1. 4. Multirate Systems and their Applications

2. We compute here … and throw away most of them here!!!! Inefficient Implementation of Downsampling

3. Most terms here are zero … and waste time to process them here!!!!$$ Inefficient Implementation of Upsampling

4. Same!!! Recall the Noble Identities

5. Example…

6. … continued

7. Example… Same

8. … continued

9. In general: Polyphase Decomposition. Take N=2: General Filter: Polyphase Decomposition with Polyphase Components

10. NOBLE IDENTITY POLYPHASE Downsampling using Polyphase Decomposition

11. This is a Serial to Parallel (a Buffer): S/P 0 0 0 1 2 3 4 5 6 7 0 2 4 6 0 1 3 5 Serial to Parallel (Buffer)

12. NOBLE IDENTITY POLYPHASE NOBLE IDENTITY Upsampling using Polyphase Decomposition

13. This is a Parallel to Serial (an Unbuffer): P/S 0 2 4 6 0 1 3 5 0 0 0 1 2 3 4 5 6 7 Parallel to Serial (Unbuffer or Interlacer)

14. Given any integer N: Example: take N=3 General Polyphase Decomposition

15. POLYPHASE Apply to Downsampling…

16. … apply Noble Identity

17. S/P 12345 1 2 6 3 4 5 6 Serial to Parallel (Buffer): Serial to Parallel (Buffer)

18. POLYPHASE Same for Upsampling…

19. NOBLE IDENTITY … apply Noble Identity

20. This is a Parallel to Serial (an Unbuffer): P/S 1 2 3 4 5 6 123456 Parallel to Serial (Unbuffer or Interlacer)

21. Processing Data by Blocks In most efficient implementations we process data by blocks, rather than one sample at a time. Real Time simulation and design software such as Simulink are designed to take advantages of block processing for two purposes: efficient computations, thus faster simulations; efficient design.

22. “Sample Based” and “Frame Based” Signals Sample Based: time They are MN distinct signals arranged in a matrix Particular Case: is like M distinct signals

23. “Sample Based” and “Frame Based” Signals Frame Based: They are N distinct signals, each one represented as a sequence of frames of length M Particular Case: one signal as a sequence of frames of length M

24. y0=Frame based y1=Sample based y2=y0=Frame based Convert “to Sample” and “to Frame” Example:

25. Serial to Parallel in Simulink Serial to Parallel is implemented by the “Buffer” Simulink block in Signal Processing Blcokset > Signal Management > Buffers: S/P Sample based Frame based

26. Example of Downsampling

27. Parallel to Serial in Simulink Parallel to Serial is implemented by the “UnBuffer” Simulink block in Signal Processing Blcokset > Signal Management > Buffers: Frame based Sample based P/S

28. Example of Upsampling