GPS World, March 2011

Signal Processing INNOVATION tsim t1 t2 t3 t4 Adjust PRNs 03 09 27 Adjust PRNs 08 11 12 27 29 30 32 Adjust PRN 16 tn exp Adjust PRNs 03 08 09 11 12 16 21 24 27 29 30 32 tsi tn exp N i t1 t2 t3 t6 PRN 03 PRN 08 PRN 09 Aggregate t4 t5 t7 FIGURE 4 Fluctuation timing models top Multi SV variant bottom Indiv SV variant and compared to the histogram of the original source data The amplitude of the blocked column and the center of mass are two simple metrics to check a more general way of comparing histograms is the two sided Kolmogorov Smirnov test see Results Timing Models The histograms described in the preceding section specify the amplitude distribution of satellite signals in a given environment but they contain no information about the temporal characteristics of those signals This models used in the current study as well as alternatives that may merit further investigation In real world conditions the temporal characteristics of a given satellite signal depend on many factors including the physical features of the test environment multipath fading and the velocity of the user during data collection Various timing models can be used to simulate those temporal characteristics in laboratory scenarios Perhaps the simplest model is one in intervals This is trivial to implement on the simulator side but it is clearly unlikely to resemble the real world conditions mentioned above A second alternative would be to generate timing intervals based on the Allan or twosample variance of individual C N 0 readings observed during data collection as a measure of the stability of the readings While this is more physically realistic than an arbitrarily chosen interval as described above it is still suggest that a timing model including some measure of randomness may represent a more realistic approach One statistical function commonly used for real world modeling of discrete events radioactive decay customers arriving at a restaurant and so on is the Poisson arrival process This process is completely described with which characterizes the rate at which random events occur Equivalently the time between successive events in such a process is itself a random variable de distribution function p t e t t 0 0 t 0 3 The resulting inter event timings described by this function are strictly non negative which is at least physically reasonable and directly controllable by varying the timing parameter as an initial attempt at temporal modeling and used to generate the results presented in this article nential timing model are considered Multi SV case a single thread deter and the power levels of one or more satellites are adjusted at each event Indiv SV case each simulator channel receives its own individual timing interleaved in constructing the tim variants are shown schematically in FIGURE 4 Constructing Scenarios Once a target histogram is available it is necessary to generate random signal amplitudes for use with a simulator scenario This is done by means of a technique known as the probability integral transform PIT This approach uses the cumulative distribution function or in the discrete case considered on the cumulative mass function of a probability distribution to transform a sequence of uniformly distributed random numbers into a sequence whose distribution matches the target function generated by the PIT process are assigned to individual simulator channels according to a set of timed events as described in the preceding section completing the randomized scenario to be used for testing Results Given a simulator scenario constructed as described above the RUTs originally included in the data collection campaign are again used to conduct acquisition tests this time driven from the simulator tuating scenario properly represents the live data it is necessary to quantify two things how well a generated histogram matches the source data and how well a receivers acquisition performance under simulated signals tatively different problems but a mathematical tool known as the two sided Kolmogorov Smirnov K S test can be used for both tasks Validation of Experimental Setup As C N 0 matches that of the reference data TABLE 1 gives the values of the two sided K S test statistic D a measure of the greatest discrepancy between a sample and the reference distribution for histograms generated with the reference www gpsworld com March 2011 GPS World 47