Power spectral density estimation in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute the power spectral density (PSD) using Welch's method.

Signal Processing

from scipy.signal import welch
frequencies, psd = welch(signal, fs=[1])

Drag options to blanks, or click blank then click option'

A10

B1000

C50

D500

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong sampling frequency causes incorrect frequency axis in PSD.

The sampling frequency fs must match the signal's sampling rate. Here, 500 Hz is the correct value.

2fill in blank

medium

Complete the code to plot the PSD on a logarithmic scale.

Signal Processing

import matplotlib.pyplot as plt
plt.semilogy(frequencies, [1])
plt.xlabel('Frequency (Hz)')
plt.ylabel('PSD (V**2/Hz)')
plt.show()

Drag options to blanks, or click blank then click option'

Apsd

Bsignal

Cfrequencies

Dnp.log(psd)

Attempts:

3 left

💡 Hint

Common Mistakes

Plotting the signal or frequencies instead of PSD.

The PSD values are plotted on the y-axis using psd. The semilogy function plots y-axis on a log scale.

3fill in blank

hard

Fix the error in the code to compute PSD with a Hamming window.

Signal Processing

from scipy.signal import welch, [1]
frequencies, psd = welch(signal, fs=500, window=[1](256))

Drag options to blanks, or click blank then click option'

Ahamming

Bhanning

Cblackman

Dbartlett

Attempts:

3 left

💡 Hint

Common Mistakes

Using a window name not imported or misspelled.

The hamming window function is imported and used to create the window for Welch's method.

4fill in blank

hard

Fill both blanks to compute PSD with segment length 512 and overlap 256.

Signal Processing

frequencies, psd = welch(signal, fs=500, nperseg=[1], noverlap=[2])

Drag options to blanks, or click blank then click option'

A512

B256

C128

D1024

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up segment length and overlap values.

The segment length nperseg is 512 samples, and the overlap noverlap is 256 samples.

5fill in blank

hard

Fill all three blanks to create a dictionary of PSD values for frequencies above 100 Hz.

Signal Processing

psd_dict = {f: p for f, p in zip(frequencies, psd) if f [1] 100}
filtered_freqs = [f for f in frequencies if f [2] 100]
filtered_psd = [p for f, p in zip(frequencies, psd) if f [3] 100]

Drag options to blanks, or click blank then click option'

A>

B<

C>=

D<=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '<' or '<=' which selects wrong frequency range.

We filter frequencies and PSD values where frequency is greater than 100 Hz using the '>' operator.

Practice

1. What does the Power Spectral Density (PSD) of a signal represent?

easy

A. How the power of a signal is distributed over different frequencies

B. The total energy of the signal in the time domain

C. The phase shift of the signal over time

D. The amplitude of the signal at a single point in time

Solution

1. Step 1: Understand the definition of PSD

   PSD shows how the power of a signal is spread across frequencies, not just total energy or amplitude.

2. Step 2: Compare options with PSD meaning

   Only How the power of a signal is distributed over different frequencies correctly describes PSD as power distribution over frequencies.

3. Final Answer:

   How the power of a signal is distributed over different frequencies -> Option A

4. Quick Check:

   PSD = power distribution over frequency [OK]

Hint: PSD shows power spread by frequency, not time or amplitude [OK]

Common Mistakes:

• Confusing PSD with total signal energy
• Thinking PSD shows signal amplitude at a time
• Mixing PSD with phase information

2. Which Python function from the scipy.signal library is commonly used to estimate the Power Spectral Density using Welch's method?

easy

A. scipy.signal.spectrogram()

B. scipy.signal.fft()

C. scipy.signal.correlate()

D. scipy.signal.welch()

Solution

1. Step 1: Identify Welch's method function

   Welch's method for PSD estimation is implemented in scipy.signal.welch().

2. Step 2: Differentiate from other functions

   fft() computes Fourier transform, correlate() computes correlation, spectrogram() shows time-frequency but not PSD by Welch.

3. Final Answer:

   scipy.signal.welch() -> Option D

4. Quick Check:

   Welch PSD = scipy.signal.welch() [OK]

Hint: Welch PSD uses scipy.signal.welch() function [OK]

Common Mistakes:

• Using fft() instead of welch() for PSD
• Confusing spectrogram with PSD estimation
• Trying correlate() for PSD calculation

3. Given the following Python code, what will be the length of the frequency array f returned by scipy.signal.welch?

import numpy as np
from scipy.signal import welch

fs = 1000  # Sampling frequency
x = np.random.randn(1024)  # Random signal
f, Pxx = welch(x, fs=fs, nperseg=256)

medium

A. 1024

B. 129

C. 512

D. 256

Solution

1. Step 1: Understand nperseg effect on frequency bins

   Welch returns frequency bins of length nperseg/2 + 1 for real signals.

2. Step 2: Calculate frequency array length

   nperseg=256, so length = 256/2 + 1 = 129.

3. Final Answer:

   129 -> Option B

4. Quick Check:

   Frequency bins = nperseg/2 + 1 = 129 [OK]

Hint: Frequency bins length = nperseg/2 + 1 for real signals [OK]

Common Mistakes:

• Assuming frequency length equals nperseg
• Using full signal length instead of segment length
• Confusing frequency bins with time samples

4. Identify the error in this code snippet estimating PSD with Welch's method:

import numpy as np
from scipy.signal import welch

fs = 500
x = np.sin(2 * np.pi * 50 * np.arange(1000) / fs)
f, Pxx = welch(x, fs=fs, nperseg=2000)

medium

A. Sampling frequency fs is too low

B. Signal x is not a numpy array

C. nperseg is larger than the signal length

D. Missing window parameter in welch()

Solution

1. Step 1: Check nperseg vs signal length

   nperseg=2000 is greater than signal length 1000, which causes a warning or error.

2. Step 2: Verify other parameters

   fs=500 is valid, x is numpy array, window parameter is optional with default.

3. Final Answer:

   nperseg is larger than the signal length -> Option C

4. Quick Check:

   nperseg > signal length causes error [OK]

Hint: nperseg must not exceed signal length [OK]

Common Mistakes:

• Setting nperseg larger than signal length
• Assuming window parameter is mandatory
• Confusing sampling frequency with segment length

5. You have a noisy signal sampled at 2000 Hz. To get a smoother PSD estimate with better frequency resolution using Welch's method, which combination of parameters should you choose?

hard

A. Increase segment length (nperseg) and use overlapping segments

B. Decrease segment length and avoid overlapping

C. Use no window and set nperseg to 1

D. Increase sampling frequency beyond 2000 Hz

Solution

1. Step 1: Understand effect of segment length and overlap

   Longer segments improve frequency resolution; overlapping segments reduce variance (smoother PSD).

2. Step 2: Evaluate options for smoothing and resolution

   Increase segment length (nperseg) and use overlapping segments correctly suggests increasing nperseg and using overlap; others reduce resolution or are impractical.

3. Final Answer:

   Increase segment length (nperseg) and use overlapping segments -> Option A

4. Quick Check:

   Longer segments + overlap = smoother, better PSD [OK]

Hint: Longer segments + overlap = smoother PSD estimate [OK]

Common Mistakes:

• Using very short segments losing frequency detail
• Ignoring overlap reduces averaging benefits
• Thinking sampling rate increase smooths PSD