As shown in Section 3, we use the objective function FHNGPS-CA(xms, yms) in (5) rather than the LS obcetive function (FLS(xms, yms)) shown in (6). To verify this choice, let us look at the objective functions FHNGPS-CA(xms, yms) and FLS(xms, yms).
For sake of simplicity, from now on we will drop the argument (xms, yms) in FLS(xms, yms) and FHNGPS-CA(xms, yms). Thus, we can write (6) as follows
(12)
where
From the definition of l
i
in (3), the function f
i
can be simplified to
(13)
Let us recall that we intend to minimize η
i
which always has a nonnegative value, i.e., η
i
≥ 0. Also, η
i
is actually a distance and function of the estimated MS coordinates , where {^} represents the estimated value of the variable, i.e., for example is the estimated value of v. Since η
i
is a positive error which increases as the estimated MS coordinates get further from the actual MS coordinates (xms, yms), then η
i
is a convex surface which has a minima (zero value) at (xms, yms). Now, because μ
i
is a noise signal it can have both negative and positive values. If it has a positive value then the surface η
i
+ μ
i
has a value which is greater than zero, i.e., η
i
+ μ
i
> 0 and has a minima at (xms, yms). Thus, f
i
= (η
i
+ μ
i
)2 will be convex in this case and FLS which is the summation of positively weighted convex functions (f
i
) for i = 1, 2, 3 will be convex in this case as well [9]. Whereas, if μ
i
is negative, i.e., μ
i
< 0, then parts of the surface η
i
+ μ
i
will be below zero and other parts will be above zero with the minimum located at (xms, yms) to be below zero. When taking the square of η
i
+ μ
i
, i.e., (η
i
+ μ
i
)2, then, the point (xms, yms) will have a local maxima and the surface surrounding it will be decreasing until it hits the zero level. Then, the surface (η
i
+ μ
i
)2 is reflected back to the increasing mode as we get further from the point (xms, yms), causing (η
i
+ μ
i
)2 to be a non-convex surface, i.e., f
i
will be non-convex in this case, causing FLS which is the summation of positively weighted non-convex functions (f
i
) for i = 1, 2, 3 will be non-convex in this case [9]. This is illustrated in Figure 2. Thus, when minimizing FLS we will reach an unwanted local minimum which is not located at (xms, yms). So, there will be an error in locating the MS.
Similarly, FHNGPS-CA in (5) can be written as follows
(14)
where
From the definition of in (4), the function g
i
can be simplified to
(15)
Let us introduce the variable h
i
which is defined as
Thus,
Because R
s
is the distance between the MS and the satellite which is much greater than zero, i.e., R
s
>> 0, and because R
s
>> μ
i
and R
s
>> μ
s
, then h
i
>> 0 for all realistic values of μ
i
and μ
s
. Thus, the function g
i
is the square of the summation of a positive surface (η
i
) which has a minimum value of zero at (xms, yms) and a positive variable (h
i
). So, g
i
is always a convex function and has a minimum value at (xms, yms). This is illustrated in Figure 3. Thus, FHNGPS-CA is convex as well because the positively weighted sum of convex functions is also convex [9] (recall that ). This will assure that when minimizing FHNGPS-CA we will not reach an unwanted local minimum as the case for minimizing FLS, but rather we will reach the true MS location (xms, yms).