DFT Leakage and Window Function Calculator
Compare window functions and see how much spectral leakage each one leaves behind for your FFT size and sample rate.
🪟 What is DFT Leakage and a Window Function?
DFT spectral leakage is a distortion that appears whenever a signal is analyzed with a Discrete Fourier Transform (DFT) or FFT and its frequency doesn't line up exactly with one of the transform's bin centers. The DFT implicitly treats the captured block of samples as one period of an infinitely repeating signal. If the true frequency isn't an exact multiple of the bin spacing, the wraparound point creates a discontinuity, and that discontinuity's energy "leaks" outward into every neighboring bin instead of staying concentrated in a single spike.
A window function is a weighting curve applied to the samples before the transform, tapering the amplitude smoothly down toward zero at both ends of the block. This removes the sharp discontinuity that causes leakage. Audio engineers use windowing before spectral analysis of recordings, RF engineers use it in spectrum analyzers to resolve nearby signals, and vibration analysts use it when computing FFTs of machinery sensor data to avoid false peaks.
Every window function makes the same fundamental trade-off: suppressing side lobes (leakage) always widens the main lobe (frequency resolution). Rectangular (effectively no window) has the narrowest main lobe but the worst leakage. Blackman has the best leakage suppression but the widest main lobe. Hann and Hamming sit in between, each favoring a different leakage profile. There is no single "best" window, only the right trade-off for the measurement being made.
This calculator computes the main lobe width, highest side lobe level, equivalent noise bandwidth (ENBW), coherent gain, and scalloping loss for Rectangular, Hann, Hamming, and Blackman windows, scaled to your actual FFT size and sampling rate, and plots all four frequency responses on one chart for direct comparison. The chart's frequency axis is in Hz, scaled by your actual bin width (sampling rate ÷ FFT size), so the curves visibly widen or narrow as you change either input, matching the main lobe width shown in the results above.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Hann Window on an Audio FFT
Hann window, N = 1,024, sampled at 48,000 Hz
Example 2 — Hamming Window on a CD-Rate Recording
Hamming window, N = 2,048, sampled at 44,100 Hz
Example 3 — Blackman Window on a Low Sample Rate Sensor
Blackman window, N = 512, sampled at 8,000 Hz
❓ Frequently Asked Questions
🔗 Related Calculators
What causes DFT spectral leakage?
Spectral leakage happens because a DFT implicitly assumes the captured samples repeat periodically forever. If the true signal frequency doesn't fall exactly on a bin center, the discontinuity at the wraparound edge spreads (leaks) energy into neighboring bins instead of showing a single clean spike.
How does a window function reduce leakage?
A window function tapers the signal smoothly to near zero at both edges of the sample block before the DFT runs, removing the sharp discontinuity that causes leakage. This lowers the side lobes at the cost of widening the main lobe, trading frequency resolution for leakage suppression.
Which window should I use, Hann or Hamming?
Hann gives a faster -18 dB/octave side-lobe roll-off, which suppresses far-away interference better over a wide span. Hamming's first side lobe is deeper (-42.7 dB vs -31.5 dB) but its side lobes roll off more slowly at -6 dB/octave, so it's better when the interference is close in frequency.
What is the difference between main lobe width and bin width?
Bin width (fs/N) is fixed by your FFT size and sampling rate alone. Main lobe width is the bin width multiplied by a window-specific factor (2 bins for Rectangular, 4 for Hann and Hamming, 6 for Blackman), representing how many bins of frequency resolution you actually lose to the window.
What does equivalent noise bandwidth (ENBW) mean?
ENBW is the width of an ideal rectangular (brick-wall) filter that would pass the same total noise power as the window in use. It is always at least the bin width and is used to correctly normalize noise-floor or power spectral density measurements taken through a windowed FFT.
Why does a Rectangular window have the worst scalloping loss?
Rectangular has the narrowest main lobe, so its peak falls off fastest as a real tone drifts away from an exact bin center. Its worst-case amplitude error (scalloping loss) is 3.92 dB, higher than any of the tapered windows, even though its main lobe width in bins is the smallest of the four.
Does the window choice change the bin width?
No. Bin width is purely fs/N, determined by the sampling rate and FFT size. The window only changes how energy near a bin spreads into neighboring bins (leakage), not the underlying bin spacing itself.
What is coherent gain and why does it matter?
Coherent gain is the DC gain of the window (its average sample value), used to correct amplitude measurements. Rectangular has a coherent gain of 1.0 (no correction needed), while Hann's is 0.5, meaning a Hann-windowed FFT amplitude must be divided by 0.5 to recover the true signal amplitude.
When should I skip windowing entirely and use a Rectangular window?
Use Rectangular (effectively no window) when your signal is already periodic within the FFT block, such as a coherently sampled test tone at an exact bin frequency, since windowing would only widen the main lobe unnecessarily and cost you frequency resolution for no leakage benefit.
How do I convert the main lobe width from bins to Hz?
Multiply the window's main lobe width in bins by the bin width (fs/N). For example, a Hann window (4 bins) with fs = 48,000 Hz and N = 1,024 gives a bin width of 46.875 Hz, so the main lobe spans 4 x 46.875 = 187.5 Hz.