Library / Signal processing and digital filters / DSP-13
Signal processing and digital filters
Blackman window sample
Coefficient n of the symmetric three-term Blackman window.
- Ref
- DSP-13
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- w[n]
The Blackman window adds a second cosine term, w = 0.42 − 0.5 cos θ + 0.08 cos 2θ with θ = 2πn/(N − 1). The extra term lowers the sidelobes to about −58 dB (from −31 dB for Hann) while broadening the main lobe by half. Both end points are zero.
Coefficient n of the symmetric three-term Blackman window. The extra cosine term lowers the sidelobes and broadens the main lobe relative to Hann and Hamming. The page needs integers with 0 ≤ n ≤ N − 1.
Calculator
- Position along the window, n/(N − 1)0.25
- Value with the periodic convention (N in the denominator)0.258
- Value at the mirror sample, w[N − 1 − n]0.34
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Sample 2 of a 9-point Blackman window is 0.34.
- The Blackman window rises more slowly from the ends than Hann: this gives lower leakage, which helps when a weak tone sits next to a strong one.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Sample index | 0 to 8 | −1.388e-17 | 1 | 1 | +1 | +128 % |
| N Window length | 9 to 1,024 samples | 1.358e-5 | 0.34 | 0.34 | +1 samples | −24.1 % |
Sweep of Sample index
| Sample index | Window coefficient |
|---|---|
| 0 | −1.388e-17 |
| 1 | 0.06645 |
| 2 | 0.34 |
| 3 | 0.7736 |
| 4 | 1 |
| 5 | 0.7736 |
| 6 | 0.34 |
| 7 | 0.06645 |
| 8 | −1.388e-17 |
Sweep one variable
Variables and how each one contributes
Window coefficient
What the equation gives you. Everything else feeds it.
Window length
The window length, in samples. A longer window has more samples across the same shape, so a given n sits nearer the start. Raising it by 10 % (to 9.9) moves the result by −24.1 %, from 0.34 to 0.258.
Sample index
The sample index, 0 to N − 1. Moves along the window from the start towards the centre and the other end. Raising it by 10 % (to 2.2) moves the result by +0 %, from 0.34 to 0.34.
Worked example with the values above
- Angle: θ = 2π × 2 / 8 = 1.5708 rad.
- Terms: 0.5 cos θ = 3.0616×10−17, 0.08 cos 2θ = −0.08.
- w = 0.42 − 3.0616×10−17 + −0.08 = 0.34.
Assumptions and limits
- A symmetric N-point window with integer N ≥ 2 and 0 ≤ n ≤ N − 1, so the end points have the same value.
- For periodic spectral analysis, replace N − 1 with N in the denominator.
Common mistakes
- Using the symmetric form for spectral analysis, which assumes a periodic window.
- Using an index outside 0 to N − 1.
- Forgetting that Hamming and Kaiser (with β > 0) end points are not zero.
Statistics and uncertainty
Window coefficients are exact formulas, but sampled windows differ slightly between software packages (symmetric against periodic form), which changes the leakage figures in the fourth decimal place.
Where it is used
- Generating window coefficients for FIR design and spectral estimation.
- Trading main-lobe width against sidelobe level in tone measurements.
Maths you need
- Spectral leakage: energy spreading into neighbouring bins
- Sidelobe level: the height of the largest lobe outside the main lobe
- Symmetric against periodic windows: N − 1 or N in the denominator