Why signal processing extracts information in SciPy - Challenge Your Understanding

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Fourier Transform on a Simple Signal

What is the output magnitude array of the Fourier Transform applied to a simple sine wave signal?

SciPy

import numpy as np
from scipy.fft import fft

# Create a sine wave signal
fs = 100  # Sampling frequency
f = 5     # Frequency of sine wave
x = np.linspace(0, 1, fs, endpoint=False)
signal = np.sin(2 * np.pi * f * x)

# Compute Fourier Transform
fft_result = fft(signal)
magnitude = np.abs(fft_result)
print(magnitude[:10])

A [0.0, 50.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

B [2.5e-15, 50.0, 1.2e-15, 1.1e-15, 1.0e-15, 1.0e-15, 1.1e-15, 1.2e-15, 1.3e-15, 1.4e-15]

C [2.5e-15, 0.0, 0.0, 0.0, 0.0, 50.0, 0.0, 0.0, 0.0, 0.0]

D [0.0, 25.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

💡 Hint

Look for the peak magnitude at the frequency of the sine wave.

The Fourier Transform of a sine wave at frequency f shows a peak magnitude at the corresponding frequency bin. The magnitude is approximately half the number of samples due to the FFT scaling.

❓ data_output

intermediate

Number of Peaks Detected in a Noisy Signal

Given a noisy signal with multiple peaks, how many peaks does the scipy.signal.find_peaks function detect?

SciPy

import numpy as np
from scipy.signal import find_peaks

np.random.seed(0)
x = np.linspace(0, 6*np.pi, 1000)
signal = np.sin(x) + 0.5 * np.sin(3*x) + 0.2 * np.random.normal(size=1000)
peaks, _ = find_peaks(signal, height=0.5)
print(len(peaks))

A 6

B 9

C 12

D 15

💡 Hint

Consider the combined frequencies and the threshold height parameter.

The find_peaks function detects peaks above the height threshold. The combined sine waves produce multiple peaks, but only those above 0.5 are counted, resulting in 9 peaks.

❓ visualization

advanced

Visualizing the Effect of a Low-Pass Filter

Which plot correctly shows the effect of applying a low-pass Butterworth filter to a noisy signal?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import butter, filtfilt

fs = 500
x = np.linspace(0, 1, fs, endpoint=False)
signal = np.sin(2*np.pi*5*x) + 0.5*np.sin(2*np.pi*50*x) + 0.3*np.random.normal(size=fs)

b, a = butter(4, 10, btype='low', fs=fs)
filtered = filtfilt(b, a, signal)

plt.figure(figsize=(10, 4))
plt.plot(x, signal, label='Noisy Signal')
plt.plot(x, filtered, label='Filtered Signal', linewidth=2)
plt.legend()
plt.title('Low-Pass Filter Effect')
plt.xlabel('Time [s]')
plt.ylabel('Amplitude')
plt.show()

A Plot shows noisy signal with high-frequency noise and filtered signal smooth with only low-frequency components.

B Plot shows filtered signal with amplified high-frequency noise compared to original.

C Plot shows filtered signal identical to noisy signal with no smoothing.

D Plot shows filtered signal with added random noise spikes.

💡 Hint

Low-pass filters remove high-frequency noise, leaving smooth low-frequency signals.

A low-pass Butterworth filter removes high-frequency components, so the filtered signal is smoother and retains only low-frequency parts of the original signal.

🧠 Conceptual

advanced

Why Signal Processing Extracts Information

Why does signal processing help extract meaningful information from raw data?

A Because it transforms data to highlight patterns and reduce noise, making features clearer.

B Because it compresses data without any loss of information.

C Because it converts data into random noise to test system robustness.

D Because it removes all data except the largest values, making analysis simpler.

💡 Hint

Think about how filtering and transformations help reveal hidden patterns.

Signal processing techniques like filtering and transforms reduce noise and emphasize important patterns, making it easier to analyze and extract useful information.

🔧 Debug

expert

Identify the Error in Signal Filtering Code

What error does the following code produce when applying a Butterworth filter, and why? import numpy as np from scipy.signal import butter, filtfilt fs = 100 x = np.linspace(0, 1, fs, endpoint=False) signal = np.sin(2 * np.pi * 10 * x) b, a = butter(4, 60, btype='low', fs=fs) filtered_signal = filtfilt(b, a, signal) print(filtered_signal[:5])

SciPy

import numpy as np
from scipy.signal import butter, filtfilt

fs = 100
x = np.linspace(0, 1, fs, endpoint=False)
signal = np.sin(2 * np.pi * 10 * x)

b, a = butter(4, 60, btype='low', fs=fs)
filtered_signal = filtfilt(b, a, signal)
print(filtered_signal[:5])

A No error, code runs and prints filtered signal values.

B TypeError: butter() got an unexpected keyword argument 'fs'.

C RuntimeWarning: Filter coefficients are unstable causing NaN values in output.

D ValueError: The critical frequency must be 0 < Wn < fs/2 because 60 is above Nyquist frequency.

💡 Hint

Check the critical frequency relative to the sampling frequency.

The critical frequency parameter must be less than half the sampling frequency (Nyquist frequency). Here, 60 is greater than fs/2=50, causing a ValueError.