Patent Publication Number: US-2009231192-A1

Title: Method and system for generating temporary ephemeris

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS/INCORPORATION BY REFERENCE 
     Not applicable 
     FIELD OF THE INVENTION 
     Certain embodiments of the invention relate to signal processing for satellite navigation systems. More specifically, certain embodiments of the invention relate to a method and system for generating temporary ephemeris. 
     BACKGROUND OF THE INVENTION 
     Global Navigation satellite systems (GNSS) receivers may normally determine their position by receiving satellite broadcast signals from a plurality of satellites. These satellites, for example 24 at any time for the Global Positioning System (GPS), may broadcast radio frequency signals that comprise information that may be exploited by the satellite receiver to determine its own position. By measuring the time the broadcast signals may travel from the satellites to the satellite receiver, and the known position of the transmitting satellite, the satellite receiver may be able to determine its own position by trilateration. In general, at least 3 satellite signals may need to be decoded at the satellite receiver in order to determine its position. 
     Further limitations and disadvantages of conventional and traditional approaches will become apparent to one of skill in the art, through comparison of such systems with some aspects of the present invention as set forth in the remainder of the present application with reference to the drawings. 
     BRIEF SUMMARY OF THE INVENTION 
     A method and/or system for generating temporary ephemeris, substantially as shown in and/or described in connection with at least one of the figures, as set forth more completely in the claims. 
     These and other advantages, aspects and novel features of the present invention, as well as details of an illustrated embodiment thereof, will be more fully understood from the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1  is a diagram illustrating an exemplary satellite navigation system, in accordance with an embodiment of the invention. 
         FIG. 2  is a diagram illustrating an exemplary satellite navigation system in a two-dimensional setting, in accordance with an embodiment of the invention. 
         FIG. 3  is a diagram illustrating an exemplary satellite navigation system in a two-dimensional setting with a timing offset, in accordance with an embodiment of the invention. 
         FIG. 3B  is a diagram illustrating an exemplary satellite positioning, in accordance with an embodiment of the invention. 
         FIG. 4  is a diagram illustrating an exemplary construction of temporary ephemeris based on a satellite&#39;s almanac position. 
         FIG. 5  is a diagram illustrating a prediction error, in accordance with an embodiment of the invention. 
         FIG. 6  is a diagram illustrating temporary ephemeris without the use of an almanac position, in accordance with an embodiment of the invention. 
         FIG. 7  is a flowchart illustrating an exemplary algorithm to compute a satellite receiver position in accordance with an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Certain embodiments of the invention may be found in a method and system for generating temporary ephemeris. Aspects of a method and system for generating temporary ephemeris may comprise determining one or more positions of a satellite receiver based on a plurality of satellite signals received from a plurality of satellites for which complete ephemeris data has been received at the satellite receiver. Temporary ephemeris data may be generated from the determined one or more positions of the satellite receiver and one or more satellite signals from one or more satellites with incomplete ephemeris data. One or more estimated positions of the satellite receiver may be determined based on the generated temporary ephemeris and a second plurality of satellite signals, wherein at least one of the second plurality of satellite signals is associated with the one or more satellites with incomplete ephemeris data. 
     The temporary ephemeris data may be generated by generating a translated satellite position and a rate of change of an associated receiver clock corrected pseudorange. The translated satellite position may be generated based on an almanac satellite position, or at a position determined by said receiver clock corrected pseudorange and/or said associated rate of change of said receiver clock corrected pseudorange. The rate of change of said associated receiver clock corrected pseudorange may be generated based on one or more Doppler frequency measurement of said plurality of satellite signals and/or said second plurality of satellite signals, or based on a plurality of pseudorange measurements. The receiver clock corrected pseudorange may be determined from at least a receiver clock offset and a pseudorange measurement. When the satellite receiver may receive complete ephemeris data from the one or more satellites from which the incomplete ephemeris data was received by the satellite receiver, utilizing this complete ephemeris data and the determined one or more positions to determine a location of the satellite receiver. 
       FIG. 1  is a diagram illustrating an exemplary satellite navigation system, in accordance with an embodiment of the invention. Referring to  FIG. 1 , there is shown a satellite navigation system  100 , comprising a satellite receiver  102  and a plurality of satellites, of which satellites  110   a ,  110   b ,  110   c ,  110   d  may be illustrated. The satellite receiver  102  may be communicatively coupled to a receiver antenna  112 . The satellite receiver  102  may comprise a Global Navigation Satellite System (GNSS) radio-frequency (RF) and intermediate-frequency (IF) front-end  104 , a processor  106  and memory  108 . 
     The satellites  110   a  through  110   d  may comprise suitable logic, circuitry and/or code that may be enabled to generate and broadcast suitable radio-frequency signals that may be received by a satellite receiver, for example satellite receiver  102 , to determine the satellite receiver  102  position. The satellite receiver  102  may comprise suitable logic, circuitry and/or code that may be enabled to receive signals broadcasted from satellites, for example satellites  110   a  through  110   d , and process the received signals to determine the position of the satellite receiver  102 . The GNSS RF/IF front-end  104  may comprise suitable logic, circuitry and/or code that may be enabled to receive satellite broadcast signals via receiver antenna  112  and process them in a desirable fashion to generate baseband signals, which may be suitable for further processing in the satellite receiver  102  and the processor  106 . The memory  108  may comprise suitable logic, circuitry and/or code that may enable storage and access to data and code suitable for the operations performed by the satellite receiver  102  and the processor  106 . 
     In  FIG. 1 , an exemplary satellite navigation scenario may be illustrated, wherein a satellite receiver  102  may receive a plurality of satellite signal from which the satellite receiver  102  may be able to extract information that may enable the satellite receiver to determine its position. The satellite receiver  102  and the satellites, for example satellites  110   a  through  110   d , may be operating in compliance with the Global Positioning System (GPS) developed and operated by the United States of America Department of Defense. In accordance with various embodiments of the invention, the invention may not be limited to application in GPS and may be applied to other GNSS systems, for example GALILEO, GLONASS, IRNSS, and BEIDOU. 
       FIG. 2  is a diagram illustrating an exemplary satellite navigation system in a two-dimensional setting, in accordance with an embodiment of the invention. Referring to  FIG. 2 , there is shown a satellite navigation system  200 , comprising a receiver  202  (illustrated by a small circle) at position p, satellites  210   a  and  210   b , an earth surface  203  illustrated by a dotted circle, and an exemplary coordinate system  206 . The exemplary coordinate system  206  may be a two-dimensional coordinate system in    2 , spanned by the unit vectors x and y, and where   may denote the set of real numbers and    n  denotes a n-dimensional space with real coordinates. There is also shown a position of satellite  210   a  in    2  denoted p( 210   a ), a position of satellite  210   b  denoted p( 210   b ), an intersection point q, a range from satellite  210   a  to the satellite receiver  202  r( 210   a ) and a range from satellite  210   b  to the satellite receiver  202  r( 210   b ). 
     To illustrate the principles involved in determining a position of the receiver  202  from the satellites, for example the satellites  210   a  and  210   b , it may be useful to consider a two-dimensional scenario as illustrated in  FIG. 2 . The three-dimensional case encountered in reality may be considered an extension to three dimensions of the principles demonstrated in the two-dimensional case. As illustrated in  FIG. 2 , the principle of determining the position p of the satellite receiver  202  may be to measure the range from the satellite receiver  202  to a plurality of satellites, for example r( 210   a ) and r( 210   b ), based on the known positions of the satellites, for example p( 210   a ) and p( 210   b ). Based on the measured ranges from the satellites  210   a  and  210   b  to the satellite receiver  202  and the known position of the satellites, each satellite may define a circle (in    2 ) of positions that lie at a given range from the satellite, as illustrated in  FIG. 2 . In the case of two satellites, there may be two intersection points: one may be the desired position p and the other may be the intersection q. As may be observed from  FIG. 2 , only p may be close to the surface of the earth. Hence, only p may be a feasible solution for the position of the satellite receiver  202 . Therefore, in the depicted two-dimensional scenario of  FIG. 2 , two satellites may suffice in principle to determine the position p. The position p may be given by one solution to the following relationships in the two-dimensional case: 
         r ( k )=∥ p ( k )− p ∥, k= 210   a , 210   b    
     In three dimensions, the circles around the satellites may become spheres and the intersection of two spheres may generate a circle of feasible solutions. By intersecting the circle with a further sphere, two possible positions will be found. Again, only one of the two solutions will be close to the surface of the earth. Therefore, in the three dimensional case, the solution may require 1 more satellite to resolve the extra dimension and the position may be resolved from the following relationship, where each k may denote a different satellite: 
         r ( k )=∥ p ( k )− p ∥, k=1, 2, 3 
     Each satellite, for example satellites  210   a  and  210   b , may broadcast a signal that may comprise information to determine the satellite&#39;s position. Once placed in orbit, a satellite&#39;s position may be predictable. This predicted position of the satellites may generally be available in an almanac at the satellite receiver and may be stored, for example, in the memory  108 . Due to certain imperfections in computing the satellite&#39;s position, a GPS ground station may monitor the satellite&#39;s exact position. In order to correct for any deviations from the almanac position, the ground station may supply the satellite with data that may allow the satellite&#39;s position to be determined to a high degree of accuracy when received by a satellite receiver. This data may be valid for a limited time only and may be referred to as ephemeris data. Its ephemeris data may be broadcast by each satellite, and may be received by the satellite receiver. The satellite position p(k,t) of satellite k, may be computed using the ephemeris data. The almanac position P(k,t) of a given satellite k may hence be related to the position p(k,t) together with a correction term Δ(k,t) from the following relationship: 
         p ( k,t )= P ( k,t )+Δ( k,t ) 
     where the variable t may denote time and indicate that the position of the satellite may change as a function of time. In accordance with various embodiments of the invention, if the correction term Δ(k,t) may be available at a satellite receiver, for example satellite receiver  202 , the exact position of the satellite k may be determined to a high degree of accuracy. The range r(k) from a satellite k to the satellite receiver  202  may be calculated by measuring the time it may take a signal to travel from the satellite to the satellite receiver  202 . The signal may be transmitted by a satellite at a pre-determined time. Each satellite may broadcast its signal comprising various information by using a spreading code unique to each satellite. The spreading codes may be known by the satellite receiver and used to correlate the received signal with a local replica of the spreading code to establish the time it took the broadcast signal to arrive at the satellite receiver  202 . Since the satellite receiver  202  clock may initially not be synchronized to the satellites clock, the measurement will only be accurate up to an additive constant b. The additive constant b may be the same for all satellites. 
       FIG. 3  is a diagram illustrating an exemplary satellite navigation system in a two-dimensional setting with a common offset, in accordance with an embodiment of the invention. Referring to  FIG. 3 , there is shown a satellite navigation system  300 , comprising a receiver  302  (illustrated by a small circle) at position p, satellites  310   a ,  310   b , and  310   c , and an exemplary coordinate system  306 . The exemplary coordinate system  306  may be a two-dimensional coordinate system in    2 , spanned by the unit vectors x and y, and where   may denote the set of real numbers and    n  denotes a n-dimensional space with real coordinates. There is also shown a position of satellite  310   a  in    2  denoted p( 310   a ), a position of satellite  310   b  denoted p( 310   b ), a position of satellite  310   c  denoted p( 310   c ), a range from satellite  310   a  to the satellite receiver  302  r( 310   a ), a range from satellite  310   b  to the satellite receiver  302  r( 310   b ), and a range from satellite  310   c  to the satellite receiver  302  r( 310   b ). The 3 ranges may be illustrated with a solid circle. There is also shown a bold dashed-line circle corresponding to each r( 310   a )+b and r( 310   b )+b, and a bold dot-dashed-line circle corresponding to r( 310   c )+b. 
     As for  FIG. 2 , it may be useful to consider a two-dimensional scenario. The three-dimensional case encountered in reality may be considered a simple extension to three dimensions of the principles demonstrated in the two-dimensional case. 
     As illustrated in  FIG. 2 , the 2 satellites may suffice to determine the position b when the clocks may be synchronized. In cases where the clocks between the satellites and the satellite receiver  302  may not be synchronized, an additional variable b may need to be solved for, requiring an additional satellite. From the bold dashed lines of the offset range circles corresponding to the satellites  310   a  and  310   b , one may observe that the intersection of the two circles may generate 2 intersection points, similar to  FIG. 2 . However, the third satellite, satellite  310   c  illustrated by the dot-dashed line may not intersect at one of the intersection points of the range plus offset circles generated from the satellites  310   a  and  310   b . However, if the offset b may be removed, as shown by the solid circles associated with the satellites  310   a ,  310   b  and  310   c , all three circles may intersect at position p. Hence, using an extra satellite, the common offset b may be determined by ensuring that at least 3 satellites may intersect at a common point. 
     This same observation may be carried over into three dimensions and using 4 satellites, it may thus be possible to resolve the offset b. Accordingly, the position and the offset b in 3 dimensions may be resolved from 4 satellites by using the following relationships: 
         r ( k )+ b=∥p ( k )− p∥+b , k=1, 2, 3, 4 
     In a GPS system, the spreading codes that may be used to determine the range through correlation with a local replica of the correlation code, may be referred to as C/A codes. The C/A codes may be unique to each satellite, and may hence identify a received signal. A C/A code may comprise 1023 chips and may take 1 ms to send. For each bit, 20 repetitions of the C/A code may be transmitted by the satellite, for example satellite  310   a , over the duration of 20 ms. In addition, 30 bits may make one word, 10 words may make a subframe (6 s), 5 subframes may make one page, and 25 pages may make a superframe (12.5 min). In most instances, it may take at least the first three subframes of a transmission to generate a position. In those instances, at least the first 18 seconds of data may need to be received. 
     As mentioned above, the range r(k) may be determined from measuring the time it may take for the transmission to travel from the satellite, for example satellite  310   a , to the satellite receiver  302 . Ideally, the clocks of the satellite  310   a  and the satellite receiver  302  may be synchronized and the travel time τ(k) may be determined by correlation as described above. In that case, the range may be given by r(k)=cτ(k), where c may be the speed of light. In practice, however, the satellite time and the receiver time may not be synchronized. Although the satellites may comprise an atomic clock, there may be some drift and/or offset that may need to be compensated. If GPS time (or absolute time) may be time t, this may describe the satellite time t s (k,t) of satellite k by the following relationship: 
         t   s ( k,t )= t+δ   s ( k,t ) 
     The offset δ s (k,t) may be estimated by a GPS ground control station and the coefficients of its Taylor expansion may be transmitted as part of the satellite data. Similarly, the receiver clock (user time) t u  may be described by the following relationship: 
         t   u ( t )= t+δ   u ( t ) 
     In accordance with various embodiments of the invention, δ s (k,t) and δ u (t) may comprise one or more delay terms, for example atmospheric delays, RF delays, and/or processing delays. A new frame may be sent at a pre-determined time t s (k,t) and may be detected at the satellite receiver, for example satellite receiver  302 , at time {circumflex over (t)}(k)=t+τ(k). The pseudorange to a particular satellite may be defined by the following relationship: 
       ρ( k,{circumflex over (t)} )= c[t   u ( {circumflex over (t)} )− t   s ( k,{circumflex over (t)} −τ( k ))] 
     which may essentially be the distance computed from the receiver time minus the satellite time. Since the receiver knows t u ({circumflex over (t)}) and it may know t s (k,t)=t s (k,{circumflex over (t)}−τ), the satellite receiver  302  may compute the pseudorange. Substituting the definitions of the satellite time t s (k,t) and the satellite receiver time t u (t) from the above equations, the following relationships may be obtained. 
       ρ( k,{circumflex over (t)} )= c[{circumflex over (t)}+δ   u ( {circumflex over (t)})−{circumflex over (t)} +τ( k )−δ s ( k,{circumflex over (t)}− τ)] 
       ρ( k,{circumflex over (t)} )= c τ( k )+ c[δ   u ( {circumflex over (t)} )−δ s ( k,t )] 
     Since the satellite receiver  302  may receive δ s (k,t) from the satellite, for example satellite receiver  310   a , the clock offset term δ s (k,t) may be corrected. The corrected pseudorange may then be given by the following relationship: 
       ρ c ( k,{circumflex over (t)} )= c τ( k )+ cδ   u ( {circumflex over (t)} ) 
     By denoting the offset b=cδ u ({circumflex over (t)}) and recalling that the range may be defined as r(k)=∥p(k)−p∥=cτ(k), a set of equations may be given by the following relationship: 
       ρ c ( k,{circumflex over (t)} ( k ))− b=r ( k )=∥ p ( k )− p ∥ k=1, 2, 3, 4 
     These equations may be four equations in 4 unknowns (3 coordinates of p and the offset b). The satellite may broadcast positional correction data in what may be referred to as ephemeris. Ephemeris may comprise the position correction factor Δ(k,t) and the satellite timing offset δ s (k,t). 
     In some instances, it may take long to obtain the ephemeris data from a particular satellite. For example, in hilly terrain, a satellite may only intermittently be in view of a moving vehicle, for example. In these scenarios, where uninterrupted reception of a satellite&#39;s broadcast may not be possible, it may take a significant amount of time to obtain a satellite&#39;s ephemeris data that may be used to compute δ s (k,t). This may also be the case when a vehicle travels in urban areas because of tunnels and high structures, for example. In these scenarios, it may be desirable to be able to use a satellite that may intermittently be received, even if its ephemeris data may not be available. For example, there may be instances when a satellite navigation as described above may be possible, for example from 4 satellites for which ephemeris may have been received at the satellite receiver. In addition, one or more satellites may intermittently be in view and their broadcast signal may be received at times, but their ephemeris may not yet be available. In some instances, for example by turning a corner and changing a vehicle&#39;s travel direction, one or more of the previously intermittently received satellites may become visible, whereas one or more of the previously visible satellites may no longer be visible. In these cases, it may be desirable to start navigation with the newly visible satellites almost immediately, before ephemeris has been received. Since no ephemeris may be available for the intermittently received satellites, the corrected pseudorange ρ c (k,t) may not be obtained. However, since other satellites are visible with ephemeris, the offset b may be known and the receiver clock corrected pseudorange ρ′(k,t) may be determined by correcting for the offset b, given by the following relationship: 
     
       
         
           
             
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     Since the satellite receiver  302  position may be known from the satellites received with ephemeris, the receiver clock corrected pseudorange may be used to ‘fix’ a position of the satellite associated with it, as is explained for  FIG. 3B  below. 
     In accordance with various embodiments of the invention, it may be desirable to determine the receiver position by Linearization. Using the Linearization approach, a system of linear equations may be solved in accordance with the following relationship: 
       p=Hx  (1) 
     where p=[ρ(1,t)−{circumflex over (ρ)}{circumflex over (p)}(1,t 2 ), . . . , ρ(n,t)−{circumflex over (ρ)}(n,t 2 )] T  may be a vector with n elements for n satellites, which may represent an estimated change in pseudorange between the pseudorange ρ(k,t) for satellite k at time t, and the estimated pseudorange {circumflex over (ρ)}(k,t 2 ) of satellite k at some future time t 2 &gt;t. The estimated pseudorange {circumflex over (ρ)}(k,t 2 ) may be estimated based on the common bias, for example offset b, and ephemeris data. The vector x may be a vector comprising an estimated change in position. For example, the vector x may be a receiver state vector x=[Δ lat (t,t 2 ), Δ lon (t,t 2 ),Δ alt (t,t 2 )Δ b (t,t 2 )] T , where Δ lat (t,t 2 ) may be an estimated difference in the latitude of the receiver position between time t and t 2 , Δ lon (t,t 2 ) may be an estimated difference in the longitude of the receiver position between time t and t 2 , Δ alt (t,t 2 ) may be an estimated difference in the altitude of the receiver position between time t and t 2 , and Δ b (t,t 2 ) may be an estimated difference in the common offset between time t and t 2 . In accordance with various embodiments of the invention, the receiver state vector x may be in any desirable coordinate system. The matrix H may be an appropriately dimensioned mixing matrix which may be referred to as “line-of-sight matrix”, or “observation matrix.” The matrix H may relate the change in pseudorange of vector p with the change in receiver state x. 
     In accordance with various embodiments of the invention, with known ephemeris, p,H may be determined and a position update may be generated from solving the linear system for x. 
       FIG. 3B  is a diagram δ illustrating an exemplary satellite positioning, in accordance with an embodiment of the invention. Referring to  FIG. 3B , there is shown a known position satellite receiver  302   a  at position p, a dashed circle range  350  centered at  302   a , and a satellite  320 . In the two-dimensional setting depicted in  FIG. 3B , by knowing the position of the satellite receiver  302   a  to be p, an effective satellite position may be determined by the dashed circle  350 , centered at the satellite receiver  302   a  and with radius ρ′( 320 ,t). Because this radius may be given by ρ′( 320 , {circumflex over (t)})=cτ( 320 )−cδ s (320, t), the radius may not correspond to the actual distance to the satellite  320  but to a corrected distance, corrected for the timing offset cδ u (t). The exact assumed position of the satellite  320  on the dashed circle  350  may be arbitrary, as may be seen from the situation depicted in  FIG. 3B , since any assumed satellite position on the dashed circle  350  may go through the correct position p of the satellite receiver. In three dimensions, the situation may be similar and the dashed circle  350  may become a sphere around the position p. The fact that an effective assumed position may be used for a satellite&#39;s position (without ephemeris) may be exploited for navigational purposes by constructing temporary ephemeris for satellites where the ephemeris may not be available, for example based on the satellite&#39;s Almanac position P(k,t). 
       FIG. 4  is a diagram illustrating an exemplary construction of temporary ephemeris based on a satellite&#39;s almanac position. Referring to  FIG. 4 , there is shown a satellite  410   a  at position p( 410   a ,t 1 ). The same satellite is also shown at time t 2  at position p( 410   a ,t 2 ) and labeled satellite  418 . These are the only two actual satellite positions, the other satellites depicted show various associated positions. There is shown an almanac satellite position  414  of satellite  410   a  at time t 1  and an almanac satellite position  424  of satellite  410   a  at time t 2 . There is a measured satellite position  412  at time t 1 , and a measured satellite position  420  at time t 2 . There is shown a transposed satellite position  416  at time t 1 , and a transposed satellite position  426  at time t 2 . There is shown a satellite receiver  402  at a known position p. There is also shown a range r( 410   a ,t 1 ) from the satellite  410   a  to the satellite receiver  402 , a range offset cδ s ( 410   a ,t 1 ) from the actual satellite position  410   a  to the measured satellite position  412 , an almanac range r A ( 410   a ,t 1 ) from the satellite receiver  402  to the almanac position  414 , an almanac correction term e( 410   a,t   1 ) between the almanac position  414  and the transposed satellite position  416 , and a distance vector Δ( 410   a,t   1 ) between the almanac position  414  and the actual satellite position  410   a . Similar quantities may be identified for time instance t 2 . 
     As described above, a satellite receiver  402  position p may be determined from 4 satellites (with ephemeris data, not shown in  FIG. 4 ). The satellite  410   a  may be a satellite that may be visible intermittently and its ephemeris data may not be available. A receiver clock corrected pseudorange 
     
       
         
           
             
               
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     may be measured for satellite  410   a  as described for  FIG. 3 . The actual position of the satellite  410   a  may be at a range r( 410   a,t   1 ) from the satellite receiver  402 . Since the range offset cδ s ( 410   a,t   1 ) may not be corrected in the absence of ephemeris data, the measured receiver clock corrected pseudorange ρ′( 410 ,t 1 ) makes the satellite  410   a  appear to be located at the measured satellite position  412 , based on the known position p. In the illustrated example of  FIG. 4 , the measured distance may be larger than the actual range r( 410   a,t   1 ). Although the satellite receiver  402  may not know the position p( 410   a,t   1 )=P( 410   a,t   1 )+Δ( 410   a,t   1 ) of the satellite  410   a  because the ephemeris data to determine Δ( 410 ,t 1 ) may not be available, the satellite receiver  402  may know the approximate position of the satellite  410   a  based on locally stored almanac data P( 410   a,t   1 ). This almanac position P( 410   a,t   1 ) may be illustrated by the almanac satellite position  414 . 
     Based on the almanac position P( 410   a,t   1 ) and the receiver clock corrected pseudorange ρ′( 410   a,t   1 ), a transposed satellite position  416  may be determined such that p T ( 410   a,t   1 ) may be at a distance given by ρ′( 410   a,t   1 )=r A ( 410   a,t   1 )+e( 410   a,t   1 ). In these instances, the transposed satellite position  416  may be on the same sphere/circle as the measured satellite position  412 . The almanac correction term e( 410   a,t   1 ) may be a range correction term similar to a position correction term Δ( 410   a,t   1 ) and may serve as a temporary ephemeris together with the almanac to determine the satellite position. As illustrated for  FIG. 3B , any position at a certain range on a sphere/circle around the known satellite receiver position may be equally desirable as a measured satellite position. The transposed satellite position  416  may hence be located at the same distance as the measured satellite position  412 , and may be defined by an almanac correction term e( 410   a,t   1 ) that may be added to the range r A ( 410   a ,t 1 ) between the known satellite receiver position  402  and the known almanac satellite position  414 . Hence, the transposed satellite position  416  may be assumed to be a hypothetical satellite position that may correct for the satellite clock timing offset δ s (t). By assuming that the satellite may be placed at the transposed satellite position  416 , the receiver clock corrected pseudo-range measurement ρ′( 410 ,t 1 ) may indicate the correct position p of the satellite receiver  402 . 
     As shown in  FIG. 3B , any translated satellite position at a desirable distance from the satellite receiver  302   a  may be useable. In the instance depicted in  FIG. 4 , any location on the circle around p at distance ρ′( 410   a,t   1 ) may be chosen in principle, for example the transposed satellite position  416 . The advantage of linking the transposed satellite position  416  to the almanac satellite position  414  via the almanac correction term e( 410   a,t   1 ) may be that the almanac position may be available for future time instances, for example almanac position  424  at time t 2 . In instances where the time difference from the initial determination of the transposed satellite position  416  may be within certain limits, it may be assumed that the almanac correction term e( 410   a ,t 1 ) may stay approximately constant since the actual satellite position may vary slowly with respect to the almanac predicted satellite position, and may be offset by a relatively small error Δ( 410   a ,t 1 ). At a future time instant t 2 , it may be the case that the previously intermittently visible satellite  410   a  may become visible, while one of the previously used satellites with ephemeris data available, may no longer be visible. In these instances, it may be that the ephemeris for satellite  410   a  may not have been acquired yet. However, because the almanac correction term e( 410   a ,t 1 ) e( 410   a ,t 2 ) may have been computed at the last known position p, at time instant t 2  an estimated transposed satellite position  426  may be predicted based on the almanac satellite position  424  at time t 2 , by the following relationship: 
         {circumflex over (p)}   T (410 a,t   2 )= P (410 a,t   2 )+ {right arrow over (e)} (410 a,t   1 ) 
         e (410 a,t   1 )≈ e (410 a,t   2 ) 
     where {right arrow over (e)}( 410   a,t   1 ) may denote a vector in an appropriate direction of length e( 410   a,t   1 ). 
     More specifically, for example where the satellite receiver  402  may remain stationary at position p, the predicted receiver clock corrected range for time instant t 2  may be {circumflex over (ρ)}′( 410   a,t   2 )=r A ( 410   a,t   2 )+e( 410   a,t   2 ), as illustrated in  FIG. 4 . Any difference from the measured receiver clock corrected pseudorange |ρ′( 410 ,t 2 )−{circumflex over (ρ)}′( 410 ,t 2 )|, which may be based on the actual satellite position  418 , may then be attributed to a movement by the satellite receiver  402 . Linking the predicted transposed position  426  to a known almanac position  424  may allow the almanac to provide a relatively accurate model of the change in range Δ e ( 410   a,t   2 −t 1 ) between the satellite receiver  402  and the position of the satellite  410   a  at different times, since the satellite  410   a  position changes little with respect to the satellite almanac position. For example, in  FIG. 4 , the change in range between the satellite receiver  402  and the satellite almanac position between time t 1  and time t 2  may be illustrated by Δ e ( 410   a,t   2 −t 1 ) at the almanac satellite position  414 , and again at time instant t 2  at the actual satellite position  418 . Because  FIG. 4  may not be to scale, the distance Δ( 410   a,t   2 ) between the actual satellite position  418  and the almanac satellite position  424  at time t 2  may not be the same as at time t 1 . 
     In summary, the set of equations to determine a position based on the approach discussed above may be as given in the following relationship, in instances where a position p(t 1 ) may have been acquired from satellites with ephemeris at least once: 
       ρ c ( k,t )− b=∥p ( k,t )− p ( t )∥ k={1, . . . , M}  (2) 
       ρ′( l,t )− e ( l,t )=∥ P ( l,t )− p ( t )∥ l={M+1, . . . N} 
     Where N satellites may be received and for M of them, the ephemeris data may be available. M may be assumed to be smaller than 4, since for M≧4 the normal position determination as described for  FIG. 2  and  FIG. 3  may be used. Since this approach may be used for M≦3, b may be assumed constant since it may not be determined for M&lt;4. Hence, with b assumed constant over a certain period, e(k,t), and P(k,t) known, the position p(t) may be determined in three dimensions from N=3 satellites, with or without ephemeris according to the above relationship. In some instances, it may also be possible to update b based on the following relationships from equation (2), for N≧4. 
       ρ c ( k,t )− b=∥p ( k,t )− p ( t )∥ k={1, . . . , M} 
       ρ( l,t )− b=∥P ( l,t )− p ( t )∥+ e ( l,t ) l={M+1, . . . N} 
     In accordance with various embodiments of the invention, it may be desirable to determine the receiver position by Linearization. Using the Linearization approach, a system of linear equations may be solved in accordance with the following relationship: p=Hx, similar to the description for  FIG. 3A . In some instances, of N received satellites, for M&lt;N, the ephemeris may be available at the satellite receiver. In these instances, for N−M satellites, {circumflex over (ρ)}(i,t 2 ) may not be computed based on ephemeris. In these instances, an estimated pseudorange based on almanac information, {circumflex over (ρ)} a (i,t 2 ), may be estimated. The estimated pseudorange based on almanac information may be related to the estimated pseudorange by the following relationship: 
           a (i,t 2 )+d{circumflex over (ρ)} a (i,t 2 ) d{circumflex over (ρ)}(i,t 2 )  (3) 
     where d{circumflex over (ρ)} a (i,t 2 ) may be the difference term between the estimated pseudorange based on almanac information and the estimated pseudorange based on ephemeris data. By substituting equation (3) into equation (1), the difference term may be described by the following relationship: 
         d{circumflex over (ρ)}   a ( i,t   2 )=ρ( i,t )−{circumflex over (ρ)} a ( i,t   2 )− H   i   x    
     where H i  may be the i-th row of matrix H. Hence, in accordance with various embodiments of the invention, d{circumflex over (ρ)} a (i,t 2 ) may be obtained when x may be available. By using Doppler measurements, the change in d{circumflex over (ρ)} a (i,t 2 ) may be estimated, and may provide d′{circumflex over (ρ)} a (i,t 2 ), the first derivative of d{circumflex over (ρ)}a(i,t 2 ). Therefore, after an initial d{circumflex over (ρ)} a (i,t 2 ) may have been estimated, d′{circumflex over (ρ)} a (i,t 2 ) may be used to extrapolate future changes for some time, without the use of ephemeris data. For example, the following pseudorange estimate may be generated based on the above quantities: 
       {circumflex over (ρ)}( i,t   3 )=   a ( i,t   2 )+ d{circumflex over (ρ)}   a ( i,t   2 )+ d′{circumflex over (ρ)}   a ( i,t   2 ) dt ( t   2   ,t   3 )  (4) 
     where dt(t 2 ,t 3 )=|t 2 −t 3 | may be the time since the last estimate of    a (i,t 2 ) and d{circumflex over (ρ)} a (i,t 2 ). Therefore, by substituting equation (4) into equation (1), an updated position of the receiver may be determined by solving the system of linear equations for x. 
       FIG. 5  is a diagram illustrating a prediction error, in accordance with an embodiment of the invention. The elements in  FIG. 5  may correspond to the elements listed in  FIG. 4 . In particular, elements  510   a  may correspond to element  410   a  in  FIG. 4 , and similarly for the other elements depicted. There is shown an additional possible predicted satellite receiver position  504  at time instant t 2 . 
     In  FIG. 5 , a scenario is depicted that may be used to illustrate possible prediction errors. In  FIG. 5 , the distance vector Δ( 510   a,t   1 ) between the almanac satellite position  514  and the actual satellite position  510   a  may remain constant at time t 2 , so that the distance between the almanac satellite position  524  and the actual satellite position Δ( 510   a ,t 2 )=Δ( 510   a ,t 1 ). In this situation, given the exemplary geometry (not to scale), the difference in range between the satellite receiver position  502  and the satellite  510   a  at time t 1  and the satellite position  518  at time t 2  may be  =|r( 510   a,t   2 )−r( 510   a,t   I )|, rather than Δ e ( 510   a,  t 2 −t 1 ) as predicted from the change in the almanac satellite position. This case may arise, for example, when the time difference between t 1  and t 2  may be significant. In such instances, the difference between Δ e ( 510   a,t   2 −t 1 ) and ó may result in an positional error that may be seen between the transposed satellite position  526  and the measured satellite position  520 . In the exemplary situation in  FIG. 5 , the measured receiver clock corrected pseudorange ρ′( 510   a,t   2 ) at time t 2  may be equivalent to the range from the measured satellite position  520  to the p(t 1 ) (assuming the satellite receiver  502  may remain stationary). However, since the satellite receiver  502  may mistakenly think that the measured pseudorange ρ′( 510   a,t   2 ) ought to correspond to a distance measured from the transposed satellite position  526 , the satellite receiver  502  may believe itself may have changed location and may place itself, for example, in the predicted position  504 . 
       FIG. 6  is a diagram illustrating temporary ephemeris without the use of an almanac position, in accordance with an embodiment of the invention. Referring to  FIG. 6 , there is shown a coordinate system  606 , satellite receiver  602  at position p, a satellite  610   a  at position p( 610   a,t   1 ), a measured satellite position  612  at time t 1 , a translated satellite position  616  at time t 1 , and a predicted translated satellite position  626  at time t 2 . There is also shown a measured receiver clock corrected pseudorange ρ′( 610   a,t   1 ) between the satellite receiver position  602  and the measured satellite position  612  comprising a range r( 610   a,t   1 ) between the actual satellite position  610   a  and the satellite receiver position  602 , and an offset cδ s ( 610   a,t   1 ) between the actual satellite position  610   a  and the measured satellite position  612 . There is also shown a predicted receiver clock corrected pseudorange {circumflex over (ρ)}′( 610   a,t   2 ). 
     In the exemplary scenario illustrated in  FIG. 6 , the position p may be known at time instant t 1  from a plurality of satellites with known ephemeris (not illustrated). As described for  FIG. 4 , the receiver clock corrected pseudorange ρ′( 610   a,t   1 ) may be measured and a measured satellite position  612  may be accordingly derived. As described for  FIG. 3B  and  FIG. 4 , a translated satellite position  616  may be defined on the circle/sphere defined by the measured satellite position  612 . In the depicted example, the translated satellite position  616  may be arbitrarily chosen, for example p T ( 610   a,t   1 ). 
     In order to predict the predicted receiver clock corrected pseudorange {circumflex over (ρ)}′( 610   a,t   2 ) at time instant t 2 , a rate of change of the measured receiver clock corrected pseudorange ρ′( 610   a,t   1 ) may be determined. This information may be available in the scenario described in  FIG. 4  because the satellite almanac position may be known in advance and hence a change in range may be computed from the change in almanac position, for example. In the scenario depicted in  FIG. 6 , there may be several means of obtaining a rate of change in the receiver clock corrected pseudorange. In accordance with one embodiment of the invention, the Doppler frequency of the received signal may be measured at the satellite receiver  602 . The Doppler frequency may be a frequency shift in the frequency of the received signal that may be induced when the satellite moves towards or away from the satellite receiver  602 . GPS signals may be transmitted at 1575.42 MHz. Proportional to the velocity at which the satellite receiver  602  and the satellite  610   a  may move toward or away from each other, the received signal frequency may be shifted by approximately +/−5 kHz. Hence, based on the Doppler shift in the frequency of the received signal at the satellite receiver  602 , the satellite receiver  602  may estimate how fast the satellite moves towards or away from it. This estimated rate of change may hence be used to predict the predicted measured receiver clock corrected pseudorange according to the following relationship: 
     
       
         
           
             
               
                 
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     may denote the estimated rate of change in ρ′( 610   a,t   1 ) based on measured Doppler shift. 
     In accordance with another embodiment of the invention, the rate of change of the measured receiver clock corrected pseudorange ρ′( 610   a,t   1 ) may be estimated by collecting pseudoranges over a period of time. A polynomial may then be fit to the collected data and the first derivative of the polynomial fit at the desired time instant, for example t 2 , may provide an estimate of the rate of change, for example 
     
       
         
           
             
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     may denote the estimated rate of change based on a series of measured receiver clock corrected pseudorange measurements. For example, in the simplest case, a straight line may be fit to the measurements ρ′( 610   a,t   0 ) and ρ′( 610   a,t   1 ). Correspondingly, the gradient of the straight line connecting the two measurements may be used to predict {circumflex over (ρ)}′( 610   a,t   2 ). Similarly, using at least 3 measurement points, a second order polynomial fit may be generated. The invention may not be limited to any particular method of generating the required rate of change, or any particular polynomial order. In another embodiment of the invention, a polynomial may be fit to a series of Doppler frequency measurements, which may also be used to predict the rate of change for a future time instant, as described above. 
     The set of equations to determine a position based on the approach discussed may be as given in the following relationship, in instances where a position p(t 1 ) may have been acquired from satellites with ephemeris at least once: 
       ρ c ( k,t )− b=∥p ( k,t )− p ( t )∥ k=1, . . . , M 
       ρ′( l,t )=∥{circumflex over (p)} T ( l,t )− p ( t )∥ l=M+1, . . . , N 
     where, {circumflex over (p)} T (l,t) may an arbitrary predicted translated satellite position, for example predicted translated satellite position  626 , N satellites may be received and for M of them, the ephemeris data may be available. Note that M may be assumed smaller than 4, since for M≧4 the normal position determination as described in  FIG. 2  and  FIG. 3  may be used. Since this approach may be used for M≦3, b may be assumed constant. Hence, with b assumed constant over a certain, the position p(t) may be determined in three dimensions from N=3 satellites, with or without ephemeris according to the above relationship. In some instances, it may also be possible to update b, similarly to the procedure described for  FIG. 5 . 
     As described for  FIG. 3 , in a GPS system, the spreading codes that may be used to determine the range through correlation with a local replica of the correlation code, are termed C/A codes. The C/A codes may be unique to each satellite, and may hence identify a received signal. A C/A code may comprise 1023 chips and may take 1 ms to send. For each bit, 20 repetitions of the C/A code may be transmitted by the satellite, for example satellite  610   a , over the duration of 20 ms. In addition, the 30 bits may make one word, 10 words may make a subframe (6 s), 5 subframes may make one page, and 25 pages may make a superframe (12.5 min). In most instances, it may take at least three subframes of a transmission to generate a position. In those instances, at least 18 seconds of data may need to be received. 
     With regards to pseudorange measurements, the smallest unit useable may be 1 ms, or one C/A code. The satellite receiver  602  may correlate the received signal with a local copy of the C/A code. Hence, the satellite receiver may first detect any offset that may occur modulo 1 ms, as it may acquire C/A code synchronization. For example, if a travel time from a satellite  610   a  to a satellite receiver  602  may take 78.2 ms, and assuming for illustrative purposes that the signal was sent at t=0 ms, the receiver may initially be able to determine the modulo-1 ms offset. In this instance, the satellite receiver may determine that satellite travel time may be offset by x C/A =78.2(mod 1 ms)=0.2. After at least 20 ms and twenty C/A codes, the satellite receiver  602  may acquire bit-level synchronization and may be able to determine the start of each bit. At this point, the satellite receiver  602  may be able to determine an offset x b =78.2(mod 20 ms)=18.2. At this point, the satellite receiver  602  may already know that the satellite timing offset may be 78.2 ms or more (98.2 ms, 110.2 ms, . . . ), since the satellite&#39;s altitude may be known by the receiver to be more than 20,000 km. In instances when the satellite may have acquired word-level synchronization after at least 600 ms, the satellite may then achieve precise timing (modulo 600 ms). In the above discussion, it may have been assumed that the satellite may have very little information. However, in practice, the satellite receiver  602  may possess quite accurate information about the satellite  610   a  position because of the almanac data. Since the signal may travel 300 km in 1 ms, and the almanac may provide a satellite position estimate that may be within 300 km from the actual satellite position  610   a , the satellite receiver  602  may be able to determine the precise timing offset already from the first, modulo 1 ms, measurement. 
       FIG. 7  is a flowchart illustrating an exemplary algorithm to compute a satellite receiver position in accordance with an embodiment of the invention. After initializing the algorithm in step  702 , the satellite may need to decide in step  704  whether M≧4 satellites with ephemeris may be available. If not, an initial position may not be obtained and hence the satellite receiver, for example satellite receiver  602 , may have to wait until M≧4 satellites with ephemeris may be available. In instances where more than 4 satellites with ephemeris may be available, the satellite receiver may generate its position by the procedures described in  FIG. 2  and  FIG. 3 , for example, in step  706 . In addition, if other satellites may be visible for which ephemeris may not be available, the satellite receiver  602  may measure pseudoranges to those satellites in step  708 . For satellites without ephemeris, in step  710  the satellite may compute a receiver clock corrected pseudorange. Using the receiver clock corrected pseudorange, known satellite receiver position, for example satellite receiver position  602 , the satellite receiver may generate a translated satellite position, for example. In some instances, the translated satellite position may be generated based on an almanac satellite position. The satellite receiver may also generate a rate of change for the pseudorange, which may be used to generate a predicted translated satellite position. In some instances, the rate of change of the pseudorange may be estimated from measurements, or from almanac data, as described for  FIG. 4  and  FIG. 6 . In step  712 , if there are M≧4 or more satellites with ephemeris, the satellite receiver may start over and return to step  704 . If, in step  712 , there may be less than M&lt;4 satellites available with ephemeris and more than N&gt;3 satellites may be visible in total, the satellite position may be computed as described in  FIG. 4  to  FIG. 6  in step  714 . In step  716 , if 4 or more satellites with ephemeris may be available, the satellite receiver may return to step  706 . If less than 4 satellites with ephemeris may be available, the satellite receiver may remain in step  714  and continue to compute its position according to step  714 . In accordance with various embodiments of the invention, temporary ephemeris may comprise a translated satellite position and a rate of change of any associated receiver clock corrected pseudorange. 
     In accordance with an embodiment of the invention, a method and system for generating temporary ephemeris may comprise determining one or more positions of a satellite receiver, for example satellite receiver  202 , based on a plurality of satellite signals received from a plurality of satellites for which complete ephemeris data has been received at the satellite receiver, as described for  FIG. 2  and  FIG. 3 . Temporary ephemeris data may be generated from the determined one or more positions p(.) of the satellite receiver and one or more satellite signals from one or more satellites with incomplete ephemeris data. One or more estimated positions of the satellite receiver may be determined based on the generated temporary ephemeris and a second plurality of satellite signals, wherein at least one of the second plurality of satellite signals is associated with the one or more satellites with incomplete ephemeris data, as described for  FIG. 3B ,  FIG. 4 ,  FIG. 5 ,  FIG. 6  and  FIG. 7 . 
     The temporary ephemeris data may be generated by generating a translated satellite position, for example a translated satellite position  416 , and a rate of change, for example Δ e , of an associated receiver clock corrected pseudorange, for example p′(.). The translated satellite position may be generated based on an almanac satellite position, for example P(.), or at a position determined by said receiver clock corrected pseudorange and/or said associated rate of change of said receiver clock corrected pseudorange. The rate of change of said associated receiver clock corrected pseudorange may be generated based on one or more Doppler frequency measurement of said plurality of satellite signals and/or said second plurality of satellite signals, or based on a plurality of pseudorange measurements, as described, for example in  FIG. 6 . The receiver clock corrected pseudorange may be determined from at least a receiver clock offset and a pseudorange measurement. When the satellite receiver may receive complete ephemeris data from the one or more satellites from which the incomplete ephemeris data was received by the satellite receiver, utilizing this complete ephemeris data and the determined one or more positions to determine a location of the satellite receiver, as described in  FIG. 2  and  FIG. 3 . The estimated position of the satellite receiver may be update based on newly generated temporary ephemeris until the satellite receiver receives complete ephemeris data from the one or more satellites for which the incomplete ephemeris data was received by the satellite receiver. The generated temporary ephemeris data may be updated based on a rate of change of an associated receiver clock corrected pseudorange and a translated satellite position. The plurality of satellites, a second plurality of satellites from which the second plurality of satellite signals are received, and the satellite receiver may conform to the Global Positioning System (GPS) standard, the GALILEO standard, the GLOSNASS standard, the IRNSS standard, or the BEIDOU standard. 
     Another embodiment of the invention may provide a machine-readable storage, having stored thereon, a computer program having at least one code section executable by a machine, thereby causing the machine to perform the steps as described above for a method and system for generating temporary ephemeris. 
     Accordingly, the present invention may be realized in hardware, software, or a combination of hardware and software. The present invention may be realized in a centralized fashion in at least one computer system, or in a distributed fashion where different elements are spread across several interconnected computer systems. Any kind of computer system or other apparatus adapted for carrying out the methods described herein is suited. A typical combination of hardware and software may be a general-purpose computer system with a computer program that, when being loaded and executed, controls the computer system such that it carries out the methods described herein. 
     The present invention may also be embedded in a computer program product, which comprises all the features enabling the implementation of the methods described herein, and which when loaded in a computer system is able to carry out these methods. Computer program in the present context means any expression, in any language, code or notation, of a set of instructions intended to cause a system having an information processing capability to perform a particular function either directly or after either or both of the following: a) conversion to another language, code or notation; b) reproduction in a different material form. 
     While the present invention has been described with reference to certain embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the scope of the present invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the present invention without departing from its scope. Therefore, it is intended that the present invention not be limited to the particular embodiment disclosed, but that the present invention will include all embodiments falling within the scope of the appended claims.