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Chapter 3. The Discrete Fourier Transform

Chapter 3. The Discrete Fourier Transform. 3.1 The Discrete Fourier Series. Definition: Periodic sequence N: the fundamental period of the sequences

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Chapter 3. The Discrete Fourier Transform

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  1. Chapter 3. The Discrete Fourier Transform

  2. 3.1 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 combination of complex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency (2pi/N). • From the frequency-domain periodicity of the DTFT, we conclude that there are a finite number of harmonics; the frequencies are {2pi/N*k,k=0,1,…,N-1}. Engineering college, Linyi Normal University

  3. Fourier series of periodic continuous signals • Ω0—period of x(t) in radian; Let T---sampling period; ω0----smpling period in radian Engineering college, Linyi Normal University

  4. So X(k) is also a periodic function with N Engineering college, Linyi Normal University

  5. DFS pair Engineering college, Linyi Normal University

  6. Properties of DFS Suppose the following 3 sequences’s period is N • Linearity Engineering college, Linyi Normal University

  7. Shifting • Symmetry Engineering college, Linyi Normal University

  8. Periodic convolution • Distinction with convolution sum Engineering college, Linyi Normal University

  9. Engineering college, Linyi Normal University

  10. 3.2 The Discrete Fourier Transform • Suppose: x(n)------finite-length sequence, N-----length; i.e. , x(n)=0 when n<0 or n>N-1 • Let x(n) be a period sequence of a periodic sequence Then we have Engineering college, Linyi Normal University

  11. It can be written as • Further, we usually express it as below Engineering college, Linyi Normal University

  12. Example Engineering college, Linyi Normal University

  13. The same as doing in the time-domain • On the other hand : Therefore: Engineering college, Linyi Normal University

  14. Note: x(n) and X(k) are not defined outside the interval: 0<=n<=N-1 and 0<=k<=N-1 Engineering college, Linyi Normal University

  15. 3.3 Properties of the DFT • Linearity If Then Note: If the length of each sequence is : Then Engineering college, Linyi Normal University

  16. While: Therefore, we should augment zero to the shorter sequence until the two sequences have the same length. Engineering college, Linyi Normal University

  17. Circular shift To a sequence x(n) with length N, its circular shifting is defined as: Engineering college, Linyi Normal University

  18. If Then Engineering college, Linyi Normal University

  19. Circular Convolution • Definition of circular convolution Suppose: two finite-duration sequences:x1(n) and x2(n) Note: The result of circular convolution is also a finite-duration sequence with the duration [0,N-1] • The operate steps can be divided into 3 main steps: • To period the two sequences with period N; • To compute the periodic convolution of the two periodic sequences • To get out the duration sequence between [0,N-1] Engineering college, Linyi Normal University

  20. Engineering college, Linyi Normal University

  21. Circular convolution as linear convolution with aliasing • Linear convolution of two finite-length sequences X1(n)---N1 points; x2(n)---N2 points Then the linear convolution of the two sequences are Because: So: x3(n)=0 when n<0 or n>N1+N2-2 i. e. the total length of x3(n) is N1+N2-1 Engineering college, Linyi Normal University

  22. Suppose When Prove: Engineering college, Linyi Normal University

  23. 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 finite-length sequences can be avoided . Example: suppose x1(n)=x2(n)=u(n)-u(n-6) (1)compute (2)compute Engineering college, Linyi Normal University

  24. Engineering college, Linyi Normal University

  25. Conjugate-Symmetry properties • Definition Engineering college, Linyi Normal University

  26. Conjugate-Symmetry properties of real sequence x(n) • sequence • Conjugate-symmetry sequence: • Conjugate-antisymmetry sequence: Engineering college, Linyi Normal University

  27. To real sequence Engineering college, Linyi Normal University

  28. Frequency selection Engineering college, Linyi Normal University

  29. DFT and z-transform Engineering college, Linyi Normal University

  30. 3.4 The Fast Fourier Transform • when the sequence length N is large, the straightforward implementation of DFT is very inefficient. %N2 complex multiplications and N(N-1) complex additions %1 complex multiplication includes: 4 real multiplications and 2 real additions. 1 complex addition includes: 2 real additions %so the total amount of 1 DFT operator:4 real multi-plications and N(4N-2) real additions Engineering college, Linyi Normal University

  31. In 1965, Cooley and Tukey showed a procedure to substantially reduce the amount of computa-tions involved in the DFT. • This led to reduce the amount of computation to N/2log2N times complex multiplications and N log2N times complex additions. • It is known as fast Fourier transform (FFT) algorithms. And we will only discuss radix-2 algorithm. • Two important classes algorithm • DIT FFT-----Decimation-in time FFT • DIF FFT-----Decimation-in frequency FFT Engineering college, Linyi Normal University

  32. 3.4.1 properties of • Twiddle factor -------WN • Periodicity in n and k • Symmetry • Transmutation • Special points Engineering college, Linyi Normal University

  33. 3.4.2 Decimation-in-time FFT • Algorithm principle suppose N=2v ---------even integer (radix-2) Engineering college, Linyi Normal University

  34. According to the inherent properties of DFT in k and n with period N/2,I.e. , Obviously, we see that a N-point DFT can be obtained by computing 2 N/2-point DFT. Because N/2 is also even, then we can divide it into 2 N/4-point. Similarly, we can divide a N-point sequence into N/2 2-point sequence. Engineering college, Linyi Normal University

  35. a a+bW a a+bW W W b a-bW a-bW b -1 -W (a) (b) • Butterfly structure Engineering college, Linyi Normal University

  36. x(0) G(0) X(0) 4点 DFT x(2) G(1) X(1) x(4) G(2) X(2) x(6) G(3) X(3) x(1) H(0) X(4) 4点 DFT -1 x(3) H(1) WN1 X(5) -1 x(5) H(2) WN2 X(6) -1 x(7) H(3) WN3 X(7) -1 Example: N=8,N/2=4 g(n)=x(2r)={x(0),x(2),x(4),x(8)}; h(n)=x(2r+1)={x(1),x(3),x(5),x(7)} a) FFT implementation of an 8-point DFT using two 4-point DFTs Engineering college, Linyi Normal University

  37. 2点 2点 2点 2点 b) FFT implementation of an 8-point DFT as two 4-point DFTs and four 2-point DFTs Engineering college, Linyi Normal University

  38. c) Full decimation-in-time FFT implementation of an 8-point DFT Engineering college, Linyi Normal University

  39. The computation times for FFT • Stages : v=log2N • Butterflies of each stage: N/2 • Each butterfly: 1 complex multiplications and 2 complex additions • N-point FFT: (N/2log2N) complex multiplications and (Nlog2N) complex additions • Computing DFT directly: N2 complex multiplications and N(N-1) complex additions Engineering college, Linyi Normal University

  40. For example: N=210=1024 DFT: complex-mul. N2=220=1048576 complex-add. N(N-1)=1024×1023=1047552 FFT: complex-mul. N/2log 2N =5120 complex-add. Nlog 2N =10240 Assume: 1 complex-mul. 100us 1 complex-add. 20us Then DFT needs 125.809s, and DFT needs 0.7168s only. Engineering college, Linyi Normal University

  41. x(0) x(4) G1(0) G1(1) • In-place computation • We needn’t to open another memory to store the output of each stage, because the former data we will not use again in later computation. • Example Engineering college, Linyi Normal University

  42. Order of input sequence x(n) Engineering college, Linyi Normal University

  43. Summarization • Let N=2v; then we choose M=2 and L=N/2 and divide x(n) into two N/2-point sequence. • This procedure can be repeated again and again. At each stage the sequences are decimated and the smaller DFTs combined. This decimation ends after v stages when we have N one-point sequences, which are also one-point DFTs. • The resulting procedure is called the decimation-in-time FFT (DIF-FFT) algorithm; Engineering college, Linyi Normal University

  44. Algorithm principle Suppose N=2v ----- even integer 3.4.3 Decimation-in-frequency FFT Engineering college, Linyi Normal University

  45. x(0) g(0) X(0) N/2点 DFT x(1) g(1) X(2) x(2) g(2) X(4) x(3) g(3) X(6) x(4) h(0) X(1) N/2点 DFT -1 WN1 x(5) h(1) X(3) -1 WN2 x(6) h(2) X(5) -1 WN3 x(7) h(3) X(7) -1 For example: N=8, the flow graph is shown below Engineering college, Linyi Normal University

  46. Repeat this operation again, we may obtain the following graphs four 2-point DFTs Engineering college, Linyi Normal University

  47. DIF-FFT flow graph for input in normal order and output in bit-reversed order Engineering college, Linyi Normal University

  48. Computation times Complex multiplication: N/2log2N complex additions: Nlog2N • Compare the DIT-FFT and DIF-FFT’s flow graphs, we may obtain the following conclusion: Two flow graphs are just transposed each other. • In-palace computations NOTE: In this algorithm, the input data is in normal sequence, but the output data is in bit-reverse sequence. Engineering college, Linyi Normal University

  49. Comparison of DIT-FFT and DIF-FFT • They have same number of complex multiplications and complex additions • They are all in-palace computation • DIT-FFT and DIF-FFT are transposed each other • They all need order sorting • They have different iterative formulas Engineering college, Linyi Normal University

  50. 3.5 Application of FFT algorithms • Computing IDFT using FFT Compare the expression of DFT and IDFT • Twiddle factor : • Constant coefficient: 1/N So we have Engineering college, Linyi Normal University

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