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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
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 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 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 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.