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Exploring Signal Processing with Haar Basis and Filter Operators

Discover the fundamental results and intuitive ideas behind signal processing using Haar complete orthonormal basis and filter operators. Explore how to replace Haar with continuous functions and control energy in filtering operations. Learn about energy-preserving filters, Fourier transforms, and frequency response functions. Ideal for understanding basic trigonometry concepts in signal processing.

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Exploring Signal Processing with Haar Basis and Filter Operators

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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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