Hello Everyone,
I am implementing tracking algorithm for Galileo satellite in MATLAB. In the tracking algorithm we generally have two loops, PLL and DLL and both of them have loop filters. And we need to find the coefficients of both these filters.
To find the coefficients, we need to know the noise bandwidth, damping ratio and gain of the loop filter. Now my question is:
1. Are there any mathematical relationships to find the values of (bandwidth, damping ratio and gain) ? That is, using signal specifications we can find the values of these parameters.
2. Or are they signal specific? I mean the noise bandwidth would be different for Galileo E1 and Galileo E5 and so on?
3. Or is it just hit and trial? that we should test different values for all these parameters and find the suitable combination?
I have read in one paper, that bandwidth must be chosen in such a way that the product BxT (T = integration time) must be less than 1. But there can be so many values of bandwidth to satisfy this condition.
I tried to explain my doubt but if someone needs any particular clarification, I would be happy to do that.
Thank you
Raza Javed
Dear Raza Javed
This link might be helpful, have a look:
https://www.mathworks.com/matlabcentral/fileexchange/68320-gps-visibility-predictor
Kind Regards
Qamar Ul Islam
Dear Raza Javed,
A digital PLL is usually composed of a proportional Integrator (PI) filter corresponding to 2nd order filter. Its corresponding Z transform is :
G(z) = alpha + beta / (1 - z^(-1))
with alpha and beta being the loop parameters..
Under the assumption of linear operation of the loop , this 2nd order loop, we have the following relationships between the real parameters of the loop (alpha, beta) and their equivalent parameters : B (the equivalent noise bandwidth of the loop) and Ke (the loop gain) :
alpha ~= (4 ksi/(ksi + 0.25 ksi)) x (B x T/ Ke)
beta ~= (4/(ksi+0.25ksi)^2) x (B x T)^2 / Ke
where ksi is the damping factor. These 2 relations are valid under the assumptions that the product B x T 0 then low noise + low speed). A good compromise between precision and speed of convergence is found with a value of ksi = 1/sqrt(2). Once parameters B x T and ksi are set, the only remaining parameter is Ke.
Defining the S-curve of the loop as :
S(phi) = E ( e[n] / phi )
and differentiating relative to phi, we get at phi = 0 :
Ke = S'(phi = 0)
I hope this will help,
Pascal.
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