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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? 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: Filter ( = convolution):
Basic Filtering: filter coefficients Control energy? . So FIRfilter: only finitely many .
Filtering as an operator: FIR for convenience delay-invariant meaning : Adjoint
Fourier Transform on : complete orthonormal family in Discrete time Fourier Transform Energy preserving ,
Adjoint DFT: Fourier transform on Fourier coefficients of : Fourier representation:
Filtering and DFT: filter: Frequency Response function: DFT diagonalizes convolution operators:
Filtering and frequencies: selects or rejects frequencies: (a) Ideal: sharp cut-offs Low pass: (b) (c) High pass:
‘Almost’ Ideal filters: for FIR filters only at points, not intervals (a) Low pass: (b) High pass:
More on filters: variations needed FIR for convenience Adjoint: Frequency Response function:
Example: basic trig basic trig!