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Fix-rate Signal Processing

Filter. Fix-rate Signal Processing. Fix rate filters - same number of input and output samples. x ( n ) 8 samples. y ( n ) = h(n) * x ( n ). y ( n ) 8 samples. Figure 1. Filter. Multirate Signal Processing. Multirate filters - different numbers of input and output samples.

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Fix-rate Signal Processing

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  1. Filter Fix-rate Signal Processing . Fix rate filters - same number of input and output samples x(n) 8 samples y(n) = h(n) * x(n) y(n) 8 samples Figure 1

  2. Filter Multirate Signal Processing . Multirate filters - different numbers of input and output samples x(n) 8 samples y(n) 4 samples Figure 2

  3. Filter Multirate Signal Processing . M samples Decimation - M>N N samples M samples Interpolation - M<N N samples Filter Figure 3

  4. Decimator x(n) y(n) M Figure 4 Basic application - reduce bit-rate by discarding samples Consequence - Distortion and Aliasing error

  5. Interpolator x(n) y(n) M Figure 4 Basic application - Insert samples between missing gaps Consequence - Restore the number of samples before decimation

  6. Decimation x(n) y(n) M x(n) e.g. M = 2 x’(n) -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 y(n) Figure 5 -2 -1 0 1 2

  7. (1) Decimation x(n) x’(n) x(n) n = 0 , +M , +2M , +3M , +4M, ... x’(n) = 0 otherwise

  8. (1) i(n) -2M -M 0 M 2M Figure 6 Decimation i(n) is a periodic impulse train that can be expressed as (2)

  9. Decimation (1) (2) (3)

  10. Decimation x(n) y(n) M x(n) e.g. M = 2 x’(n) -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 y(n) -2 -1 0 1 2

  11. According to equation (3) hence (5) Decimation x(n) x’(n) y(n) y(n) = x’(Mn) (4)

  12. Consider the z transform of x(n) and x’(n) (6) According to equation (3) Decimation Distortion due to Decimation can be seen in the frequency domain

  13. (7) Decimation According to equation (3)

  14. where p = Mn (8) (9) Decimation According to equation (4), y(n) = x’(Mn)

  15. Key equations of Decimation x(n) x’(n) y(n) y(n) = x’(Mn)

  16. z transform Convert to DFT with z = ej Key equations of Decimation x(n) x’(n) y(n)

  17. Spectral Changes in Decimation x(n) x’(n) y(n) Figure 7a /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4 

  18. Spectral Changes in Decimation x(n) x’(n) y(n) Figure 7a /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  images images Figure 7b M=4 /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4 

  19. Spectral Changes in Decimation x(n) x’(n) y(n) Figure 7a /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  Figure 7b M=4 /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  Figure 7c /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4 

  20. Spectral Changes in Decimation x(n) x’(n) y(n) Figure 7a /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  Figure 7b M=4 /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  Figure 7c /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4 

  21. x’(n) y(n) 3. Decimation by a factor of M stretches the width of the spectrum by M times Spectral Changes in Decimation x(n) x’(n) 1. Retaining one out of M samples in x(n) generates M replicated images of the original spectrum. 2. The spacing of images is 2/M

  22. y(n) -2 -1 0 1 2 y(n) -2 -1 0 1 2 Interpolation x(n) y(n) M Figure 8 e.g. M = 2 x’(n) -4 -3 -2 -1 0 1 2 3 4 e.g. M = 3 x’(n) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  23. x(n/M) n = 0 , +M , +2M , +3M , ... y(n) = (10) 0 otherwise (11) (12) Interpolation x(n) y(n) M

  24. (13) Spectral Changes in Interpolation x(n) y(n) M

  25. Spectral Changes in Interpolation x(n) y(n) M Figure 9a /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  Figure 9b M=4 /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4 

  26. Spectral Changes in Interpolation x(n) y(n) M Figure 9a /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  Figure 9b M=4 /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4 

  27. Spectral Changes in Interpolation x(n) y(n) 1. Interpolation by a factor of M compresses the width of the spectrum by M times 2. M images separating from each other by a spacing of 2/M are generated

  28. Decimation & Interpolation (M=4) Input Sequence -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 M -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 Output Sequence M -4 -3 -2 -1 0 1 2 3 4 Figure 10

  29. Decimation & Interpolation (M=4) Bandwidth - /8 Figure 11 /4 0 /4 /2 /4  /4 /2 /4  /4 /2 /4  M=4 M M

  30. It seems that the bitrate can be reduced simply by decimation Decimation & Interpolation

  31. Decimation & Interpolation But something is wrong,what’s the problem?

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