Let's assume your model is sufficient for your application, and we can really describe the behavior of the device simply in additive terms, one for the per-measurement stochastic behavior and one for the overall bias. Three observations before we get started:
- I assume you are running the sensor on a 3V supply. If not, you can use the values in the datasheet to adjust the calculations.
- These terms are expressed in the same units as \$\bar{a}\$, which in this case is volts, not \$g\$.
- You will actually need three such equations, one for each axis. So 6 terms in total.
For the bias \$b_a\$ we can turn to the charts in figures 5,6, and 7 on page 6 of the datasheet, titled "{X,Y,Z}-axis zero g bias". In a perfect world the zero g output would be 1.5V, but as we can see from the charts the actual value varies between parts. To select your \$b_a\$ for a particular simulated device for a particular axis, you can draw a random sample from that distribution, and use the offset from the expected value of 1.5 as your value for \$b_a\$ for that axis.
Let's look for example at the X-axis term for a particular device. Eyeballing the distribution's parameters I would model it as a Gaussian with \$\mu = 1.53V\$ and \$\sigma=0.01V\$. This means that the distribution for your bias \$b_a\$ for that axis (0g output - expected 0g output of 1.5V) is also a Gaussian, but with \$\mu = 0.03V\$ and \$\sigma=0.01V\$.
In order to assess the random noise, we need to stipulate some sort of output filtering. As is mentioned in the data sheet, by reducing the bandwidth you also significantly reduce noise on the output. I am going to assume a bandwidth of 100Hz just to make the math easier, but feel free to substitute your own values. There is a fairly extensive treatment of this topic in the datasheet under the heading "Design trade-offs for selecting filter characteristics".
With a bandwidth of 100HZ we can expect, according to the datasheet, a noise around 280*10 \$\mu g\$= 2.8 \$mg\$ RMS for the x-axis. We need to convert this to volts in order be able to add it to the formula. The expected sensitivity is about 300 mV/g, so we're execpting a noise of about 0.8 mV RMS. Note that RMS is exactly equal to the standard deviation of the distribution, so you can draw your per-measurement noise samples \$\mu_a\$ directly from a gaussian with \$\mu=0\$ and \$\sigma=0.0008 V\$.
So, for an output filtering of 100HZ: \$\mu_a\ \sim \mathcal{N}(0,0.0008)\$ and \$b_a \sim \mathcal{N}(0.03,0.01)\$, with the stipulation that \$\mu_a\$ is sampled at every measurement, and \$b_a\$ is sampled once for every device.
A factor that we neglected to consider is the variation in sensitivity between devices. This can be accounted for in a manner similar to our treatment of \$b_a\$, but since it's a multiplicative factor, it is not easily captured in your additive model.
When talking about abstract quantities, we generally stop talking about "noise" and instead start talking about probability distribution functions (PDFs) and confidence levels.
You might be able to say, for example, that the answer (position) is correct to within 0.1 meters 95% of the time. You need to establish what your correctness criteria are for your application, and analyze how the PDF of the raw data gets transformed into the PDF of the result.
For a more detailed description, do some research into how the results are described for GPS receivers.
Best Answer
The sensor specifies its noise in terms of the input signal (acceleration), so the first thing you need to do is convert that to equivalent voltage noise, taking into account the gain settings you're using in the sensor and the interface circuitry.
The signal power in a series of normally-distributed samples is equal to the variance, which is the square of the standard deviation (sigma, σ). However, the numbers in your simulation represent voltages (or currents), not power. Therefore, you want to set σ equal to the equivalent noise voltage you calculated above, and then add these numbers to your simulated acceleration signal.