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
Delay:
Energy-preserving:
Filter ( = convolution):
operators on
Filtering:
Basic
filter coefficients
Control energy?
.
So
FIR filter: only finitely many
.
Filtering as an
operator:
delay-invariant meaning
Adjoint
:
FIR for convenience
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:
(b) Low pass:
(c) High pass:
sharp cut-offs
‘Almost’ Ideal
filters:
for FIR filters
(a) Low pass:
(b) High pass:
only at points, not intervals
More on filters:
variations needed
FIR for convenience
Adjoint:
Frequency Response function:
Example: basic
trig
basic trig!