【正文】 nt classes algorithm ? DIT FFTDecimationin time FFT ? DIF FFTDecimationin frequency FFT Engineering college, Linyi Normal University ? Twiddle factor WN ? Periodicity in n and k ? Symmetry ? Transmutation ? Special points ( ) ( )k n k n N k N nN N NW W W++== properties of knNW() ()k N n k n k nN N NW W W *==//k n m k n k n mN m N N mW W W=0 / 4 / 23 / 41111NNN N NN N k N N nN N N NW W j WW j W W W= = = = = = =Engineering college, Linyi Normal University Decimationintime FFT ? Algorithm principle suppose N=2v even integer (radix2) 10/ 2 1 / 2 12 ( 2 1 )00/ 2 1/ 2 / 20( 2 ) , int( ) , 0 / 2( 2 1 ) , int( ) ( ) ( ) ( )( 2 ) ( 2 1 )( 2 ) ( 2 1 )Nn k n k n kN N Nn n e v e n n o d dNNrk r kNNrrNrk k rkN N Nrrx r e v e n num be re d po sx n r Nx r odd num be re d po sX k x n W x n W x n Wx r W x r Wx r W W x r W= 撾+====＃+= = +邋 ?= + +邋= + +229。239。239。239。239。239。239。239。239。= +邋= +邋=* ＃ 229。Engineering college, Linyi Normal University ? Circular shift To a sequence x(n) with length N, its circular shifting is defined as: ( ) (( )) ( )NNy n x n m R n=+Engineering college, Linyi Normal University If Then ( ) ( )D F Tx n X k171。238。%1111( ) [ ( m o d ) ] ( ( ) )。=+229。 . , 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。171。= ?= W = ===229。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。229。171。= ?236。%%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。 int , 0 1( ( ) ) 。===%%%1010( ) ( )1( ) ( )NnkNnNnkNnX k x n Wx n X k WN=== 229。(( )) ( )( ) (( ))DFTnkNNDFTnlNNx n m W X kW x n X k l+??Engineering college, Linyi Normal University ? Circular Convolution ? Definition of circular convolution Suppose: two finiteduration sequences:x1(n) and x2(n) Note: The result of circular convolution is also a finiteduration sequence with the duration [0,N1] ? The operate steps can be divided into 3 main steps: ① To period the two sequences with period N。Engineering college, Linyi Normal University Conclusion: If , then the circular convolution of x1(n) and x2(n) is equal to the linear convolution of the two sequences, and time aliasing in the circular convolution of two finitelength sequences can be avoided . Example: suppose x1(n)=x2(n)=u(n)u(n6) (1)pute (2)pute 12( ) ( ) 。=+ 239。=239。= 239。= 239。== 237。= 229。239。237。/ 2 10/ 2 1/20/ 2 1/20( ) ( ) 0 , 1 , 2 , , 1( ) [ ( 2 ) ] ( 2 ) 。239。 LEngineering college, Linyi Normal University /2/20/2/20( ) ( ) ( / 2) 。 y(n) Let: x(n)real part。[G(k)G *(Nk )] Engineering college, Linyi Normal University Example: x(n)=[1,2,0,1]。 。,d。 G*(Nk)=[4j6, j2, 2,2] Therefore: X(k)=[4,1j, 2,1+j] Y(k)=[6,1j, 0,1+j] Engineering college, Linyi Normal University ? Computing 2Npoint real sequence using Npoint FFT First we depose x(n) (2Npoint) into 2 Npoint sequences according to radix2 DITFFT, .: X1(n)=x(2n)。[g(n)+g*(n)]。( 2 1 ) ( ) [ ( ) ] , 0 , 1 , 2 , , / 2 1NnrNnNn n r nN N Nnl e t g n x n