values proportional to DES Coefficients x小]=∑X[小 2z/N) k=0 A()x2+2 2兀 2丌k Xk16|
31 8.3 The Fourier Transform of Periodic Signal ◆We can represent Periodic sequences as sums of complex exponentials: DFS ◆We can combine DFS and Fourier transform ◆Fourier transform of periodic sequences: Periodic impulse train with values proportional to DFS coefficients ( ) 1 0 2 2 - 2 N j r k X k k e r N N X − =− = = + ( ) 1 0 1 2 N k j n N k x X N n k e − = = - 2 2 - k k X k N N = = ⎯⎯→ FT k k k jw n x n = a e ( ) 2 2 ( ) k k jw r k X e w a w r =− = − +
8.3 The Fourier Transform of Periodic signal 2丌 2兀k J Xrd k N ◆ This is periodic with2π since dFs is periodic with period N, and the impulses are spaced at integer multiples of 2 /N The inverse transform can be written as 2丌-E on ∑3.2p 2丌J0-6 2T 30-6 k N 2兀k k 2-8 on do=∑X[小 N ∑[l] 0-E k k=0 32
32 8.3 The Fourier Transform of Periodic Signal ◆This is periodic with 2 since DFS is periodic with period N, and the impulses are spaced at integer multiples of 2/N. ( ) - 2 2 - j k k X e X k N N = = ( ) 2 - 2 - 0- 0- - 1 1 2 2 - 2 2 j j n j n k k X e e d X k e d N N = = -1 2 2 - 0 - - 0 1 2 1 - N k j n j n N k k k X k e d X k e N N N = = = = ◆The inverse transform can be written as = x n