1. 1 Signals and Systems Lecture 23 DT Processing of CT Signals Digital Differentiator and Half-Sample Delay DT Decimation and Interpolation

2. 2 DT Processing of Band-limited CT Signals Chapter 7 Sampling Why do this? — Inexpensive, versatile, and higher noise margin. How do we analyze this system? — We will need to do it in the frequency domain in both CT and DT — In order to avoid confusion about notations, specify ω— CT frequency variable Ω — DT frequency variable (Ω =ωΤ) Step 1: Find the relation between x c (t) and x d [n], or X c (jω) and X d (e jΩ )

3. 3 Time-Domain Interpretation of C/D Conversion Chapter 7 Sampling

4. 4 Frequency-Domain Interpretation Chapter 7 Sampling CT----periodic with period ω s =2π/T DT----periodic with period 2π Note: ω s 2π CT DT (Ω = ωT)

5. 5 Illustration of C/D Conversion in the Frequency-Domain Chapter 7 Sampling

6. 6 D/C Conversion: Reverse of the process of C/D conversion Chapter 7 Sampling Again, Ω = ωT Reverse frequency scaling bandlimited

7. 7 Now the whole picture Chapter 7 Sampling Overall system is time-varying if sampling theorem is not satisfied; It is LTI if the sampling theorem is satisfied, i.e. for bandlimited inputs x c (t), with ω M <2ω s /2 When the input x c (t) is band-limited (X(jω) = 0 at |ω|>ω M ) sampling theorem is satisfied (ω s > 2ω M ), then LTI

8. 8 Frequency-domain Illustration of DT Processing of CT Signals Chapter 7 Sampling Original signal After sampling After C/D conversion #step 1-C/D: -periodic - π < Ω < π if no aliasing ω=Ω/T

9. 9 Frequency-domain Illustration of DT Processing of CT Signals Chapter 7 Sampling DT processing After D/C conversion Equivalent CT system #step 2-Digital Filter: Ω=ωTΩ=ωT - π < Ω < π #step 3-D/C: ω s /2<ω<ω s /2

10. 10 Design Digital Filter Chapter 7 Sampling In practice, first specify the desired H c (jω), then design H d (e jΩ ).

11. 11 Example: Digital Differentiator Chapter 7 Sampling

12. 12 Construction of Band-limited Digital Differentiator Chapter 7 Sampling Desired: Set ω s =2ω c =>Assume ω M <ω c →Nyquist rate is met Choice for H d (e jΩ ):

13. 13 Band-limited Digital Differentiator (continued) Chapter 7 Sampling

14. 14 Digital Differentiator in the Time-Domain Chapter 7 Sampling

15. 15 Example 7.2 Chapter 7 Sampling

16. 16 Half-Sample Delay Chapter 7 Sampling

17. 17 Half-Sample Delay in the Frequency Domain Chapter 7 Sampling

18. 18 Half-Sample Delay in the Time Domain Chapter 7 Sampling

19. 19 Impulse-Train Sampling Chapter 7 Sampling

20. 20 Illustration of Impulse-Train Sampling in Time Chapter 7 Sampling

21. 21 Impulse-Train Sampling in Frequency Domain Chapter 7 Sampling

22. 22 Illustration of Impulse-Train Sampling in Frequency Chapter 7 Sampling

23. 23 Recovery of a Discrete-Time Signal Chapter 7 Sampling

24. 24 Discrete-Time Decimation Chapter 7 Sampling

25. 25 Discrete-Time Decimation in Frequency Domain Chapter 7 Sampling

26. 26 Prepare of Downsampling Chapter 7 Sampling

27. 27 Upsampling Chapter 7 Sampling

28. 28 Sampling Rate Conversion Chapter 7 Sampling

29. 29 Summary DT Processing of CT Signals Digital Differentiator and Half-Sample Delay DT Decimation and interpolation

30. 30 Problem Set P567 7.31