1. Rate Conversion

2. 2 Outline Problem statement Standard approach Decimation by a factor D Interpolation by a factor I Sampling rate conversion by a rational factor I/D Sampling rate conversion by an arbitrary factor Orthogonal projection re-sampling General theory Spline spaces Oblique projection re-sampling General theory Spline spaces

3. 3 Problem statement Given samples of a continuous-time signal taken at times, produce samples corresponding to times that best represent the signal. Applications: Conversion between audio formats Enlargement and reduction of images

4. 4 Digital Filtering Viewpoint In the sequel:

5. 5 Digital Filtering Viewpoint Reconstruction filter Anti-aliasing filter

6. 6 Standard Approach Decimation by a Factor D Standard choice (for avoiding aliasing):

7. 7 Standard Approach Decimation by a Factor D

8. 8 Standard Approach Interpolation by a Factor I Standard choice (for suppressing replicas):

9. 9 Standard Approach Interpolation by a Factor I

10. 10 Standard Approach Conversion by a Rational Factor I/D If the factor is not rational then conventional rate conversion cannot be implemented using up-samplers, down-samplers and digital filters. To retain efficiency, it is custom to resort to non-exact methods such as first and second order approximation.

11. 11 Orthogonal Projection Re-Sampling Reinterpretation of Standard Approach Reconstruction filter Anti-aliasing filter The prior and re-sampling spaces are related by a scaling of the generating function.

12. 12 Orthogonal Projection Re-Sampling General Spaces

13. 13 Orthogonal Projection Re-Sampling General Spaces

14. 14 Orthogonal Projection Re-Sampling General Spaces

15. 15 Orthogonal Projection Re-Sampling General Spaces

16. 16 Orthogonal Projection Re-Sampling Summary PrefilterRate conversionPostfilter For splines, there is a closed form for each of the components.

17. 17 Orthogonal Projection Re-Sampling Splines PrefilterPostfilter

18. 18 Orthogonal Projection Re-Sampling Examples

19. 19 Orthogonal Projection Re-Sampling Splines

20. 20 Orthogonal Projection Re-Sampling Interpretation PrefilterPostfilterReconstruction filter Anti-aliasing filter Problem: The exact formula for the conversion block gets very hard to implement for splines of degree greater than 1. Solution: Use a simple anti-aliasing filter, which is not matched to the reconstruction space, and compensate by digital filtering. Thus, instead of orthogonally projecting the reconstructed signal onto the reconstruction space, we oblique-project it.

21. 21 Oblique Projection Re-Sampling PrefilterPostfilterReconstruction filter Anti-aliasing filter

22. 22 Oblique Projection Re-Sampling PrefilterPostfilterReconstruction filter Anti-aliasing filter

23. 23 Orthogonal Projection Re-Sampling Examples

24. 24 Orthogonal Projection Re-Sampling Examples

25. 25 Orthogonal Projection Re-Sampling Examples