Power spectral density in SciPy - Step-by-Step Execution

Concept Flow - Power spectral density

Start with time signal

↓

Apply Fourier Transform

↓

Calculate magnitude squared

↓

Normalize by frequency

↓

Output Power Spectral Density (PSD)

The process starts with a time signal, transforms it to frequency domain, computes power at each frequency, normalizes it, and outputs the PSD.

Execution Sample

SciPy

import numpy as np
from scipy.signal import welch

fs = 1000
x = np.sin(2*np.pi*50*np.arange(1000)/fs)
freqs, psd = welch(x, fs)

This code computes the power spectral density of a 50 Hz sine wave sampled at 1000 Hz using Welch's method.

Execution Table

Step	Action	Input/Variable	Output/Result
1	Create time array	np.arange(1000)	Array of 1000 time points from 0 to 999
2	Generate sine wave	50 Hz sine with fs=1000	Array x with sine values
3	Call welch(x, fs)	x array, fs=1000	Compute PSD using Welch's method
4	Split signal into segments	x array	Segments of length 256 (default)
5	Apply window and FFT to each segment	Segments	FFT of each segment
6	Calculate magnitude squared and average	FFT results	Estimate power at each frequency
7	Normalize by sampling frequency	Power values	PSD values
8	Return frequencies and PSD	freqs, psd arrays	freqs from 0 to 500 Hz, psd values
9	Plot or analyze PSD	freqs, psd	Visual or numerical insight into signal power distribution
10	End		PSD computed successfully

💡 Welch method completes after processing all segments and returns frequency and PSD arrays.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 6	Final
x	undefined	Array of sine wave values	Same array passed to welch	Used in FFT segments	Unchanged
freqs	undefined	undefined	Calculated frequencies	Calculated frequencies	Array from 0 to 500 Hz
psd	undefined	undefined	Power values per freq	Power values per freq	Normalized PSD values

Key Moments - 3 Insights

Why does the PSD output only show frequencies up to half the sampling rate?

Because of the Nyquist theorem, the highest frequency that can be represented is half the sampling rate. The execution_table row 8 shows freqs from 0 to 500 Hz when fs=1000 Hz.

Why do we split the signal into segments before FFT in Welch's method?

Splitting into segments reduces noise in the PSD estimate by averaging. Execution_table row 4 shows segmenting, and row 6 shows averaging power across segments.

What does the normalization step in PSD calculation do?

Normalization scales the power values to correct units considering sampling frequency. Execution_table row 7 describes normalizing power values to get PSD.

Visual Quiz - 3 Questions

Look at the execution_table at step 8, what is the maximum frequency in the freqs array?

A 1000 Hz

B 50 Hz

C 500 Hz

D 256 Hz

💡 Hint

Refer to execution_table row 8 where freqs range is described.

At which step does the code average the power from multiple segments?

A Step 6

B Step 5

C Step 4

D Step 7

💡 Hint

Check execution_table row 6 about calculating magnitude squared and averaging.

If the sampling frequency fs is doubled, how does the PSD frequency range change?

A Maximum frequency halves

B Maximum frequency doubles

C Maximum frequency stays the same

D PSD values double

💡 Hint

Refer to Nyquist limit explained in key_moments and execution_table row 8.

Concept Snapshot

Power Spectral Density (PSD) shows how signal power is distributed over frequency.
Use scipy.signal.welch(x, fs) to compute PSD.
Welch splits signal, applies FFT, averages power, and normalizes.
Output frequencies go from 0 to fs/2 (Nyquist frequency).
PSD helps analyze signal frequency content and noise.

Full Transcript

Power spectral density (PSD) analysis starts with a time-domain signal. We apply Fourier transform to convert it to frequency domain. Then we calculate the squared magnitude of the transform to get power at each frequency. Welch's method splits the signal into overlapping segments, applies windowing and FFT to each, averages the power spectra, and normalizes by sampling frequency. The output is frequency values from zero up to half the sampling rate (Nyquist frequency) and corresponding PSD values. This helps us understand how signal power is distributed across frequencies. The example code generates a 50 Hz sine wave sampled at 1000 Hz and computes its PSD using scipy.signal.welch. The execution table traces each step from signal creation, segmentation, FFT, power calculation, normalization, to final PSD output. Key moments clarify why frequencies only go up to half the sampling rate, why segmentation and averaging reduce noise, and the role of normalization. The visual quiz tests understanding of frequency range, averaging step, and effect of changing sampling frequency.

Practice

1. What does the power spectral density (PSD) represent in signal analysis?

easy

A. The phase shift of a signal at various points

B. The total energy of a signal over time

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

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

Solution

1. Step 1: Understand the definition of PSD

   Power spectral density shows how the power of a signal is spread across frequencies, not just total energy or amplitude at one time.

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 C

4. Quick Check:

   PSD = power distribution over frequency [OK]

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

Common Mistakes:

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

2. Which of the following is the correct way to import the Welch method for PSD calculation from scipy?

easy

A. from scipy.signal import welch

B. import scipy.welch

C. from scipy import welch

D. import scipy.signal.welch as welch

Solution

1. Step 1: Recall scipy.signal module structure

   The Welch method is inside scipy.signal, so we import it from there.

2. Step 2: Check import syntax correctness

   Only 'from scipy.signal import welch' is valid Python syntax for this function.

3. Final Answer:

   from scipy.signal import welch -> Option A

4. Quick Check:

   Correct import = from scipy.signal import welch [OK]

Hint: Welch is in scipy.signal, import directly from there [OK]

Common Mistakes:

• Trying to import welch directly from scipy
• Using incorrect import syntax
• Assuming welch is a standalone module

3. What will be the output type of the following code snippet?

from scipy.signal import welch
import numpy as np
fs = 100
x = np.sin(2 * np.pi * 10 * np.arange(1000) / fs)
freqs, psd = welch(x, fs)
print(type(freqs), type(psd))

medium

A. (, )

B. (, )

C. (, )

D. (, )

Solution

1. Step 1: Understand welch output types

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

2. Step 2: Check printed types

   Printing type(freqs) and type(psd) will show numpy.ndarray for both.

3. Final Answer:

   (<class 'numpy.ndarray'>, <class 'numpy.ndarray'>) -> Option D

4. Quick Check:

   welch returns arrays = numpy.ndarray [OK]

Hint: Welch returns frequency and PSD as numpy arrays [OK]

Common Mistakes:

• Expecting lists instead of numpy arrays
• Thinking output is a tuple of floats
• Confusing output with single float values

4. Identify the error in this code snippet that calculates PSD:

from scipy.signal import welch
import numpy as np
fs = 50
x = np.random.randn(500)
freqs, psd = welch(x)
print(freqs)

medium

A. The print statement should be print(psd) instead

B. Missing sampling frequency (fs) argument in welch call

C. Random data cannot be used with welch

D. Import statement for welch is incorrect

Solution

1. Step 1: Check welch function call parameters

   The welch function needs the sampling frequency (fs) to correctly compute frequencies.

2. Step 2: Identify missing argument

   Here, fs=50 is defined but not passed to welch, causing incorrect frequency calculation.

3. Final Answer:

   Missing sampling frequency (fs) argument in welch call -> Option B

4. Quick Check:

   Pass fs to welch to get correct frequencies [OK]

Hint: Always pass fs to welch for correct frequency output [OK]

Common Mistakes:

• Forgetting to pass fs argument
• Assuming random data is invalid input
• Confusing print output with error

5. You have a noisy signal sampled at 200 Hz. You want to find the dominant frequency component using scipy's Welch method. Which parameter adjustment helps improve frequency resolution in the PSD estimate?

hard

A. Increase the segment length (nperseg) to use longer data chunks

B. Decrease the sampling frequency (fs) to reduce data size

C. Use a smaller window type to reduce computation

D. Set detrend to False to keep original signal shape

Solution

1. Step 1: Understand frequency resolution in Welch method

   Frequency resolution improves with longer segments (nperseg) because it provides finer frequency bins.

2. Step 2: Evaluate options for improving resolution

   Increasing nperseg increases segment length, improving resolution. Decreasing fs reduces max frequency but not resolution. Window type affects leakage, not resolution. Detrend affects baseline, not resolution.

3. Final Answer:

   Increase the segment length (nperseg) to use longer data chunks -> Option A

4. Quick Check:

   Longer nperseg = better frequency resolution [OK]

Hint: Longer segments (nperseg) give finer frequency detail [OK]

Common Mistakes:

• Reducing sampling frequency to improve resolution
• Confusing window type with resolution effect
• Ignoring segment length impact on frequency bins