[10 pts] You are given two datasets ( 1. case1.mat & case2.mat). Each .mat file contains two vectors of data, namely x and y. Both x and y are sampled versions of the same random noise signal. One of...

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[10 pts] You are given two datasets ( 1. case1.mat & case2.mat). Each .mat file contains two vectors of data, namely x and y. Both x and y are sampled versions of the same random noise signal. One of these cases was simultaneously sampled (though different channels were used), while the other was acquired with a multiplexed DAQ. All data were sampled at 5 kHz (per channel). For each case, subtract the respective mean values from the data, and compute and plot the cross correlation coefficient functions, Rx0y0(τ ). Do not use the ’coeff ’ scaling argument in the xcorr() command to do the scaling for you; use the appropriate scaling argument and normalize the result with the proper quantities after xcorr() is used. Use your plots to determine which data set (case 1 or 2) was simultaneously sampled. Justify your answer by plotting the data in a different form (not using the cross-correlation function) that represents what the linear correlation coefficients are for each case. Determine what the linear correlation coefficients, ρxy, are for each case as well. What insight does this give you about simultaneous vs multiplexed sampling? [20 pts] When acquiring experimental data, the analog signal is digitized through an A/D converter. 2. Consequentially, all of the values calculated or derived from those data are only estimations of the true values (recall that DAQ devices have quantization errors). Consider an analog signal that is a sine wave with a frequency of f = 2 Hz, an amplitude A = 1.5, and a phase lag that is equivalent to τ = 0.8 seconds. Let’s call this analog signal x(t). (a) Determine the autocorrelation for x(t) analytically (i.e., evaluate the following equation for some value “t0” by hand) Rxx(t0, τ ) = E[x(t0)x(t0 + τ )] (b) What effect does t0 have on the autocorrelation? What are the implications of this? (c) Discretize x(t) using a sampling frequency of fs = 100 Hz for 0 ≤ t ≤ 2 s. This will simulate you having acquired the data with no quantization errors. (d) Using the discretized signal, compute the autocorrelation of x(n∆t) using both the ‘biased’ and ‘unbiased’ scaling options for the xcorr() command in MATLAB. (e) Plot all three autocorrelations (your analytical solution, the biased output, and the unbi- ased output) on the same figure (one single axis), showing only the results for 0 ≤ lags ≤ 200 [points]. Note: You also have to discretize the analytical solution from part (a). You can convert between τ and lags using the following equation: τ = lags/fs. (f) Explain your results. Why is there a difference between the three plots? [10 pts] Write a script to calculate both the power spectrum and power spectral density of a signal 3. using the formulas from the lecture notes. Do not use the pwelch(), cpsd(), or spectrogram() commands in your script. Use a single block of data (i.e., no overlap) and a rectangular window for this analysis. (a) Plot the results (power spectrum and power spectral density) for the following signal with an integer number of periods in the record length you choose, T. x(t) = 2.3 sin (2π150t XXXXXXXXXXsin (2π60t XXXXXXXXXXb) What values did you choose for fs, T, N, and ∆f? Before proceeding, if you have ∆f = 1 Hz, change your values so ∆f 6= 1 Hz and list out your new values. (c) Tabulate the quantities of the computed power spectrum and power spectral density at DC, at 60 Hz, and at 150 Hz. (d) Verify that Parseval’s theorem holds. Adjust the mean value and amplitudes of the terms to verify that your code works properly. (e) Verify the PSD of the signal in part (a) by producing an additional plot using the pwelch() command. (Make a new plot comparing your power spectral density and that of the pwelch() output)
Answered 6 days AfterMay 09, 2022

Answer To : [10 pts] You are given two datasets ( 1. case1.mat & case2.mat). Each .mat file contains two vectors...

Lalit answered on May 16 2022
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