Patent Publication Number: US-8971921-B2

Title: Method and computer system for obtaining weighted geometric dilution of precision closed form

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method and a computer system for obtaining a weighted geometric dilution of precision (WGDOP) closed form, and more particularly, to a method and a computer system for obtaining a WGDOP closed form via a weighted geometric matrix to position a mobile device. 
     2. Description of the Prior Art 
     In the development of wireless communications and the mobile device, the prior art has provided various wireless position algorithms accompanying a computer system for estimating a position of a mobile device, wherein a weighted geometric dilution of precision (WGDOP) closed form has been widely utilized to position the mobile device. However, during a calculating period of the WGDOP closed form, the computer system is necessary to deal with complicated inverse matrix computations such that more hardware resources of the computer system are occupied and a longer computational period is spent as well. If more base stations are inevitable or the mobile device is continuously moving and the computer system has to correspondingly obtain an accurate computation of the mobile device, the WGDOP closed form has the ability to provide a precise estimated position of the mobile device with more inverse matrix computations, which results in a heavy burden of the computer system and limits the application range of the WGDOP closed form utilized in a complex wireless communication system. 
     SUMMARY OF THE INVENTION 
     A method and a computer system are disclosed for obtaining a weighted geometric dilution of precision (WGDOP) closed form to position a mobile device. 
     According to an aspect of the disclosure, a method, which utilized in a wireless communication system comprising a mobile device and a plurality of base stations, comprises obtaining a geometric matrix according to a plurality of relative distances between the mobile device and the plurality of base stations, obtaining a weighted matrix according to the mobile device and the plurality of base stations, obtaining a weighted geometric dilution of precision according to the weighted matrix and the geometric matrix, so as to obtain a weighted geometric dilution of precision closed form, and choosing a plurality of selected base stations from the plurality of base stations according to the weighted geometric dilution of precision closed form to position the mobile device. 
     According to another aspect of the disclosure, a computer system is provided. The computer system comprises a central processing unit, a detection module coupled to the central processing unit for detecting a plurality of base stations neighboring the computer system, and a storage device coupled to the central processing unit for storing a programming code, and the programming code is utilized to instruct the central processing unit to process a method for a wireless communication system. 
     According to further another aspect of the disclosure, a method utilized in a wireless communication system comprising a mobile device and a plurality of base stations (i=1−n) is disclosed. The method comprises obtaining a multiplied geometric matrix according to a plurality of relative distances between the mobile device and the plurality of base stations, obtaining a plurality of parameters as P=4+p, Q=4+q, M=4+m, N=4+n, a and c, individually dividing the parameters P, Q, M and N by a plurality of diagonal elements k 1 -k 4  of the multiplied geometric matrix, to correspondingly sum results of the division and then multiply by two, so as to form a numerator value, multiplying a first sum of the parameters P, Q, M and N by two to obtain a product, adding the parameters a and c to the product to obtain a second sum, and subtracting 16 from the second sum to form a denominator value, dividing the numerator value by the denominator value and processing a square root operation to obtain a weighted geometric dilution of precision (WGDOP) closed form, and positioning the mobile device according to the WGDOP closed form. 
     These and other objectives of the present invention will no doubt become obvious to those of ordinary skill in the art after reading the following detailed description of the preferred embodiment that is illustrated in the various figures and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a schematic diagram of a computer system according to an embodiment of the invention. 
         FIG. 2  illustrates a flow chart of a positioning process according to an embodiment of the invention. 
         FIG. 3  illustrates a flow chart of a simplification process according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     The specification and the claims of the present invention may use a particular word to indicate an element, which may have diversified names named by distinct manufacturers. The present invention distinguishes the element depending on its function rather than its name. The phrase “comprising” used in the specification and the claims is to mean “is inclusive or open-ended but not exclude additional, un-recited elements or method steps.” In addition, the phrase “electrically connected to” or “coupled” is to mean any electrical connection in a direct manner or an indirect manner. Therefore, the description of “a first device electrically connected or coupled to a second device” is to mean that the first device is connected to the second device directly or by means of connecting through other devices or methods in an indirect manner. 
     Please refer to  FIG. 1 , which illustrates a schematic diagram of a computer system  10  according to an embodiment of the invention. The computer system  10  is utilized for a wireless communication system  12 , and the wireless communication system  12  comprises a plurality of base stations BS 1 -BSn and a mobile device MD. As shown in  FIG. 1 , the computer system  10  comprises a central processing unit  100 , a detection module  102  and a storage module  104 . The central processing unit  100  is coupled to the detection module  102  and the storage module  104 . After the computer system  10  initiates, the detection module  102  correspondingly detects the available plurality of base stations BS 1 -BSn neighboring the computer system  10  via a control signal (not shown in the figure) of the central processing unit  100 , to store information S_IN corresponding to the plurality of base stations BS 1 -BSn, such as parameters related to relative distances or transmission rates between the plurality of base stations BS 1 -BSn and the mobile device MD, in the storage device  104 , such that the central processing unit  100  can be operated as follows. The storage device  104  further comprises a programming code (not shown in the figure) corresponding to a weighted geometric dilution of precision (WGDOP) process, and the central processing unit  100  process the programming code to obtain a WGDOP closed form corresponding to the WGDOP process, so as to position the mobile device MD. 
     Under such circumstances, the wireless communication system  12  of the embodiment utilizes the central processing unit  100  for processing the WGDOP process to obtain a simplified WGDOP and a WGDOP closed form thereof, such that the computer system  10  can select a plurality of selected base stations from the wireless communication system  12  to position the mobile device MD. For example, the embodiment selects four base stations BS 1 -BS 4  from the plurality of base stations for positioning. Additionally, the wireless communication system  12  of the embodiment is utilized in a three-dimensional coordinate system comprising the X-axis, the Y-axis, the Z-axis and a plurality of coordinates. Certainly, the wireless communication system  12  of the embodiment is utilized in a two-dimensional coordinate system comprising the X-axis, the Y-axis and a plurality of coordinates. For detailed descriptions, the embodiment of the invention is focused on the wireless communication system  12  in the three-dimensional coordinate system, and the following assumptions of the X-axis, the Y-axis, the Z-axis and the plurality of coordinates are considered. Of course, those skilled in the art can modify/adjust the utilized coordinate system and the coordinates thereof to comply with different requirements, which is also in the scope of the invention. 
     Preferably, the computer system  10  processing the WGDOP to obtain the WGDOP closed form for positioning the mobile device MD can be derived into a positioning process  20 , as shown in  FIG. 2 . The positioning process  20  includes the steps as follows: 
     Step  200 : Start. 
     Step  202 : Obtaining a geometric matrix H according to a plurality of relative distances of the mobile device MD and the plurality of base stations BS 1 -BSn. 
     Step  204 : Obtaining a weighted matrix W according to the mobile device MD and the plurality of base stations BS 1 -BSn. 
     Step  206 : Obtaining the WGDOP according to the geometric matrix H and the weighted matrix W, to obtain the WGDOP closed form. 
     Step  208 : Selecting the plurality of selected base stations BS 1 -BS 4  from the plurality of base stations BS 1 -BSn according to the WGDOP closed form, to position the mobile device MD. 
     Step  210 : End. 
     The positioning process  20  is compiled as the programming code to be stored in the storage device  104 , and the central processing unit  100  processes the programming code accordingly. In step  202 , the computer system  10  obtains the available base stations BS 1 -BSn and the information S_IN thereof via the detection module  102 , and the related information of the mobile device MD is obtained via the base stations BS 1 -BSn as well. Thus, the computer system  10  has the relative distances r i  between the mobile device MD and the base stations BS 1 -BSn, as shown in equation (1):
 
 r   i =√{square root over (( x−X   i ) 2 +( y−Y   i ) 2 +( z−Z   i ) 2 )}{square root over (( x−X   i ) 2 +( y−Y   i ) 2 +( z−Z   i ) 2 )}{square root over (( x−X   i ) 2 +( y−Y   i ) 2 +( z−Z   i ) 2 )}+ C·t   b   +v   ri   (1),
 
wherein the coordinate (x,y,z) represents the position of mobile device MD, the coordinate (X i ,Y i ,Z i ) represents the position of the i th  base station, C represents the light speed, t b  represents a time offset, and v ri  represents a pseudo-range measurements noise. Further, equation (1) can be linearized by taking Taylor series expansion around an approximate mobile device position ({circumflex over (x)},ŷ,{circumflex over (z)}) and correspondingly neglecting higher order terms, such that equation (2) is obtained:
 
Δ r=r   i   −{circumflex over (r)}   i   ≅e   i1 δ x   +e   i2 δ y   +e   i3 δ z   +C·t   b   +v   ri   (2),
 
wherein (δ x δ y ,δ z ) are coordinate offsets of coordinate (x,y,z), respectively, symbols shown in equation (2) are
 
                 e     i   ⁢           ⁢   1       =         x   ^     -     X   i           r   ^     i         ,     
     ⁢       e     i   ⁢           ⁢   2       =         y   ^     -     Y   i           r   ^     i         ,     
     ⁢       e     i   ⁢           ⁢   3       =         z   ^     -     Z   i           r   ^     i         ,         
and {circumflex over (r)} i =√{square root over (({circumflex over (x)}−X i ) 2 +(ŷ−Y i ) 2 +({circumflex over (z)}−Z i ) 2 )}, and (e i1 ,e i2 ,e i3 ) with i=1, 2, . . . , n can represent the LOS vectors between the mobile device and the base stations. Next, applying z=Hδ+v with
 
               z   =     [             r   1     -       r   ^     1                   r   2     -       r   ^     2               ⋮               r   n     -       r   ^     n             ]       ,     
     ⁢     δ   =     [           δ   x               δ   y               δ   z               c   ·     t   b             ]       ,     
     ⁢     v   =         [           v     r   ⁢           ⁢   1                 v     r   ⁢           ⁢   2               ⋮             v   rn           ]     ⁢           ⁢   to   ⁢           ⁢   have   ⁢           ⁢   H     =     [           e   11           e   12           e   13         1             e   21           e   22           e   23         1           ⋮       ⋮       ⋮       ⋮             e     n   ⁢           ⁢   1             e     n   ⁢           ⁢   2             e     n   ⁢           ⁢   3           1         ]         ,         
equation (3) of the GDOP is obtained:
 
 GDOP =√{square root over ( tr ( H   T   H ) −1 )}  (3).
 
     In step  204 , considering current positions (circumstance parameters) of the base stations BS 1 -BSn as well as estimated computational accuracy (individual parameters) of each of the base stations or processing the positioning process  20  in different wireless communication systems, a covariance as σ i   2  is obtained while estimating the relative distance r i  between the mobile device MD and the base stations BS 1 -BSn. Accordingly, every base station has different covariances to obtain the weighted matrix W, as shown in equation (4): 
     
       
         
           
             
               
                 
                   W 
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                               1 
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                                 σ 
                                 1 
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     In step  206 , equation (4) and equation (3) are combined to rewrite the equation for the WGDOP, as shown in equation (5):
 
 WGDOP =√{square root over ( tr ( H   T   WH ) −1 )}  (5).
 
Under such circumstances, the WGDOP processes a matrix computation operation for the geometric matrix H and the weighted matrix W, and accordingly, the WGDOP closed form is obtained.
 
     In step  208 , the computer system  10  selects four selected base stations BS 1 -BS 4  from the base stations BS 1 -BSn of the wireless communication system  12  to position the mobile device MD. In detail, the computer system  10  utilizes the related information S_IN of the base stations BS 1 -BSn and the WGDOP closed form to obtain a plurality of computational results, and the computational results corresponding to the base stations BS 1 -BSn are sorted to select the smallest four values as well as the base stations thereof for positioning the mobile device MD. In comparison with the conventional GDOP closed form, the WGDOP closed form is operated with less complicated computations to lessen computational burdens of the computer system  10 . Details of the simplified computational processes can be understood as follows. 
     In the positioning process  20 , step  206  as obtaining the WGDOP according to the geometric matrix H and the weighted matrix W to obtain the WGDOP closed form can also be derived into a simplification process  30 , as shown in  FIG. 3 . The simplification process  30  includes the steps as follows: 
     Step  300 : Start. 
     Step  302 : Processing a transposition operation for the geometric matrix H to obtain a transposed geometric matrix H T . 
     Step  304 : Sequentially processing a matrix multiplication operation for the geometric matrix H, the transposed geometric matrix H T  and the weighted matrix W to obtain a multiplied geometric matrix M. 
     Step  306 : Processing a trace computation operation for the multiplied geometric matrix M to obtain an adjoint matrix of a plurality of multiplied geometric matrix elements on a principal diagonal of the multiplied geometric matrix M and a determinant value of the multiplied geometric matrix M, so as to process a deflation operation for obtaining the WGDOP closed form corresponding to the WGDOP. 
     Step  308 : End. 
     In step  302 , the geometric matrix H is processed via the transposition operation to obtain the transposed geometric matrix H T , and the transposition operation should be well known to those skilled in the art, which means each of the matrix elements of the geometric matrix H is relocated from row to column as well as from column to row, and is not described herein for brevity. 
     In step  304 , the geometric matrix H, the transposed geometric matrix H T  and the weighted matrix W are sequentially processed via the matrix multiplication operation, wherein the matrix multiplication operation is 
                 HH   T     ⁢   W     =         [           e   11           e   12           e   13         1             e   21           e   22           e   23         1             e   31           e   32           e   33         1             e   41           e   42           e   43         1         ]     ⁡     [           e   11           e   21           e   31           e   41               e   12           e   22           e   32           e   42               e   13           e   23           e   33           e   43             1       1       1       1         ]       ⁡     [           k   1         0       0       0           0         k   2         0       0           0       0         k   3         0           0       0       0         k   4           ]             
and the multiplied geometric matrix is
 
             M   =     [           2   ⁢           ⁢     k   1               k   2     ⁢     B   12               k   3     ⁢     B   13               k   4     ⁢     B   14                   k   1     ⁢     B   12             2   ⁢           ⁢     k   2               k   3     ⁢     B   23               k   4     ⁢     B   24                   k   1     ⁢     B   13               k   2     ⁢     B   23             2   ⁢           ⁢     k   3               k   4     ⁢     B   34                   k   1     ⁢     B   14               k   2     ⁢     B   24               k   3     ⁢     B   34             2   ⁢           ⁢     k   4             ]           
with B ij =e i1 e j1 +e i2 e j2 +e i3 e j3 +1, 1i≦i&lt;j≦4. Preferably, the embodiment utilizes the matrix computation to rewrite the WGDOP as WGDOP=√{square root over (tr(H T WH) −1 )}=√{square root over (tr(HH T W) −1 )}=√{square root over (tr(M) −1 )} before processing step  306 , which means that the WGDOP can be rewritten as the multiplication geometric matrix for obtaining a simplified WGDOP as S_WGDOP=√{square root over (tr(M) −1 )}.
 
     In step  306 , the multiplied geometric matrix M is processed via the trace computation operation as 
                   tr   ⁡     (   M   )         -   1       =         ∑     i   =   1     4     ⁢           ⁢       (   M   )       i   ,   i       -   1         =         tr   ⁡     [     adj   ⁡     (   M   )       ]         det   ⁢           ⁢     (   M   )         =       ∑     i   =   1     4     ⁢           ⁢         cof     i   ,   i       ⁡     (   M   )         det   ⁢           ⁢     (   M   )                 ,         
to simplify the adjoint matrix adj(M) or the cofactor cof i,j (M) of the plurality of multiplied geometric matrix elements on the principal diagonal of the multiplied geometric matrix M and the determinant value det(M) of the multiplied geometric matrix M, so as to process the deflation operation, wherein the cofactor on the i th  row and i th  column are shown as
 
cof 1,1 (HH T W)=k 2 k 3 k 4 [8+2(B 23 B 24 B 34 −(B 23   2 +B 24   2 +B 34   2 ))],
 
cof 2,2 (HH T W)=k 1 k 3 k 4 [8+2(B 13 B 14 B 34 −(B 13   2 +B 14   2 +B 34   2 ))],
 
cof 3,3 (HH T W)=k 1 k 2 k 4 [8+2(B 12 B 14 B 24 −(B 12   2 +B 14   2 +B 24   2 ))] and
 
cof 4,4 (HH T W)=k 1 k 2 k 3 [8+2(B 12 B 13 B 23 −(B 12   2 +B 13   2 +B 23   2 ))], and the determinant value of the multiplied geometric matrix M is
 
               det   ⁢           ⁢     (       HH   T     ⁢   W     )       =       k   1     ⁢     k   2     ⁢     k   3     ⁢     k   4     ⁢       {     16   +     2   ⁡     [         B   23     ⁢     B   24     ⁢     B   34       -     (       B   23   2     +     B   24   2     +     B   34   2       )       ]       +     2   ⁡     [         B   13     ⁢     B   14     ⁢     B   34       -     (       B   13   2     +     B   14   2     +     B   34   2       )       ]       +     2   ⁡     [         B   12     ⁢     B   14     ⁢     B   24       -     (       B   12   2     +     B   14   2     +     B   24   2       )       ]       +     2   ⁡     [         B   12     ⁢     B   13     ⁢     B   23       -     (       B   12   2     +     B   13   2     +     B   23   2       )       ]       +       (         B   12     ⁢     B   34       +       B   13     ⁢     B   24       -       B   14     ⁢     B   23         )     2     -     4   ⁢           ⁢     B   12     ⁢     B   34     ⁢     B   13     ⁢     B   24       +     2   ⁡     [         B   12     ⁡     (         B   13     ⁢     B   23       +     B   14     +     B   24       )       +       B   34     ⁡     (         B   13     ⁢     B   14       +       B   23     ⁢     B   24         )         ]         }     .             
Accordingly, the simplified WGDOP can be represented with the cofactor on the i th  row and i th  column after the deflation operation and the determinant value of the multiplied geometric matrix M. Preferably, a plurality of parameters as a, c, p, q, m and n can be obtained as
 
a=(B 12 B 34 +B 13 B 24 −B 14 B 23 ) 2 −4B 12 B 34 B 13 B 24 , c=2[B 12 (B 13 B 23 +B 14 B 24 )+B 34 (B 13 B 14 +B 23 B 24 )],
 
p=[B 23 B 24 B 34 −(B 23   2 +B 24   2 +B 34   2 )], q=[(B 13 +B 14 +B 34 −(B 13   2 +B 14   2 +B 34   2 )],
 
m=[B 12 B 14 B 24 −(B 12   2 +B 14   2 +B 24   2 )] and n=[B 12 B 13 B 23 −(B 12   2 +B 13   2 +B 23   2 )], and a WGDOP closed form corresponding to the simplified WGDOP is obtained as
 
             S_WGDOP   =           2   ·     [         1     k   1       ·     (     4   +   p     )       +       1     k   2       ·     (     4   +   q     )       +       1     k   3       ·     (     4   +   m     )       +       1     k   4       ·     (     4   +   n     )         ]         a   +   c   -   16   +     2   ·     [       (     4   +   p     )     +     (     4   +   q     )     +     (     4   +   m     )     +     (     4   +   n     )       ]             .           
If parameters can be rewritten as P=4+p, Q=4+q, M=4+m and N=4+n, another WGDOP closed form is obtained as
 
               S_WGDOP   =         2   ·     (         1     k   1       ·   P     +       1     k   2       ·   Q     +       1     k   3       ·   M     +       1     k   4       ·   N       )         a   +   c   -   16   +     2   ·     (     P   +   Q   +   M   +   N     )               ,         
accordingly.
 
     In the embodiment of the invention, the computer system  10  utilizes the central processing unit  100  to process the programming codes corresponding to the positioning process  20  and the simplification process  30  for obtaining the WGDOP closed form. Accordingly, the simplified WGDOP closed form can also be stored in the storage device  104 . Preferably, the computer system  10  in the embodiment can continuously process an update operation for simplifying the WGDOP closed form within a predetermined period to cooperate with the continuously moving mobile device, i.e. the mobile device is not fixed and will continuously search for the appropriate base station to process the positioning operation as well as the wireless transmission process. Noticeably, the WGDOP is obtained from a combination of the GDOP and the weighted matrix, and certainly, those skilled in the art can simultaneously store the programming codes corresponding to the WGDOP of the embodiment and the GDOP of the prior art, such that both programming codes can be integrated for positioning the mobile device or either one of the programming codes can be adaptively selected by the user according to practical requirements in order to provide more accurate positioning efficiency as well as shorter calculating periods, which is also in the scope of the invention. Accordingly, the embodiment of the invention selects the four selected base stations BS 1 -BS 4  from the plurality of base stations BS 1 -BSn in the wireless communication system  12  to position the mobile device MD, and those skilled in the art can adaptively select a particular number of the selected base stations for positioning according to different requirements, which is not limiting the scope of the invention. 
     For example, if the user wants to select four base stations as a base station set from the N number of base stations to position the mobile device MD, the calculation of C 4   N  becomes more complex when the N number of the base stations is larger, which means that the computer system  10  requires longer calculating periods as well as more hardware resources for processing the programming codes of the positioning process  20  as well as the simplification process  30 . Thus, by processing the WGDOP closed form of the embodiment, it can effectively reduce the complex calculation of the computer system  10  or the stored programming code therein to correspondingly improve the application range of the computer system  10 . 
     Noticeably, when the computer system  10  detects that the wireless communication system  12  is in the three-dimensional coordinate system and the plurality of covariances corresponding to the plurality of base stations are different, the embodiment of the invention renders the WGDOP closed form to comprise 45 multiplication computations, 49 addition computations, a division computation and a square root computation. When the computer system  10  detects that the wireless communication system  12  is in the two-dimensional coordinate system and the plurality of covariances corresponding to the plurality of base stations are different, the embodiment of the invention renders the WGDOP closed form to comprise 39 multiplication computations, 43 addition computations, a division computation and a square root computation. In other words, the WGDOP closed form of the embodiment only comprises four fundamental computations of arithmetic (i.e. the addition, the subtraction, the multiplication and the division), and the calculating burden of the computer system  10  can be efficiently reduced in comparison with the prior art of the GDOP, which utilizes a much more complicated inverse matrix calculation. 
     From a different perspective, the above descriptions can be summarized as a positioning method for a wireless communication system comprising a mobile device and a plurality of base stations (i=1-n). The method comprises the following steps. A multiplied geometric matrix M is obtained according to a plurality of relative distances between the mobile device and the plurality of base stations. A plurality of parameters are obtained as P=4+p, Q=4+q, M=4+m, N=4+n, a and c. The parameters P, Q, M and N are individually divided by a plurality of diagonal elements k1-k4 of the multiplied geometric matrix, to correspondingly sum results of the division and then multiply by two, so as to form a numerator value. A first sum of the parameters P, Q, M and N are multiplied by two to obtain a product, the parameters a and c are added to the product to obtain a second sum, and 16 is subtracted from the second sum to form a denominator value. The numerator value is divided by the denominator value and a square root operation is processed to obtain a weighted geometric dilution of precision (WGDOP) closed form as 
             S_WGDOP   =           2   ·     (         1     k   1       ·   P     +       1     k   2       ·   Q     +       1     k   3       ·   M     +       1     k   4       ·   N       )         a   +   c   -   16   +     2   ·     (     P   +   Q   +   M   +   N     )             .           
Lastly, the mobile device is positioned according to the WGDOP closed form.
 
     Moreover, when one of the plurality of base stations is a serving base station (such as the base station BS 1 ), a covariance of the serving base station corresponds to a first value, and covariances of other base stations of the plurality of base stations (such as the base stations BS 2 -BSn) correspond to a second value, which means that there are only two covariances while obtaining the relative distances r i  of the base stations BS 1 -BSn. Under such circumstances, when the computer system  10  detects that the wireless communication system  12  is in the three-dimensional coordinate system, the embodiment of the invention renders the WGDOP closed form to comprise 42 multiplication computations, 45 addition computations, a division computation and a square root computation. When the computer system  10  detects that the wireless communication system  12  is in the two-dimensional coordinate system, the embodiment of the invention renders the WGDOP closed form to comprise 36 multiplication computations, 39 addition computations, a division computation and a square root computation. In other words, the WGDOP closed form of the embodiment only comprises four fundamental computations of arithmetic (i.e. the addition, the subtraction, the multiplication and the division), and the calculating burden of the computer system  10  can be efficiently reduced. 
     Moreover, the computer system  10 , the positioning process  20  and the simplification process  30  of the embodiment can be utilized or cooperated with other algorithms or related hardware devices to be applied to a global positioning system (GPS), a wireless sensor network (WSN) or a femtocell, which is not limiting the scope of the invention. Additionally, the mentioned positioning process  20  and the simplification process  30  can be realized via different embodiments. In one embodiment, a non-transitory computer readable recording medium is utilized for instructing a processor (i.e. a central processing unit) to process the positioning process  20  and the simplification process  30  shown in  FIG. 2  and  FIG. 3 . Also, the non-transitory computer readable recording medium can be in the form of ROM, flash memory, floppy disk, hard disk, CD, USB, magnetic tape, a database accessible by the network or any other similar storage medium familiar with those skilled in the art. In other embodiments, the positioning process  20  and the simplification process  30  shown in  FIG. 2  and  FIG. 3  can also be realized in a form of a computer programming product, which can be functionally operated once a computer has installed the computer programming product to process the plurality of instructions, so as to process the positioning process  20  and the simplification process  30  as mentioned. Preferably, the computer programming product can be stored inside a non-transitory computer readable recording medium or transmitted via the network, which is also in the scope of the invention. 
     In summary, the embodiments of the invention provide a method and a computer system to obtain a weighted geometric dilution of precision (WGDOP) closed form, so as to simplify a calculating process of the WGDOP for positioning a mobile device. As can be seen, the WGDOP of the embodiments of the invention only comprises the four fundamental computations of arithmetic (i.e. the addition, the subtraction, the multiplication and the division). In comparison with the prior art processing the complicated inverse matrix calculation, the embodiments of the invention has efficiently reduced the calculating burden/complexity of the computer system, and programming codes corresponding to the WGDOP as well as the GDOP can be adaptively switched by the user according to different requirements, which means that the embodiments of the invention can be utilized in the conditions as the continuously moving mobile device or the fixed mobile device. A great number of the base stations can also be inputted into the computer system of the embodiments of the invention without complicated inverse matrix calculation for positioning the mobile device, which can significantly improve the application range of the method and the computer system rendered in the embodiments of the invention. 
     Those skilled in the art will readily observe that numerous modifications and alterations of the device and method may be made while retaining the teachings of the invention. Accordingly, the above disclosure should be construed as limited only by the metes and bounds of the appended claims.