Motivation:
Wavelets are building blocks that can
quickly decorrelate data
1. what type of data?
2. each signal written as (possibly infinite) sum
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
Basis:
Energy:
real:
complex:
real:
complex:
Geometry via inner
products
dot product, inner product
real:
complex:
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
,
defn
Energy:
in
.
Basic Example:
Inner product:
Energy:
Standard basis:
Standard representation:
.
Basic Example:
Addition structure on
.
:
defn
modular addition.
Set
,
Roots of unity:
Multiplication structure on
:
Basic Example:
Notation:
.
denotes all functions
With inner product
becomes inner product space:
Fundamental Theorem:
(Standard Basis)
is orthonormal basis for
.
.
and DFT
Important idea for DFT: each
in
defines
function
such that
.
Fundamental Theorem:
(Fourier Basis)
is orthonormal basis for
DFT: Standard basis
.
Fourier basis
DFT
.
use signal analysis notation
tion:
Fourier representation:
ier Transform:
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?