# White noise specification

1. Jul 28, 2007

### reg_free

hello,
I am stuck at understanding the units of noise specification of accelerometers and gyros. I am referring to the datasheet of 3DM-GX1 which has accelerometers and gyro sensors.
the pdf is ..
http://www.microstrain.com/pdf/3DM-GX1%20Detailed%20Specs%20-%20Rev%201%20-%20070723.pdf [Broken]

My doubt is, the accelerometer white noise is specified as 0.4 mg/(root Hz) .. mg here is milli g (acceleration due to gravity).. so the units are meters per sec-square/(root hertz) .... how do i get standard deviation of the noise in meters/sec-square ??

Moreover, the similar entry for gyro output(Angular rate in the pdf) is given as Random Walk Noise = 3.5 degrees/(root hour) ... how do i get standard deviation of the white noise of gyro output ?? its units should be, ofcourse, degrees/sec or radians/sec

Please, help me, I am not into stochastics and I am stuck at this point since two weeks!!!

Last edited by a moderator: May 3, 2017
2. Jul 28, 2007

### D H

Staff Emeritus
A gyro (or an accelerometer) is a "smart" sensor: It has an internal processor that takes measurements at a very high rate. The sensor output is sampled at a much lower rate. The output is an aggregate of the noisy raw measurements: a random walk with random step sizes (i.e., Brownian motion). The sampled data are best characterized as Brownian noise rather than white noise. Brownian noise is characterized by an angular random walk (ARW) coefficient ($$3.5^\circ/\surd{\text{hr}}$$).

So, how to convert the spec value to a value you can use to simulate sensor output? The sampled data can be treated as a white noise process, but with the variance depending on the sampling interval: $$\sigma_{\omega} = \text{ARW}/\sqrt{\Delta t}$$. Suppose you sample at one Hertz. $$\sigma_{\omega}|_{\Delta t = 1 \text{sec}} = 3.5^\circ/\surd{\text{hr}}/\surd{1\text{sec}} = 0.058^\circ/\text{sec}$$. Sampling at 10 Hertz increases the noise to $$0.184^\circ/\text{sec}$$.

Last edited: Jul 28, 2007
3. Jul 29, 2007

### reg_free

thanks a ton DH :)

i suppose the formula you specified is compatible with allan variance method since that is what is specified in the datasheet.

regarding the accelerometer specification, is it a different variation of representation of power spectral density(psd) ? if it was psd, the units should be mg/Hz. and the psd equals square of standard deviation in case of white noise! Am i missing something here? i'd highly appreciate any help regarding this..

thanks again DH..

Last edited: Jul 29, 2007
4. Jul 30, 2007

### D H

Staff Emeritus
A simple way to look at IMUs is that they downsample high-rate signals by averaging to produce a lower rate output. (This treatment is a bit naive, but it is close enough for many purposes.) If the underlying process is a white noise process with variance $\sigma^2$, the output will be a white noise process but with a reduced variance $\frac {\delta t}{\Delta t} \sigma^2$ where $\delta t$ is the internal clock interval and $\Delta t$ is the output clock interval. The output rate is user-selectable, so how to characterize the noise? Answer: remove the $\Delta t$, leaving $\delta t \sigma^2$. Taking the square root leads to the goofy square root time stuff.

5. Feb 22, 2008

### DHe

I'm trying to simulate the gyro noise specified in the paper MicroStrain (cf. 1st post of reg_free) under Simulink.
I use the Band Limited White Noise, with a noise power of (3.5°/Vh)^2, and a sample time of 0.01s (corresponding to the update rate of 100Hz).
The output signal is a white noise with a range of roughly +/-6000°/h (3sigma), whereas the bias is given at 0.1°/s, i.e. 360°/h. I'm very surprised that the white noise is so much greater than the bias. Is this normal?
Thanks a lot...

6. Feb 22, 2008

### D H

Staff Emeritus
The noise spec is in terms of "angle random walk" with a 3 sigma value of $3.5^{\circ}/\sqrt{\text{hour}}$. Sampling faster increases the noise. At 100 Hz, you should be getting 2100o/hour (3 sigma), not 6000. Why so much more noise than bias? It's because you are sampling at a fairly high rate and because it's a fairly good sensor.

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