Rate Conversion

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Outline

Problem statementStandard approach

Decimation by a factor DInterpolation by a factor ISampling rate conversion by a rational factor I/DSampling rate conversion by an arbitrary factor

Orthogonal projection re-samplingGeneral theorySpline spaces

Oblique projection re-samplingGeneral theorySpline spaces

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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 formatsEnlargement and reduction of images

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Digital Filtering Viewpoint

In the sequel:

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Digital Filtering Viewpoint

Reconstruction filter

Anti-aliasing filter

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Standard ApproachDecimation by a Factor D

Standard choice (for avoiding aliasing):

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Standard ApproachDecimation by a Factor D

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Standard ApproachInterpolation by a Factor I

Standard choice (for suppressing replicas):

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Standard ApproachInterpolation by a Factor I

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Standard ApproachConversion 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.

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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.

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Orthogonal Projection Re-Sampling General Spaces

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Orthogonal Projection Re-Sampling General Spaces

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Orthogonal Projection Re-Sampling General Spaces

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Orthogonal Projection Re-Sampling General Spaces

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Orthogonal Projection Re-Sampling Summary

Prefilter Rate conversion

Postfilter

For splines, there is a closed form for each of the components.

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Orthogonal Projection Re-Sampling Splines

Prefilter Postfilter

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Orthogonal Projection Re-Sampling Examples

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Orthogonal Projection Re-Sampling Splines

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Orthogonal Projection Re-Sampling Interpretation

Prefilter PostfilterReconstruction 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.

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Oblique Projection Re-Sampling

Prefilter PostfilterReconstruction filter

Anti-aliasing filter

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Oblique Projection Re-Sampling

Prefilter PostfilterReconstruction filter

Anti-aliasing filter

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Orthogonal Projection Re-Sampling Examples

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Orthogonal Projection Re-Sampling Examples

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Orthogonal Projection Re-Sampling Examples