Motivation: Wavelets are building blocks that can quickly decorrelate data 2. each signal written as (possibly infinite) sum 1. what type of data? 3. new coefficientsprovide 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. 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:

Now 2 x 3 x 8 multiplications. See any patterns?