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- 1. Pharos University in Alexandria Faculty of Engineering Computer Engineering Department Digital Signal Processing EC362 Spring 2022-2023
- 2. Chapter 2 Discrete Fourier Transform(DFT)
- 3. Discrete Fourier Transform(DFT) 𝐗𝐗 𝒌𝒌 = � 𝒏𝒏=𝟎𝟎 𝑵𝑵−𝟏𝟏 𝒙𝒙(𝒏𝒏)𝒆𝒆 −𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋 𝑵𝑵 Where k=0,1,2…….N-1 • Requires N2 complex multiplies and N(N-1) complex additions
- 5. 2 3
- 8. DFT with Twiddle Factor • The forward DFT, frequency domain output in the range 0≤k≤N-1 is given by: 𝐗𝐗 𝒌𝒌 = � 𝒏𝒏=𝟎𝟎 𝑵𝑵−𝟏𝟏 𝒙𝒙(𝒏𝒏)𝑾𝑾𝑵𝑵 𝒏𝒏𝒏𝒏 𝑾𝑾𝑵𝑵 𝒏𝒏𝒏𝒏 = 𝒆𝒆 −𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋 𝑵𝑵 Twiddle Factor
- 12. DFT in Matrix Form X(0) X(1) X(2) X(3) = 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 1 𝑊𝑊4 2 𝑊𝑊4 3 𝑊𝑊4 0 𝑊𝑊4 2 𝑊𝑊4 4 𝑊𝑊4 6 𝑊𝑊4 0 𝑊𝑊4 3 𝑊𝑊4 6 𝑊𝑊4 9 𝑥𝑥(0) 𝑥𝑥(1) 𝑥𝑥(2) 𝑥𝑥(3) X(0) X(1) X(2) X(3) = 1 1 1 1 1 −j −1 j 1 −1 1 −1 1 j −1 −j 𝑥𝑥(0) 𝑥𝑥(1) 𝑥𝑥(2) 𝑥𝑥(3) For N=4
- 13. Matrix Relations can be expanded as NXN DFT matrix = − − − − − − = ∑ 2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 4 2 ) 1 ( 2 1 1 0 1 1 1 1 1 1 1 N N N N N N N N N N N N N N N k nk N W W W W W W W W W W ∑ − = 1 0 N k nk N W
- 14. For 8 point DFT: N=8 W8 0x0 W8 0x1 W8 0x2 W8 0x3 W8 0x4 W8 0x5 W8 0x6 W8 0x7 W8 1x0 W8 1x1 W8 1x2 W8 1x3 W8 1x4 W8 1x5 W8 1x6 W8 1x7 W8 2x0 W8 2x1 W8 2x2 W8 2x3 W8 2x4 W8 2x5 W8 2x6 W8 2x7 W8 3x0 W8 3x1 W8 3x2 W8 3x3 W8 3x4 W8 3x5 W8 3x6 W8 3x7 W8 4x0 W8 4x1 W8 4x2 W8 4x3 W8 4x4 W8 4x5 W8 4x6 W8 4x7 W8 5x0 W8 5x1 W8 5x2 W8 5x3 W8 5x4 W8 5x5 W8 5x6 W8 5x7 W8 6x0 W8 6x1 W8 6x2 W8 6x3 W8 6x4 W8 6x5 W8 6x6 W8 6x7 W8 7x0 W8 7x1 W8 7x2 W8 7x0 W8 7x4 W8 7x5 W8 7x6 W8 7x7
- 15. The characteristic matrix of 8-point DFT is: 1 1 1 1 1 1 1 1 1 1 j − 2 j − 1 j + ( ) − 2 1 − 1 j − ( ) − 2 j 1 j + 2 1 j − 1 − j 1 j − 1 − j 1 1 j + ( ) − 2 j 1 j − 2 1 − 1 j + 2 j − 1 j − ( ) − 2 1 1 − 1 1 − 1 1 − 1 1 − 1 1 j − ( ) − 2 j − 1 j + 2 1 − 1 j − 2 j 1 j + ( ) − 2 1 j 1 − j − 1 j 1 − j − 1 1 j + 2 j 1 j − ( ) − 2 1 − 1 j + ( ) − 2 j − 1 j − ( ) 2
- 16. Example1: Find the DFT for 𝑥𝑥(𝑛𝑛)={1,1,0,1}
- 17. Example2: Find the DFT using matrix form for 𝑥𝑥(𝑛𝑛)={2,3,4,5}
- 19. Properties of Twiddle Factor 1. Periodicity 2. Symmetry 3. Exponential
- 28. Inverse Discrete Fourier Transform(IDFT) 𝒙𝒙 𝒏𝒏 = 𝟏𝟏 𝑵𝑵 � 𝒌𝒌=𝟎𝟎 𝑵𝑵−𝟏𝟏 𝐗𝐗(𝒌𝒌)𝒆𝒆 𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋𝒋 𝑵𝑵 Where n=0,1,2…….N-1
- 29. IDFT with Twiddle Factor • While the Inverse DFT, time domain output, again, in the range 0≤k≤N-1 is denoted by 𝒙𝒙 𝒏𝒏 = 𝟏𝟏 𝑵𝑵 � 𝒌𝒌=𝟎𝟎 𝑵𝑵−𝟏𝟏 𝐗𝐗(𝒌𝒌)𝑾𝑾𝑵𝑵 −𝒏𝒏𝒏𝒏
- 30. IDFT matrix form 𝑥𝑥(0) 𝑥𝑥(1) 𝑥𝑥(2) 𝑥𝑥(3) = 1 𝑁𝑁 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 0 𝑊𝑊4 −1 𝑊𝑊4 −2 𝑊𝑊4 −3 𝑊𝑊4 0 𝑊𝑊4 −2 𝑊𝑊4 −4 𝑊𝑊4 −6 𝑊𝑊4 0 𝑊𝑊4 −3 𝑊𝑊4 −6 𝑊𝑊4 −9 X(0) X(1) X(2) X(3) 𝑥𝑥(0) 𝑥𝑥(1) 𝑥𝑥(2) 𝑥𝑥(3) = 1 4 1 1 1 1 1 j −1 −j 1 −1 1 −1 1 − j −1 j X(0) X(1) X(2) X(3) For N=4
- 31. DFT and its Inverse Conversion in system matrices of DFT and IDFT obtained by replacing by j by –j.. 1 1 1 1 1 1 1 1 1 1 j − 2 j − 1 j + ( ) − 2 1 − 1 j − ( ) − 2 j 1 j + 2 1 j − 1 − j 1 j − 1 − j 1 1 j + ( ) − 2 j 1 j − 2 1 − 1 j + 2 j − 1 j − ( ) − 2 1 1 − 1 1 − 1 1 − 1 1 − 1 1 j − ( ) − 2 j − 1 j + 2 1 − 1 j − 2 j 1 j + ( ) − 2 1 j 1 − j − 1 j 1 − j − 1 1 j + 2 j 1 j − ( ) − 2 1 − 1 j + ( ) − 2 j − 1 j − ( ) 2 1 8 1 1 1 1 1 1 1 1 1 1 j + 2 j 1 j − 2 − 1 − 1 j + 2 − j − 1 j − 2 1 j 1 − j − 1 j 1 − j − 1 1 j − 2 − j − 1 j + 2 1 − 1 j − 2 j 1 j + 2 − 1 1 − 1 1 − 1 1 − 1 1 − 1 1 j + 2 − j 1 j − 2 1 − 1 j + 2 j − 1 j − 2 − 1 j − 1 − j 1 j − 1 − j 1 1 j − 2 j − 1 j + 2 − 1 − 1 j − 2 − j 1 j + 2 ⋅ DFT IDFT
- 32. Example3: Find the IDFT using matrix form for X(𝑘𝑘)={3,1,-1,1}