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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete Fall Term, 2008 Quiz #2 (Take Home) Due in class (1pm) on December 9, 2008 Notes: • This is a take-home quiz. • Collaboration (with anybody) is expressly forbidden. • Do not spend excessive time on this quiz! • There are 4 problems, answer them all. • Partial credit will be given. • You may ask the teaching staff for help -but that help will be generally limited to ensuring that you have access to the required data files. 1M     Problem 1: (20 points) Here is a simple way to design a digital high-pass filter from a low-pass design (this is not a method we talked about in class). Consider a prototype causal digital low-pass filter H¯lp(ej ω). The figure below shows how a digital low-pass filter may be transformed to a high-pass filter by simply translating (shifting) its frequency response H(ej ω)by π, that is we create the high-pass filter from the prototype: Hhp(ej ω)= Hlp(ej(ω−π)). | H ( e ) |l p j M M F0 | H ( e ) |h p j M F 0 p r o t o t y p e l o w - p a s s f i l t e r h i g h - p a s s f i l t e r (a) Assume that the original low-pass filter has a recursive difference equation N M y(n)= − aky(n − k)+ bkx(n − k) k=1 k=0 corresponding to the discrete-time transfer function M bkz−k k=0Hlp(z)= N1+ k=1 akz−k Show that the difference equation of the new high-pass filter is N M y(n)= − (−1)k aky(n − k)+ (−1)kbkx(n − k) k=1 k=0 In other words, a high-pass filter may be constructed by simply alternating the signs of the coefficients in the low-pass difference equation! (b) Show that the two discrete-time impulse responses are related by hhp(n)=(−1)nhlp(n) (c) How is the phase response  Hhp(ej ω) of the high-pass filter related to the phase response  Hlp(ej ω) of the prototype low-pass filter? Problem 2: (25 points) An acoustic micro-GPS system. An experiment on human movement requires that we monitor the position of a person’s head during the experiment. The experiment is to be conducted in a room 20×20 ft. with a 10 ft. ceiling. Four loudspeakers have been installed in the top four corners of the room as shown below: 2) s 3 ( t ) ( 0 , 0 , 1 0 ) ( 2 0 , 0 , 1 0 ) ( 2 0 , 2 0 , 1 0( 0 , 2 0 , 1 0 ) x y z 4 - c h a n n e l t r a n s m i t t e r s ( t ) s ( t ) s ( t ) 1 2 4 l o u d s p e a k e r m i c r o p h o n e P ( x , y , z ) r e c e i v e rr ( t ) ( x , y , z ) The human subject wears a microphone mounted on a helmet. • At regular intervals the transmitter broadcasts four uncorrelated wide-band acoustic signals si(t), i =1 ... 4, simultaneously - each to one of the loudspeakers as shown above. • The sampling rate used in both the transmitted and received signals is 100,000 sam-ples/sec. • The microphone signal r(t) is fed to the receiver, which computes and stores the carte-sian coordinates x, y, z of the subject’s head. • The transmitter and receiver have precision clocks that are synchronized, in other words the receiver knows the time origin of each transmitted signal. • The speed of sound is 1125 ft/s. • There is “significant” noise in the room. The Problem: You are given a file Q2Prob2.mat containing two MATLAB data ar-rays: (1) A two-dimensional array s(4,500) containing the four transmitted waveforms s, and (2) a one-dimensional array r containing a single data record from the micro-phone. Your task is to determine the cartesian coordinates (x, y, z) of the location of the microphone at this time. Notes: • The system is (deliberately) over-determined. In fact you only need three speakers to solve this problem. Just as in the GPS system the use of redundant sources allows for situations when one source might be occluded from the microphone. Your solution should use all four speakers. Use a least-squares approach to estimate the coordinates x, y, z. Suggestion: use MATLAB’s lsqnonlin(). 3• Since this is not a course in numerical optimization, most of the grade will be based on setting up the data to pass to the least-squares solver. • MATLAB’s lsqnonlin() requires you to write a function that returns an error vec-tor (not the objective function). For example x = lsqnonlin(@myfun, x0) uses the function myfun(). See the help. Problem 3: (25 points) System Identification We talked about the correlation method of system identification briefly in class, that is Hˆ(j Ω) = Rfy(j Ω) Rff(j Ω) where Hˆ(j Ω) is the estimated frequency response, Rff(j Ω) = F{ρff(τ)} is the auto-power spectrum at the input, and Rfy(j Ω) = F{ρfy(τ)} is the cross-power spectrum between input and output. This is all very nice in theory but, as we will see in this problem, things don’t always work out quite so well in practice. We have an unknown plant with frequency response H(j Ω) and impulse response h(t). Our goal is to find estimates Hˆ(j Ω) and hˆ(t) of these quantities from input-output measure-ments. The system was driven with a wide-band input f (t), and the input and output y(t) were sampled. f ( t ) y ( t ) H ( j 9 ) , h ( t ) " U n k n o w n " L i n e a r S y s t e m The MATLAB file Q2Prob3.mat contains two data records (each of length 4096), recorded at 10,000 samples/sec. The vector f contains the input, and y contains the output. (a) Use the two data vectors to estimate H(j Ω) using the correlation method. Make plots of the magnitude and phase responses. (They are a bit messy!) (b) From your Hˆ(j Ω) find an estimate of the impulse response hˆ(n). Make a plot of the first 40 samples, using the stem() function. (c) Now repeat (a) and (b), but this time truncate the correlation functions so as to save only the first 100 lags (a total of 201 samples). Apply a Hamming window of length 201 to the new correlation sequences ρff(n), and cross-correlation ρfy(n), −100 ≤ n ≤ 100, before computing the power spectra. (d) Discuss any improvements you see in the estimates of the frequency response and/or impulse response


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