# noise calculation in the gps-simulator

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Siyu on 22 Jun 2023
Commented: Ryan Salvo on 26 Jun 2023
Hi,
I am trying to understand the principle of gps sensor simulator in Matlab. According to the documentation, the horizontal position accuracy specifies the standard deviation of the noise in the horizontal position measurement and the vertical position accuracy specifies the standard deviation of the noise in the vertical position measurement. From my understanding, the measurement noise matrix be
. Where I assumed that the the standard deviation in x and y directions are the same.
But after reading the source code of matlab implementation, I am wondering where comes from the factor 2. The relevant source code is shown in the below.
dt = 1 ./ obj.SampleRate;
% in case of white noise process the decayfactor is 0.
decayFactor = obj.DecayFactor;
% tau = dt for white noise process
tau = dt / (1-decayFactor);
horzSigma = obj.HorizontalPositionAccuracy;
vertSigma = obj.VerticalPositionAccuracy;
sigmas = [horzSigma horzSigma vertSigma];
% updated standard deviation
% it equals to obj.pSigmaScaled = sigmas .* sqrt(2);
obj.pSigmaScaled = sigmas .* sqrt(2.*dt./tau);
obj.pPositionErrorFilterNum = 1;
obj.pPositionErrorFilterDen = [1 -decayFactor];
According to the implementation, the measurement noise is:
.
I don't know where I got it wrong. Is there someone could help me about it?
Thanks for any help and hints in advance.
Best regards

Ryan Salvo on 23 Jun 2023
Hi Siyu,
For the noise calculation, the standard deviation is scaled by a bandwidth equal to 1/2 of the sampling rate. That results in the multiplication of the standard deviation by 2*dt.
Thanks,
Ryan

Siyu on 26 Jun 2023
Hi Ryan,
I can understand that acoording to the Nyquist Sampling Theorem the sampling rate is 2 times of the noise bandwidth. But I am still not clear what is the relationship between the noise bandwidth and the rescaling. Further, why should the term be divided to calculate the scaled sigma? Maybe I am missing some background information. Could you provide me more detailed information about it?
Best
Siyu
Ryan Salvo on 26 Jun 2023
Hi Siyu,
We use the generic sensor noise model detailed in the TSim manual to model this random noise in the GPS sensor. See the Derivation of Modified Rate Random Walk Noise Model section for the complete details.
Thanks,
Ryan

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