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20(DFT)FFTX(k)=DFT[x(n)]=∑−=10)(NnknNWnx0≤k≤N-1x(n)=IDFT[X(k)]=N1∑−=−10)(NkknNWkX0≤n≤N-1DFTDFT[a)(1nx+b)(2nx]=aDFT[)(1nx]+bDFT[)(2nx]))((Nnx−=⎩⎨⎧−)()0(nNxx1,,2,10−==NnnKDFT[))((Nnx−]=))((NkX−=X(N-k)DFT[x*(n)]=))((*NkX−=X*(N-k)DFT[))((Nsnx−]=ksNWDFT[x(n)]DFT[snNW−x(n)]=))((NskX−)(3kX=)(1kX)(2kX)(3nx=∑−=−1021))(()(NmNmnxmx=)(1nx)(2nx=)(2nx)(1nx)(3nx=)(1nx)(2nx)(3kX=N1)(1kX)(2kX=N1)(2kX)(1kX)(1nxn1N)(2nxn2NL=1N+2N1)(1nx)(2nxL2-1NDFT0nNx(n)=1x(n)=δ(n)21x(n)=δ(n-0n)00nNx(n)=2ax(n)=Nmnje/2π0mNx(n)=cos(2πmn/N)0mNX(k)=∑−=−10/2NnNknjeπk=0,X(k)=∑−=101Nn=Nk≠0,X(k)=∑−=−10/2NnNknjeπ=Nkjkjee/2211ππ−−−−=0X(k)=Nδ(k)X(k)=DFT[δ(n)]=1X(k)=DFT[δ(n-0n)]=0knNW=02knNjeπ−X(k)=∑−=102NnknNWa=N2aδ(k){}X(k)=∑−=−10/2/2NnNknjNmnjeeππ=∑−=−−10/)(2NnNnmkjeπ=Nδ(k-m)x(n)=cos(2πmn/N)=(Nmnje/2π+Nmnje/2π−)/2DFT)(2kX)(2kX∑−=−10/2NnknNNmnjWeπ=∑−=+10)(NnnmkNWNmk)(+=0,)(2kX=N.0≤kN,0mN,0k+m2N,Nmk)(+=0k=N-m,k)(2kX=0m≠N/2X(k)=N[δ(k-m)+δ(k+m-N)]/2N=1024DFT10kHzHz22kfk=0f=0k=Nf=SfSf/N=10000/1024≈10Hz[2-3]DFT∑−=102|)(|Nnnx=N1∑−=102|)(|NkkX∑−=102|)(|Nnnx=∑−=10)(*)(Nnnxnx=∑∑−=−=−1010)(*])(1[NnNkknNnxWkXN,,∑−=102|)(|Nnnx=N1∑−=10)(NkkX∑−=−10)(*NnknNWnx=N1∑−=10)(NkkX∑−=10*))((*NnknNWnx=N1∑−=10)(NkkX∑−=10*])([NnknNWnx=N1∑−=10)(NkkXX*(k)=N1∑−=102|)(|NkkX[2-4]X(k)=DFT[x(n)],)(1nx=∑−=10)(NkknNWkX,)(1nxx(n))(1nx=∑∑−=−=1010])([NkknNNrkrNWWrx=∑−=10)(Nrrx∑−=+10)(NknrkNW[1-1](6)∑−=+10)(NknrkNW=⎩⎨⎧0NnnNr−=)(1nx=Nx(N-n)[2-5]X(k)=DFT[x(n)],y(n)=⎩⎨⎧0)(nx110−≤≤−≤≤rNnNNnDFT[y(n)]X(k)rrY(k)=DFT[y(n)]rNX(k)NY(k)X(k)23Y(k)=DFT[y(n)]=∑−=10)(rNnknrNWny=∑−=10)(NnknrNWnx0≤k≤rN-1X(k)=∑−=10)(NnknNWnx=∑−=10)(NnrnkrNWnx0≤k≤N-1X(k)=Y(rk)(FFT)FFTknNWknNW)(NnkNW+nNkNW)(+)(nNkNW−knNW−*)(knNWFFTN=M2MDITFFTx(n)nDIFFFTx(n)X(k)kFFTMN/2FFTDITN=4)(1rx=x(2r),)(2rx=x(2r+1)r=0,1)(1kX=∑=1021)(rrkWrx,)(2kX=∑=1022)(rrkWrxk=0,12W=-1)(kX=)(1kX+kW4)(2kX)2(+kX=)(1kXkW4)(2kXk=0,1)(1kX)(2kX)(kXx(0)X(0)x(1)X(2)x(2)X(1)-1x(3)X(3)-1-104W04W04W14W24DIFN=4X(2r)=∑=++10)]2()([nnxnxnrW2X(2r+1)=∑=+−104)]2()([nnWnxnxnrW2r=0,1x(0)X(0)x(2)X(1)-1x(1)X(2)-1x(3)X(3)-1-1[2-6]4FFTDIFX(2r)=∑=++10)]2()([nnxnxnrW2n=0,1X(2r+1)=∑=+−104)]2()([nnWnxnxnrW2r=0,1[]x(0)X(0)x(1)X(2)-1x(2)X(1)-1x(3)X(3)-1-1[2-7]DIF8FFTDIFX(2r)=∑−=++120)]2()([NnNnxnxnrNW2/04W04W14W04W04W04W14W04W25X(2r+1)=∑−=+−120)]2()([NnnNWNnxnxnrNW2/48x(0)X(0)x(4)X(1)-1x(2)X(2)-1x(6)X(3)-1-1x(1)X(4)-1x(5)X(5)-1-1x(3)X(6)-1-1x(7)X(7)-1-1-1[2-8]x(n)DFTX(k)DFTX(k)=DFT[x(n)]=∑−=10)(NnknNWnxx(n)=N1∑−=−10)(NkknNWkX=N1∑−=10*))((NkknNWkX=N1∑−=10*])(*[NkknNWkXX*(k)DFTNx(n)[2-9]x(n)MzX(z)=∑−=−10)(NnnznxX(z)N)(kzXkz=Nkje/2π,k=0,1,K,N-1N≤MNMNFFT)(kzX0NW2NW0NW0NW1NW3NW2NW0NW0NW0NW0NW2NW26)(kzX=∑=−MnNknjenx0/2)(πN=M,)(kzX=X(k)=DFT[x(n)]NM,x(n),)(0nx=⎩⎨⎧0)(nx110−≤≤−≤≤NnMMn)(kzX=DFT[)(0nx],NFFTNM,kzN,(r1)N≤MrN)(kzX=∑=−MnNknjenx0/2)(π=∑−=−10/2)(NnNknjenxπ+∑−=−12/2)(NNnNknjenxπ+K+∑−−=−1)1(/2)(MNrnNknjenxπNknje/2π−,)(kzX=∑−=−10/2)(NnNknjenxπ+∑−=−+10/2)(NnNknjeNnxπ+K+∑−−−=−−+1)1(0/2])1([NrMnNknjeNrnxπMx(n)rN)(nyi,1,,1,0−=riK)(0ny=)()(nRnxN,)(1ny=)()(nRNnxN+,K,)(nyi=)()(nRiNnxN+,K,)(1nyr−=⎩⎨⎧−+0])1([Nrnx1)1(1)1(0−≤≤−−−−−≤≤NnNrMNrMn)(0nx=∑−=10)(riiny,)(kzX=DFT[)(0nx],NFFT[2-10]X(k),k=0,1,K,2N-12Nx(n)DFTX(k)x(n)NIFFTFFT2NNDFT)(1kX)(2kX27X(k)=)(1kX+kNW2)(2kXX(N+k)=)(1kXkNW2)(2kX0≤k1−≤N)(1kX=[X(k)+X(N+k)]/2)(2kX=kNW−2[X(k)X(N+k)]/20≤k1−≤NNG(k)=)(1kX+j)(2kXNIFFTg(n)=)(1nx+j)(2nxDIT)(1nx=x(2r),)(2nx=x(2r+1),0≤r1−≤Nx(2n)=Re[g(n)],x(2n+1)=Im[g(n)],0≤n1−≤N