Windowing before FFT in SciPy - Mini Project: Build & Apply

Windowing before FFT

📖 Scenario: You have recorded a short sound signal and want to analyze its frequency content. To get better results, you will apply a window function before computing the Fast Fourier Transform (FFT).

🎯 Goal: Learn how to apply a window function to a signal before performing FFT to reduce spectral leakage and visualize the frequency spectrum.

📋 What You'll Learn

Create a signal array with exact values

Create a window array using a Hamming window

Multiply the signal by the window

Compute the FFT of the windowed signal

Print the magnitude of the FFT result

💡 Why This Matters

🌍 Real World

Windowing before FFT is used in audio processing, vibration analysis, and any signal processing to get clearer frequency results.

💼 Career

Understanding windowing and FFT is essential for roles in data science, signal processing, and engineering fields involving time-series data.

1

Create the signal array

Create a NumPy array called signal with these exact values: [0, 1, 0, -1, 0, 1, 0, -1].

SciPy

import numpy as np
# Create the signal array called signal with values [0, 1, 0, -1, 0, 1, 0, -1]
# Your code here

Hint

Use np.array to create the array with the exact values.

2

Create the Hamming window

Create a variable called window that holds a Hamming window of the same length as signal using np.hamming(len(signal)).

SciPy

import numpy as np
signal = np.array([0, 1, 0, -1, 0, 1, 0, -1])
# Create the Hamming window called window with length equal to signal
# Your code here

Hint

Use np.hamming with the length of signal to create the window.

3

Apply the window and compute FFT

Create a variable called windowed_signal by multiplying signal and window element-wise. Then create a variable called fft_result by computing the FFT of windowed_signal using np.fft.fft(windowed_signal).

SciPy

import numpy as np
signal = np.array([0, 1, 0, -1, 0, 1, 0, -1])
window = np.hamming(len(signal))
# Multiply signal and window element-wise to create windowed_signal
# Compute FFT of windowed_signal and store in fft_result
# Your code here

Hint

Multiply arrays element-wise with *. Use np.fft.fft to compute the FFT.

4

Print the magnitude of the FFT result

Print the magnitude (absolute value) of fft_result using print(np.abs(fft_result)).

SciPy

import numpy as np
signal = np.array([0, 1, 0, -1, 0, 1, 0, -1])
window = np.hamming(len(signal))
windowed_signal = signal * window
fft_result = np.fft.fft(windowed_signal)
# Print the magnitude of fft_result
# Your code here

Hint

Use np.abs() to get the magnitude of complex FFT results.

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