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An Introduction to Fourier and Wavelet Analysis: Part I Norman C. Corbett Thursday, July-16-15 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 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) 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? s ) of tissue along a curvilinear 3. The density rr((s) path through a human brain 4. The long profile of a river E(d) 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: s(t ), s(t ) • It’s easy to make an a-periodic signal s1 (t ) sin(3.1t ) sin(t ) s1 s2 (t ) sin( t ) sin(t ) s2 Sinusoids • • The search for periodicity is the search for the sinusoidal components of a signal General sinusoid s (t ) A cos(2 ft f ) 1. Amplitude A (units of observed quantity) 2. Frequency f (Hz: cycles/s) or wave number n (1/m) – Angular frequency w 2 f (radians/s) 3. Period T (s) or wavelength l (m) 4. Phase angle f (radians) An Example • What sinusoidal components do you see in the graph of? All real s1 s1 (t ) sin(4 t ), • That was easy. What about this one? s2 world signals contain noise! s2 (t ) 0.5 sin(4 t ) 0.5 sin(80 t ) n(t ), • Picking out the periodic components of a general signal is very difficult Fourier to the Rescue! • The continuous Fourier transform (CFT) sˆ( f ) ei 2 ft s(t ) dt where e i 2 ft cos(2 ft ) i sin(2 ft ), i 1 • “Compare” s(t) to a family of complex exponentials indexed by frequency f • CFT of noisy sinusoid Fs2 Another Example • Define s(t) by 1, 0 t 0.5 s(t ) 0, otherwise • The CFT is sin( f ) cos( f ) 1 sˆ( f ) i 2 f 2 f rectangular pulse Complex Numbers • • sˆ( f ) is complex number. The CFT s(t) A complex number z i is a point ( , ) in the complex plane Im z z i • • • • ( , ) z f Re z Real part Imaginary part Magnitude | z | 2 2 Argument tan(f ) Spectrum • Since the CFT is complex, we compute 1. Amplitude spectrum sˆ( f ) 2 2. Spectral density (energy/power) sˆ( f ) – • Sometimes look at phase spectrum f ( f ) Amplitude spectrum of the rectangular pulse 1 cos( f ) sˆ( f ) 2 2 f 2 AS Inverse Fourier Transform • The original signal can be recovered via s(t ) i 2 ft e sˆ( f ) df • Implies that s(t) can be expressed as a “weighted sum” of complex exponentials – The magnitude of the weight is the amplitude spectrum Computation • In practice we work with samples of s(t) on a finite interval [0,T] s(kTs ), k 0,1, – • ,N Ts=T/N is the sampling period Discrete Fourier transform (DFT) S[l ] k 0 s(kTs )e k N 1 i 2 kl N , l 0,1, , N 1 Computation • Under suitable conditions T sˆ(l T ) S[l ], l 0, N • ,N 2 We can use the DFT to estimate the CFT at the discrete frequencies: 1 2 fl 0, , , T T N , (Hz) est 2T Some Real World Signals • Look at the spectra of river profiles in terms of the periods (wavelength) 2T Tl , N • T T , , , T (m) 3 2 Zim Bur A smackrel of weather data: – Winnipeg monthly mean temperatures for the years Wpg 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. Yet Another Example • The two signals s2 s3 • Have similar amplitude spectra Fs2 Fs3 • Remove the noise and look at the underlying signals cs2 cs3 • The frequency of the second signal changes – Can’t see this by looking at spectra • Wavelets to the rescue! (TBC) Questions?