Introduction To Fourier Series

Math 250B, Spring 2010

03.02.10

Blackbird (Turdus merula)

Spectrogram

Time Waveform

time

frequency

amplitude

http://www.birdsongs.it/

Me

matic

sa

th

Mathematics: Relationship Between Taylor and Fourier Series

Imagine a periodic time-series (w/ period 2) described by the following function:

Taylor series about t=0

OR

Fourier series for t=[-,]

- Taylor series expands as a linear combination of polynomials

- Fourier series expands as a linear combination of sinusoids

Trigonometry review Sinusoids (e.g. tones)

A sinusoid has 3 basic properties:i. Amplitude - height of waveii. Frequency = 1/T [Hz]iii. Phase - tells you where the

peak is (needs a reference)

Why Use Fourier Series?

2. Taylor series can give a good local approximation (given you are within the radius of convergence); Fourier series give good global approximations

1. Many phenomena in nature repeat themselves (e.g., heartbeat, songbird singing)

Might make sense to ‘approximate them by periodic functions’

4. Fourier series gives us a means to transform from the time domain to frequency domain and vice versa (e.g., via the FFT)

Can be easier to see things in one domain as opposed to another

3. Still works even if f (t) is not periodic

0. Idea put forth by Joseph Fourier (early 19’th century); his thesis committee was not impressed [though Fourier methods have revolutionized many fields of science and engineering]

time waveform recorded from ear canal

... zoomed in

Fourier transform

Time Domain Spectral Domain

One of the ear’s primary functions is to act as a Fourier ‘transformer’

Tone-like sounds spontaneously emitted by the ear

Example: Square Wave

For periodic function f with period b, Fourier series on t =[-b/2, b/2] is:

where

(these are called the Fourier coefficients)

Example: Square Wave (cont.)

When the smoke clears....

include first two terms only (red)

include first three terms only (black dashed)

include first four terms only (green)

Note that approximation gets better as the number of higher order terms included increases

SUMMARY

- Taylor series expands as a linear combination of polynomials

- Fourier series expands as a linear combination of sinusoids

- Idea is that a function (or a time waveform) can effectively be represented as a linear combination of basis functions, which can be very useful in a number of different practical contexts

Fini

NOTE: different vertical scales! (one is logarithmic)

Why might the ear emit sound? An Issue of Scale

decibels (dB)

0 dB = x1

10 dB x3

20 dB = x10

40 dB = x100

60 dB = x1000

80 dB = x10000

100 dB = x100000

…

• a dB value is a comparison of two numbers

[dB= 20 log(x/y)] • A means to manage

numbers efficiently But why do we need

to use a dB scale?

Humans hear over a pressure range of 120 dB

Dynamic Range

[that’s a factor of a million]

‘The ear is capable of processing soundsover a remarkably wide intensity range, encompassing at least a million-fold change in energy….’ - Peter Dallos

VS

x5

Energy is related to the square of pressure …

WRONG ANALOGY

‘To appreciate this range … we represent a similar range of potential energies by contrasting the weight of a mouse with that of five elephants.’

VS

human threshold curve

SOAEs byproduct of an amplification mechanism?

SOAEs & Threshold

Mathematical Model: coupled resonators (2nd order filters)

Model Schematic

Each resonator has a unique tuning bandwidth [Q(x)] and spatially-defined characteristic frequency [(x)]

Equation of Motion

Assumptions-inner fluids are incompressible and the pressure is uniform within each scalae-papilla moves transversely as a rigid body (rotational modes are ignored)- consider hair cells grouped together via a sallet, each as a resonant element (referred to as a bundle from here on out) - bundles are coupled only by motion of papilla (fluid coupling ignored) - papilla is driven by a sinusoidal force (at angular frequency )- system is linear and passive- small degree of irregularity is manifest in tuning along papilla length

An Emission Defined

[SFOAE is complex difference between ‘smooth’ and ‘rough’ conditions]

Phase-Gradient Delay

Analytic Approximation

-To derive an approximate expression for the model phase-gradient delay, we make several simplifying assumptions (e.g., convert sum to integral, assuming bundle stiffness term is approximately constant, etc.)

given the strongly peaked nature of the integrand and by analogy to coherent reflection theory, we expect that only spatial frequencies close to some optimal value will contribute

opt

Analytic Approximation (cont.)

Model and Data Comparison

Model can be used to help us better understand physiological processes at work in the ear giving rise to emissions, leading to new science and clinical applications

Bundle = Force Generator?

- bundle can oscillate spontaneously

- exhibits nonlinear and negative stiffness

Martin (2008)

Simple Model to Explain SOAEs?: Part I

- ear is composed of resonant filters (e.g. a second-order filter such as a harmonic oscillator)

e.g., an individual hair cell or a particular location along the length of the basilar membrane

- consider just one of these filters:

massterm

dampingterm

stiffnessterm

drivingterm

Frishkopf & DeRosier (1983)

NOTE: quantities are complex so to describe both magnitude and phase

http://en.wikipedia.org/wiki/Image:VanDerPolOscillator.png

- active term?

- nonlinear?

negative damping

yup!

Model Idea I: SOAEs arise due to self-sustained oscillations of individual resonators (e.g. a limit cycle)

- one can readily envision adding in a driving term (e.g. stochastic force due to thermal noise)

- need some sort of ‘active’ term for self-sustained oscillation (e.g. van der Pol)

Simple Model to Explain SOAEs?: Part I (cont.)

Hair cell = ‘mechano-electro’ transducer

1. Mechanical stimulation deflects bundle, opening transduction channels

2. HC membrane depolarizes

3. Vesicle release triggers synapsed neuron to fire

non-linear (saturation)

+80 mV

-60 mV

Martin (2008)