basis functions

Concept

Signals as vectors

• signal space is a hilbert space
• a sound can be defined by up to an infinite number of frequencies

by definition of a signal as a summation, amplitude is not important

Inner product of signals

• similarity between two functions
• behaves like the dot product for discrete functions

ignore the complex conjugate, we are only concerned with the real component

Basis signals

• any signal can be thought of as a linear combination of the basis signals, usually simple sinusoidal
• choosing a set of orthogonal bases, allows the separation of component frequencies by projection

alternatively it can be an orthonormal basis since we aren’t concerned with amplitude

• orthonormal with the kronecker delta

Isolating a basis from a signal

• apply the inner product