Windowing before FFT in SciPy

Introduction

Windowing helps reduce unwanted effects when analyzing signals with FFT. It smooths the edges to get clearer frequency results.

When analyzing sound recordings to find different tones.

When studying vibrations in machines to detect faults.

When processing EEG brain signals to identify brain waves.

When working with any signal that is cut into pieces before frequency analysis.

Syntax

SciPy

from scipy.signal import windows
window = windows.hann(N)
windowed_signal = signal * window
fft_result = np.fft.fft(windowed_signal)

Use a window function like Hann to reduce edge effects.

Multiply your signal by the window before applying FFT.

Examples

Apply a Hann window of length 100 to the first 100 samples of your signal.

SciPy

from scipy.signal import windows
window = windows.hann(100)
windowed_signal = signal[:100] * window

Use a Hamming window matching the signal length for smoother edges.

SciPy

from scipy.signal import windows
window = windows.hamming(len(signal))
windowed_signal = signal * window

Blackman window gives stronger edge smoothing for better frequency resolution.

SciPy

from scipy.signal import windows
window = windows.blackman(len(signal))
windowed_signal = signal * window

Sample Program

This code creates a noisy 5 Hz sine wave, applies a Hann window, and compares the FFT magnitude of the original and windowed signals. The window reduces spectral leakage, making the 5 Hz peak clearer.

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import windows

# Create a sample signal: 5 Hz sine wave + noise
fs = 100  # Sampling frequency
T = 1.0   # seconds
N = int(T * fs)  # Number of samples

t = np.linspace(0, T, N, endpoint=False)
signal = np.sin(2 * np.pi * 5 * t) + 0.5 * np.random.randn(N)

# Create a Hann window
window = windows.hann(N)

# Apply window to the signal
windowed_signal = signal * window

# Compute FFT of original and windowed signals
fft_orig = np.fft.fft(signal)
fft_win = np.fft.fft(windowed_signal)

# Frequency axis
freqs = np.fft.fftfreq(N, 1/fs)

# Plot magnitude spectra
plt.figure(figsize=(10,5))
plt.plot(freqs[:N//2], np.abs(fft_orig)[:N//2], label='Original Signal')
plt.plot(freqs[:N//2], np.abs(fft_win)[:N//2], label='Windowed Signal')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Magnitude')
plt.title('FFT Magnitude: Original vs Windowed Signal')
plt.legend()
plt.grid(True)
plt.show()

Important Notes

Windowing reduces spectral leakage, which is when energy spreads to nearby frequencies.

Choosing the right window depends on your signal and what you want to see.

Always multiply the signal by the window before FFT, not after.

Summary

Windowing smooths signal edges to improve FFT results.

Apply window functions like Hann or Hamming before FFT.

This helps see true frequencies more clearly by reducing leakage.

Practice

1. What is the main purpose of applying a window function to a signal before performing an FFT?

easy

A. To increase the signal amplitude

B. To reduce spectral leakage by smoothing signal edges

C. To convert the signal from time domain to frequency domain

D. To remove noise from the signal

Solution

1. Step 1: Understand spectral leakage in FFT

   Spectral leakage occurs when the signal edges are abrupt, causing spreading of frequency components.

2. Step 2: Role of window functions

   Window functions smooth the edges of the signal to reduce this leakage before FFT.

3. Final Answer:

   To reduce spectral leakage by smoothing signal edges -> Option B

4. Quick Check:

   Windowing reduces leakage = B [OK]

Hint: Windowing smooths edges to reduce leakage before FFT [OK]

Common Mistakes:

• Thinking windowing increases amplitude
• Confusing windowing with noise removal
• Believing windowing converts domains

2. Which of the following is the correct way to create a Hann window of length 256 using scipy?

easy

A. window = scipy.signal.hann(256)

B. window = scipy.get_window('hann', 256)

C. window = scipy.signal.window('hann', 256)

D. window = scipy.signal.get_window('hann', 256)

Solution

1. Step 1: Identify the correct scipy module and function

   The function to create windows is get_window inside scipy.signal.

2. Step 2: Check syntax for get_window

   Correct syntax is scipy.signal.get_window(window_name, length).

3. Final Answer:

   window = scipy.signal.get_window('hann', 256) -> Option D

4. Quick Check:

   Use scipy.signal.get_window('hann', length) = D [OK]

Hint: Use scipy.signal.get_window('hann', length) to create Hann window [OK]

Common Mistakes:

• Using scipy.get_window instead of scipy.signal.get_window
• Calling nonexistent scipy.signal.window function
• Using scipy.signal.hann which does not exist

3. What is the output of the following code snippet?

import numpy as np
from scipy.signal import get_window

signal = np.ones(4)
window = get_window('hann', 4)
windowed_signal = signal * window
print(np.round(windowed_signal, 3))

medium

A. [0. 0.75 0.75 0. ]

B. [0. 0.5 0.5 0. ]

C. [0. 1. 1. 0. ]

D. [1. 1. 1. 1.]

Solution

1. Step 1: Calculate the Hann window values for length 4

   The Hann window for length 4 is [0, 0.75, 0.75, 0].

2. Step 2: Multiply the signal by the window element-wise

   Signal is all ones, so windowed_signal equals the window values.

3. Final Answer:

   [0. 0.75 0.75 0. ] -> Option A

4. Quick Check:

   Hann window at length 4 = [0, 0.75, 0.75, 0] [OK]

Hint: Hann window starts and ends at 0, peaks near middle [OK]

Common Mistakes:

• Assuming window values are 0.5 instead of 0.75
• Forgetting to multiply element-wise
• Confusing Hann with rectangular window

4. Identify the error in the following code that applies a Hamming window before FFT:

import numpy as np
from scipy.signal import get_window

signal = np.array([1, 2, 3, 4, 5])
window = get_window('hamming', 5)
windowed_signal = signal + window
fft_result = np.fft.fft(windowed_signal)
print(np.round(np.abs(fft_result), 2))

medium

A. Window should be multiplied, not added, to the signal

B. The window length must be larger than the signal length

C. np.fft.fft cannot be used on windowed signals

D. Hamming window cannot be created with length 5

Solution

1. Step 1: Check how window is applied to signal

   The code adds the window to the signal instead of multiplying element-wise.

2. Step 2: Correct application of window

   Windowing requires element-wise multiplication to smooth edges before FFT.

3. Final Answer:

   Window should be multiplied, not added, to the signal -> Option A

4. Quick Check:

   Windowing = element-wise multiply, not add [OK]

Hint: Always multiply signal by window, never add [OK]

Common Mistakes:

• Adding window instead of multiplying
• Assuming window length must exceed signal length
• Thinking FFT can't be applied after windowing

5. You have a noisy signal sampled at 1000 Hz with 1024 points. You want to analyze its frequency content using FFT with minimal spectral leakage. Which steps correctly apply windowing before FFT using scipy?

hard

A. Create a Hamming window of length 512, multiply it with the first half of the signal, then compute FFT

B. Create a rectangular window, add it to the signal, then compute FFT

C. Create a Hann window of length 1024, multiply it element-wise with the signal, then compute FFT

D. Compute FFT directly without windowing to preserve original signal

Solution

1. Step 1: Choose appropriate window and length

   Hann window is common for reducing spectral leakage; length must match signal length (1024).

2. Step 2: Apply window and compute FFT

   Multiply signal element-wise by window, then compute FFT on the windowed signal.

3. Final Answer:

   Create a Hann window of length 1024, multiply it element-wise with the signal, then compute FFT -> Option C

4. Quick Check:

   Window length = signal length, multiply before FFT = A [OK]

Hint: Match window length to signal, multiply before FFT [OK]

Common Mistakes:

• Adding window instead of multiplying
• Using wrong window length
• Skipping windowing and accepting leakage