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Discrete Fourier Transform: Properties and Applications

Explore the properties and applications of the Discrete Fourier Transform (DFT) in signal processing. Learn about Fourier series, circular convolution, and the Fast Fourier Transform (FFT). Understand key concepts such as linearity, shifting, symmetry, and frequency selection.

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Discrete Fourier Transform: Properties and Applications

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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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