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`EROVISIONAL APPLICATION FOR PA TENT COVER SHEET (Small Entity)
`This is a request for filing a PROVISIONAL APPLICATION FOR PATENT under 37 CFR 1.53 (c).
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`I Additional inventors are being named on page 2 attached hereto
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`TITLE OF THE INVENTION (280 characters max)
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`"DATA FUSION FOR WIRELESS LOCATION-BASED APPLICATIONS"
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`P19$MALLIREVO4
`Location Labs Exhibit 1106 Page 1
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`Location Labs Exhibit 1106 Page 1
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`

`

`PROVISIONAL APPLICATION FOR PA TENT COVER SHEET (Small Entity)
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`Given Name (first and middle [if any])
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`Family Name or Surname
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`Residence (city and either State or Foreign Country)
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`lNVENTOR(S)/APPLICANT(S)
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`II'1H1HH
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`Certificate ofMailing by Express Mail
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`
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`this provisional patent application cover sheet,
`I certify that
`provisional patent application and fee is being deposited on
`November 3, 1998
`with the US. Postal Service as
`"Express Mail Post Office to Addressee" service under 37
`CF.
`. 1.10 and is addre
`to the Assistant Commissioner
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`for
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` igitll/LZ
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`‘ents, Washingt
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`.20231.
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`USE ONLY FOR FILING A PROVISIONAL APPLICATION FOR PATENT
`SEND TO: Box Provisional Application, Assistant Commissionerfor Patents, Washington, DC 20231
`P1 QSMALL/REVO4
`
`[Page 2 of 2]
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`Location Labs Exhibit 1106 Page 2
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`Location Labs Exhibit 1106 Page 2
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`

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`DATA FUSION FOR WIRELESS LOCATION—BASED APPLICATIONS
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`FIELD OF THE INVENTION
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`The present invention relates in general to wireless location—based applications and,
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`in particular, to a method and apparatus for use in processing multiple location finding
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`5
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`equipment inputs and making the resulting location information available to wireless
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`location-based applications.
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`BACKGROUND OF THE INVENTION
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`Wireless communications networks generally allow for communication between
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`wireless stations, e. g., wireless telephones (analog, digital cellular and PCS), pagers or data
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`10
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`terminals that communicate using RF signals. In recent years, a number of location—based
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`service systems have been implemented or proposed for wireless networks. Such systems
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`generally involve determining location information for a wireless station and processing the
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`location information to provide an output desired for a particular application.
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`Examples of such existing or proposed applications include emergency or "911"
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`15
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`applications, location dependent call billing and vehicle tracking. In 911 applications, the
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`location of a wireless station is determined when the station is used to place an emergency
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`call. The location is then transmitted to a local emergency dispatcher to assist in responding
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`to the call. In typical location dependent call billing applications, the location of a wireless
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`station is determined, for example, upon placing or receiving a call. This location is then
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`transmitted to a billing system that determines an appropriate billing value based on the
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`location of the wireless station. Vehicle tracking applications are used, for example, to track
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`the location of stolen vehicles.
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`In this regard, the location of a car phone or the like in a
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`stolen vehicle can be transmitted to the appropriate authorities to assist in recovering the
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`vehicle.
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`From the foregoing, it will be appreciated that location—based service systems involve
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`location finding equipment (LFE) and location—related applications. To some extent, the
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`LFEs and applications have developed independently. In this regard, a number of types of
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`LFEs exist and/or are in development. These include so—called angle of arrival (AOA) time
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`delay of arrival (TDOA), handset global positioning system (GPS) and cell/sector equipment.
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`#3 8849
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`Location Labs Exhibit 1106 Page 3
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`Location Labs Exhibit 1106 Page 3
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`The types of equipment employed and the nature of the information received from such
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`equipment vary in a number of ways. First, some of these equipment types, like GPS, are
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`wireless station-based whereas others are "ground—based", usually infrastructure—based.
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`Some can determine a wireless station‘s location at any time via a polling process, some
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`5
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`require that the station be transmitting on the reverse traffic channel (voice channel), and
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`others can only determine location at call origination, termination, and perhaps registration.
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`Moreover, the accuracy with which location can be determined varies significantly from case
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`to case. Accordingly, the outputs from the various LFE's vary in a number of ways including
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`data format, accuracy and timeliness.
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`The nature of the information desired for particular applications also varies. For
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`example, for certain applications such as 911, accuracy and timeliness are important. For the
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`applications such as vehicle tracking, continuous or frequent monitoring independent of call
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`placement is a significant consideration. For other applications, such as call billing, location
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`determination at call initiation and call termination is generally sufficient.
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`Heretofore, developers have generally attempted to match available LFEs to
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`particular applications in order to obtain the location information required by the application.
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`This has not always resulted in the best use of available LFE resources for particular
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`applications. Moreover, applications designed to work with a particular LFE can be disabled
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`when information from that LFE is unavailable, e. g., due to limited coverage areas,
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`malfunctions or local conditions interfering with a particular LFE modality. In addition, the
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`conventional query and response mode of operation between applications and the associated
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`LFEs has resulted in the use by applications of LFE dependent data formats, LFE limited data
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`contents, and single LFE input location determinations.
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`SUMMARY OF THE INVENTION
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`25
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`The present invention is directed to a method and apparatus for using multiple LFE
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`inputs to enhance the location information made available to wireless location-based
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`applications.
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`The invention allows wireless location-based applications access to
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`information based inputs from LFEs of different types, thereby enhancing the timeliness,
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`accuracy and/or reliability of the requested location information. Moreover, in accordance
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`with the present invention, applications are independent of particular LFEs and can access
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`#38849
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`Location Labs Exhibit 1106 Page 4
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`Location Labs Exhibit 1106 Page 4
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`multiple LFEs receiving a location request from a wireless location application seeking LFE
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`independent location data (i.e., location data having a content and format independent of any
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`particular location finding technology) and responding to the location request based on LFE
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`dependent location data. The process implemented by the processing system may further
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`5
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`involve generating and storing LFE independent location data based on the LFE dependent
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`data. The processing system may be resident on the location finding controllers associated
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`with each LFE, on a separate platform and/or the processing system functionality may be
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`distributed over multiple platforms.
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`According to a still further aspect of the present invention, multiple LFE inputs, are
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`utilized to make a location determination regarding a wireless station. The corresponding
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`method involves the steps of receiving a first location input from a first LFE including first
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`location information and first uncertainty information, receiving a second location input from
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`a second LFE including second location information and second uncertainty information and
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`combining the first and second location inputs to provide a combined location input
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`15
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`including combined location information and uncertainty information based on the first and
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`second inputs. Preferably, the first and second inputs include raw location and uncertainty
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`information obtained from LFE measurements prior to aggregation and related processing.
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`One or both of the first and second inputs may constitute partial information, insufficient on
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`its own to yield a location and uncertainty regarding the wireless station within the
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`20
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`requirements of the wireless location application. For example, in the case of LFEs that
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`determine location based on readings obtained from two or more cell sites, a reading from
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`one of the cell sites may be used in conjunction with, e.g., cell sector information to make
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`a location determination.
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`According to another aspect of the present invention, multiple LFE inputs, obtained
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`25
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`at different times from the same or different LFEs, are utilized to derive tracking information
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`improved location determination accuracy. The associated method includes the steps of
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`receiving a first LFE input including first location information and first corresponding time
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`information for a particular wireless station, receiving a second LFE input including second
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`location information and second time information for the wireless station, and using the first
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`30
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`and second inputs to derive tracking information for the wireless station. The tracking
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`information preferably includes information regarding the mobile station's speed of travel
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`#38849
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`" 4 —
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`Location Labs Exhibit 1106 Page 5
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`Location Labs Exhibit 1106 Page 5
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`and direction of travel. This tracking information can be used in conjunction with
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`subsequent LFE inputs for the wireless station to improve location determination accuracy
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`and can also be used to interpolate wireless station location between location determinations,
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`or to project future wireless station locations as may be desired for some applications. It will
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`5
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`be appreciated that this tracking function and other functions are facilitated by the provision
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`of a system for receiving inputs from one or more LFEs, standardizing such inputs with
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`regard to data content and format, and storing such information.
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`In particular, such
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`standardized and stored information can be readily analyzed to yield derivative information
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`regarding wireless station position as well as statistical information for wireless stations of
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`10
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`interest in the service area.
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`A system constructed in accordance with the present invention includes an input
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`facility for receiving inputs from multiple LFEs, a memory such as a cache for storing
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`information from the LFE inputs (e. g., a wireless station identification, a location, a time
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`associated with that location, an uncertainty for that location, and travel speed and bearing),
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`15
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`an interface for receiving location requests from wireless location applications and providing
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`responses to such requests, and a processing subsystem for processing the LFE inputs and
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`location requests. The apparatus may also include a facility for prompting LFEs to make
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`location measurements in response to location requests. Among other things, the processing
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`subsystem may convert the LFE inputs into a standard format, direct storage of data in the
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`20
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`memory, derive tracking or other derivative information from multiple inputs, analyzing
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`stored information relative to received location requests to determine whether the stored
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`information includes information responsive to the requests and selectively directing the
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`LFEs to make location measurements. The system may be resident on a single or multiple
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`platform and the functionality may be spread among multiple applications.
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`25
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`BRIEF DESCRIPTION OF THE DRAWINGS
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`For a more complete understanding of the present invention and further advantages
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`thereof, reference is now made to the following detailed description taken in conjunctions
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`with the drawings in which:
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`Figure 1 is a schematic diagram of a wireless network implementing a location
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`30
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`finding system in accordance with the present invention;
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`#38349
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`‘ 5 —
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`Location Labs Exhibit 1106 Page 6
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`Location Labs Exhibit 1106 Page 6
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`

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`Figure 2 is a schematic diagram illustrating a wireless location—based services system
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`in accordance with the present invention;
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`Figures 3a—3e illustrate various location finding technologies that may be utilized in
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`the context of the present invention;
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`5
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`Figure 4 is a graphical illustration ofthe use ofmultiple LFE inputs to reduce location
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`uncertainty in accordance with the present invention;
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`Figure 5 is a graphical depiction of a location uncertainty analysis in accordance with
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`the present invention; and
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`Figures 6—9 illustrate various wireless location interface signaling sequences in
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`10
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`accordance with the present invention.
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` 15
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`DETAILED DESCRIPTION
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`In the following description, particular embodiments and implementations of the
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`present invention are set forth in the context of an intelligent telecommunications network.
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`It will be appreciated however, that various aspects of the invention are more broadly
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`applicable to other location based services environments.
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`Referring to Figure 1, an intelligent Wireless telecommunications network
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`implementing the present invention is generally identified by the reference numeral 100.
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`Generally, the network includes a mobile switching center (MSC) 112 for use in routing
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`20
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`wireless communications to or from wireless stations 102, an intelligent network platform
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`114 associated with the MSC 112 for implementing a variety of subscriber or network
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`service functions, and a variety of location finding equipment (LFE) systems 104, 106, 108
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`and 110.
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`In the illustrated embodiment, the intelligent network platform is used to run a
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`Location Finding System (LFS)16 in accordance with the present invention and a number of
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`25
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`wireless location applications 118. Although the illustrated location finding system 116 and
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`wireless location applications 118 are illustrated as being resident on the intelligent network
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`platform 114, it will be appreciated that the elements 116 and 118 may be located elsewhere
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`in the network 100, may be resident on separate platforms, or the functionality of each of
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`#38849
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`_ 6 _
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`Location Labs Exhibit 1106 Page 7
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`Location Labs Exhibit 1106 Page 7
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`

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`these elements 116 and 118 may be spread over multiple platforms.
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`In addition, other
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`applications not depicted in Figure 1 may be resident on the platform 114.
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`As shown in Figure 1, multiple LFE systems 104, 106, 108 and 110 may be
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`associated with the network 100. These LFE systems 104, 106, 108 and 110 may employ a
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`5
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`variety of location finding technologies such as AOA, TDOA, GPS and cell/sector
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`technologies. Three of the illustrated systems 104, 106 and 108 operate separate from the
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`MSC. Examples of such systems include infrastructure based systems such as AOA and
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`TDOA and external systems such as GPS. Generally, infrastructure based system such as
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`AOA and TDOA determine the location of a wireless station 102 based on communications
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`10
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`between the wireless station and the cell site equipment of multiple cell sites. For example,
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`and as will be described in more detail below, such systems may receive information
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`concerning a directional bearing of the wireless station 102 or a distance of the wireless
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`station 102 relative to each of multiple cell sites. Based on such information, the location
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`of
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`the wireless
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`station
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`102
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`can be determined by triangulation or
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`similar
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`15
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`geometric/mathematic techniques. External systems such as GPS systems, determine the
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`wireless station location relative to an external system.
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`In the case of GPS systems, the
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`wireless station 102 is typically provided with a GPS transceiver for determining geographic
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`position relative to the GPS satellite constellation. This location information is then
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`transmitted across an air interface to the network 100.
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`20
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`The illustrated cell sector system 110 may be associated with cell site equipment for
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`communicating with the wireless station 102.
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`In this regard, the cell site equipment may
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`include three or more directional antennas for communicating with wireless stations within
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`subsections ofthe cell area. These directional antennas can be used to identify the subsection
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`of a cell where the wireless station 102 is located. In addition, ranging information may be
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`obtained to identify a radius range from the cell site equipment where the wireless station
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`102 is located, thereby yielding a wireless station location in terms of a range of angles and
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`a range of radii relative to the cell site equipment. This cell/sector location information can
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`be transmitted to the LFS 116 via the MSC 112.
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`As shown, the LFS 116 receives location information from the various LFE systems
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`30
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`104, 106, 108 and 110. The nature of such information and handling of such information is
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`described in more detail below. Generally, however, such information is processed by the
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`#38849
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`T 7 —
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`Location Labs Exhibit 1106 Page 8
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`Location Labs Exhibit 1106 Page 8
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`

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`LFS 116 to provide location outputs for use by any of various wireless location applications
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`1 18 in response to location requests fiom the application 1 18. Such applications may include
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`any Wireless location services applications such as 911, vehicle tracking and location-based
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`billing programs.
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`5
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`Figure 2 illustrates a location—based services system 200 in accordance with the
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`present invention. An important aspect of the present invention relates to the operation of
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`the LFS 214 to receive inputs from multiple LFEs 202, 204 and 206 and provide location
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`outputs to multiple applications 226, 228 and 230. In accordance with the present invention,
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`the LFEs 202, 204 and 206 may be based on different technologies, and may therefore
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`10
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`provide different types of location information, in different data formats, with different
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`accuracies based on different signals.
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`iéi
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`A number of different location finding technologies are depicted in Figures 3a—3d for
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`purposes of illustration. Figure 3a generally shows the coverage area 300 of a cell sector.
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`As noted above, the cell site equipment for a particular cell of a wireless telecommunications
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`15
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`system may include a number, e.g., three or more, of directional antennas. Each antenna thus
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`covers an angular range relative to the cell site bounded by sides 302. In the case of a three
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`sector cell, each antenna may cover about 120° - 150° relative to the cell site. In addition
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`the coverage range for the antenna defines an outer perimeter 304 of the coverage area 300.
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`As shown, the range varies with respect to angle defining a somewhat jagged outer perimeter
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`20
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`304. Accordingly, the actual uncertainty regarding the location of a wireless station located
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`in the illustrated cell sector is defined by the coverage area 300. The location determination
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`output from a cell/sector LFE is therefore effectively defined by the coordinates of the
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`coverage area 300.
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`Figure 3b depicts a TOA based LFE. In this case, the wireless station's range from
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`25
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`a cell sector antenna is determined, based on time of signal arrival or signal transit time to
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`within a radius range, e.g., about 1000 meters. Accordingly, the wireless station‘s location
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`can be determined to be within an area bounded by sides 306 (based on the angular range of
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`the cell sector antenna) and inner 308 and outer 310 arcs (defined by the ranging uncertainty).
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`The output from a TOA based LFE is effectively defined by the coordinates of the sides 306
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`30
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`and the axes 308 and 310.
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`#38849
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`_ 8 _
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`Location Labs Exhibit 1106 Page 9
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`Location Labs Exhibit 1106 Page 9
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`

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`An AOA based LFE is generally illustrated in Figure 3c. AOA based LFEs determine
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`the location of a wireless station based on the angle of arrival of signals, generally indicated
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`by rays 312 and 314, from the Wireless station as measured by two or more cell sites 316 and
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`318. Each angle measurement has an angular uncertainty generally indicated by line
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`5
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`segments 320 and 322. Consequently,
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`the uncertainty region for a given location
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`determination is defined by a polygon having 2n sides, where n is the number of cell sites
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`316 and 318 involved in the measurement.
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`Figure 3d illustrates a TDOA based LFE.
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`In TDOA systems, multiple cell sites
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`measure the time of arrival of signals from a wireless station. Based on such measurements,
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`10
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`each cell site can provide information regarding wireless station location in terms of a
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`hyperbola 324 or 326 and an uncertainty, generally indicated by segments 328 and 330. The
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`resulting uncertainty region is defined by a multi—sided region (where each wall is curved)
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`having 2n walls, where n is the number of cell sites involved in the determination.
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`Figure 3e illustrates a GPS based LFE. In GPS systems, the wireless station includes
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`15
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`a GPS transceiver for receiving signals indicating the wireless station's location relative to
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`multiple satellites in the GPS constellation. Based on these signals, the geographic
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`coordinates of the wireless station's location is determined to an accuracy of perhaps 20
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`meters as generally indicated by circle 332. This information is then transmitted to the
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`wireless network across an air interface.
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`20
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`Referring again to Figure 2, each of the LFEs 202, 204 or 206 outputs location
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`information to its respective LFC 208, 210 or 212. The nature of this "raw" LFE output
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`depends in part on the type of LFE involved. For example, in the case of a cell sector system
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`the output may be a sector identifier or coordinates; in the case of a TOA system, the output
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`may be a sector identifier or coordinates and a radius; in an AOA system the output may be
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`angular measurements and corresponding cell site identifiers/coordinates; in TDOA systems
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`the output may define multiple hyperbolae; and in GPS systems the output may be
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`geographic coordinates.
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`The LFCs 208, 210 and 212 collect and aggregate the "raw" location into a standard
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`format which is then sent to the location cache (LC) 220 of the LFS 214 for storage.
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`30
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`Aggregation involves using the raw data to determine a wireless station location and
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`uncertainty. For some LFE systems, such as GPS systems, this process is simple because
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`mmw
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`‘9—
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`iiiiii
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`Location Labs Exhibit 1106 Page 10
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`Location Labs Exhibit 1106 Page 10
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`

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`location coordinates are reported and the uncertainty is known. For other LFE systems,
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`aggregation is more involved. For example, in the case of TDOA, aggregation may involve
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`receiving multiple hyperbola definitions and using these definitions to define a wireless
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`station location and a multi—sided uncertainty region. The LFCs 208, 210 and 212 may be
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`provided by the LFE vendors or their functionality may be incorporated into a subsystem of
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`the LFS 214.
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`In the context of the present invention, it is useful to express the location information
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`in a standard format. Accordingly, the LFCs 208, 210 and 212 or a cooperating subsystem
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`of the LFS 214 associated with the LC 220, may implement a conversion facility for
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`10
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`converting the determined (processed) location information of the LFCs 208, 210 and 212
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`into standardized location information expressed, for example, as geographical location
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`coordinates and an uncertainty radius.
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`In the illustrated embodiment, two dimensional
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`location coordinates are defined (e.g., latitude and longitude) together with an uncertainty
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`radius applied relative to the location coordinates. It will be appreciated that the standard
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`format may allow for altitude coordinates, non-circular uncertainty regions and other
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`parameters.
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`Referring again to Figures 3a—3e, examples of these coordinates and circular
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`uncertainty regions are graphically depicted. In particular, in each case, a location "L" and
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`standardized uncertainty region "C" are geometrically defined such that the standardized
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`20
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`uncertainty region C circumscribes the actual uncertainty region associated with that location
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`finding technology.
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`In this regard, the location L may be defined first (e.g., as the
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`intersection of rays 312 and 314 in Figure 3c) and then the minimum radius circle C may be
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`defined to circumscribe the actual uncertainty region; the standardized uncertainty region C
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`may be defined first (e. g., as the minimum radius circle required to circumscribe the actual
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`25
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`uncertainty region) and then L be defined as the center of the circle C; or any other
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`appropriate geometric solutions/approximations may be employed.
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`This standardized location information is then stored in a database in LC 220.
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`Specifically, the location coordinates for a wireless station and corresponding uncertainties
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`can be stored in a field, in a relational database, or can otherwise be indexed to a wireless
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`3O
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`station identifier, e. g., a cellular telephone Electronic Serial Number/Mobile Identification
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`Number (ESN/MIM. The coordinates and uncertainty may be expressed in terms of any
`
`#38849
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`_ l O _
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`Location Labs Exhibit 1106 Page 11
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`Location Labs Exhibit 1106 Page 11
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`

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`appropriate units. For example, the coordinates may be expressed as latitude and longitude
`
`values in units of 10'6 degrees and the uncertainty may be expressed in units of meters.
`
`The stored, standardized information can be used to perform a number of multiple
`
`input analyses. Three examples of such facilities are generally indicated by the velocity 216,
`
`5
`
`multi-input processing 217 and tracking 218 facilities of LFS 214. The velocity facility 216
`
`involves determining and storing speed information and direction (bearing) information for
`
`a wireless station based on multiple LFE inputs for the station. Because of the standardized
`
`format, such determinations can be easily made relative to inputs from the same or different
`
`LFEs 104, 106 and/or 108. The velocity information can be obtained based on knowledge
`
`10
`
`of the change in position and the change in time (determined by way of the time stamps
`
`associated with the location information) and may be expressed in terms of latitudinal and
`
`longitudinal velocity components in units of meters per second, together with velocity
`
`uncertainty terms. The direction information can be directly obtained from the location
`
`information, or can be based on a ratio of the velocity components, using standard
`
`15
`
`trigonometric principles. It will be appreciated that such speed and direction information
`
`may be useful for a variety of applications such as vehicle tracking.
`
`The multi-input processing facility 217 can be used to improve location accuracy
`
`based on multiple inputs from the same or, more preferably, different LFEs 202, 204 and/or
`
`206. That is, if two locations with two uncertainties can be obtained for a given wireless
`
`20
`
`station at a given time, a reduced uncertainty can be calculated as the overlap of the two
`
`original uncertainties. A complicating factor is that the locations and uncertainties stored in
`
`the LC 220 for a given wireless station will typically not represent location determinations
`
`for the same time. Because wireless stations are generally mobile, an additional element of
`
`uncertainty is introduced.
`
`25
`
`The illustrated multi-input processing facility 217 takes time into account. This is
`
`
`
`accomplished by:
`
`l.
`
`accessing the LC 220 to obtain two (or more) sets of location information for
`
`a given Wireless station;
`
`2.
`
`3.
`
`identifying a location, uncertainty and time for each set of information;
`
`determining a time difference between the times of the information sets;
`
`— l l ‘
`
`Location Labs Exhibit 1106 Page 12
`
`30
`
`#38849
`
`Location Labs Exhibit 1106 Page 12
`
`

`

`4.
`
`calculating an element of location uncertainty associated with the time
`
`difference; and
`
`5 .
`
`applying the calculated element of location uncertainty to the earlier location
`
`information to obtain time translated location information.
`
`5
`
`This time translated location information can then be compared to the later location
`
`information in an uncertainty overlap analysis, as described below, to obtain a reduced
`
`uncertainty.
`
`Various processes can be employed to calculate the additional, time—related element
`
`of location uncertainty. A simple case involves assuming a maximum rate of travel. For
`
`10
`
`example, a maximum rate of travel of 70 miles per hour may be assumed to account for
`
`travel of a mobile phone in a vehicle. The uncertainty associated with an earlier location
`
`determination may then be expanded by a value determined by multiplying the maximum
`
`rate of travel by the time difference between the two measurements to be compared.
`
`Different maximum travel rates may be assumed for different conditions, for example, a
`
`15
`
`lower rate may be assumed for city locations than for suburban locations, a lower rate may
`
`be assumed for peak traffic periods, or a lower rate may be assumed for mobile stations that
`
`are not generally used on fast moving vehicles. Also, wireless station speed and direction
`
`information as described above or other tracking information as described below may be
`
`used to reduce the time—related element of uncertainty.
`
`20
`
`Once such a time translation process has been employed to normalize multiple LFE
`
`inputs relative to a given time, an uncertainty overlap analysis can be implemented. Such
`
`an analysis is graphically illustrated in Figures 4 and 5. Referring first to Figure 4, the
`
`smaller circle represents a location and uncertainty associated with a later LFE input taken
`
`to be at time t]. The larger circle 402 represents a location and uncertainty associated with
`
`25
`
`a time translated location information based on an earlier LFE input taken to be at time to.
`
`Circle 402 is illustrated as having a larger uncertainty than circle 400 to account for the
`
`additional time and travel related element of uncertainty associated with the time translation.
`
`The shaded overlap area 404 represents the reduced uncertainty achieved by using multiple
`
`inputs. That is, statistically, if circle 400 represents a 95% confidence level regarding the
`
`30
`
`position of the station at t1 and circle 402 represents a nearly 95% confidence level regarding
`
`
`
`#33349
`
`— 12 ‘
`
`Location Labs Exhibit 1106 Page 13
`
`Location Labs Exhibit 1106 Page 13
`
`

`

`
`
`the position of the station at t1, the position of the station can be determined to be in the
`
`shaded area 404 with a high level of confidence.
`
`Figure 5 illustrates a mathematical process for combining the original uncertainties
`
`to obtain a more accurate position and uncertainty. Mathematically, the problem is to
`
`5
`
`compute the intersection of the circular uncertainty regions, and express the result as a
`
`location with an uncertainty (e.g., a circular uncertainty circumscribing the intersection
`
`region). To simplify the mathematics, the geometric arrangement of Figure 4 is translated
`
`to provide a first axis (x in Figure 5) that extends through the centerpoints of the circular
`
`uncertainty regions 500 and 502 (generally, the coordinates of the originally determined
`
`10
`
`locations) and an orthogonal axis (y) intersecting the center of the larger (in this case later)
`
`circular uncertainty region 502. The mathematical equations for the boundaries of circular
`
`uncertainty regions 500 and 502 are:
`
`fi+¢=fi
`
`(n
`
`(X-Xo)2 + Y2 = r22
`
`(2)
`
`15
`
`It will be appreciated that the values of r1, r2 and x0 are known as these are the uncertainty of
`

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