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

Concept Flow - Power spectral density

Start with time signal

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Apply Fourier Transform

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Calculate magnitude squared

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Normalize by frequency

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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.