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An Introduction to Fourier and Wavelet Analysis: Part I Norman C. Corbett Sunday, June 1, 2014

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Motivation Search for patterns, periodicity or recurrent features in nature –An important element in the (stable) behaviour of many physical systems Examine the long profile of rivers –Contribute to the debate about the periodicity of step-pool sequences

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Signals Variety of definitions Real valued functions of one or more independent variables Serve as mathematical models for quantities that vary in time and/or space Denote a generic signal by s(t)

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Signals Some examples: 1.The voltage v(t) in an electrical circuit as a function of time t 2.A black and white image I(x,y) What about a colour image? 3.The density (s) of tissue along a curvilinear path through a human brain 4.The long profile of a river E(d)

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A Diatribe of Periodicity No real world signal is strictly periodic! –Noise, friction, etc. crash the party. Check periodicity by constructing a phase plane plot: Its easy to make an a-periodic signal s1s1 s2s2

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Sinusoids The search for periodicity is the search for the sinusoidal components of a signal General sinusoid 1.Amplitude A (units of observed quantity) 2.Frequency f (Hz: cycles/s) or wave number (1/m) –Angular frequency f (radians/s) 3.Period T (s) or wavelength (m) 4.Phase angle (radians)

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An Example What sinusoidal components do you see in the graph of? That was easy. What about this one? Picking out the periodic components of a general signal is very difficult s1s1 s2s2 All real world signals contain noise!

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Fourier to the Rescue! The continuous Fourier transform (CFT) where Compare s(t) to a family of complex exponentials indexed by frequency f CFT of noisy sinusoid Fs 2

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Another Example Define s(t) by The CFT is rectangular pulse

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Complex Numbers The CFT s(t) is complex number. A complex number is a point in the complex plane Real part Imaginary part Magnitude Argument

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Spectrum Since the CFT is complex, we compute 1.Amplitude spectrum 2.Spectral density (energy/power) –Sometimes look at phase spectrum ( f ) Amplitude spectrum of the rectangular pulse AS

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Inverse Fourier Transform The original signal can be recovered via Implies that s(t) can be expressed as a weighted sum of complex exponentials –The magnitude of the weight is the amplitude spectrum

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Computation In practice we work with samples of s(t) on a finite interval [0,T] –T s =T/N is the sampling period Discrete Fourier transform (DFT)

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Computation Under suitable conditions We can use the DFT to estimate the CFT at the discrete frequencies: est

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Some Real World Signals Look at the spectra of river profiles in terms of the periods (wavelength) A smackrel of weather data: –Winnipeg monthly mean temperatures for the years BurZim Wpg

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A Problem with Fourier Real world signals are often nonstationary –The spectra of such signals are very complex Detrending ensures first order stationarity –The mean of the residuals is zero Second and higher order moments may still evolve: –Variance, (auto)correlation, etc.

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Yet Another Example The two signals Have similar amplitude spectra Remove the noise and look at the underlying signals The frequency of the second signal changes –Cant see this by looking at spectra Wavelets to the rescue! (TBC) s3s3 s2s2 Fs 3 Fs 2 cs 3 cs 2

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