Power spectral density estimation in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of PSD estimation using Welch's method

What is the shape of the output array psd after running Welch's method on a 1D signal of length 1024 with nperseg=1024?

Signal Processing

import numpy as np
from scipy.signal import welch

fs = 1000  # Sampling frequency
signal = np.random.randn(1024)
frequencies, psd = welch(signal, fs=fs, nperseg=1024)
print(psd.shape)

A (513,)

B (512,)

C (1024,)

D (256,)

💡 Hint

Welch's method returns half the FFT length plus one for the frequency bins.

Welch's method with nperseg=1024 uses an FFT length of 1024 (same as signal length). The output PSD array length is nfft/2 + 1 = 512 + 1 = 513.

❓ data_output

intermediate

Number of frequency bins in periodogram

Given a signal of length 2048 sampled at 500 Hz, what is the number of frequency bins returned by the periodogram method with default FFT length?

Signal Processing

import numpy as np
from scipy.signal import periodogram

fs = 500
signal = np.random.randn(2048)
frequencies, psd = periodogram(signal, fs=fs)
print(len(frequencies))

A 2048

B 2049

C 1025

D 1024

💡 Hint

Periodogram returns nfft/2 + 1 frequency bins for real signals.

The default FFT length is the signal length (2048). The number of frequency bins is nfft/2 + 1 = 1024 + 1 = 1025.

❓ visualization

advanced

Identifying the correct PSD plot

Which plot correctly shows the power spectral density of a noisy sine wave at 50 Hz using Welch's method?

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import welch

fs = 500
T = 2
t = np.linspace(0, T, int(T*fs), endpoint=False)
signal = np.sin(2*np.pi*50*t) + 0.5*np.random.randn(len(t))
frequencies, psd = welch(signal, fs=fs, nperseg=256)

plt.figure(figsize=(8,4))
plt.semilogy(frequencies, psd)
plt.xlabel('Frequency (Hz)')
plt.ylabel('PSD (V**2/Hz)')
plt.title('Power Spectral Density using Welch')
plt.grid(True)
plt.show()

A A plot with a peak near 100 Hz instead of 50 Hz

B A flat line with no peaks

C Multiple peaks at random frequencies with no dominant peak

D A plot with a clear peak near 50 Hz and noise floor elsewhere

💡 Hint

The sine wave frequency should appear as a peak in the PSD.

The noisy sine wave at 50 Hz produces a PSD with a clear peak near 50 Hz. Noise contributes to a flat noise floor.

🧠 Conceptual

advanced

Effect of segment length on PSD resolution

How does increasing the segment length (nperseg) in Welch's method affect the frequency resolution and variance of the PSD estimate?

A Increases frequency resolution but increases variance

B Decreases frequency resolution and decreases variance

C Increases frequency resolution and decreases variance

D Decreases frequency resolution but increases variance

💡 Hint

Longer segments give finer frequency bins but fewer averages.

Longer segments increase frequency resolution (smaller frequency bins) but reduce the number of segments averaged, increasing variance.

🔧 Debug

expert

Identifying error in PSD calculation code

What error will this code raise when calculating PSD using Welch's method?

import numpy as np
from scipy.signal import welch

fs = 1000
signal = np.random.randn(500)
frequencies, psd = welch(signal, fs=fs, nperseg=1024)
print(psd)

A No error, prints PSD array

B ValueError: nperseg must not be greater than input length

C IndexError: index out of range

D TypeError: unsupported operand type(s) for /: 'int' and 'str'

💡 Hint

Check if segment length is valid for the signal length.

Welch's method requires nperseg to be less than or equal to the signal length. Here, nperseg=1024 > 500 causes ValueError.

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