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Chapter 10 fourier analysis of signals using discrete fourier transform

Chapter 10 fourier analysis of signals using discrete fourier transform. 10.1 fourier analysis of signals using the DFT 10.2 DFT analysis of sinusoidal signals 10.3 the time-dependent fourier transform. Figure 10.1. 10.1 fourier analysis of signals using the DFT.

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Chapter 10 fourier analysis of signals using discrete fourier transform

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  1. Chapter 10 fourier analysis of signals using discrete fourier transform 10.1 fourier analysis of signals using the DFT 10.2 DFT analysis of sinusoidal signals 10.3 the time-dependent fourier transform

  2. Figure 10.1 10.1 fourier analysis of signals using the DFT

  3. true frequency spectrum frequency response of antialiasing filter error caused by ideal filter error caused by quantization and aliasing frequency spectrum of window sequence error caused by windowing in time domain and spectral sampling Figure 10.2

  4. 10.2 DFT analysis of sinusoidal signals 10.2.1 the effect of windowing 10.2.2 the effect of spectral sampling

  5. 10.2.1 the effect of windowing Before windowing:

  6. After windowing:

  7. EXAMPLE Example 10.3 effects of windowing： (1) spectral lines are broadened to the mainlobe of window frequency spectrum (2) produce sidelobe, its attenuation is equal to sidelobe attenuation of window frequency spectrum after windowing before windowing Figure 10.3(a)(b)

  8. broadened spectral lines lead to: difficult to confirm the frequency； reduced resolution mainlobe leads to ： produce false signal； flood small signal. leakage: （magnitudes of two frequency components interact each other） Figure 10.3(c)(d)(e)

  9. EXAMPLE L=32 L=42 L=54 conservative definition （exercise10.7）： resolution= main-lobe width L=64 Figure 10.10 Conclusion: increase L can increase resolution

  10. EXAMPLE changing the shape of window can change resolution Conclusion: shape of window has effect on frequency resolution

  11. Determin window’s shape and length （1） for kaiser windows: （2）for blackman window: look up the table

  12. EXAMPLE Before sampling After sampling Figure 10.5(a)(b)(f) 10.2.2 the effect of spectral sampling can not reach peak value N=128 N=64

  13. N=64 N=128 Figure 10.7 EXAMPLE only reach the peak value and zero values N=64 Figure 10.6 conclusion : increase N can improve effect N=128

  14. N=32 N=64 N=128 N=1024 N=32 EXAMPLE N=64 N=128 Conclusion: increase N can’t increase resolution N=1024 Figure 10.9

  15. analyze effects of window length to DFT using MATLAB EXAMPLE

  16. L1=64; L2=128; N=128; T=1/64 n1=0:L1-1; x1=cos(2*pi*2*n1*T)+ cos(2*pi*2.5*n1*T) n2=0:L2-1; x2=cos(2*pi*2*n2*T)+ cos(2*pi*2.5*n2*T) k=0:N-1; X1=fft(x1,N); X2=fft(x2,N) stem(k,abs(X1)); hold on; stem(k,abs(X2),’r.’); consideration：how to get DFS of periodic sequence with period N using windowing DFT？

  17. 10.3 the time-dependent fourier transform 1.definition

  18. Figure 10.12 Figure 10.11

  19. summary 10.1 fourier analysis of signals using the DFT 10.2 DFT analysis of sinusoidal signals conclusion: effects of windowing and spectral sampling 10.3 the time-dependent fourier transform apply the conclusion in 10.2 in analysis of general signals

  20. requirements： effects of windowing and sampling to DFT line; concept of frequency resolution，relationship among the shape and length of window , and the points of DFT; DFT analysis of sinusoidal signals .

  21. exercises and experiment 10.21 10.31(a)-(c) experiment 31 34 and 35（draw them together） 36 37（B） 39（A）

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