Windowing before FFT in SciPy - Step-by-Step Execution

Concept Flow - Windowing before FFT

Start: Raw Signal

↓

Apply Window Function

↓

Windowed Signal

↓

Compute FFT

↓

Frequency Spectrum Output

The raw signal is first multiplied by a window function to reduce edge effects, then the FFT is computed on the windowed signal to get a cleaner frequency spectrum.

Execution Sample

SciPy

import numpy as np
from scipy.signal import windows
from scipy.fft import fft

signal = np.array([1, 2, 3, 4, 5, 6, 7, 8])
window = windows.hann(len(signal))
windowed_signal = signal * window
spectrum = fft(windowed_signal)

This code applies a Hann window to a simple signal and then computes its FFT.

Execution Table

Step	Variable	Value/Action	Result/Output
1	signal	[1 2 3 4 5 6 7 8]	Raw input signal array
2	window	[0. 0.1882551 0.61126047 0.95048443 0.95048443 0.61126047 0.1882551 0. ]	Hann window values
3	windowed_signal	[0. 0.3765102 1.8337814 3.8019377 4.7524221 3.6675628 1.3187857 0. ]	Signal after element-wise multiplication with window
4	spectrum	fft(windowed_signal)	Complex frequency spectrum array
5	spectrum (magnitude)	[15.751 -6.371+0.j -2.059-1.927j -0.25 -0.25j -0.25 -0.25j -0.25 +0.25j -0.25 +0.25j -2.059+1.927j]	Magnitude and phase of FFT output

💡 FFT computed on windowed signal to reduce spectral leakage

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
signal	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]
window	N/A	[0. 0.1882551 0.61126047 0.95048443 0.95048443 0.61126047 0.1882551 0. ]	[0. 0.1882551 0.61126047 0.95048443 0.95048443 0.61126047 0.1882551 0. ]	[0. 0.1882551 0.61126047 0.95048443 0.95048443 0.61126047 0.1882551 0. ]	[0. 0.1882551 0.61126047 0.95048443 0.95048443 0.61126047 0.1882551 0. ]
windowed_signal	N/A	N/A	[0. 0.3765102 1.8337814 3.8019377 4.7524221 3.6675628 1.3187857 0. ]	[0. 0.3765102 1.8337814 3.8019377 4.7524221 3.6675628 1.3187857 0. ]	[0. 0.3765102 1.8337814 3.8019377 4.7524221 3.6675628 1.3187857 0. ]
spectrum	N/A	N/A	N/A	[15.751 -6.371+0.j -2.059-1.927j -0.25 -0.25j -0.25 -0.25j -0.25 +0.25j -0.25 +0.25j -2.059+1.927j]	[15.751 -6.371+0.j -2.059-1.927j -0.25 -0.25j -0.25 -0.25j -0.25 +0.25j -0.25 +0.25j -2.059+1.927j]

Key Moments - 3 Insights

Why do we multiply the signal by a window before FFT?

Multiplying by a window reduces edge effects and spectral leakage, as shown in step 3 of the execution_table where the signal is modified before FFT.

What happens if we skip the window and compute FFT directly?

Without windowing, the FFT may show spectral leakage causing less clear frequency peaks. The execution_table shows the windowed_signal differs from the raw signal, which improves FFT results.

Why is the window array zero at the first and last points?

The Hann window tapers the signal edges to zero to smooth transitions, visible in step 2 of execution_table where window values at edges are zero.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what is the value of windowed_signal at index 3?

A 4.7524221

B 3.8019377

C 1.8337814

D 0.3765102

💡 Hint

Check the 'windowed_signal' value in step 3 of execution_table at the 4th element (index 3).

At which step is the FFT actually computed?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look for the step where 'spectrum' is assigned using fft(windowed_signal) in execution_table.

If we used a rectangular window (all ones) instead of Hann, how would windowed_signal change at step 3?

A It would be the same as the original signal

B It would be all zeros

C It would be half the original signal

D It would be the square of the original signal

💡 Hint

Recall windowing multiplies signal by window values; rectangular window is all ones, so signal stays unchanged.

Concept Snapshot

Windowing before FFT:
- Multiply signal by window (e.g., Hann) to reduce edge effects
- Window tapers signal edges to zero
- Compute FFT on windowed signal
- Results in cleaner frequency spectrum
- Use scipy.signal.windows and scipy.fft for implementation

Full Transcript

Windowing before FFT means we first take our raw signal and multiply it by a window function like the Hann window. This step reduces edge effects and spectral leakage that happen when we directly compute FFT on raw data. The window function tapers the signal edges smoothly to zero. After applying the window, we compute the FFT on this modified signal. This process results in a cleaner frequency spectrum output. In the example, we start with a simple signal array, create a Hann window of the same length, multiply them element-wise to get the windowed signal, and then compute the FFT. The execution table shows each step's values, including the window values, the windowed signal, and the final FFT spectrum. This approach is important to get accurate frequency analysis in real-world signals.

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