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Fourier Analysis

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Joseph Fourier ( ) “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 FREQUENCY 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

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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 Nov 15;529 Pt 1: 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 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 Oct;79(4):

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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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