Homework 8

Due: Mar 29 by 11:59 pm eastern standard time

Submit a single file called name_08.py to Brightspace/OWL where name is replaced with your last name, e.g. gribble_08.py. If you would rather use a jupyter notebook that is OK too, upload your .ipynb file, e.g. gribble_08.ipynb.

Signal Processing Exercises

Use the following code to generate a noisy signal composed of sinusoids at several frequencies, sampled at 1000 Hz:

import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

np.random.seed(42)  # for reproducibility

fs = 1000  # sampling rate (Hz)
t = np.arange(0, 1, 1/fs)  # 1 second of data

y1 = np.sin(2*np.pi*30*t)          # 30 Hz component
y2 = 0.8 * np.sin(2*np.pi*125*t)   # 125 Hz component
y3 = 0.9 * np.sin(2*np.pi*300*t)   # 300 Hz component
noise = np.random.randn(len(t)) * 0.5  # Gaussian noise
y = y1 + y2 + y3 + noise

Plot the signal y in the time domain. Label the axes and include a title.
Compute and plot the power spectral density of y using scipy.signal.welch() with nperseg=256. Label the axes and include a title. In a code comment, briefly describe what you see—can you identify the three frequency components? (1–2 sentences)
Using a Butterworth low-pass filter (see the plg_lowpass function from the course notes), low-pass filter the signal at a cutoff frequency of 80 Hz. Make a 2-panel figure: (a) the top panel should show both the original signal and the filtered signal overlaid in the time domain (use different colors and a legend), and (b) the bottom panel should show the PSD of the filtered signal. Label all axes and include titles.
Now apply a high-pass filter at 80 Hz to the original unfiltered signal y. Plot the PSD of the high-passed signal. In a code comment, describe what frequency components remain. (1–2 sentences)