Short-Time Fourier Transform (STFT) in Signal Processing - Mini Project: Build & Apply

Short-Time Fourier Transform (STFT) Analysis

📖 Scenario: You have recorded a short audio signal and want to analyze how its frequency content changes over time. This is useful in music, speech, and many other sound applications.

🎯 Goal: Build a simple program to compute the Short-Time Fourier Transform (STFT) of a signal using a sliding window and visualize the frequency changes over time.

📋 What You'll Learn

Create a sample signal as a list of numbers

Define a window size for the STFT

Compute the STFT by applying Fourier Transform on each window segment

Print the magnitude of the STFT result for each window

💡 Why This Matters

🌍 Real World

STFT is used in audio processing to analyze how sounds change over time, such as in music, speech recognition, and noise detection.

💼 Career

Understanding STFT helps in roles like audio engineer, data scientist working with time-series data, and anyone involved in signal processing.

1

Create the sample signal

Create a list called signal with these exact values: 0, 1, 0, -1, 0, 1, 0, -1, 0, 1

Signal Processing

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

Hint

Think of the signal as a simple wave repeating values.

2

Set the window size for STFT

Create a variable called window_size and set it to 4

Signal Processing

signal = [0, 1, 0, -1, 0, 1, 0, -1, 0, 1]
# Set the window size to 4
# Your code here

Hint

The window size decides how many samples you analyze at once.

3

Compute the STFT magnitude for each window

Import fft from numpy.fft. Create an empty list called stft_magnitudes. Use a for loop with variable start to slide over signal in steps of 1 until len(signal) - window_size + 1. Inside the loop, take a slice segment of signal[start:start + window_size], compute its Fourier Transform using fft(segment), then append the absolute values of the result to stft_magnitudes.

Signal Processing

signal = [0, 1, 0, -1, 0, 1, 0, -1, 0, 1]
window_size = 4
# Import fft and compute STFT magnitudes
# Your code here

Hint

Use a loop to slide the window and apply fft on each segment.

4

Print the STFT magnitudes

Use a for loop with variable i and magnitudes to iterate over enumerate(stft_magnitudes). Inside the loop, print the string "Window {i}: {magnitudes}" using an f-string.

Signal Processing

import numpy as np
from numpy.fft import fft

signal = [0, 1, 0, -1, 0, 1, 0, -1, 0, 1]
window_size = 4

stft_magnitudes = []
for start in range(len(signal) - window_size + 1):
    segment = signal[start:start + window_size]
    fft_result = fft(segment)
    magnitudes = np.abs(fft_result)
    stft_magnitudes.append(magnitudes)

# Print the STFT magnitudes for each window
# Your code here

Hint

Print each window number and its magnitude array using an f-string.

Practice

1. What is the main purpose of the Short-Time Fourier Transform (STFT) in signal processing?

easy

A. To compress a signal into fewer data points

B. To analyze how frequencies change over time in a signal

C. To remove noise from a signal completely

D. To convert a signal from digital to analog form

Solution

1. Step 1: Understand STFT function

   STFT breaks a signal into small time segments and finds frequency content in each segment.

2. Step 2: Identify main goal

   This helps us see how frequencies change over time, unlike a normal Fourier transform which shows overall frequencies.

3. Final Answer:

   To analyze how frequencies change over time in a signal -> Option B

4. Quick Check:

   STFT = frequency changes over time [OK]

Hint: STFT = frequency analysis in short time windows [OK]

Common Mistakes:

• Confusing STFT with noise removal
• Thinking STFT compresses data
• Mixing STFT with analog conversion

2. Which of the following is the correct Python function call to compute STFT using the scipy library on a signal array named signal with a window size of 256?

easy

A. scipy.stft(signal, nperseg=256)

B. scipy.stft(signal, window=256)

C. scipy.signal.fft(signal, window=256)

D. scipy.signal.stft(signal, nperseg=256)

Solution

1. Step 1: Recall scipy STFT syntax

   The STFT function is in scipy.signal as stft(), and the window size is set by nperseg parameter.

2. Step 2: Match correct syntax

   scipy.signal.stft(signal, nperseg=256) uses scipy.signal.stft(signal, nperseg=256), which is correct syntax.

3. Final Answer:

   scipy.signal.stft(signal, nperseg=256) -> Option D

4. Quick Check:

   Correct function and parameter = scipy.signal.stft(signal, nperseg=256) [OK]

Hint: Use scipy.signal.stft with nperseg for window size [OK]

Common Mistakes:

• Using scipy.stft instead of scipy.signal.stft
• Confusing fft() with stft()
• Passing window size as 'window' instead of 'nperseg'

3. Given the following Python code using scipy.signal.stft, what is the shape of the output array Zxx?

import numpy as np
from scipy.signal import stft
signal = np.sin(2 * np.pi * 5 * np.linspace(0, 1, 100))
f, t, Zxx = stft(signal, nperseg=20)
print(Zxx.shape)

medium

A. (11, 6)

B. (20, 5)

C. (50, 2)

D. (10, 10)

Solution

1. Step 1: Calculate frequency bins

   With nperseg=20, frequency bins = nperseg//2 + 1 = 10 + 1 = 11 rows.

2. Step 2: Calculate time segments

   Signal length=100, window=20, default no overlap means about 6 segments (100/20 rounded up).

3. Final Answer:

   (11, 6) -> Option A

4. Quick Check:

   Zxx shape = (freq bins, time segments) = (11, 6) [OK]

Hint: Output shape = (nperseg/2+1, number of windows) [OK]

Common Mistakes:

• Confusing nperseg with output shape rows
• Miscounting time segments
• Assuming output shape equals input length

4. Identify the error in this STFT code snippet and choose the correct fix:

from scipy.signal import stft
signal = [1, 2, 3, 4, 5]
f, t, Zxx = stft(signal, nperseg=10)
print(Zxx)

medium

A. Error: nperseg is larger than signal length; fix by setting nperseg=5

B. Error: signal must be a numpy array; fix by converting signal to np.array(signal)

C. Error: stft requires window parameter; fix by adding window='hann'

D. No error; code runs fine

Solution

1. Step 1: Check nperseg vs signal length

   Signal length is 5, but nperseg=10 is larger, which causes a warning or error in scipy.

2. Step 2: Fix nperseg value

   Set nperseg to 5 or less to match signal length and avoid error.

3. Final Answer:

   Error: nperseg is larger than signal length; fix by setting nperseg=5 -> Option A

4. Quick Check:

   nperseg ≤ signal length to avoid error [OK]

Hint: nperseg must not exceed signal length [OK]

Common Mistakes:

• Ignoring nperseg size error
• Thinking signal type causes error
• Adding unnecessary window parameter

5. You want to analyze a music clip with fast-changing notes using STFT. Which combination of window size and overlap will give you better time resolution but less frequency resolution?

hard

A. Large window size with high overlap

B. Large window size with low overlap

C. Small window size with high overlap

D. Small window size with low overlap

Solution

1. Step 1: Understand window size effect

   Small window size improves time resolution but reduces frequency resolution.

2. Step 2: Understand overlap effect

   High overlap keeps time segments close, improving time detail further.

3. Final Answer:

   Small window size with high overlap -> Option C

4. Quick Check:

   Better time resolution = small window + high overlap [OK]

Hint: Small window + high overlap = better time detail [OK]

Common Mistakes:

• Choosing large window for better time resolution
• Ignoring overlap effect
• Assuming low overlap improves time resolution