ESE 531: Digital Signal Processing

Transcription

1 ESE 531: Digital Signal Processing Lec 9: February 13th, 2018 Downsampling/Upsampling and Practical Interpolation

2 Lecture Outline! CT processing of DT signals! Downsampling! Upsampling 2

3 Continuous-Time Processing of Discrete-Time! Useful to interpret DT systems with no simple interpretation in discrete time 3

4 Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 δ[n] 1 δ[n n d ] e jωn d 4

5 Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 5

6 Example: Non-integer Delay 6

7 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t T Δ) 7

8 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t T Δ) y[n] = y c (nt ) = x c (nt T Δ) 8

9 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t T Δ) y[n] = y c (nt ) = x c (nt T Δ) x c (t) = x[k] h r (t kt ) k = x[k] sinc t kt T k 9

10 Reminder: Reconstruction in Time Domain 10

11 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t T Δ) y[n] = y c (nt ) = x c (nt T Δ) x c (t) = x[k] h r (t kt ) k = x[k] sinc t kt T k 11

12 Reminder: Reconstruction in Time Domain 12

13 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y[n] = y c (nt ) = x c (nt T Δ) x c (t) = x[k] sinc t kt T k 13

14 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y[n] = y c (nt ) = x c (nt T Δ) x c (t) = x[k] sinc t kt T k x c (nt T Δ) = k x[k] sinc nt T Δ kt T = k ( ) x[k] sinc n Δ k 14

15 Example: Non-integer Delay! The block diagram is for interpretation/analysis only y[n] = y c (nt ) = x c (nt T Δ) x c (t) = x[k] sinc t kt T k x c (nt T Δ) = k x[k] sinc nt T Δ kt T = k ( ) x[k] sinc n Δ k 15

16 Example: Non-integer Delay! Delay system has an impulse response of a sinc with a continuous time delay y[n] = k ( ) x[k] sinc n Δ k ( ) = x[n] sinc n Δ 16

17 Example: Non-integer Delay! Delay system has an impulse response of a sinc with a continuous time delay y[n] = k ( ) x[k] sinc n Δ k ( ) = x[n] sinc n Δ h[n] = sinc(n Δ) 17

18 Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 δ[n] 1 δ[n n d ] e jωn d 18

19 Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 19

20 Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 20

21 Downsampling! Definition: Reducing the sampling rate by an integer number 21

22 Downsampling! Similar to C/D conversion " Need to worry about aliasing " Use anti-aliasing filter to mitigate effects 22

23 Downsampling! Similar to C/D conversion " Need to worry about aliasing " Use anti-aliasing filter to mitigate effects! If your discrete time signal is finely sampled almost like a CT signal " Downsampling is just like sampling (C/D conversion) 23

24 Downsampling 24

25 Downsampling! Want to relate X d (e jω ) to X(e jω ) not X c (jω)! Separate sum into two sums fine sum and coarse sum (i.e like counting minutes within hours) 25

26 Downsampling! k=rm+i " i = 0, 1,, M-1 " r = -,..., 26

27 Downsampling k = rm + i 27

28 Downsampling 28

29 Downsampling 29

30 Example 30

31 Example 4π 31

32 Example 2π 4π 32

33 Example 6π 33

34 Example 2π 6π 34

35 Example 2π 4π 6π 35

36 Example 36

37 Example 37

38 Upsampling! Much like D/C converter! Upsample by A LOT # almost continuous! Intuition: " Recall our D/C model: x[n] # x s (t)#x c (t) " Approximate x s (t) by placing zeros between samples " Convolve with a sinc to obtain x c (t) 38

39 Upsampling! Definition: Increasing the sampling rate by an integer number x[n] = x c (nt ) x i [n] = x c (nt ') 39

40 Upsampling x i [n] 40

41 Upsampling 41

42 Frequency Domain Interpretation 42

43 Frequency Domain Interpretation 43

44 Frequency Domain Interpretation 44

45 Frequency Domain Interpretation 45

46 Example 46

47 Example 47

48 Example 48

49 Example 49

50 Example 50

51 Example 51

52 Practical Interpolation! Interpolate with simple, practical filters " Linear interpolation samples between original samples fall on a straight line connecting the samples " Convolve with triangle instead of sinc 52

53 Practical Interpolation! Interpolate with simple, practical filters " Linear interpolation samples between original samples fall on a straight line connecting the samples " Convolve with triangle instead of sinc 53

54 Frequency Domain Interpretation 54

55 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 55

56 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 56

57 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 57

58 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 58

59 Big Ideas! CT processing of DT signals " Allows for interpretation of DT systems! Downsampling " Like a C/D converter! Upsampling " Like a D/C converter! Practical Interpolation " Linear interpolation " Approximate sinc function with triangle 59

60 Admin! HW 4 due Friday 60