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What have we seen?

What have we seen?. Fundamental result: (Haar). complete orthonormal basis for. Intuitive ideas:. (a). fine detail space at resolution. (b). ,. Where are we going?. Replace box, Haar by continuous functions. Signal processing ideas. operators on. Delay:. Energy-preserving:.

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What have we seen?

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  1. What have we seen? Fundamental result: (Haar) complete orthonormal basis for . Intuitive ideas: (a) fine detail space at resolution . (b) ,

  2. Where are we going? Replace box, Haar by continuous functions.

  3. Signal processing ideas operators on Delay: Energy-preserving: Filter ( = convolution):

  4. Basic Filtering: filter coefficients Control energy? . So FIRfilter: only finitely many .

  5. Filtering as an operator: FIR for convenience delay-invariant meaning : Adjoint

  6. Fourier Transform on : complete orthonormal family in Discrete time Fourier Transform Energy preserving ,

  7. Adjoint DFT: Fourier transform on Fourier coefficients of : Fourier representation:

  8. Filtering and DFT: filter: Frequency Response function: DFT diagonalizes convolution operators:

  9. Filtering and frequencies: selects or rejects frequencies: (a) Ideal: sharp cut-offs Low pass: (b) (c) High pass:

  10. ‘Almost’ Ideal filters: for FIR filters only at points, not intervals (a) Low pass: (b) High pass:

  11. More on filters: variations needed FIR for convenience Adjoint: Frequency Response function:

  12. Example: basic trig basic trig!

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