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Fourier Series in Mathematics: Relationship Between Taylor and Fourier Series

Learn about the relationship between Taylor series and Fourier series and how Fourier series can provide global approximations even for non-periodic functions. Explore the concept of sinusoids and their properties in representing periodic phenomena. Discover how Fourier series can transform time-domain data to frequency-domain data for easier analysis.

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Fourier Series in Mathematics: Relationship Between Taylor and Fourier Series

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  1. Introduction To Fourier Series Math 250B, Spring 2010 03.02.10

  2. Spectrogram frequency time amplitude Time Waveform Blackbird (Turdus merula) http://www.birdsongs.it/

  3. mat M th s ic a e

  4. Taylor series about t=0 Fourier series for t=[-,] Mathematics: Relationship Between Taylor and Fourier Series Imagine a periodic time-series (w/ period 2) described by the following function: OR - Taylor series expands as a linear combination of polynomials - Fourier series expands as a linear combination of sinusoids

  5. Trigonometry review Sinusoids (e.g. tones) A sinusoid has 3 basic properties: • Amplitude - height of wave • Frequency = 1/T [Hz] • Phase - tells you where the peak is (needs a reference)

  6. 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 Why Use Fourier Series? 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] 2. Taylor series can give a good local approximation (given you are within the radius of convergence); Fourier series give good global approximations 3. Still works even if f (t) is not periodic

  7. Time Domain Spectral Domain time waveform recorded from ear canal Tone-like sounds spontaneously emitted by the ear Fourier transform ... zoomed in  One of the ear’s primary functions is to act as a Fourier ‘transformer’

  8. 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

  9. include first two terms only (red) Example: Square Wave (cont.)  When the smoke clears....

  10. 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

  11. 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

  12. Fini

  13. NOTE: different vertical scales! (one is logarithmic) Why might the ear emit sound?  An Issue of Scale

  14. 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? decibels (dB)

  15. Dynamic Range Humans hear over a pressure range of 120 dB [that’s a factor of a million]

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

  17. x5 VS 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.’ • Energy is related to the square of pressure …

  18. VS

  19. human threshold curve SOAEs & Threshold SOAEs byproduct of an amplification mechanism?

  20. Model Schematic Mathematical Model: coupled resonators (2nd order filters)  Each resonator has a unique tuning bandwidth [Q(x)] and spatially-defined characteristic frequency [(x)]

  21. 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 islinear and passive • - small degree of irregularity is manifest in tuning along papilla length

  22. An Emission Defined [SFOAE is complex difference between ‘smooth’ and ‘rough’ conditions]

  23. opt 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

  24. Analytic Approximation (cont.)

  25. 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

  26. Bundle = Force Generator? • bundle can oscillate spontaneously • exhibits nonlinear and negative stiffness Martin (2008)

  27. driving term damping term stiffness term mass term Frishkopf & DeRosier (1983) 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: NOTE: quantities are complex so to describe both magnitude and phase

  28. http://en.wikipedia.org/wiki/Image:VanDerPolOscillator.png Simple Model to Explain SOAEs?: Part I (cont.) - need some sort of ‘active’ term for self-sustained oscillation (e.g. van der Pol) negative damping yup! • active term? • nonlinear? • one can readily envision adding in a driving term • (e.g. stochastic force due to thermal noise) Model Idea I: SOAEs arise due to self-sustained oscillations of individual resonators (e.g. a limit cycle)

  29. Martin (2008) Hair cell = ‘mechano-electro’ transducer Mechanical stimulation deflects bundle, opening transduction channels HC membrane depolarizes Vesicle release triggers synapsed neuron to fire non-linear(saturation) +80 mV -60 mV

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