Short-Time Fourier Transform (STFT) in Signal Processing - Step-by-Step Execution

Concept Flow - Short-Time Fourier Transform (STFT)

Input Signal

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Split into Overlapping Windows

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Apply Window Function to Each Segment

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Compute FFT of Each Windowed Segment

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Collect FFT Results

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Form Time-Frequency Matrix (Spectrogram)

STFT splits a signal into small overlapping parts, applies a window to each, then finds frequency content using FFT, producing a time-frequency view.

Execution Sample

Signal Processing

import numpy as np
from scipy.signal import stft

x = np.sin(2*np.pi*5*np.linspace(0,1,100))
f, t, Zxx = stft(x, nperseg=20)

This code computes the STFT of a 5 Hz sine wave sampled 100 times over 1 second using windows of length 20.

Execution Table

Step	Action	Input Segment Indices	Window Applied	FFT Output Shape	Notes
1	Split signal into first window	0-19	Hann window	11 complex values	First 20 samples selected and windowed
2	Compute FFT of first window	0-19	Hann window	11 complex values	FFT shows frequency content of first segment
3	Split signal into second window	10-29	Hann window	11 complex values	Window shifted by 10 samples (50% overlap)
4	Compute FFT of second window	10-29	Hann window	11 complex values	FFT of second segment
5	Repeat for all windows	varies	Hann window	11 complex values each	Process continues until end of signal
6	Collect all FFT results	all windows	Hann window	Matrix shape (freq_bins x time_slices)	Final STFT matrix formed
7	Exit	N/A	N/A	N/A	All windows processed, STFT complete

💡 All signal segments processed with overlapping windows, STFT matrix fully computed

Variable Tracker

Variable	Start	After 1	After 2	After 3	Final
x	Full signal array (length 100)	Unchanged	Unchanged	Unchanged	Unchanged
segment_indices	N/A	0-19	10-29	20-39	Last window indices
windowed_segment	N/A	Segment 0-19 * Hann	Segment 10-29 * Hann	Segment 20-39 * Hann	Last windowed segment
FFT_output	N/A	11 complex values	11 complex values	11 complex values	All FFTs collected in matrix
STFT_matrix_shape	N/A	N/A	N/A	N/A	(freq_bins=11, time_slices=9)

Key Moments - 3 Insights

Why do we use overlapping windows instead of non-overlapping?

Overlapping windows (see rows 1 and 3 in execution_table) ensure smooth transitions and better time resolution by capturing signal changes between segments.

What does the window function do to each segment?

The window (like Hann) reduces edge effects by tapering segment ends, preventing sharp jumps that distort FFT results (rows 1 and 3).

Why is the FFT output complex numbers?

FFT outputs complex values representing both amplitude and phase of frequencies, essential for reconstructing signal frequency content (rows 2 and 4).

Visual Quiz - 3 Questions

Look at the execution_table, what is the input segment indices for the third window processed?

A 20-39

B 0-19

C 10-29

D 30-49

💡 Hint

Check the 'Input Segment Indices' column for step 5 or infer from overlapping windows shifting by 10 samples

At which step does the STFT matrix get fully formed?

A Step 4

B Step 6

C Step 2

D Step 1

💡 Hint

Look for the step mentioning 'Final STFT matrix formed' in the 'Notes' column

If the window length (nperseg) was increased, how would the FFT output shape change?

A No change in FFT output size

B Fewer frequency bins (smaller FFT output)

C More frequency bins (larger FFT output)

D FFT output becomes real numbers only

💡 Hint

FFT output length equals window length; increasing window length increases frequency resolution

Concept Snapshot

Short-Time Fourier Transform (STFT):
- Splits signal into overlapping windows
- Applies window function (e.g., Hann) to each segment
- Computes FFT on each windowed segment
- Produces time-frequency matrix (spectrogram)
- Balances time and frequency resolution via window size and overlap

Full Transcript

The Short-Time Fourier Transform (STFT) breaks a signal into small overlapping parts. Each part is multiplied by a window function to reduce edge effects. Then, the FFT is computed on each windowed segment to find its frequency content. These FFT results are collected into a matrix showing how frequencies change over time. Overlapping windows help capture smooth transitions. The FFT outputs complex numbers representing amplitude and phase. Increasing window size increases frequency resolution but reduces time resolution. This process creates a time-frequency view of the signal.