【正文】 選 ? 3: 。y(n)=[2,2,1,1] Solution: Let g(n)=x(n)+jy(n)=[1+j2, 2+j2, j, 1+j] Then according to the radix2 DITFFT, we get G(k)=[4+j6, 2,2,j2]。 y(n)imaginary part ., g(n)=x(n)+jy(n) Then x(n)=189。 ( ) ( ) ( / 2)( 2 ) ( ) [ ( ) ] 。 + = 239。( ) [ ( 2 1 ) ] ( 2 1 ) 0 , 1 , 2 , , / 2 1NkNNrkNrNrkNrG k W H k k Nw he re G k D FT x r x r WH k D FT x r x r W k N==229。239。254。239。239。239。254。239。239。 6x n x n N?12( ) ( ) 。 ② To pute the periodic convolution of the two periodic sequences ③ To get out the duration sequence between [0,N1] 11 2 1 20( ) ( ) ( ) ( ( ) ) , 0 1NNmx n x n x m x n m n N=? ＃ 229。= 229。 ( ( ) ) ( )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)N x xxxxx= = == ? \ = \ = = ? \ = \ =%%%%Q%%QEngineering college, Linyi Normal University ? The same as doing in the timedomain ? On the other hand : Therefore: ( ) , 0 1()0,( ) [ ( m od ) ] ( ( ) )( ) ( ) ( )NNX k k NXkothe rw iseX k X k N X kor X k X k R n236。 ＃ 239。 ＃ 239。%%%%%%[ ]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 ( ) 。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。 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。= ?165。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。Engineering college, Linyi Normal University Engineering college, Linyi Normal University The Discrete Fourier Transform ? Suppose: x(n)finitelength sequence, Nlength。238。238。239。= ＃ 229。= ?? 229。239。?+253。239。237。 =239。239。= 229。239。238。 =+239。 Engineering college, Linyi Normal University Decimationinfrequency FFT ? Algorithm principle ? Suppose N=2v even integer 1 / 2 1 10 0 / 2/ 2 1 / 2 1( / 2 )00/ 2 1 / 2 12 2 ( / 2 )00/ 2 1/20( ) ( ) ( ) ( )( ) ( / 2): ( 2 ) ( ) ( / 2)[ ( ) ( / 2) ] , 0 , 1 , 2 , ,N N Nnk nk nkN N Nn n n NNNnk n N kNNnnNNnr n N rNNnnNnrNnX k x n W x n W x n Wx n W x n N Wso X r x n W x n N Wx n x n N W r = = =+==+==== = +邋 ?= + +邋= + +邋= + + =229。,n k n kNNWW101( ) ( )N nkNkx n X k WN**=輊= 229。[G(k)+G *(Nk )]。 Examples Engineering college, Linyi Normal University 參考書閱讀和作業(yè) ? Textbook: ~172 ? Chinese reference book: ~37, ~82 ,~109 ? Exercises: ? 1: ,d。 。 ? Convolutiontime parison of using FFT versus DFT。[g(n)g*(n)]。 LFor example: N=8, the flow