Library / Signal processing and digital filters / DSP-15
Signal processing and digital filters
Bartlett window sample
Coefficient n of the symmetric triangular Bartlett window, with zero end points, valid for N ≥ 2.
- Ref
- DSP-15
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- w[n]
The Bartlett window w[n] = 1 − |2n/(N − 1) − 1| is a triangle that rises linearly from zero at the first sample to one at the centre and falls back to zero. It is the convolution of two rectangles, so its spectrum is a squared sinc. An even-length symmetric taper has two equal centre samples below one.
Coefficient n of the symmetric triangular Bartlett window, with zero end points, valid for N ≥ 2. An even-length symmetric taper has two equal centre samples below one. 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)0.4444
- Value at the mirror sample, w[N − 1 − n]0.5
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 Bartlett window is 0.5, rising in a straight line from zero at the first sample.
- The straight-line shape gives sidelobes that fall faster than a rectangle’s but start higher than Hann’s, so it is used mainly for simple smoothing and spectral estimation.
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 | 0 | 1 | 1 | +1 | +50 % |
| N Window length | 9 to 1,024 samples | 0.00391 | 0.5 | 0.4961 | +1 samples | −11.1 % |
Sweep of Sample index
| Sample index | Window coefficient |
|---|---|
| 0 | 0 |
| 1 | 0.25 |
| 2 | 0.5 |
| 3 | 0.75 |
| 4 | 1 |
| 5 | 0.75 |
| 6 | 0.5 |
| 7 | 0.25 |
| 8 | 0 |
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 −11.1 %, from 0.5 to 0.4444.
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 0.5 to 0.5.
Worked example with the values above
- Position: 2n/(N − 1) = 2 × 2 / 8 = 0.5.
- Distance from the centre: |0.5 − 1| = 0.5.
- w = 1 − 0.5 = 0.5.
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