ME6449_HW2

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Georgia Institute Of Technology *

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6449

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Mechanical Engineering

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Dec 6, 2023

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G.W. WOODRUFF SCHOOL OF MECHANICAL ENGINEERING ME 6449 HW #2 (Due 11:59pm September 28 th , 2023) 1. Sampling exercise: Consider the signal 𝑥𝑥 ( 𝑡𝑡 ) = cos (400 𝜋𝜋𝑡𝑡 ) from 𝑡𝑡 = 0 𝑡𝑡𝑡𝑡 𝑡𝑡 = 0.1 𝑠𝑠 . a) Sample this signal at 2kHz, i.e. 2kSample/s rate, for 0.1s starting from t=0, to generate 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) and take the FFT of 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) . Plot the magnitude of the result with proper frequency and amplitude axes. Comment on the result. b) Sample this signal at 400Hz, i.e. 400 Sample/s rate, for 0.1s starting from t=0, to generate 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) and take the FFT of 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) . Plot the magnitude the result with proper frequency and amplitude axes. Comment on the result. c) Sample this signal at 200Hz, i.e. 200 Sample/s rate, for 0.1s starting from t=0, to generate 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) and take the FFT of 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) . Plot the magnitude of the result with proper frequency and amplitude axes. Comment on the result. d) Zero pad this signal to increase its length to 1s and sample it at 2kHz starting from t=0, to generate 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) and take the FFT of 𝑥𝑥 𝑠𝑠 ( 𝑡𝑡 ) . Plot the magnitude and phase of the result with proper frequency and amplitude axes. Compare the result with part a). Include all your MATLAB codes in your solution. 2. FFT and filtering exercise: Load the time sequence x[n] from the file HW2_Q2_data.mat to MATLAB. This sequence is sampled at 44.1 kHz. a. Take the FFT of the signal in MATLAB and plot its spectrum, i.e. its magnitude as a function of frequency in dB scale. What are the frequencies of the signal components? b. Filter the signal x[n] with a 6 th order Butterworth low pass filter with a cut-off frequency 1kHz above the frequency of the first signal component. Plot the time domain output signal and also its frequency spectrum in dB scale. Comment on results. c. Filter the signal x[n] with a 6 th order Butterworth band pass filter with a 3-dB bandwidth of 0.5kHz around the frequency of the second signal component. Plot the time domain output signal and also its frequency spectrum in dB scale. Comment on results. Include all your MATLAB codes in your solution. 3. Generate a zero mean random noise signal of length 1024 with standard deviation of 2, using x=2*randn(1024,1) command in MATLAB. a) Calculate the rms value of this signal and compare with the standard deviation. b) Calculate and plot the one sided power spectral density of the noise, S x (f) using the approach described in the notes. c) Verify your power spectral density calculation by comparing your result to the rms value.

d) Window the same signal using a Hann window function and calculate the rms value of the resulting output signal. Is the answer the same as part a)? Comment on the result. 4. Signal to noise ratio calculation and averaging: Load the sequences x1[n] to x10[n] from the file HW2_Q4_data.mat to MATLAB. These sequences contain tone burst voltage signals corrupted by noise. a. Calculate the SNR (in dB) of x1[n], x4[n] and x8[n]. Noise power can be found from the region of the sequence where there is only noise (from 1300 to 4000 for example). How would you find the signal power during the tone burst? Note that noise is present at all times. b. Calculate the average signal x_a[n] using all the sequences. Perform point by point averaging and plot x_a[n]. c. Repeat the SNR calculation for x_a[n] as in part a). Comment on the results. Include all your MATLAB codes in your solution.

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