【正文】 Chapter 3. The Discrete Fourier Transform Engineering college, Linyi Normal University The Discrete Fourier Series ? Definition: Periodic sequence ? N: the fundamental period of the sequences ? From FT analysis we know that the periodic functions can be synthesized as a linear bination of plex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency (2pi/N). ? From the frequencydomain periodicity of the DTFT, we conclude that there are a finite number of harmonics。 the frequencies are {2pi/N*k,k=0,1,… ,N1}. knkNnxnx ,),(~)(~ ???Engineering college, Linyi Normal University ? Fourier series of periodic continuous signals ? Ω0— period of x(t) in radian。 Let Tsampling period。 ω0smpling period in radian 00( ) ( )jk tkx t X k e165。W= ?=W229。22000222( ) ( )()NNj k nTNkj k nkTTTNx n X k eX k epppppw165。= ?165。= ?= W = ===229。229。Engineering college, Linyi Normal University 00022 2 2200010001011()00101( ) ( ), , ,1( ) ( )11( ) ( ) ( )1( ) ( )NN N NNTjk tNTnNjk nnNNj k m N n jk n jm N nnnNjk nnX k x t e dtTt nT T NT dt TX k x n eNX k m N x n e x n e eNNx n e X kNpp p ppW== + ===W==+ = ===242。229。242。229。邋229。QSo X(k) is also a periodic function with N Engineering college, Linyi Normal University ? DFS pair ?,1,0,)(~1)(~102??? ???nekXNnxNkknj N ??,1,0,)(~)(~102 ??? ? ??? kenxkX Nnknj N??????????????1010)(~1)](~[I D F S)(~)(~)](~[D F S)(~L e t2NknkNNnnkNjNWkXNkXnxWnxnxkXeW N?Engineering college, Linyi Normal University ? Properties of DFS Suppose the following 3 sequences’s period is N ? Linearity 112233( ) ( )( ) ( )( ) ( )D F SD F SD F Sx n X kx n X kx n X k171。171。171。%%%%%%[ ]1 2 1 2( ) ( ) ( ) ( )D FS ax n bx n aX k bX k+ = +%%%%Engineering college, Linyi Normal University ? Shifting ? Symmetry [ ]( ) ( )( ) ( )mkNnlND F S x n m W X kI D F S X k l W x n+=輊 +=犏臌%%% %[ ]{ }[ ]{ }[ ]{ }******( ) ( )( ) ( )11R e ( ) ( ) ( ) ( ) ( )2211I m ( ) ( ) ( ) ( ) ( )22:( ) R e ( ) 。 ( ) I meoDFS x n X kDFS x n X kDFS x n DFS x n x n X k X N kDFS j x n DFS x n x n X k X N kde f i neX k DFS x n X k DFS j**輊 =犏臌輊 =犏臌輊輊= + = + 犏犏臌臌輊輊= = 犏犏臌臌==%%%%%%% % %%%% % %%% %%[ ]{ }()xnEngineering college, Linyi Normal University ? Periodic convolution ?Distinction with convolution sum 3 1 213 3 1 2013 3 1 20( ) ( ) ( )( ) [ ( ) ] ( ) ( )( ) [ ( ) ] ( ) ( )NmNmif X k X k X kthe n x n I D F S X k x m x n mo r x n I D F S X k x n m x m==== = 229。= = 229。% % %%% % %%% % %1 2 1 2( ) ( ) ( ) ( )mx n x n x m x n m165。= ?* = 229。Engineering college, Linyi Normal University Engineering college, Linyi Normal University The Discrete Fourier Transform ? Suppose: x(n)finitelength sequence, Nlength。 . , x(n)=0 when n0 or nN1 ? Let x(n) be a period sequence of a periodic sequence Then we have ()xn%( ) , 0 1()0,( ) ( )rx n n Nxnot he r w is eandx n x n r N165。= ?236。 ＃ 239。239。= 237。239。239。238。=+229。%%Engineering college, Linyi Normal University ? It can be written as ? Further, we usually express it as below ( ) ( ) ( )in t1 , 0 1()0,NNx n x n R nth e re onNRno th e rwise=236。 ＃ 239。239。=237。239。239。238。%1111( ) [ ( m o d ) ] ( ( ) )。 int , 0 1( ( ) ) 。 ( ( ) ) ( )NNNx n x n N x nnif m n m e g e r n NNthe n x n n so x n x n=== 危 ?==%LEngineering college, Linyi Normal University ? Example 888 , ( 18 ) ? , ( 4) ?18 2 8 2 , ( ( 18 ) ) 2 , ( 18 ) ( 2)4 ( 1 ) 8 4 , ( ( 4)) 4 , ( 4) ( 4