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?