Positioning
A positioning method comprising: solving a geometric algebraic expression that relates: position vectors for a plurality of antennas relative to a shared origin, a putative position vector for the shared origin and phase information measured at each of the plurality of antennas, to determine a position vector for the shared origin; and positioning an apparatus using the determined position vector for the shared origin.
1. A positioning method for positioning an electronic apparatus comprising:
solving a geometric algebraic expression that relates:
1) position vectors for a plurality of signals received from either a) a plurality of antennas or b) a plurality of receivers relative to a shared origin,
2) a putative position vector for the shared origin and
3) phase information measured at each of the plurality of a) the antennas or b) the receivers,
to determine a position vector for the shared origin; and
positioning the electronic apparatus using the determined position vector for the shared origin,
wherein solving the geometric algebraic expression comprises determining a putative position vector for the shared origin that is parallel to a difference vector defined as the difference between an averaged origin dependent vector and an averaged phase dependent vector wherein
the averaged origin dependent vector is formed from a summation of geometric products, each of which involves two position vectors for a respective antenna and the putative position vector for the shared origin and
wherein the averaged phase dependent vector is formed from a summation of geometric products involving two position vectors for a respective antenna or receiver, wherein one of the two position vectors is scaled using the phase information for the respective antenna or receiver.
2. A positioning method as claimed in claim 1 , wherein the apparatus houses the plurality of antennas.
3. A positioning method as claimed in claim 1 , wherein the apparatus houses at least one but not all of the plurality of antennas.
4. A positioning method as claimed in claim 1 , wherein position vectors for a respective antenna or receiver are predetermined and stored in memory.
5. A positioning method as claimed in claim 4 , wherein position vectors are determined by an iterative process.
6. A positioning method as claimed in claim 5 , wherein the phase information for an antenna or receiver is dependent upon a measured phase for that antenna or receiver compensated by an offset for that antenna or receiver.
7. A positioning method as claimed in claim 6 , wherein the offsets for the antennas or receivers are pre-determined and stored in a memory.
8. A positioning method as claimed in claim 7 , wherein the offsets are determined. . . . each time position vectors change.
9. A positioning method as claimed in claim 1 , wherein the averaged phase dependent vector is defined by:
γ
→
=
∑
n
=
1
Γ
∑
k
=
1
P
(
n
)
(
1
2
β
n
,
k
r
→
n
)
2
where
n is the index identifying a respective one of the plurality of antennas or receivers
{right arrow over (r )} n is a vector positioning an antenna or receiver to a shared origin
k is the index identifying a phase measurement
β n,k is phase information relating to a phase measurement at instant k at antenna or receiver n and is expressed in the dimensions of distance
Γ is the number of antennas or receivers
P(n) is the number of phase measurements made at the antenna or receiver n.
10. A positioning method as claimed in claim 9 , wherein the averaged origin dependent vector is defined by:
∂
[
R
→
⋀
s
,
m
]
=
∑
n
=
1
Γ
∑
k
=
1
P
(
n
)
(
1
4
r
→
n
R
→
⋀
s
,
m
r
→
n
)
where
n is the index identifying a respective one of the plurality of antennas or receivers
{right arrow over (r )} n is a vector positioning an antenna or receiver n with respect to a shared origin
k is the index identifying a phase measurement
Γ is the number of antennas or receivers
P(n) is the number of phase measurements made at the antenna or receiver n
{right arrow over ({circumflex over (R)} s,m is a putative unit vector between source and the shared origin from which {right arrow over (r)} n are measured, where s is an index identifying the source of the received signals.
11. A positioning method as claimed in claim 10 wherein the geometric algebraic expression may be expressed as
(∂[ {right arrow over ({circumflex over (R)} s,m ]−{right arrow over (γ)}) ^ {right arrow over ({circumflex over (R)} s,m =0.
12. A positioning method as claimed in claim 10 , wherein solving the geometric algebraic expression involves solving by iteration:
∂[ {right arrow over ({circumflex over (R)} s,m ]{right arrow over ({circumflex over (R)} s,m −{right arrow over ({circumflex over (R)} s,m ∂[{right arrow over ({circumflex over (R)} s,m ]−{right arrow over (γ)}∂[{right arrow over ({circumflex over (R)} s,m ]={right arrow over ({circumflex over (R)} s,m {right arrow over (γ)}=0
where
∂
[
R
→
⋀
s
,
m
]
=
∑
n
=
1
Γ
∑
k
=
1
P
(
n
)
(
1
4
r
→
n
R
→
⋀
s
,
m
r
→
n
)
γ
→
=
∑
n
=
1
Γ
∑
k
=
1
P
(
n
)
(
1
2
β
n
,
k
r
→
n
)
2
where
n is the index identifying a respective one of the plurality of antennas or receivers,
{right arrow over (r )} n is a vector positioning an antenna n with respect to a shared origin
k is the index identifying a phase measurement,
β n,k is phase information relating to a phase measurement at instant k at antenna n and is expressed in the dimensions of distance,
Γis the number of antennas or receivers,
P(n) is the number of phase measurements made at the antenna or receiver n, and
{right arrow over ({circumflex over (R)} s,m is a putative unit vector between source and the shared origin from which {right arrow over ({circumflex over (R)} n are measured, where s is an index identifying the source of the received signals.
13. A positioning method as claimed in claim 10 , wherein solving the geometric algebraic expression involves solving by iteration any one or more of the following iterative geometric algebraic expressions:
{right arrow over ({circumflex over (R)} s,m+1 =∂[{right arrow over ({circumflex over (R)} s,m ] −1 ( {right arrow over ({circumflex over (R)} s,m ∂[{right arrow over ({circumflex over (R)} s,m ]+{right arrow over (γ)}∂[{right arrow over ({circumflex over (R)} s,m ]−{right arrow over ({circumflex over (R)} s,m {right arrow over (γ)})
{right arrow over ({circumflex over (R)} s,m+1 =(∂[ {right arrow over ({circumflex over (R)} s,m ]{right arrow over ({circumflex over (R)} s,m −{right arrow over (γ)}∂[{right arrow over ({circumflex over (R)} s,m ]+{right arrow over ({circumflex over (R)} s,m {right arrow over (γ)})∂[ {right arrow over ({circumflex over (R)} s,m ] −1
{right arrow over ({circumflex over (R)} s,m+1 =(−∂[ {right arrow over ({circumflex over (R)} s,m ]{right arrow over ({circumflex over (R)} s,m +{right arrow over ({circumflex over (R)} s,m ∂[{right arrow over ({circumflex over (R)} s,m ]+{right arrow over (γ)}∂[{right arrow over ({circumflex over (R)} s,m ]){right arrow over (γ)} −1
{right arrow over ({circumflex over (R)} s,m+1 ={right arrow over (γ)} −1 (∂[ {right arrow over ({circumflex over (R)} s,m ]{right arrow over ({circumflex over (R)} s,m −{right arrow over ({circumflex over (R)} s,m ∂[{right arrow over ({circumflex over (R)} s,m ]+{right arrow over ({circumflex over (R)} s,m {right arrow over (γ)})
where
∂
[
R
→
⋀
s
,
m
]
=
∑
n
=
1
Γ
∑
k
=
1
P
(
n
)
(
1
4
r
→
n
R
→
⋀
s
,
m
r
→
n
)
γ
→
=
∑
n
=
1
Γ
∑
k
=
1
P
(
n
)
(
1
2
β
n
,
k
r
→
n
)
2
n is the index identifying a respective one of the plurality of antennas or receivers,
{right arrow over (r )} n is a vector positioning an antenna or receiver n with respect to a shared origin
k is the index identifying a phase measurement,
β n,k is phase information relating to a phase measurement at instant k at antenna n and is expressed in the dimensions of distance,
Γis the number of antennas or receivers,
P(n) is the number of phase measurements made at the antenna or receivers n, and
{right arrow over ({circumflex over (R)} s,m is a putative unit vector between source and the shared origin from which {right arrow over (r)} n are measured, where s is an index identifying the source of the received signals.
14. An apparatus comprising:
processing circuitry configured to solve a geometric algebraic expression that relates: position vectors for a plurality of antennas or receivers relative to a shared origin, a putative position vector for the shared origin and phase information for each of the plurality of antennas or receivers, to determine a position vector for the shared origin and configured to perform positioning of the apparatus using the determined position vector for the shared origin; and
an interface configured to provide phase measurements,
wherein the processing circuitry configured to solve the geometric algebraic expression comprises the processing circuitry configured to determine a putative position vector for the shared origin that is parallel to a difference vector defined as the difference between an averaged origin dependent vector and an averaged phase dependent vector wherein
the averaged origin dependent vector is formed from a summation of geometric products, each of which involves two position vectors for a respective antenna and the putative position vector for the shared origin and
wherein the averaged phase dependent vector is formed from a summation of geometric products involving two position vectors for a respective antenna or receiver wherein one of the two position vectors is scaled using the phase information for the respective antenna or receiver.
15. An apparatus as claimed in claim 14 , wherein the interface is configured to provide, from storage, a set of position vectors for the plurality of antennas or receivers.
16. An apparatus as claimed in claim 14 , wherein the interface is configured to provide, from storage, an initial putative position vector of the shared origin and the processing circuitry is configured to solve the geometric algebraic expression by iteration starting with the initial putative position vector.
17. An apparatus as claimed in claim 14 , wherein the interface is configured to provide, from storage, a set of predetermined antenna dependent phase offsets, wherein the phase information for each of the plurality of antennas or receivers is dependent upon a measured phase for a respective antenna or receiver compensated by a phase offset for the respective antenna or receiver.
18. A computer program product comprising a non-transitory computer readable medium having a computer program stored thereon, which when executed by a processor causes an apparatus to perform the method of claim 1 .
19. The computer program product of claim 18 , wherein the apparatus is configured to receive a plurality of signals from dispersed reception points, and perform the method of claim 1 utilizing the plurality of received signals.