Library / Structural dynamics and vibration testing / SDY-43
Structural dynamics and vibration testing
Window coherent gain
Average of the window samples.
- Ref
- SDY-43
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- G
The coherent gain G = Σwⱼ/N is the average of the window. It is the factor by which the window scales the amplitude of a tone sitting on a bin centre, so dividing the spectral peak by G recovers the tone’s amplitude. This entry uses the generalised cosine window w = a₀ − (1 − a₀)cos(2πj/N), where a₀ = 0.5 is Hann, 0.54 is Hamming and 1 is rectangular.
Average of the window samples. It is the factor by which a window scales the amplitude of a tone that sits on a bin centre. This entry uses the generalised cosine window w = a₀ − (1 − a₀)cos(2πj/N): a₀ = 0.5 is Hann.
Calculator
- Amplitude correction factor, 1/G2
- Window at the centre sample1
- Window at the first sample0
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.
- A window with a₀ = 0.5 has a coherent gain of 0.5, so tone amplitudes read 50 % low until corrected.
- The gain is simply a₀ for any whole-period window of three or more samples, which is why Hann is 0.5 and Hamming 0.54.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| a0 Window coefficient a₀ (0.5 Hann, 1 rectangular) | 0.5 to 1 | 0.5 | 1 | 0.5 | +10% | +10 % |
| N Window length | 4 to 4,096 | 0.4995 | 0.5001 | 5.321e-4 | +1 | +2.22e-14 % |
Sweep of Window coefficient a₀ (0.5 Hann, 1 rectangular)
| Window coefficient a₀ (0.5 Hann, 1 rectangular) | Coherent gain |
|---|---|
| 0.5 | 0.5 |
| 0.5625 | 0.5625 |
| 0.625 | 0.625 |
| 0.6875 | 0.6875 |
| 0.75 | 0.75 |
| 0.8125 | 0.8125 |
| 0.875 | 0.875 |
| 0.9375 | 0.9375 |
| 1 | 1 |
Sweep one variable
Variables and how each one contributes
Coherent gain
What the equation gives you. Everything else feeds it.
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 +8 %, from 0.5 to 0.54.
Window length
The window length in samples. Longer windows narrow the bandwidth in proportion. Raising it by 8 (to 16) moves the result by +0 %, from 0.5 to 0.5.
Worked example with the values above
- Window w = 0.5 − 0.5 cos(2πj/8) for j = 0 to 7.
- Sum of the samples: Σw = 4.
- G = 4 / 8 = 0.5.
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