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.12

2 Impulse Response Impulse response (this fully characterizes the system ) Input: Incoming signal (arbitrary) Output: filtered signal (combinacon of the system and the input) Buzug (2008)

3 Ex. AcousCc Impulse Response Room response (g) filters an input sound (s) Room response (g) is just convolucon between s and room s impulse response (h) à All the relevant bits of the room s acousccs are contained in h (which we can easily measure!) Pulkki & Karjalainen (2015)

4 hsp://

5 LTI (Linear Time-Invariant) Systems Pulkki & Karjalainen (2015)

6 Recall: Why sample with pulses? Buzug (2008)

7 % ### EXconvolution2.m ### CB (updated )!! % Example code to perform convolution between a (discrete) sinusoid (wf1) and narrow! % digital pulse (wf2); should see that the convolved signal is just the! % original sinusoid!! % Note:! % o reqs. (custom-code) convolve1.m! % o allows user (via "method") to specify whether a (CB) custom-coded! % convolution code (convolve1.m) or Matlab's built-in conv.m is used! clear! % ! SR= 44100; % sample rate [Hz]! Npoints= 100; % length of window (# of points) {8192}! f= ; % wf1 Frequency (for waveforms w/ tones) [Hz]! CLKbnd= [50 51]; % wf2: indicies at which pulse turns 'on' and then 'off' {[ ]]! method= 0; % boolean to specify whether to use custom convolution code (0) or Matlab's (1) {0}! % ! % +++! t=[0:1/sr:(npoints-1)/sr]; % create an array of time points! % +++! % create two waveforms (same dimensions)! wf1= cos(2*pi*f*t);! clktemp1= zeros(1,npoints);! clktemp2= ones(1,clkbnd(2)-clkbnd(1));! wf2= [clktemp1(1:clkbnd(1)-1) clktemp2 clktemp1(clkbnd(2):end)];! % +++! % Use custom code (convolve1.m) or Matlab's built-in function? [should return identical answers]! if (method==0), C= convolve1(wf1,wf2); % custom code! else C= conv(wf1,wf2); end % Matlab's built-in function! % +++! figure(1); clf;! subplot(211)! h1= plot(t,wf1,'b'); hold on; grid on;! h2= plot(t,wf2,'r.-');! legend([h1 h2],'wf1 (sinusoid)','wf2 (impulse)');! xlabel('time [s]'); ylabel('amplitude'); title('two waveforms (wf1 and wf2)');! subplot(212); plot(c,'k'); hold on;! xlabel('sample index'); ylabel('amplitude'); title('convolution between wf1 and wf2');! EXconvoluCon2.m

8 convolve1.m function y= convolve1(wf1,wf2); % CB! % convolve two 1-D row vectors (should work similar to Matlab's conv.m)! % +++! error(nargchk(2, 2, nargin)), error(nargoutchk(0, 1, nargout))! if ~isvector(wf1) ~isvector(wf2)! error('parameters must be vectors.')! end! % ensure they are row vectors! if (~isrow(wf1)), wf1= wf1'; end! if (~isrow(wf2)), wf2= wf2'; end! m = length(wf1); n = length(wf2); % extract relevant dimensions! % create new arrays as needed for operation! g= fliplr(wf2); % flipped wf2! f= [zeros(1,n) wf1 zeros(1,n)];! NN= m+n-1;! for k=1:nn! % Note: It took me awhile to get this code right!! y(k)= sum(f.*[zeros(1,k) g zeros(1,m-k+n)]); % shifted wf2! end! return!

9 à Signal convolved w/ an impulse is (more or less) itself! EXconvoluCon2.m

10 Can we slow down for a moment? What (intuicvely?) is a convolucon?

11 (Important) Tangent: CorrelaCons ex. neural raster plot Ø How do we find paserns in signals? Ø PeriodiciCes? à Look for correlacons Izhikevich (IEEE 2003)

12 Cross-CorrelaCon Note: We focus on 1-D here for clarity, but these ideas generalize to higher dimensions (e.g., 2-D for images) à Cross-correlaCon (between y 1 and y 2 ) is a Cme-shi`ed sum of their overlap as a funccon of said shi` Hobbie & Roth

13 à Think about how a correlacon measure would tell you something along the lines of reliability

14 Usually useful to consider an average value (i.e., change the limits of integracon): 1 = lim T 2T T T y 1 (t)y 2 (t + τ) dt. (11.41) à For a given value of τ, the cross-correlacon takes a single (scalar) value [dashed lines] Hobbie & Roth

15 Auto-CorrelaCon Cross-correlate a signal with itself: à Useful for reducing noise when there is a periodic/phase-locked signal (temporal averaging) Hobbie & Roth

16 Auto-CorrelaCon 1 = lim T 2T T T y 1 (t)y 2 (t + τ) dt. (11.41) ex. pure sinusoid [keep this in mind re the next slide] Hobbie & Roth

17 Auto-CorrelaCon ß à Fourier Transform à Deep conneccon between the two paths (this has big implicacons as we ll see later on) Note: This interrelaconship Ces into the Central SecCon Theorem raised by Nishimura (Sec.2.3.2; we ll likely come back to this once we get to MRI) Hobbie & Roth

18 Recall: Two-Dimensional Fourier-Based ReconstrucCon Methods Similar flavor of idea (i.e., interrelaconaship between Fourier decomposicon and other sorts of transforms ) Buzug (2008)

19 ConvoluCon Similar in spirit to a cross-correlacon (with some addiconal strings asached) Through the lens of LTI systems: g(t) output f(t) input h(t) impulse response à A sampled signal is the original (concnuous) signal convolved with a train of impulses Hobbie & Roth

20 Recall: Why sample with pulses? Buzug (2008)

21 Various InterrelaConships Note: In addicon to Fourier transforms, Laplace transforms also commonly used to go back and forth between temporal and spectral domains Wikipedia

22 ConvoluCon Theorem Fourier Transform Sine/Cosine Transform Theorem Also applies to 2-D (and higher) Hobbie & Roth