Biophysical Techniques (BPHS 4090/PHYS 5800)

Transcription

1 Biophysical Techniques (BPHS 49/PHYS 58) Instructors: Prof. Christopher Bergevin Schedule: MWF :3-2:3 (CB 22) Website: York University Winter 27 Lec.7

2 Goal Develop knowledge and intuiaon dealing w/ 2-D Fourier transforms 2-D case (e.g., image) Why Fourier (i.e., spectral) analysis? And why 2-D?

3 Reminder Franklin & Gosling (953) Sodium thymonucleate fibres give two disanct types of X-ray diagram. The first corresponds to a crystalline form... Watson & Crick (953)

4 Crystal What is a crystal? Something w/ a periodic structure (we ll return to this later in the semester) Halliday & Resnick

5 Crystals made up of complex materials In a nutshell: Thus, the diffracaon payern of a protein crystal is the Fourier transform of the unit cell Ames the Fourier transform of the crystal lazce. The layer is called reciprocal la*ce à Should make sense that we might want to use a theoreacal foundaaon with a set of periodic basis funcaons... NolAng

6 Goal Develop knowledge and intuiaon dealing w/ 2D Fourier transforms 2-D case (e.g., image).8 -D case time waveform Ame waveform recorded from mic Pressure [arb] Time [s] à Will first explore (from a computaaonal viewpoint) -D Fourier transforms... Note: Following background notes taken from PHYS 23 W6

7 % ### EXbuildImpulse.m ###.3.4! % Code to visually build up a signal by successively adding higher and! % higher frequency terms from corresponding FFT! clear; clf;! % ! SR= 44; Npoints= 892; % sample rate [Hz]! % length of fft window (# of points) [should ideally be 2^N]! % [time window will be the same length]! INDXon= ; % index at which click turns 'on' (i.e., go from to )! INDXoff= ; % index at which click turns 'off' (i.e., go from to )! % !! dt= /SR; % spacing of time steps! freq= [:Npoints/2]; % create a freq. array (for FFT bin labeling)! freq= SR*freq./Npoints;! t=[:/sr:(npoints-)/sr]; % create an appropriate array of time points! % build signal! clktemp= zeros(,npoints); clktemp2= ones(,indxoff-indxon);! signal= [clktemp(:indxon-) clktemp2 clktemp(indxoff:end)];! % ! % *******! % plot time waveform of signal! if ==! figure(); clf; plot(t*,signal,'ko-','markersize',5)! grid on; hold on; xlabel('time [ms]'); ylabel('signal'); title('time Waveform')! end! % *******! % now compute/plot FFT of the signal! sigspec= rfft(signal);! % MAGNITUDE! figure(2); clf;! subplot(2); plot(freq/,db(sigspec),'ko-','markersize',3)! hold on; grid on; ylabel('magnitude [db]'); title('spectrum')! % PHASE! subplot(22); plot(freq/,cycs(sigspec),'ko-','markersize',3)! xlabel('frequency [khz]'); ylabel('phase [cycles]'); grid on;! % *******! % now make animation of click getting built up, using the info from the FFT! sum= zeros(,numel(t)); % (initial) array for reconstructed waveform! figure(3); clf;! for nn=:numel(freq)! sum= sum+ abs(sigspec(nn))*cos(2*pi*freq(nn)*t + angle(sigspec(nn)));! plot(t,sum); xlabel('time [s]');! legend(['highest freq= ',num2str(freq(nn)/),' khz'])! pause(2/(nn))! end! EXbuildImpulse.m

8 6 x 4 Highest freq= khz 4 x 3 Highest freq= khz Time Waveform Temporal 7 Spectrum EXbuildImpulse.m Spectral Signal is an impulse (i.e., a delta function) Magnitude [db] Signal Time [ms] Phase [cycles] Spectral representation has flat amplitude and a group delay Frequency [khz] 5 4 Reconstruct waveform by adding sinusoids (only lowest frequency here) Now the first 5 terms are included Time [s] Time [s] à Eventually all the sinusoids add up such that things cancel out everywhere except at the point of the impulse!

9 % ### EXspecREP3.m ###.29.4! % Example code to just fiddle with basics of discrete FFTs and connections! % back to common real-valued time waveforms! % --> Demonstrates several useful concepts such as 'quantizing' the frequency! % Requires: rfft.m, irfft.m, cycs.m, db.m, cyc.m! % ! % Stimulus Type Legend! % stimt= - non-quantized sinusoid! % stimt= - quantized sinusoid! % stimt= 2 - one quantized sinusoid, one un-quantized sinusoid! % stimt= 3 - two quantized sinusoids! % stimt= 4 - click I.e., an impulse)! % stimt= 5 - noise (uniform in time)! % stimt= 6 - chirp (flat mag.)! % stimt= 7 - noise (Gaussian; flat spectrum, random phase)! % stimt= 8 - exponentially decaying sinusoid (i.e., HO impulse response)!! clear; clf;! % ! SR= 44; % sample rate [Hz]! Npoints= 892; % length of fft window (# of points) [should ideally be 2^N]! % [time window will be the same length]! stimt= 8; % Stimulus Type (see legend above)! f= 258.; % Frequency (for waveforms w/ tones) [Hz]! ratio=.22; % specify f2/f2 ratio (for waveforms w/ two tones)! % Note: Other stimulus parameters can be changed below! % ! dt= /SR; % spacing of time steps! freq= [:Npoints/2]; % create a freq. array (for FFT bin labeling)! freq= SR*freq./Npoints;! % quantize the freq. (so to have an integral # of cycles in time window)! df = SR/Npoints;! fq= ceil(f/df)*df; % quantized natural freq.! t=[:/sr:(npoints-)/sr]; % create an array of time points, Npoints long! % ----! % compute stimulus! if stimt== % non-quantized sinusoid! signal= cos(2*pi*f*t);! disp(sprintf(' \n *Stimulus* - (non-quantized) sinusoid, f = %g Hz \n', f));! disp(sprintf('specified freq. = %g Hz', f));! elseif stimt== % quantized sinusoid! signal= cos(2*pi*fq*t);! disp(sprintf(' \n *Stimulus* - quantized sinusoid, f = %g Hz \n', fq));! disp(sprintf('specified freq. = %g Hz', f));! disp(sprintf('quantized freq. = %g Hz', fq));! elseif stimt==2 % one quantized sinusoid, one un-quantized sinusoid! signal= cos(2*pi*fq*t) + cos(2*pi*ratio*fq*t);! disp(sprintf(' \n *Stimulus* - two sinusoids (one quantized, one not) \n'));! elseif stimt==3 % two quantized sinusoids! fq2= ceil(ratio*f/df)*df;! signal= cos(2*pi*fq*t) + cos(2*pi*fq2*t);! disp(sprintf(' \n *Stimulus* - two sinusoids (both quantized) \n'));! elseif stimt==4 % click! CLKon= ; % index at which click turns 'on' (starts at )! CLKoff= ; % index at which click turns 'off'! clktemp= zeros(,npoints);! clktemp2= ones(,clkoff-clkon);! signal= [clktemp(:clkon-) clktemp2 clktemp(clkoff:end)];! disp(sprintf(' \n *Stimulus* - Click \n'));! elseif stimt==5 % noise (flat)! signal= rand(,npoints);! disp(sprintf(' \n *Stimulus* - Noise \n'));! elseif stimt==6 % chirp (flat)! fs= 2.; % if a chirp (stimt=2) starting freq. [Hz] [freq. swept linearly w/ time]! fe= 4.; % ending freq. (energy usually extends twice this far out)! fsq= ceil(fs/df)*df; %quantize the start/end freqs. (necessary?)! feq= ceil(fe/df)*df;! % LINEAR sweep rate! fswp= fsq + (feq-fsq)*(sr/npoints)*t;! signal = sin(2*pi*fswp.*t)';! disp(sprintf(' \n *Stimulus* - Chirp \n'));! EXspecREP3.m elseif stimt==7 % noise (Gaussian)! Asize=Npoints/2 +;! % create array of complex numbers w/ random phase and unit magnitude! for n=:asize! theta= rand*2*pi;! N2(n)= exp(i*theta);! end! N2=N2';! % now take the inverse FFT of that using Chris' irfft.m code! tnoise=irfft(n2);! % scale it down so #s are between - and (i.e. normalize)! if (abs(min(tnoise)) > max(tnoise))! tnoise= tnoise/abs(min(tnoise));! else! tnoise= tnoise/max(tnoise);! end! signal= tnoise;! disp(sprintf(' \n *Noise* - Gaussian, flat-spectrum \n'));! elseif stimt==8 % exponentially decaying cos! alpha= 5;! signal= exp(-alpha*t).*sin(2*pi*fq*t);! disp(sprintf(' \n *Exponentially decaying (quantized) sinusoid* \n'));! end!! % ! % *******! figure(); clf % plot time waveform of signal! plot(t*,signal,'k.-','markersize',5); grid on; hold on;! xlabel('time [ms]'); ylabel('signal'); title('time Waveform')! % *******! % now plot rfft of the signal! % NOTE: rfft just takes /2 of fft.m output and nomalizes! sigspec= rfft(signal);! figure(2); clf; % MAGNITUDE! subplot(2)! plot(freq/,db(sigspec),'ko-','markersize',3)! hold on; grid on;! ylabel('magnitude [db]')! title('spectrum')! subplot(22) % PHASE! plot(freq/,cycs(sigspec),'ko-','markersize',3)! xlabel('frequency [khz]'); ylabel('phase [cycles]'); grid on;! % ! % play the stimuli as an output sound?! if (==), sound(signal,sr); end! % ! % compute inverse Fourier transform and plot?! if ==! figure();! signalinv= irfft(sigspec);! plot(t*,signalinv,'rx','markersize',4)! legend('original waveform','inverse transformed')! end

10 Fourier transforms of basic (-D) waveforms stimt= - non-quantized sinusoid! EXspecREP3.m SR= 44; % sample rate [Hz]! Npoints= 892; % length of fft window! Time domain Spectral domain Time Waveform Spectrum Magnitude [db] Signal Time [ms] Phase [cycles] Frequency [khz] Ø Magnitude shows a peak at the sinusoid s frequency Note: The phase is unwrapped in all the spectral plots

11 Fourier transforms of basic (-D) waveforms EXspecREP3.m stimt= 3 - two quantized sinusoids! Time domain Spectral domain 2 Time Waveform Spectrum.5.5 Magnitude [db] 2 3 Signal Phase [cycles] Time [ms] Frequency [khz] Ø Magnitude shows two peaks (note the beating in the time domain)

12 Fourier transforms of basic (-D) waveforms EXspecREP3.m stimt= 4 - click (i.e., an impulse)! Time domain Spectral domain Time Waveform 7 Spectrum Magnitude [db] Signal Phase [cycles] Time [ms] Frequency [khz] Ø Click has a flat magnitude (This is also a good place to mention the concept of a group delay )

13 Fourier transforms of basic (-D) waveforms EXspecREP3.m stimt= 5 - noise (uniform distribution)! Time domain Spectral domain Time Waveform 2 Spectrum.9 Signal Magnitude [db] Time [ms] Phase [cycles] Frequency [khz] Ø Magnitude is flat-ish (on log scale), but actually noisy. Phase is noisy too.

14 Fourier transforms of basic (-D) waveforms EXspecREP3.m stimt= 7 - noise (Gaussian distribution)! Time domain Spectral domain Time Waveform 44 Spectrum Magnitude [db] Signal Phase [cycles] Time [ms] Frequency [khz] Ø Magnitude is flat just like an impulse (i.e., flat), but the phase is random

15 Fourier transforms of basic (-D) waveforms Impulse Noise Time Waveform Time Waveform Time domain Signal.5.4 Signal Time [ms] Time [ms] 7 Spectrum 44 Spectrum Magnitude [db] Magnitude [db] Spectral domain Phase [cycles] Frequency [khz] Phase [cycles] Frequency [khz] à Remarkable that the magnitudes are idenacal (more or less) between two signals with such different properaes. The key difference here is the phase: Timing is a cri/cal piece of the puzzle!

16 Fourier transforms of basic (-D) waveforms EXspecREP3.m stimt= 6 - chirp (flat mag.)! Time domain Spectral domain Time Waveform 2 Spectrum Magnitude [db] Signal Time [ms] Hard to see on this timescale, but frequency is changing (increasing) with time Phase [cycles] Frequency [khz]

17 Fourier transforms of basic (-D) waveforms EXspecREP3.m stimt= 8 - exponentially decaying sinusoid! Time domain Spectral domain Time Waveform 2 Spectrum Magnitude [db] Signal Phase [cycles] Time [ms] Frequency [khz] Ø This seems to look familiar...

18 ConnecAon back to the harmonic oscillator! Time Waveform Signal Time [ms] 2 Spectrum Magnitude [db] Phase [cycles] Frequency [khz] Ø The steady-state response of the sinusoidally-driven harmonic harmonic oscillator acts like a band-pass filter Ø DisAncAon between steady-state response & impulse response [we ll come back to this]

19 2-D Fourier analysis.8 -D case time waveform Ame waveform recorded from mic.6.4 Pressure [arb] D case (e.g., image) Time [s] Ø Ø Same basic idea between -D and 2-D (though math becomes a bit more complicated) Just as a waveform can have temporal frequencies, an image can have spa/al frequencies

20 2-D Fourier analysis Note: Independent variables (x and y here) can represent any physical quanaty, but for 2-D we typically use posiaon in Cartesian plane (rather than Ame)

21 NotaAon re Fourier analysis -D 2-D

22 2-D Fourier analysis A. Zisserman (Oxford)

23 2-D Fourier analysis Note: Only ½ of the informaaon is being shown for the spectral domain (i.e., just the magnitude) A. Zisserman (Oxford)

24 2-D Fourier analysis Spectral domain SpaAal domain Note: Only ½ of the informaaon is being shown for the spectral domain (i.e., just the magnitude) Hobbie & Roth

25 2-D Fourier analysis A. Zisserman (Oxford)

26 2-D Fourier analysis Swap phases, then inverse transform to get the image back A. Zisserman (Oxford)