a. Using [Ala06b] model, determine the mean and variance of TOA-based ranging error for distances of 3, 5, and 15 m if we have a 1 GHz bandwidth and a LOS condition with no UDP condition.
b. Repeat (a) for bandwidths of 100, 10, and 1 MHz at a distance of 5 m.
c. Repeat (a) and (b) when we have the UDP conditions with the same statistics as in the model.
Let us suppose that there are N estimates di of the distance of a MS from N known locations with coordinates (xi,yi) for i = 1, 2, 3, . . . , N. We then have N equations of the form: fi(x, y) = (xi − x) 2 + (yi − y) 2 − d2 i = 0 where (x, y) is the unknown location of the MS. The least squares technique provides a method of estimating x and y when there are errors in the estimates di. The technique works as follows. Let F = [f1(x, y) f2(x, y) ... fN(x, y)]T. First, we have to construct the Jacobian matrix given by:
Next we pick an estimate of the solution U = [x∗ y∗] and we determine the error in the solution as E = – (JTJ) −1JT F evaluated at the estimate U. The new solution is U + E. Iteratively, the error in the solution is reduced by computing a new error that is added to the previous solution to obtain a new solution till a point is reached where the solution does not change. Let the known locations of reference points be (10, 10), (0, 15), and (–5, 5). The measured distances from these reference points are 15, 16, and 5 m. Use the least squares approach to determine the estimate of the location by using the location (2, 2) as the initial estimate.