LAF component block

In section 3.1it was introduced the theory of the LAF, that is valid for generic signals. Here we use the LAF scheme, depicted in Fig. 3.3, by adopting the method suggested in [48] and [49], where d[n] is the measured correlation sequence and u[n] is the ideal correlation sequence

u[n] = ^Xk

m=−k

C_m^∗(m · ∆sp)Cm((m + n) · ∆sp)

where ∆sp is the resolution of the multicorrelator, that is in general different from Ts. With this scheme we are able to fit the measured correlation sequence with an approximated correlation y[n], which is a weighted sum of delayed versions of the ideal correlation. An example is in Fig. 3.4, which shows a correlation obtained with two delayed components of u[n].

The LAF method allows the estimation of the filter coefficient vector w, which can be used to decide about the presence of possible MPs. Therefore a key point of the algorithm is the computation of the vector ˆw, which can be

3.3 Multipath Distance Detector algorithm 59

done by using (3.10), where

Φ= U^HU z= U^HD

and D is the windowed measured correlation from index M to N. The first point to note is that the length M of theLAFfilter also impacts on the length of D. Since D = [d[M],d[M + 1],··· ,d[N]]^T, more taps we choose, less points of the measured correlation are used by the LAF algorithm. In our application we use a portion of the signal containing Ns samples and we measure 2Ns+ 1 correlation points in the region around the peak. Since only the correlation samples from M to N = 2Ns+ 1 are used, M has to be lower than (2Ns+ 1)/2, in order to preserve the samples close to the peak. The value of M also determines the computational load due to the fact that the matrix to invert becomes bigger, as the filter becomes longer. The M-to-N ratio defined as

MNR= M

2N + 1

must be chosen ≪ 50%. For instance, if M = 10 and N = 20, MNR = 0.2439.

Notice that in our application we are not interested in the single components of y[n], since our goal is not to mitigate MP effects, but rather to exclude satellites with severe MPs.

Energy analysis

First of all, it’s important to underline what we expect from the w vector, that is, what we can infer from the vector about MP and more in general the presence of an undesired distortion in the measured correlation.

In an ideal static situation between transmitter and receiver with no Doppler, no MPand no noise effects, w will contain only the first component (w[1]) and the other coefficients are all zero. This because the measured correlation D exactly coincides with the ideal correlation except for a normalization term w_LOS. In all the other cases the presence of undesired components in the received SIS modifies all the elements of w. In the presence of LOS and a single ray of MPwith a specific delay and no noise, w[1] and the element of w corresponding to the delay will be different from zero.

Noise complicates the identification of MP or other distortions, because spreads energy on all the elements of w. In order to mitigate this effect we use an averaged correlation sequence, as it will be explained in section 3.3.3.

Another consideration to take into account is that the MP delay is in general a quantity ∈ R, while in the discrete-time domain only delay ∈ N can be represented. This means that the expression y[n] =^Pw^∗_ku[n − k] used by the LAF to represent the measured correlation can properly model only a finite set of possible delays, therefore the method has not an infinite resolution.

For example a delay of 1 correlation sample is properly modeled by u[n − 1].

Instead, it is probable that a real delay is not proportional to the resolution of the LAFfilter or, in other words, the delay between two correlation samples cannot be modeled by a single replica of u[n].

The delay is represented in that case by more than one elements of the w vector.

It is evident that the information about the presence of MP is contained in the signal:

y_side[n] = y[n] − wLOS· u[n]

that is the curve of side components, obtained by subtracting the approximated correlation and the 0-delayed ideal correlation weighted with the calculated LOS coefficient.

In order to have an idea of the information on MP possibly embedded in y_side[n] some simulation experiments have been performed. AGPS signal with C/N₀= 45 dBHz has been generated with no MP in the initial part, and a 1 ray MP added in the time interval (t1, t₂), where T1= 15s and t2= 20s, with MSR (MSR) = −6dB. The simulation results are shown in Fig. 3.5 which shows the energy of the vectors at theLAF output. In the top left there is the evolution in time of the energy of theLOS component; in top right the evolution of the energy of the side components; in lower left and lower right respectively the energy of approximated correlation and measured one. As expected, the energy of the side components increases during the MP stage. We can say that in the correlation domain the energy of y[n] well approximates the energy of d[n].

3.3 Multipath Distance Detector algorithm 61

Fig. 3.5 Energy analysis ofLAFinput and output signals.

The energy of the error introduced in (3.5) is normalized over the energy of d[n] and the expression is

Ee[n] = ^Xk

n=−k

|d[n] − y[n]|²

|d[n]|²

This parameter is represented in Fig. 3.6, where we note that the energy of the relative error is on the order of 10⁻⁴ and it maintains constant its order of magnitude for both MP and noMP stage.