Library / Structural dynamics and vibration testing / SDY-44
Structural dynamics and vibration testing
Equivalent noise bandwidth of a window
Width of the ideal rectangular filter that would pass the same noise power as the window.
- Ref
- SDY-44
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Be [Hz]
The equivalent noise bandwidth Be = fₛΣw²/(Σw)² is the width of the ideal rectangular filter that passes the same noise power as the window. For a rectangular window it is the bin spacing fₛ/N; for Hann it is 1.5 bins. Use it to turn a windowed spectrum of noise into a density. This entry uses the generalised cosine window of the previous entry.
Width of the ideal rectangular filter that would pass the same noise power as the window. It is the factor to use when converting a windowed spectrum to a density. The window is the generalised cosine window of the previous entry.
Calculator
- Bin spacing, fₛ/N10 Hz
- Bandwidth in bins, Bₑ/Δf1.5
- Sum of squared window values, Σw²3
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.
- The window passes noise as a filter 15 Hz wide, 1.5 times the bin spacing.
- A wider noise bandwidth blurs closely spaced modes more, but gives lower leakage; this is the trade-off in choosing a window.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fs Sample rate | 10 to 10,000 Hz | 1.875 | 1,875 | 1,873 | +10% | +10 % |
| N Window length | 4 to 4,096 | 0.0293 | 30 | 29.97 | +1 | −11.1 % |
| a0 Window coefficient a₀ (0.5 Hann, 1 rectangular) | 0.5 to 1 | 10 | 15 | 5 | +10% | −11 % |
Sweep of Window coefficient a₀ (0.5 Hann, 1 rectangular)
| Window coefficient a₀ (0.5 Hann, 1 rectangular) | Equivalent noise bandwidth (Hz) |
|---|---|
| 0.5 | 15 |
| 0.5625 | 13.02 |
| 0.625 | 11.8 |
| 0.6875 | 11.03 |
| 0.75 | 10.56 |
| 0.8125 | 10.27 |
| 0.875 | 10.1 |
| 0.9375 | 10.02 |
| 1 | 10 |
Sweep one variable
Variables and how each one contributes
Equivalent noise bandwidth
What the equation gives you. Everything else feeds it.
Sample rate
The sample rate, in Hz: samples taken per second. Sets the frequency span and scales the bandwidth in proportion. Raising it by 10 % (to 88) moves the result by +10 %, from 15 to 16.5 Hz.
Window coefficient a₀ (0.5 Hann, 1 rectangular)
The window coefficient a₀ in w = a₀ − (1 − a₀)cos(2πj/N): 0.5 is Hann, 0.54 is Hamming and 1 is a rectangular window. A larger a₀ gives a flatter window with a narrower noise bandwidth but more leakage. Raising it by 0.04 (to 0.54) moves the result by −9.14 %, from 15 to 13.63 Hz.
Window length
The window length in samples. Longer windows narrow the bandwidth in proportion. Raising it by 8 (to 16) moves the result by −50 %, from 15 to 7.5 Hz.
Worked example with the values above
- Sums: Σw = 4, Σw² = 3.
- Ratio: Σw²/(Σw)² = 0.1875.
- Bₑ = 80 × 0.1875 = 15 Hz.
Assumptions and limits
- A uniformly sampled, stationary record, with the window and scaling conventions stated for each entry.
- A one-sided spectrum, with the factor 2 applied only to bins that have a distinct negative-frequency partner.
Common mistakes
- Forgetting to divide by the window’s noise bandwidth or power, which scales the spectrum wrongly.
- Applying the factor 2 at DC and at the Nyquist bin.
- Assuming that zero padding adds resolution: it only interpolates the spectrum.
- Treating overlapping segments as independent: they are correlated.
Statistics and uncertainty
Windowing and averaging trade resolution against variance and leakage. Record length sets the bin spacing, and the number of segments sets how much random error remains in the estimated spectrum.
Where it is used
- Setting the record length, window and averaging for a vibration test so that nearby modes can be told apart.
- Scaling windowed FFT output to correct amplitudes and spectral densities.
Maths you need
- Discrete Fourier transform: frequency content of a sampled record
- Window function: a taper applied to reduce spectral leakage
- Power spectral density: power per unit frequency