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Data: digital & analogue:

Data: digital & analogue:. Coarse:. Coarse + Fine:. Splitting:. coarse at resn 3 = coarse at resn 2 + fine detail at resn 2. coarse at resn 3 = coarse at resn 2 + fine detail at resn 2. Splitting:. Data: sequence. Fix FIR filter. Vetterli Condition. Interpret?. Now Down-sample:.

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Data: digital & analogue:

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  1. Data: digital & analogue:

  2. Coarse:

  3. Coarse + Fine: Splitting: coarse at resn 3 = coarse at resn 2 + fine detail at resn 2 coarse at resn 3 = coarse at resn 2 + fine detail at resn 2

  4. Splitting: Data: sequence Fix FIR filter Vetterli Condition Interpret? Now Down-sample: HALVE NUMBER of TERMS

  5. Orthonormal families: Define Synthesizing operator by: Another Synthesizing operator: Theorem: energy-preserving when Vetterli satisfied. So: , ORTHOGONAL ORTHONORMAL FAMILIES

  6. Interpret Splitting: Analyzed sequences: Analyze and Synthesize: coarse detail fine detail

  7. Fix FIR filter Analogue case: Choose scaling function , associated wavelet , orthonormal families for each k. Important property: othogonal, i.e. perpendicular Level 1 resolution function:

  8. Coarse + Fine detail: Coarse detail : use scaling function family at level 0 , Fine detail : use wavelet family at level 0 , Relate , to filter coefficients?

  9. 2-scale equation again: Exploit as before Coarse detail coefficients:

  10. 2-scale equation again: Fine detail coefficients: Analyze and Synthesize: coarse detail fine detail

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