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Fourier Analysis

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Joseph Fourier (1768-1830) “Yesterday was my 21st birthday, and at that age, Newton and Pascal had already acquired many claims to immortality.” – Fourier in a letter to CL Bonard

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What is the goal of Fourier Analysis? To express a function in terms of its component frequencies. Time domain Frequency domain

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“fourier”: 29,246 “spectral analysis”: 27,205 “filter”: 29,025 “RNAi”: 7,019 “drosophila”:57,523 Why should I care about Fourier analysis?

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But what is Fourier analysis useful for? 1.CALCULATING THINGS: Often, the frequency domain is much simpler than the time domain. –For example: filtering (convolving) becomes trivial in the frequency domain 2.DESCRIBING THINGS: –The cochlea transforms a time domain signal into a frequency signal. –Many brain regions have oscillations of a particular frequency.

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Last week: Vector Spaces Space of vectors Each vector could be completely described by –Basis set of vectors –Linear combinations of basis vectors [ 1 0 ] [ 0 1 ] [2 4]

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Today: Frequency Space Space of functions Each function can be completely described by: –Basis set of sine & cosine waves –Linear combination of basis waves SINE COSINE

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Every function can be completely expressed as a sum of sines & cosines of various amplitudes & frequencies.

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First let’s review the sine wave: A A = amplitude 1/ = frequency 1/ = period = 1/frequency = phase (offset) f t

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A cosine wave is just a sine wave shifted in phase by 90 o ( =90 o ). degrees

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Every function can be completely expressed as a sum of sines & cosines of various amplitudes & frequencies.

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SQUARE WAVE Mr. Square (waves) Mr. Men Mr. Tickle

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Mr. Square (waves) Every function can be completely expressed as a sum of sines & cosines of various amplitudes & frequencies. SQUARE WAVE

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Function Component sine waves Every function can be completely expressed as a sum of sines & cosines of various amplitudes & frequencies. SQUARE WAVE

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Fourier Series These things can be calculated. When I want to calculate them, I will find a handy computer program to calculate them for me. But how did I know which sines and cosines should be summed to create the square wave?

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Fourier Series Back to the square wave example ….

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So far, we have talked about how to write a function in terms of sums of sines and cosines. Once you know the component sines and cosines, it’s easy to rewrite the function in the “frequency domain”.

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TIME DOMAIN FREQUENCY DOMAIN Frequency (Hz) Time (sec) Power (or Energy) F ω1ω1 ω2ω2 ω1ω1 ω2ω2

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Power spectrum: Graph of the power (square of the amplitude) of the signal at each frequency Frequency (Hz) Power (or Energy) ω1ω1 ω2ω2 ω1ω1 ω2ω2 FREQUENCY DOMAIN

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Back to the square wave example …. SQUARE WAVE 0 1 2 3 4 5 FREQUENCY 0 1 2 3 4 5 FREQUENCY

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Euler equation: Using complex exponentials instead of sines and cosines Complex exponentials are sums of sines and cosines. Geometric interpretation: ANY function can be written as a sum of complex exponentials rather than a sum of sines and cosines.

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Fourier Transform time domain freq domain: freq domain time domain: DON’T WORRY: these formulas will always be calculated on the computer. Your job is to understand how to interpret the output. The extension of the Fourier series for non-periodic functions

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Fourier Transform Amplitude or Magnitude: Phase: The “power” or “energy” is the amplitude squared. a(ω) b(ω) F(ω)= a (ω)+ib(ω) Real Imaginary

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A function can be described in the frequency domain in terms of: 1) The amplitude of sine and cosines of various frequencies OR 2) The amplitude and phase of complex exponentials of various frequencies http://falstad.com/fourier/

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Any signal can be expressed in the time domain OR the frequency domain You can go back and forth between these 2 representations. –The fourier transform goes from the time domain to the frequency domain. –The inverse fourier transform goes the other way. Now, some neuroscience applications …

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Parkinson Tremors Moore GP, Ding L, Bronte-Stewart HM. Concurrent Parkinson tremors. J Physiol. 2000 Nov 15;529 Pt 1:273-81. Time domain Frequency domain

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Fourier Transforms in the nervous system Hair cells vibrate in response to sound and are “tuned” for different frequencies

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Tonotopy in the cochlea Sound: change in air pressure over time The filtering properties of the cochlea decompose the incoming signal by frequency, thereby taking a FOURIER TRANSFORM. High frequency tuning Low frequency tuning

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Spectrogram = A power spectrum taken at subsequent time bins. “Hey” + music frequency time http://www-users.cs.york.ac.uk/~alistair/research/dphil/enm/asa/complex.html A good approximation of firing rates in a tonotopic nucleus.

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Spatial Frequency changes in luminance across space

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Power spectrum 2-D Power Spectra FFF x x x y ω x ω x ω x ω yω y Spaital grating

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Retinal ganglion cells are tuned to specific spatial frequencies What gives them their spatial frequency tuning?

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What we’ve covered so far Frequency lingo –Amplitude, frequency, period, phase Fourier’s Theorem Power spectra Next week: putting it all together to make predictions about data

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Noise (terminology) “white noise” “colored noise” TIME FREQUENCY

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Acoustics Pitch fork TrumpetBassoon Sound: change in air pressure over time

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Acoustics

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2-D Fourier Decomposition

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Fourier Imaging Correlation Spectroscopy Margineantu D, Capaldi RA, Marcus AH. Dynamics of the mitochondrial reticulum in live cells using Fourier imaging correlation spectroscopy and digital video microscopy. Biophys J. 2000 Oct;79(4):1833-49.

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Sines & cosines are orthogonal They form a basis set that spans all functions. EVEN: f(x)=f(-x) ODD: f(x)=-f(x) Why can you create any function by combining sines and cosines?

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Tonotopy in the cochlea Small & stiff: high frequency resonance Wide & floppy: low frequency resonance Tonotopy is maintained through many layers of auditory processing

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Power spectrum: Graph of the amount of power (square of the amplitude) at each frequency FREQ DOMAIN TIME DOMAIN

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