Biophysical Techniques (BPHS 4090/PHYS 5800)

Transcription

1 Biophysical Techniques (BPHS 4090/PHYS 5800) Instructors: Prof. Christopher Bergevin Schedule: MWF 1:30-2:30 (CB 122) Website: York University Winter 2017 Lec.6

2 To build up a foundation to understand how this (MRI) image was acquired, we need to first clarify some key ideas (chiefly within the vein of signal processing ) Key idea: Periodicity Wikipedia

3 Fourier analysis Ø Backbone of modern signal processing and linear systems theory Ø Basic idea: Represent signal as a sum of sinusoids Note: Images in the basic sense can be considered 2-D, but we can first gain the necessary insight by initially focusing on 1-D examples Joseph Fourier ( )

4 Connec3ons to the EM spectrum... à Fairly natural to think about a spectral decomposition of light... Wikipedia

5 Focus on one (extremely useful/very basic) idea: Ø Just as visible light is composed of photons of many different wavelengths, an image is comprised different spatial wavelengths (or conversely, frequencies) à Put another way, we can make use of the notion of periodicity to describe an image in spectral terms

6 Aside: Periodicity/oscilla3ons commonly manifests throughout biology Ø Hearing Ø Circadian rhythms (and associated hormonal regula3on) Ø Min oscilla3ons in E. coli Ø Nega3ve feedback in gene3c expression: ( resul3ng from a transcrip3on factor repressing the promotor of its own gene. The feedback involves the produc3on of mrna as an intermediate. Because of the 3me lag between repression and mrna produc3on, the protein level can change periodically. ) Kruse & Julicher (2005)

7 ex. Spectrogram à Consider sound pressure varia3ons (as a 3me waveform) picked up by a microphone Blackbird (Turdus merula) Spectrogram frequency 3me amplitude Time Waveform hvp://

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9 Fourier analysis Intui3ve connec3on back to Taylor series: [see H&R Appendix D] Ø Taylor series à Expand as a (infinite) sum of polynomials Ø Different Idea: Fourier series à Expand as a (infinite) sum of sinusoids

10 Fourier analysis Note: Independent variable (t here) can represent any physical quan3ty, but for 1-D we typically use 3me Complex #s are much more compact and easier to deal with

11 Trigonometry review 2 A sinusoid has 3 basic proper3es: i - Amplitude height of wave ii - Frequency how oden does it repeat? 1/T [Hz] iii - Phase (f) where is the peak? needs a reference!

12 Trigonometry review à What does phase tell us? - Phase tells time information - Analogous to angle around circle - Arbitrary outside range [0,2π] ( phase unwrapping ambiguity ) - 1 cycle = 360 o = 2π radians Reference here is t = 0 Connection to complex numbers??

13 EXbeats.m % ### EXbeats.m ### CB (updated )!! % Plots sum of two sinusoids with different amplitudes/frequencies/phase! % offsets to demonstrate variety of "beating" patterns!! clear! % =================================! % Parameters for each of the two sinusoids! A1= 1.0; % amplitude 1! A2= 1.0; % amplitude 2! w1= 1.0; % angular frequency 1! w2= 1.2; % angular frequency 2! phi1= pi/2; % phase offset 1 (>=0 and <= 2*pi)! phi2= pi/2; % phase offset 2!! tmax= 120; % max. time value {6}! N= 1000; % # of points to compute {200}! % =================================!! % ---! t=linspace(0,tmax,n); % time variable! s1= A1*cos(w1*t- phi1);! s2= A2*cos(w2*t- phi2); % calculate both sinusoids and sum! A= s1+ s2;! % ---! % other (analytically-derived) expressions for the sum!! B= cos((w1+w2)/2*t).*(a1*cos((w1-w2)/2*t) + A2*cos((w1-w2)/2*t)); % from Serdyuk (which is incorrect)! % 2. CB calculated version! C= cos((w1+w2)/2*t).*(a1*cos((w1-w2)/2*t) + A2*cos((w1-w2)/2*t) ) +...! sin((w1+w2)/2*t).*(-a1*sin((w1-w2)/2*t) + A2*sin((w1-w2)/2*t) );! % 3. CB calculated version (factored)! D= (A1+A2)*cos((w1+w2)/2*t).*cos((w1-w2)/2*t) +...! (A2-A1)*sin((w1+w2)/2*t).*sin((w1-w2)/2*t);! % ---! figure(1); clf;! h1= plot(t,s1,'r'); hold on; grid on;! h2= plot(t,s2,'b--');! h3= plot(t,a,'k','linewidth',3);! xlabel('amplitude [arb]'); ylabel('time [arb]');! legend([h1 h2 h3],'sinusoid 1','sinusoid 2','sum');! % ---! if 1==1! figure(2); clf;! plot(t,a,'k','linewidth',2); hold on; grid on;! %plot(t,b,'r--'); plot(t,c,'go');! plot(t,d,'r.');! end!

14 Beats: Sum of (two) sinusoids f 1 =1, f 2 =1.1 A 1 =1, A 2 =1 φ 1 =0, φ 2 =0 f 1 =1, f 2 =1.2 A 1 =1, A 2 =1 φ 1 =0, φ 2 =0 à Different frequency combinations yield different patterns f 1 =1, f 2 =1.3 A 1 =1, A 2 =1 φ 1 =0, φ 2 =0

15 Beats: Sum of (two) sinusoids f 1 =1, f 2 =1.1 A 1 =1, A 2 =1 φ 1 =0, φ 2 =0 à Changing (relative) phase affects summation f 1 =1, f 2 =1.1 A 1 =1, A 2 =1 φ 1 =π/2, φ 2 =0 à Changing (relative) amplitudes affects summation f 1 =1, f 2 =1.1 A 1 =2, A 2 =1 φ 1 =0, φ 2 =0

16 Nota3on re Fourier analysis Hobbie & Roth version à Analy3c nota3on u3lized can vary widely across texts... Fourier coefficients Note the error above... Complex #s

17 Nota3on re Fourier analysis Fourier coefficients Cosine/Sine Transforms Fourier Transform Note: It s a two-way street (w/ no loss of info) Forward transform (3me à frequency) Inverse transform (frequency à 3me)