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I have a few questions about sensor specifications and its implementation in a Kalman Filter and simulation of gyroscope/accelerometer output.

Abbreviation used:

d - discrete

c - continuous

Q1:

From book: Aided Navigation - Farrell (you don't need the book to understand the question).

Section 7.2 Methodology: Detailed Example:

Here

[itex]\sigma_1[/itex], [itex]\sigma_{bu}[/itex] and [itex]\sigma_{by}[/itex] are root PSD of continuous white noise.

[itex]\sigma_2[/itex] is std. deviation of discrete white noise.

[itex]Qc[/itex] = diag([[itex]\sigma_{bu}^2[/itex] [itex]\sigma_{by}^2[/itex] [itex]\sigma_1^2[/itex] ])

[itex]Qd[/itex] = function of [itex]Qc[/itex] and sampling time

[itex]Rd[/itex] = [itex]\sigma_2^2[/itex]

This I understand, but not section 7.4.

Section 7.4 An Alternative Approach:

Same sigma values as above, plus

[itex] sigma_3[/itex] is root PSD of continuous white noise.

Qc = diag([[itex]\sigma_{bu}^2[/itex] [itex] \sigma_{by}^2[/itex] [itex] \sigma_3^2[/itex]])

Qd = function of Qc and sampling time

Rd =[itex] \sigma_1^2[/itex]

and later

Rd = diag([[itex]\sigma_2^2[/itex] [itex] \sigma_1^2[/itex]])

a)

Can you just use [itex]\sigma[/itex] in Rd no matter if it's std. deviation for discrete white noise ([itex]\sigma_2[/itex]) or root PSD of continuous white noise([itex]\sigma_1[/itex])?

b)

Shouldn't [itex]\sigma_1[/itex] be "discretizised" or something?

Q2:

If you get noise specification for the sensor noise from PSD method, Allan Variance method or data sheet, it is for continuous white noise. Is this correct?

Q3:

How do you simulate the output of a sensor in e.g. Matlab/simulink if you have noise specification in continuous time?

Q4:

When we discretizise a stochastic linear system, we get:

Qd = f(Qc, A, T)

Rd = Rc

(http://en.wikipedia.org/wiki/Discretization)

I don't understand why Rd = Rc?

Q5:

In the book: Optimal State Estimation - Kalman, Hinf and Nonlinear Approaches - 2006 - Dan Simon, section 8.1 DISCRETE-TIME AND CONTINUOUS-TIME WHITE NOISE, it says:

"... discrete-time white noise with covariance Qd in a system with a sample period of T, is equivalent to continuous-time white noise with covariance Qc*[itex]\delta[/itex] (t) (*[itex]\delta[/itex] (t): dirac delta function), where Qc = Qd / T."

"... Rd = Rc / T ... This establishes the equivalence between white measurement noise in discrete time and continuous time. The effects of white measurement noise in discrete time and continuous time are the same if

v(k) ~ (0,Rd)

v(t) ~ (0,Rc)

"

How does this relate to my other questions? It seems to suggest that Rd [itex]\neq[/itex] Rc as Q4 suggests.

Best Regard

Jonas

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# Kalman, White Noise, Sensor Specification, Discretization?

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