Power spectral density estimation in Signal Processing - Mini Project: Build & Apply

Power Spectral Density Estimation

📖 Scenario: You have recorded a simple signal from a sensor over time. You want to understand how the signal's power is spread across different frequencies. This helps in identifying dominant frequencies or noise in the signal.

🎯 Goal: Build a small program to estimate the power spectral density (PSD) of a given signal using the periodogram method. You will create the signal data, set the sampling frequency, compute the PSD, and then display the frequency and power values.

📋 What You'll Learn

Create a time-domain signal as a list of values

Define the sampling frequency as a variable

Use the periodogram method to compute the power spectral density

Print the frequency array and the corresponding power spectral density values

💡 Why This Matters

🌍 Real World

Power spectral density estimation is used in engineering and science to analyze signals like sound, vibrations, or electrical signals to find dominant frequencies or noise.

💼 Career

Understanding PSD helps in roles like signal processing engineer, data analyst, or any job involving sensor data analysis and noise filtering.

1

Create the signal data

Create a list called signal with these exact values: 0.0, 1.0, 0.0, -1.0, 0.0, 1.0, 0.0, -1.0

Signal Processing

# Create the signal list with the given values
# Your code here

Hint

This signal represents a simple repeating pattern over time.

2

Set the sampling frequency

Create a variable called fs and set it to 4 to represent the sampling frequency in Hertz

Signal Processing

signal = [0.0, 1.0, 0.0, -1.0, 0.0, 1.0, 0.0, -1.0]
# Set the sampling frequency fs to 4
# Your code here

Hint

The sampling frequency tells how many samples are taken per second.

3

Compute the power spectral density

Import periodogram from scipy.signal and use it with signal and fs to compute frequencies and power

Signal Processing

signal = [0.0, 1.0, 0.0, -1.0, 0.0, 1.0, 0.0, -1.0]
fs = 4
# Import periodogram and compute frequencies and power
# Your code here

Hint

The periodogram function returns two arrays: frequencies and power spectral density values.

4

Display the power spectral density

Print the frequencies array and the power array each on a separate line

Signal Processing

from scipy.signal import periodogram

signal = [0.0, 1.0, 0.0, -1.0, 0.0, 1.0, 0.0, -1.0]
fs = 4

frequencies, power = periodogram(signal, fs)
# Print frequencies and power arrays
# Your code here

Hint

Printing these arrays shows how power is distributed across frequencies.

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