Motivation:
Wavelets are building blocks that canquickly decorrelate data
2. each signal written as (possibly infinite) sum
1. what type of data?
3. new coefficients provide more ‘compact’representation. Why need?
4. switch representations in time proportional to size of data
Inner product spaces and the DFT
Familiar 3-space real:
Basis:
complex:
Energy:
real:
complex:
Geometry via inner products
real:
complex:
dot product, inner product
capture basic geometry of 3-space
correlation:
parallel
perpendicular
Inner product space .
capture linear combinations and geometry
vector space (over reals or complex numbers)
such that
for all in , in .
Energy:defn
Basic Example: .
Standard basis:
Standard representation:
Inner product:
Energy:
Basic Example: .
Addition structure on :defn
modular addition.
Set , Roots of unity:
Multiplication structure on :
Basic Example: .
With inner product
becomes inner product space:
Notation: denotes all functions
Fundamental Theorem:
is orthonormal basis for
.
(Standard Basis)
. and DFT
Important idea for DFT: each in defines
function
such that .
Fundamental Theorem:
is orthonormal basis for
.
(Fourier Basis)
DFT: Standard basis Fourier basis
DFT: Standard basis Fourier basis
DFT .
function:
use signal analysis notation
Fourier Transform:
Fourier representation:
where
measures correlation of with each
DFT as Matrix
But there are multiplications here.
What happened to the idea of doing things quickly?
Fast Fourier Transform: FFT
Fourier Matrix
N = 2:
Examples: N = 4 = 2x2:
still 16 multiplications, but it looks promising!
Examples: N=8=2x2x2:
Examples: N=8=2x2x2:
Now 2 x 3 x 8 multiplications. See any patterns?
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