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Up Sampling:

Up Sampling:. rate changes: ‘scaling’ on. Up: insert 0’s between consecutive terms. Decrease period in frequency:. Down Sampling:. Discard odd-indexed terms. Averaging in frequency:. because. Properties:. Adjoint:. Synthesizing:. Analyzing and Synthesizing:. Even part.

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Up Sampling:

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  1. Up Sampling: rate changes: ‘scaling’ on Up: insert 0’s between consecutive terms Decrease period in frequency:

  2. Down Sampling: Discard odd-indexed terms Averaging in frequency: because

  3. Properties: Adjoint: Synthesizing: Analyzing and Synthesizing: Even part How can we pick out Odd part to split ?

  4. Filter Banks:

  5. Vetterli’s Conditions:

  6. Conjugate Mirror filters: associate new filter with given filter : Haar example: Fourier transforms: ,

  7. New example: Tent function centered at : Scaling property:

  8. More on tents: Filter coefficients: Text

  9. Still more: So what’s the difference with Haar case? No orthogonality of translates !

  10. Daubechies 4-tap: also Factor, use trig identities: Set:

  11. Expand as Fourier series: Does there exist function such that If so, properties? What about ?

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