Window Functions in FFT: Visual Comparison and Practical Signal Analysis
The Fast Fourier Transform (FFT) is one of the most widely used techniques in signal processing for identifying frequency components within a signal.
However, when performing FFT analysis on real signals, engineers often observe unexpected frequency spread or distorted spectral peaks in the frequency spectrum. This phenomenon is called spectral leakage and commonly occurs when the signal under analysis does not perfectly match the FFT observation window.
To minimize this problem, signal processing workflows typically apply a window function before calculating the FFT.
This article explains why window functions are important, how they work, and how they help improve FFT analysis in real-world engineering applications.
Reasons why spectrum leakage occurs
FFT assumes that the signal under analysis is periodic within the observation period .
In reality, it is rare for actual signals to satisfy these conditions. If a signal starts or ends abruptly at the boundary of the observation window, the FFT interprets these discontinuities as additional frequency components.
This causes spectrum leakage , where signal energy is not concentrated at a single frequency but spreads to adjacent frequency bands.
Spectral leakage can make it difficult to accurately identify the dominant frequency in vibration signals, acoustic measurements, or experimental data.
When the truncation matches an integer number of periods (consistent sampling),
DTFT displays a sinc-shaped spectrum due to the window effect.
On the other hand, in FFT, energy is concentrated only in the discrete frequency band corresponding to the main lobe, and no leakage occurs.
If the truncation does not match the number of integer periods (inconsistent sampling),
Due to the windowing effect, the DTFT maintains a spectrum in the form of sinc.
On the other hand, FFT disperses energy beyond the main lobe into adjacent frequency bins, causing spectral leakage.
How do window functions solve this problem?
The window function reduces spectral leakage by gradually decreasing the signal amplitude near the boundaries of the observation window. This helps the FFT generate a cleaner and easier-to-interpret frequency spectrum.
Instead of abruptly cutting off the signal, the window function smoothly decreases the signal to 0 at the edges.
This reduces discontinuities and improves the accuracy of the FFT spectrum.
In signal processing, a windowed signal is calculated by multiplying the original signal by a window function.
![Window signal equation showing that the window signal x_w[n] is obtained by multiplying the window function w[n] by the original signal x[n].](https://cdn.imweb.me/upload/S20240103ccaa8a36496d4/621acb8756e1c.png)
x [ n ] is the original discrete-time signal.
w [ n ] is a window function.
x w [ n ] is the window signal used for FFT analysis.
After applying the window function, calculate the FFT for the windowed signal.
![Discrete Fourier Transform equation applied to a window signal showing FFT calculation using x[n] multiplied by the window function w[n].](https://cdn.imweb.me/upload/S20240103ccaa8a36496d4/fa0887cf285ad.png)
This process reduces spectrum leakage and generates a clearer frequency spectrum.
Example: Windowing reduces spectrum leakage.
Common window functions used in FFT
Various window functions are commonly used in signal processing.
Hann/Hanning Window
The Hann window is widely used in general FFT analysis. It provides an excellent balance between frequency resolution and spectral leakage suppression.
Hamming window
Hamming windows slightly reduce leakage current compared to a single window and are commonly used in the field of digital signal processing.
Blackman Windows
The Blackman window offers superior leakage suppression but may result in reduced frequency resolution due to a wider spectral peak width.
Selecting an appropriate analysis period depends on the analysis objective.
Comparison of window functions in MALMIJAL
Different window functions shape the signal differently before FFT calculation.
Each window has unique characteristics that affect spectral leakage, frequency resolution, and amplitude accuracy .
The figure below shows some commonly used window functions, including rectangular, triangular, Bartlett, Hamming, Hann, Blackman, Blackman-Harris, flattop, and Nuttol windows.
Comparison of window functions in the time domain ( refer to Samples/window types.mmj)
Comparison of window functions in the frequency domain (refer to Samples/window types.mmj )
Key trade-offs between general FFT window functions
The window function modifies the signal near the boundaries of the observation window. This can reduce discontinuities that cause spectral leakage during FFT analysis. However, if the window function is different, the advantages and disadvantages also differ.
| Window type | Main lobe width (≈ kΔf) | First lateral lobe (dB) | Attenuation rate (dB/dec) | Amplitude accuracy | Frequency resolution | Leakage control |
|---|
| Rectangular | 2Δf (±Δf) | -13 | 20 | poor | best | poor |
| Triangular | 4Δf (±2Δf) | -26 | 40 | fair | good | ordinary |
| Bartlett | 4Δf (±2Δf) | -25 | 40 | fair | good | ordinary |
| Hann | 4Δf (±2Δf) | -31 | 60 | good | good | good |
| Hamming | 4Δf (±2Δf) | -42 | 20 | Better than Hann | good | First lateral lobe attenuation |
| Blackman | 6Δf (±3Δf) | -58 | 60 | good | ordinary | charming |
| Blackman-Harris | 8Δf (±4Δf) | -92 | 60 | charming | lower | excellent |
| Flat Top | 10Δf (±5Δf) | -93 | 60 | best | poor | excellent |
| Nuttall | 8Δf (±4Δf) | -93 | 60 | charming | lower | excellent |
Relationship Between Window Functions and Overlap
When performing FFT-based spectral analysis such as PSD, STFT, FRF, or coherence analysis, window functions are often used together with overlap processing.
Many window functions gradually reduce the signal amplitude near the edges of the observation window to suppress spectral leakage. However, this also reduces the contribution of samples near the boundaries. To compensate for this effect, adjacent FFT segments are often partially overlapped.
For example
- Triangular / Bartlett / Hann / Hamming window → typically 50% overlap
- Blackman window → typically 67% overlap
- Blackman-Harris / Flat-top / Nuttall → typically 75% overlap
This overlap increases the number of averaged FFT segments and helps generate smoother and more stable spectral estimates. However, overlap does not improve the actual frequency resolution.
Quick Selection Guide
Window selection is a compromise between main lobe width (frequency resolution) and side lobe level (spectral leakage).
| Purpose | Recommended Windows |
|---|
| Maximum frequency resolution | Rectangular window |
| General FFT Analysis | Hann window |
| PSD and Harmonic Analysis | Hamming window optimized to suppress leakage around strong tones |
| Leakage control | Blackman window |
| Detecting very small signals | Blackman-Harris / Nuttall window |
| Accurate amplitude measurement | Flat Top window |
Why do engineers compare window functions in FFT analysis?
In actual signal analysis, the selection of the window function can have a significant impact on the FFT results.
Using signal processing tools like MALMIJAL , you can easily explore various window functions and visualize the impact of these functions on FFT analysis results.
The selection of the window function is closely related to spectral analysis methods, including STFT (Short-Time Fourier Transform), power spectrum, crossover spectrum, PSD (Power Spectral Density), CSD (Crossover Spectral Density), FRF (Frequency Response Function), coherence, and order spectrum.
Select Spectrum Analysis Method and Corresponding Window in MALMIJAL
Conclusion
Achieving coherent sampling with real-world data is very difficult. Therefore, spectral leakage is almost inevitable, making the use of window functions essential in most FFT analyses. By understanding the impact of various window types on spectral leakage and frequency resolution, engineers can select the window best suited for signal analysis tasks.
Using tools like MALMIJAL allows users to easily visualize and compare window functions, helping engineers perform reliable frequency domain analysis.
Additional references
#The following topics may also be helpful:
Window Functions in FFT: Visual Comparison and Practical Signal Analysis
The Fast Fourier Transform (FFT) is one of the most widely used techniques in signal processing for identifying frequency components within a signal.
However, when performing FFT analysis on real signals, engineers often observe unexpected frequency spread or distorted spectral peaks in the frequency spectrum. This phenomenon is called spectral leakage and commonly occurs when the signal under analysis does not perfectly match the FFT observation window.
To minimize this problem, signal processing workflows typically apply a window function before calculating the FFT.
This article explains why window functions are important, how they work, and how they help improve FFT analysis in real-world engineering applications.
Reasons why spectrum leakage occurs
FFT assumes that the signal under analysis is periodic within the observation period .
In reality, it is rare for actual signals to satisfy these conditions. If a signal starts or ends abruptly at the boundary of the observation window, the FFT interprets these discontinuities as additional frequency components.
This causes spectrum leakage , where signal energy is not concentrated at a single frequency but spreads to adjacent frequency bands.
Spectral leakage can make it difficult to accurately identify the dominant frequency in vibration signals, acoustic measurements, or experimental data.
When the truncation matches an integer number of periods (consistent sampling),
DTFT displays a sinc-shaped spectrum due to the window effect.
On the other hand, in FFT, energy is concentrated only in the discrete frequency band corresponding to the main lobe, and no leakage occurs.
If the truncation does not match the number of integer periods (inconsistent sampling),
Due to the windowing effect, the DTFT maintains a spectrum in the form of sinc.
On the other hand, FFT disperses energy beyond the main lobe into adjacent frequency bins, causing spectral leakage.
How do window functions solve this problem?
The window function reduces spectral leakage by gradually decreasing the signal amplitude near the boundaries of the observation window. This helps the FFT generate a cleaner and easier-to-interpret frequency spectrum.
Instead of abruptly cutting off the signal, the window function smoothly decreases the signal to 0 at the edges.
This reduces discontinuities and improves the accuracy of the FFT spectrum.
In signal processing, a windowed signal is calculated by multiplying the original signal by a window function.
x [ n ] is the original discrete-time signal.
w [ n ] is a window function.
x w [ n ] is the window signal used for FFT analysis.
After applying the window function, calculate the FFT for the windowed signal.
![Discrete Fourier Transform equation applied to a window signal showing FFT calculation using x[n] multiplied by the window function w[n].](https://cdn.imweb.me/upload/S20240103ccaa8a36496d4/fa0887cf285ad.png)
This process reduces spectrum leakage and generates a clearer frequency spectrum.
Example: Windowing reduces spectrum leakage.
Common window functions used in FFT
Various window functions are commonly used in signal processing.
Hann/Hanning Window
The Hann window is widely used in general FFT analysis. It provides an excellent balance between frequency resolution and spectral leakage suppression.
Hamming window
Hamming windows slightly reduce leakage current compared to a single window and are commonly used in the field of digital signal processing.
Blackman Windows
The Blackman window offers superior leakage suppression but may result in reduced frequency resolution due to a wider spectral peak width.
Selecting an appropriate analysis period depends on the analysis objective.
Comparison of window functions in MALMIJAL
Different window functions shape the signal differently before FFT calculation.
Each window has unique characteristics that affect spectral leakage, frequency resolution, and amplitude accuracy .
The figure below shows some commonly used window functions, including rectangular, triangular, Bartlett, Hamming, Hann, Blackman, Blackman-Harris, flattop, and Nuttol windows.
Key trade-offs between general FFT window functions
The window function modifies the signal near the boundaries of the observation window. This can reduce discontinuities that cause spectral leakage during FFT analysis. However, if the window function is different, the advantages and disadvantages also differ.
Relationship Between Window Functions and Overlap
When performing FFT-based spectral analysis such as PSD, STFT, FRF, or coherence analysis, window functions are often used together with overlap processing.
Many window functions gradually reduce the signal amplitude near the edges of the observation window to suppress spectral leakage. However, this also reduces the contribution of samples near the boundaries. To compensate for this effect, adjacent FFT segments are often partially overlapped.
For example
This overlap increases the number of averaged FFT segments and helps generate smoother and more stable spectral estimates. However, overlap does not improve the actual frequency resolution.
Quick Selection Guide
Window selection is a compromise between main lobe width (frequency resolution) and side lobe level (spectral leakage).
Why do engineers compare window functions in FFT analysis?
In actual signal analysis, the selection of the window function can have a significant impact on the FFT results.
Using signal processing tools like MALMIJAL , you can easily explore various window functions and visualize the impact of these functions on FFT analysis results.
The selection of the window function is closely related to spectral analysis methods, including STFT (Short-Time Fourier Transform), power spectrum, crossover spectrum, PSD (Power Spectral Density), CSD (Crossover Spectral Density), FRF (Frequency Response Function), coherence, and order spectrum.
Select Spectrum Analysis Method and Corresponding Window in MALMIJAL
Conclusion
Achieving coherent sampling with real-world data is very difficult. Therefore, spectral leakage is almost inevitable, making the use of window functions essential in most FFT analyses. By understanding the impact of various window types on spectral leakage and frequency resolution, engineers can select the window best suited for signal analysis tasks.
Using tools like MALMIJAL allows users to easily visualize and compare window functions, helping engineers perform reliable frequency domain analysis.
Additional references
#The following topics may also be helpful: