Homework 1
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1. can figure out what is wrong with this system The user s manual shows that the system uses a mirror to reflect photons f t into the sensor To make it oscillate faster the manufacturer had to make the mirror lighter which also allowed some of the incident light to penetrate the mirror and add back into the main signal but with a to 2 fs femtosecond 10715 s time delay Therefore fo is a time shifted copy of f except for the lower magnitude f2 0 2 fil a Ignoring system effects i e g t fi t write an expression for net output g t in terms of fi t and fo t See the figure below b Calculate analytically the power spectrum squared magnitude of the Fourier transform G u using the information above Plot your final answer and show that you obtain the scalloped spectrum shown in Fig 1 12 Fig 17 In Example 1 9 we somehow made measurements without the problem of noise Repeat the exercise in the example but add noise using the Poisson random number generator f A poissrnd L size t The constants A and L are the amplitude and variance of the noise added Play with those parameters and give me an example where noise changes my conclusions in that Example Compute the mean square error as in the example to explain how noise affects your fitting strategy A new flow cytometer was purchased to size microscopic particles suspended in fluids While calibrating you observe the calibration signal and a little rand2. correct structure If you understand what I did above it should be clear that you need to multiply two of the above matrices and you will get a new 800 x 100 matrix let s call it X of un normalized correlation functions Find the index corresponding to the peak values for the 100 correlation functions using C I max X Histogram the index values using BIOE 504 Fall 2009 43 hist I nbins where you pick the number of bins that gives a reasonable looking plot Print out your program code and the histograms Also display on the plots the mean value and the standard deviation of distribution b Change the amplitude suppressed region of the template to be only 1 second long and repeat part a 4 MATLAB Exercise Generate a 10 s ramp function given by g t t 10 for 0 lt t lt 10s Compute the Fourier series and FFT up to 5 Hz Plot the absolute values of both on top of each other clearly indicating each and show that they give the same result 5 Use closed form integration to show zi exp a x ve exp 17u a 6 Use the result above change the appropriate variables and then use the shift theorem for Fourier transforms to find Pini where h x exp x xo 20 7 Prove the modulation theorem 1 F h x cos 2ruoz s EG uo H u uo 8 Compute d t F rect di sx two ways a First graph by hand no MATLAB the rect function and its derivative Then compute
3. the transform of the result b Now try the approach of applying the derivative theorem and then computing the transform The two results should be equal and the answer should involve a sine function Tell me when you use theorems in your solution 9 Use closed form integration to show that the 2 D Fourier transform of the circular aperture function Fopcire r a ra jinc ap The shorthand A circ r a just means a circle of radius a where all values inside the circle at r lt a equal A and values outside equal zero The circle is centered on the origin For this problem dust off your ability to integrate in polar coordinates The first thing to consider is transforming the usual rectilinear spatial variables x y and the corresponding frequency variables u v into polar coordinates to take advantage of the symmetry You know how to do this r yx y 0 arctan y x p Vu 02 y arctan v u x r cos 8 y rsind u pcos y v psiny BIOE 504 Fall 2009 44 OK we have our notation Now you ll need these facts ee o0 2T ee J dy dx d0 rdr oo oo 0 0 1 20 0 3 dO eta cos e Jola A f e J 488 5018 anlo 0 Jn is a Bessel function of the first kind nth order Finally jinc p 2J1 2mp 27p a Compute the 2 D transform to obtain the first line above b What special name does this transform see below have G p 20 dr r g r Jo 2mrp O25 5 10 15 20 time s Fi
4. BIOE 504 Fall 2009 41 Problems for Chapter 1 1 a Compute mathematically the autocorrelation function 4 7 for g t exp t 207 b Plot g t and r on top of each other using MATLAB c Compute analytically the full width at half maximum FWHM values for both g t and g r and find their ratio Note that FWHM value is the width of the function in time at half of its maximum value 2 a Fig 1 12 Fig 15 illustrates two rectangular function gi t and go t Draw by hand the cross correlation function 2 7T Carefully label all the features in variables as I did below b How does the result from part a change if you convolve g and g2 i t j i o gb J arec T x t T 2 t 7 i2 0 M4 27 P g iN b rect a f t T t T Figure 15 Illustration of rectangular function from problem 2 3 MATLAB Exercise Use the following code to generate a matrix of 100 waveforms Call it matrix G which has dimensions 1001 x100 as illustrated below t 0 0 01 10 Y sampling interval T 0 01s and duration TO 10s hi exp t 5 72 2 0 0772 narrowband pulse for j 1 100 f poissrnd 1 size t Poisson noise is generated here gg conv hi f 0 01 G j abs gg 501 1501 noise blurred by pulse end BIOE 504 Fall 2009 42 g 1 1 g 1 2 g 1 100 92 1 g 2 2 2 100 G k g 1001 1 g 1001 2 g 1001 100 The first index indicates samples along the time axi
5. another input function make it interesting Also add a modest amount of noise via n a randn size function where a is the standard deviation of the normal random noise variable If you re not sure try it and plot everything to make sure you follow how this signal added to noise thing works Give me the same plots I gave in Example 1 9 don t bother with plot a showing the polynomials Finally look at the appendix and do an SVD analysis on the same data Comparing and interpret results by discussing how the polynomial fitting errors compare with the SVD values
6. for convolution that reads gt f ahe EE WO O co BIOE 504 Fall 2009 47 15 16 I try to always write the shorthand notation a different way using h f t because of the following situation You are using an LTI optical system to estimate the concentration of a substance in the blood as a function of time A light beam is positioned over a vein and the backscattered energy is recorded to give the measurement g t The system is defined by its impulse response h t and the substance in the blood as f t The substance is injected as a bolus so that f t looks like a rectangular function immediately after injection but spreads out in the blood stream according to f at b where a and b are constants This model suggests that the substance is shifted in time b and scaled by the factor a Therefore our measurements can be expressed using co g t dt h t t f at b 00 and so the notation h t f t doesn t make much sense Beginning with the equation above derive the Fourier transform G u by first replacing h and f by expressions of the inverse transforms and then proceeding In Section 1 9 we saw that if g t h f t then G u H u F u because of the Fourier convolution theorem Using similar arguments show that if G u H F u then g t h t f t The equation for any instrument classified as linear time invariant can be written in matrix form as g Hf e where f is an N x 1 c
7. gure 16 Illustration of the free induction decay signal in problem 10 10 One particular magnetic resonance MR free induction decay FID signal may be expressed as the sum of three exponentially decaying sinusoids plus a constant 3 FID t Mo X Mn e cos Qnt x step t where Qn 2run n 1 BIOE 504 Fall 2009 45 T 12 13 and u is temporal frequency in Hz a Plot the function M cos Q t e by hand no MATLAB and label the graph to show approximate values for My 7 and Ty 27 4 b Perform the integration to find the Fourier transform of FID t c Explain by examining the equation for FID why the transform is complex d Find the real part of the result in part b e Now use MATLAB and assume Mi 1 71 1 s 01 27 uy 1 Hz M2 0 5 mn 2 s ug 2 Hz M3 0 333 T2 3 s u3 3 Hz Plot the result from part d i e ReF FID t from 0 lt u lt 5 Hz using MATLAB and carefully label the axes You borrowed a spectrophotometer a photometer that measures optical radiation intensity as a function of frequency or wavelength from the lab next door because it is 10 times faster than the one in your lab To calibrate it before use you illuminate its sensor with a 600 nm wavelength 5 x10 4 Hz source known to give a Gaussian shaped spectrum Expecting to see the smooth spectrum in the figure below left you are surprised to observer the scalloped spectrum on the right Let s see if we
8. lecting a different amount of shift for the circular convolution Once f and H are constructed then find the 484 x 1 vector g from the product Hf as in Fig 11c Reorder the resulting vector to reconstruct its 2 D form g m n and display the 22 x 22 pixel image It should resemble f m n but blurry You might need to trim the result to make it 22 x 22 pixels Once it all works use the MATLAB functions tic and toc to time the execution BIOE 504 Fall 2009 49 b 18 19 20 Use the Fourier convolution theorem to achieve the same effect but do not use fft2 or ifft2 to compute transforms Instead construct a Q matrix and use matrix methods to complete the forward and inverse transforms You may need to trim the result to compare with a Also time the computation Finally use conv2 and fft2 to achieve the same results as a and b above For each part of this problem show me the MATLAB code the images and the computation time You ll find that g m n may not be scaled in amplitude the same as f m n which means somehow your system matrix is amplifying or attenuating your object signal Any ideas on how to get rid of the annoying scaling Compute analytically F u v given that f x y rect x xo Xo rect y yo Yo MATLAB Exercise Prove Eq 8 is correct by numerically computing Qaf and fa where a rect t 0 5 and f 1 t step t step 1 t MATLAB Exercise Follow Example 1 9 but select
9. olumn vector of values representing the signal input into the instrument g is the corresponding N x 1 vector of output values H is the N x N system matrix that describes how input values linearly map into output values and e is an N x 1 vector of additive noise Assume H is circulant and that Hf is a convolution We can analyze properties of the instrument from eigenvalues A of H H qk Ak Qk where qx is the kth eigenvector of dimension N x 1 that corresponds to Az Let s select a specific eigenvector e 27 0 k N ei2n 1 k N a 1 ei2n 2 k N VN ei2m N 1 k N and create an N x N matrix out of columns of eigenvectors Q qo qx Qv 1 The elements of the matrix are Qn e 2 N JN a Nice properties of Q stem from orthogonality Show that Q is an orthogonal matrix b Use properties of an orthogonal matrix to find an expression for elements of its inverse 1 Qin BIOE 504 Fall 2009 48 c We can decompose the system matrix as follows H QDQ7 Explain how matrix D is related to the eigenvalues of H d Begin with the linear systems equation g Hf e Rewrite this equation using the eigen decomposition of the system matrix given in part c and multiply the result by Qt e Express the term Q g from the result of part d as an equation involving a summation over integer n f What is another name for the equation that results from part e What does the integer index k represent 17 MATLAB E
10. om noise as expected but you also observe a cosine wave that you suspect the system is receiving from radiation emitted by fluorescent lights There is nothing you can do about it today and you need to measure samples so you make the measurements anyway Now you must decide if the cosine signal has contaminated your experiment to the point that results are meaningless or if the data can be used BIOE 504 Fall 2009 46 spectrophotometer g t o output a o output a 2 4 6 8 2 4 6 8 10 frequency 1014 frequency x 1014 Figure 17 Spectrophotometer experiment from Problem 11 a To isolate the nuisance cosine signal g t you flush particle free fluid through the system The cosine signal has a frequency of uo 60 Hz and an amplitude of A 10 mV Compute the frequency spectrum given by the modulus of the Fourier transform G w b Assume the desired calibration signal and nuisance cosine signal sum linearly The cali bration signal is c t 0 _ 12 20 V 210 and therefore its frequency spectrum is 2 2 we C u Coe Plot the calibration spectrum along with your result from part a for positive frequencies between 0 lt u lt 5uo assuming that o 3 uo State your decision as to whether the cosine signal s contribution to the frequency spectrum can be ignored if your experimental data have the same bandwidth as the calibration data 14 There is a common shorthand notation
11. s t nT n x 0 01 sec and the second index indicates a different waveform presumably from another measurement a Create a vector template call it h t consisting of a 1001 point sequence of values that are all 1 except values between 401 lt n lt 601 that is the central 2 s where the amplitude is 0 5 Multiply the template h t by each waveform in matrix G Here s something that works for j 1 100 G j GC j h Zoooh bad programming practice end This new G matrix represents a series of 100 measurement waveforms described as noisy narrowband signals with a region between 4 and 6 seconds that has 50 of the amplitude of the surrounding regions Plot a few waveforms to be sure that you formed them correctly Now devise a simple algorithm that finds the location of the low amplitude region Use h as a matched filter on the 100 waveforms to locate the positions of the low amplitude regions Of course you already know they are all centered at 5 seconds or index number 501 Specifically your program must correlate the template with each waveform and identify the time of the correlation peak value To do the correlation I recommend that you create a circulant matrix H 800 1001 where each row is a shifted copy of vector h For example H ones 800 1001 for j 1 800 H j j j 200 0 2 end To be sure everything smells right display H using imagesc H colormap gray axis square just to be sure it has the
12. xercise Assume we have the simple 2 D object f n m with dimensions 22 x 22 pixels See the figure on the right below Here I placed a 12 x 12 square of 1 s in the middle of a field of 0 s Next to f n m on the left is the 2 D impulse response of the imaging system h n m that also has dimensions 22 x 22 However the only nonzero pixels are near the center where we find a 2 x 2 block of 1 s Do not use the native MATLAB functions xcorr2 conv2 fft2 or ifft2 for this problem until part c IT T iat i E i a 1 T ji T T E E i T E MR EER LOB DE ME Di D E ial Tt ER Ws E TT 1 T fe ial ay E S E a fa EE E E E T TEE T T IES E E l T TTT TT T T Figure 18 Figure for Problem 17 a Reorder f m n into a 484 x 1 vector f as illustrated in Fig 12 Also figure out a way to reorder h m n into a vector that will become the rows of circulant system matrix H Recall that the task for reordering h m n into rows of H is to find an ordering where multiplication of f by each row of H is equivalent to multiplying the 2 D functions h m n and f m n Selecting a different row is the same as se
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