Patent Publication Number: US-2006014545-A1

Title: Using multiple receive antenna to determine the location of a transmitter with respect to a receiver in ultra wideband systems

Description:
RELATED APPLICATIONS  
      This application claims the benefit of the filing dates of U.S. Provisional Application No. 60/433,920 entitled “USING MULTIPLE RECEIVE ANTENNAS TO ESTIMATE PROPAGATION DISTANCES BETWEEN TRANSMITTERS AND RECEIVERS IN WIRELESS COMMUNICATIONS SYSTEMS” filed Dec. 16, 2002 and U.S. Provisional Application No. 60/451,506 entitled “USING MULTIPLE RECEIVE ANTENNAS TO ESTIMATE POSITIONS OF IMAGES OF TRANSMITTERS IN A UWB COMMUNICATION SYSTEM” filed Mar. 3, 2003, the contents of each incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION  
      The present invention relates to wireless communications systems and, more particularly, to methods and apparatus for determining the location of a transmitter or an image of the transmitter with respect to a receiver having multiple antennas.  
     BACKGROUND OF THE INVENTION  
      Wireless communication systems such as wireless personal area network systems (e.g. PAN systems) are becoming increasingly popular. PAN systems are based on ad hoc networks. In a typical ad hoc network, the individual nodes within a group of nodes that make up the network are mobile (e.g., portable wireless devices). Routing is performed at the network level and entails having each node maintain routing information about every other node. The nodes can dynamically hand off from one sub-network to another when they move. A good measurement or estimation of the distance between the mobile terminal and the sub-networks is desirable to make the hand off management both effective and efficient.  
      In current wireless communications systems, distances between transmitters and receivers are estimated by measuring the strength of received signals. The measurements, however, may be inaccurate due to unreliable wireless channels.  
      Ultra Wideband (UWB) technology is presently being introduced in radar systems and ad hoc networking. UWB uses base-band pulses of very short duration spread over a wide band of frequencies to spread the energy of transmitted signals very thinly from near zero Hz to several GHz. The techniques for generating UWB signals are well known. UWB technology has been used for military applications for many years. Commercial applications will soon become possible due to a recent decision announced by the Federal Communications Commission (FCC) that permits the marketing and operation of certain new types of consumer products incorporating UWB technology. The key motivation for the FCC&#39;s decision to allow commercial applications is that no new spectrum is required for UWB transmissions because, when they are properly configured, UWB signals can coexist with other application signals in the same spectrum with negligible mutual interference. In addition, the use of UWB in radar systems is expected to provide improvements in resolution.  
      Recently Multiple Input &amp; Multiple Output (MIMO) technology has attracted attentions in wireless applications. MIMO systems use multiple transmitter antennas and/or receiver antennas to achieve diversity gain, spectral efficiency gain and interference suppression. MIMO technology has been proposed to UWB systems to resolve multi-path and multi-user problems in wireless systems. An exemplary MIMO system is described in an article by L. Yang et al. entitled “Space-Time Coding for Impulse Radio,” 2002  IEEE Conference on Ultra Wideband Systems and Technologies , May 2002. MIMO systems, however, are subject to the same limitations as the wireless communication system described above with respect to determining distances between transmitters and receivers.  
      There is an ever present desire for better wireless networks and radar systems. One way of improving these systems is to more accurately determine the location of transmitters with respect to receivers. Accordingly, methods and systems are needed for more accurately determining the location of a transmitter relative to a receiver that are not subject to the above limitations and are compatible with UWB application. The present invention fulfill this need among others.  
     SUMMARY OF THE INVENTION  
      The present invention is embodied in an apparatus, system, method, and computer program product for determining a location of at least an image of a transmitter transmitting a signal. The location of at least the image of the transmitter is determined by receiving a signal transmitted by the transmitter at a plurality of receiver antennas separated by known distances. Differences in time are then determined between receipt of the signal at one of the plurality of antennas and at least two other antennas. The known distances and the determined differences in receipt times are then processed to determine the location of the transmitter. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      The invention is best understood from the following detailed description when read in connection with the accompanying drawings, with like elements having the same reference numerals. When a plurality of similar elements are present, a single reference numeral may be assigned to the plurality of similar elements with a small letter designation referring to specific elements. When referring to the elements collectively or to a non-specific one or more of the elements, the small letter designation may be dropped. Included in the drawings are the following figures:  
       FIG. 1  is a topological diagram showing the relative positions between a receiver and a transmitter.  
       FIG. 2  is a topological diagram which is useful for describing problems associated with reflection of signals in estimating a distance between a receiver and a transmitter.  
       FIGS. 3A and 3B  are topological diagrams which are useful for describing location ambiguity when multiple receiver antennas are used to receive a signal transmitted by a single transmitter.  
       FIGS. 4, 5 , and  6  are topological diagrams showing relative positions of a transmitter and three receiver antennas that are useful for describing the operation of an exemplary embodiment of the invention.  
       FIG. 7  is a topological diagram showing possible positions of a transmitter relative to the three receiver antennas calculated according to the present invention.  
       FIGS. 8, 9 , and  10  are topological diagrams showing possible positions of a transmitter relative to a receiver having four antennas in accordance with the present invention.  
       FIG. 11  is a block diagram of a network in accordance with the present invention.  
       FIG. 12  is a flow chart of exemplary steps for determine a location of a transmitter in accordance with the present invention.  
       FIG. 13  is an illustrative representation of network including sub-networks in accordance with the present invention.  
       FIG. 14  is an illustrative representation of a personal area network system in accordance with the present invention. 
    
    
     DETAILED DESCRIPTION  
       FIG. 11  depicts an exemplary communication system  100  capable of determining the location (e.g., distance and/or position) of a transmitter  102  with respect to a receiver  104 , e.g., signal propagation distances between the transmitter  102  and the receiver  104 , in accordance with the present invention. In general overview, a signal transmitted by the transmitter  102  via a transmitter antenna  106  is received at the receiver  104  via multiple receiver antennas  108   a - c , which have a predefined relationship to one another. A location of at least an image of the transmitter  102  is then determined by processing differences in receive times of the signal by the multiple receiver antennas  108   a - c  and the predefined distances therebetween. If the receiver antennas  108   a - c  are in a single device, or in relatively close proximity to one another, a centralized timer (not shown) may provide necessary timing information. Optionally, if the antennas are relatively far apart, global positioning system (GPS) transmitters  110   a - d  may be used to synchronize local time bases in the receiver  104  and/or predetermine the distances between receiver antennas  108   a - c . The communication system  100  is now described in detail.  
      The transmitter  102  transmits signals through an antenna  106 . As described in further detail below, reflections of the signals (for example, by a wall) result in the transmitter appearing to be located in a different location than where it is physically located. These apparent locations are referred to as images of the transmitter. In an exemplary embodiment, the transmitter is a Ultra Wideband (UWB) transmitter that transmits UWB pulse signals. It is contemplated that in addition to UWB, the present invention may be practiced with essentially any wireless communication system or radar system in which it is desirable to determine the distance or position of a transmitter (or transmitter image) relative to a receiver as long as the wireless communication system can provide adequate timing resolution for the intended application.  
      In an alternative exemplary embodiment, the transmitter  102  is a reflective body (not shown). For example, in a radar system, signals are directed into an area and reflections from reflective bodies (e.g., a boat hull or a human being) within that area are assimilated to determine the location of those reflective bodies. The reflective bodies reflect signals as if they are the transmission source and, thus, behave as a transmitter.  
      The receiver  104  receives signals from the transmitter  102  via the plurality of receiver antennas  108   a - c . The distances between one of the receiver antennas  108  and at least two other receiver antennas  108  is known. In addition, the receiver is configured to associate a respective time at which the signal was received by each antenna. For example, a timer (not shown) within the receiver  104  that is controlled by a processor  112  may be used to determine the respective time, which the processor  104  associates with a particular antenna. As described in further detail below, the processor  112  processes the respective times and the known distances to determine a location of the transmitter  102  with respect to the receiver  104 . In an exemplary embodiment, the receiver  104  is a UWB receiver with the antennas  108  and processor  112  of the receiver  104  configured to process UWB signals. In alternative exemplary embodiments, essentially any wireless communication or radar medium may be used. A suitable receiver for use in the present invention will be understood by those skilled in the art from the description herein.  
      In the illustrated embodiment, there are three receiver antennas  106   a - c , which are substantially in a straight line. In an exemplary embodiment, the receiver antennas are omni-directional antennas. In an alternative exemplary embodiment, one or more of the antennas may be directional antennas. The distances between one of the receiver antennas and each of the other antennas are known. For example, the distances between a first receiver antenna, e.g., receiver antenna  106   a , and each of the remaining antennas  106   b  and  106   c  may be determined. In an exemplary embodiment, the distances between the antennas are fixed at the time of manufacture or deployment. In alternative exemplary embodiments, the distance between antennas may be adjustable or may vary, but are known at the time measurements are made to determine the location of the transmitter.  
      The receiver  104  may be a single receiver with a plurality of antennas  108 . Alternatively, the receiver  104  may be multiple receivers (represented by dashed lines dividing the receiver  104  into three parts, i.e., representing three receivers). If multiple receivers are used, each receiver  104  includes its own processor (further represented by the dashed line passing through the processor  112 . The multiple receivers are desirably synchronized prior to determining transmitter locations. In addition, if the multiple receivers vary in position with respect to one another, the multiple receivers are synchronized prior to determining the distances between the receivers.  
      In an exemplary embodiment, each receiver may include a GPS receiver  114   a - c  for receiving GPS time signals from known GPS transmitters  110   a - d . The GPS time signals may be use to synchronize the local time bases in the receiver(s). In addition, assuming adequate resolution, the processor can determine the distances between the receiver antennas based on GPS location information gathered by each receiver for assembly by a common processor.  
      A conventional display  116  may be coupled to the receiver to present determined location information, e.g., a numeric or graphical representation. For example, in a wireless network, the locations (e.g., distance and/or direction) of a plurality of sub-networks that are based on transmissions received from the sub-networks may be presented to a user to enable the user to select a particular sub-network in the direction the user is traveling. In another example, the location (e.g., position) of a reflective body with respect to the receiver in a radar system may be displayed to a user so that the user may identify the position of the reflective body.  
       FIG. 12  depicts a flow chart  200  of exemplary steps described with reference to  FIG. 11  for determining the location of a transmitter  102  with respect to a receiver  104 .  
      At block  202 , the receiver  104  receives a signal transmitted by the transmitter  102 . The receiver receives the signal at a plurality of antennas  108  that are separated by known distances. As described above, the known distance may be defined at the time the receiver is manufactured or deployed or may be determined by the processor  112 , e.g., based on internal timers (not shown) or based on timing and/or location information received through GPS receivers  114  from GPS transmitters  110 .  
      At block  204 , the processor  112  determines differences in time between receipt of the transmitted signal by the plurality of antennas  108 . When a signal is received at the antennas  108  of the receiver  104 , a respective time for the receipt is associated with the antenna  108  through which it was received. For example, the processor  112  may determine the difference in time between a signal received at a first antenna and each of a second and third antenna. In addition, differences in time between these antennas and other antennas, such as a fourth antenna, may also be determined. In an exemplary embodiment, the times are referenced to a synchronized local time base in the receiver  104 .  
      At block  206 , the processor  112  processes the known distances between antennas and the determined differences in time to find the location of at least an image of the transmitter. If the distance between one of the receive antennas and each of two other receive antennas is known and the difference in time of receipt of a signal by each of the antennas is known, the distance to at least an image of the transmitter may be determined. Greater resolution in determining the distance or in determining the position of the image may be achieved through the use of additional antennas and processing respective times and distances associated with those antennas.  
      As described herein, the transmitter  102  is assumed to be substantially co-located with the transmitter antenna  106 . Thus, determining the location of the transmitter antenna  106  determines the location of the transmitter  102  as well. In addition, where one receiver is used, or multiple receiver that are coupled together, the receiver(s)  104  is assumed to be substantially co-located with the receiver antennas  108 . Thus, determining the location of the transmitter antenna  106  with respect to the receiver antennas  108  effectively determines the location of the transmitter  102  with respect to the receiver  104 . Extending the present invention to encompass situations were the transmitter  102  and/or receiver  104  and their respective antennas  106 ,  108  are not co-located will be understood by those of skill in the art.  
      At block  208 , the processor  112  manages network handoffs or presents location information (e.g., via the display  116 ) based on the determined location. In an exemplary embodiment, the determined location is used for network management, which is described below with reference to  FIG. 13 . In an alternative exemplary embodiment, the present invention may be used in ad-hoc networks, radar systems, or essentially any system in which it may be useful to determine the distance between a transmitter or transmitter image and a receiver.  
       FIG. 13  is an illustrative diagram of an exemplary use of the present invention. In the illustrated embodiment, a mobile transmitter  102 , e.g., a cellular telephone in an automobile, is in communication with a first receiver  104   a , e.g., a cellular telephone tower. The transmitter  102  and the receiver  104   a  together form a first sub-network  150 . As the transmitter  102  moves away from the first receiver  104   a  toward a second receiver  104   b  and a third receiver  104   c , it is desirable for the transmitter  102  to establish a new connection with one of the other receivers to form a new sub-network. In an exemplary embodiment of the present invention, the first receiver  104   a  determines the location of the transmitter  102  based on the receipt times of a signal at each antenna in that receiver  104   a . Based on stored information in the first receiver  104   a , the first receiver  104   a  then determines if the location of the transmitter  102  is closer to the second receiver  104   b  or the third receiver  104   c . Assuming the second receiver  104   b  is determined to be closer, the first receiver  104   a  hands off communication to the second receiver  104   b  (forming a new sub-network  152 ) rather than requiring the exchange of signals between the transmitter  102  and each receiver  104  in the area. Various other embodiments will be understood by those of skill in the art from this embodiment and the remainder of the detailed description.  
       FIG. 14  is an illustrative diagram of another exemplary use of the present invention. In the illustrated embodiment, mobile wireless communication devices  160   a - d , such as cellular telephones or portable computers, are capable of communicating with one another to establish personal area networks (PAN). Each of the wireless communication devices may include an antenna apparatus  162  including at least a first antenna  108   a , a second antenna  108   b , and a third antenna  108   c . At least one of the antennas  108  may be used for transmission and at least three antennas  108  may be used for reception.  
      In an exemplary embodiment, at least one of the communication devices, e.g., communication device  160   a , is able to determine the location of one or more of the other communication devices  160   b - d . In accordance with this embodiment, to determine the location of the other communication devices  160   b - d , communication device  160   a  behaves as a receiver  104  having a plurality of receiver antennas  108  and the other communication devices behave as transmitters  102 . In an exemplary embodiment, communication device  160   a  periodically monitors the location of the other communication device  160   b - d  and establishes a PAN with the communication device that is closest in the direction the communication device is moving. For example, communication device  106   a  may be in a PAN  164  including communication device  160   b . As the communication device  160   a  moves, the communication device determines the location of the closest communication device in the direction it is traveling, e.g., communication device  160   c . The communication device  160   a  may then establish a new PAN  166  including communication device  160   c.    
      Additional technical support for determining the location of a transmitter with respect to a receiver having multiple receiver antennas is now described in further detail. Wireless signals are propagated in air from transmitters, T, to receivers, R. The propagation path may be direct as shown in  FIG. 1  or it may be reflected when obstacles block direct propagation as shown in  FIG. 2 . In  FIG. 1 , the propagation distance is the distance between the transmitter, T, and the receiver, R. In  FIG. 2 , the propagation distance is the distance between R and T″ (i.e., the image of T′ reflected from axis L 2  at point A″, where T′ is the image of transmitter T reflected from axis L 1  at point A′). The total distance between the transmitter, T, and the receiver, R, is the summation of T-A′, A′-A″ and A″-R. This relationship can be expressed by equation (1). 
 
 dist   R-T″   =dist   R-A″   +dist   A″-T′   =dist   R-A″   +dist   A″-A′   +dist   A′-T′   (1) 
 
       FIGS. 3A and 3B  show two possible positions of a transmitter T(xt,yt) relative to a receivers R 1 , R 2 , and R 3 . The difference is whether T(xt,yt) is above R 2  in its y-coordinate, as shown in  FIG. 3A , or below R 2  as shown in  FIG. 3B . By properly choosing the coordinates, the terms xt and yt may both be made positive, as shown in  FIG. 3A  in which T(xt,yt) is always above R 2 .  
      The propagation distance from T to R 1  and R 3  is now calculated. The coordinate system shown in  FIG. 4  is chosen. In  FIG. 4 , T 1  is the transmitter and R 1  and R 3  are two antennas at the receiver. Point A is marked so that dist T1-A =dist T1-R3 .  
      Then a circle can be drawn which has its center is at the location of antenna R 1  and has a radius d 1 , defined by equation (2) 
 
 d   1   =dist   R1-A   =dist   T1-R1   −dist   T1-A   (2) 
 
 In this problem, the values c and d 1  are known. The value c is the distance between R 1  and R 3 , and the value d 1  is the difference in the signal propagation time between the transmitter and the two receiver antennas, R 1  and R 3 . It is also noted that d 1 ≦c. When T 1 , R 1  and R 3  are on a line, or when T 1  is on the y axis with xt=0, d 1 =c. 
 
      The circle can be expressed by equation (3) 
 
 x   2   +y   2   =d   1   2   (3) 
 
 Line l 1  passes through R 3 (0,c) and A(x 1 ,y 1 ). It can be expressed as shown in equation (4).  
             y   =             y   1     -   c       x   1       ⁢   x     +   c             (   4   )             
 
      Point B is the middle point between R 3 (0,c) and A(x 1 , y 1 ), therefore its location is  
         (         x   1     2     ,       c   +     y   1       2       )     .       
 
 Line l 2  is perpendicular to line l 1  at point B. Hence it can be expressed by equation (5)  
                   y   =           x   1       c   -     y   1         ⁢   x     +         c   2     -     x   1   2     -     y   1   2         2   ⁢     (     c   -     y   1       )                       =           x   1       c   -     y   1         ⁢   x     +         c   2     -     d   1   2         2   ⁢     (     c   -     y   1       )                         (   5   )             
 
 Line l 3  passes through R 1 (0,0) and A(x 1 , y 1 ). It can be expressed by equation (6)  
             y   =         y   1       x   1       ⁢   x             (   6   )             
 
 Point T 1  is at the intersection of l 2  and l 3 . Its location can be derived from equations (7) and (8):  
             y   =         x   1       c   -     y   1         ⁢   x   ⁢         c   2     -     d   1   2         2   ⁢     (     c   -     y   1       )                   (   7   )               y   =         y   1       x   1       ⁢   x             (   8   )             
 
      The solution to these equations is given by the equations (9) is:  
             {             x   T1     =           c   2     -     d   1   2       2     ⁢       x   1         cy   1     -     d   1   2                         y   T1     =           c   2     -     d   1   2       2     ⁢       y   1         cy   1     -     d   1   2                           (   9   )             
 
 Because x 1  and y 1  can take any value on the circle described by equation (3), the above solution in (9) is not unique. This can also be illustrated in  FIG. 5 . Moving point A to point A 2  on the circle, T 1  moves to T 2  so that 
 
dist T1-R3 =dist T1-A  
 
dist T2-R3 =dist T2-A2  
 
 which means that propagation difference between T 1  to R 1  and R 3  is the same as that between T 2  to R 1  and R 3  because A and A 2  are on the same circle described by equation (3). A curve can be drawn between T 1  and T 2  to represent every possible location of the transmitter that would result in equal propagation differences between the transmitter to R 1  and to R 3 . It is expected that if another receiver antenna is used, e.g., R 2 , another curve can be drawn that represents every possible location of the transmitter that would result in equal propagation differences between the transmitter to R 1  and to R 2 . The position of the transmitter, or image of the transmitter, is then found at the intersection of the two curves. 
 
      In order to measure differences in propagation time between the transmitter and the receiver antennas, it may be desirable for the antennas to have a well-defined temporal relationship. If each antenna is coupled to a respective receiver which receives its signal separately and the time at which the signal is received is conveyed to receivers coupled to the other antennas, it may be desirable for each receiver to synchronize to a common reference, for example, signals from four or more global positioning satellites. Alternatively, the signals may be received at a single receiver from multiple antennas. In this instance, it may be desirable to measure the signal propagation time from each antenna to the receiver in order to be able to accurately estimate the times at which the various signals are received by the various antennas.  
      Signal propagation from T to R 1  and R 2  is now described in which R 2  is an antenna positioned between R 1  and R 3 . For the sake of simplicity, only R 1  and R 2  are shown in  FIG. 6  while only R 1  and R 3  are shown in  FIG. 5 . Similar to the analysis presented above, it is noted that, with reference to  FIG. 6 , c/2 is the distance between R 1  and R 2 , and d 2  is the difference in signal propagation time between the transmitter and the two receiver antennas. It is also noted that d 2 ≦c/2. When T 1 , R 1 , and R 2  are on a line, or T 1  is on the y axis with xt=0, d 2 =c/2.  
      This circle can be expressed by equation (10) 
 
 x   2   +y   2   =d   2   2   (10) 
 
 Line l′ 1  passes R 2 (0,c/2) and A′(x 2 , Y 2 ). It can be expressed by equation (11)  
             y   =             y   2     -     c   2         x   2       ⁢   x     +     c   2               (   11   )             
 
 Point B′ is the middle point of R 2 (0,c/2) and A′(x 2 ,y 2 ). Therefore its location is  
         (         x   2     2     ,         c   /   2     +     y   2       2       )     .       
 
 Line l′ 2  is perpendicular to line l′ 1  and passes point B′. Hence it can be expressed by equation (12)  
             y   =           x   2         c   /   2     -     y   2         ⁢   x     +           c   2     /   4     -     d   2   2         c   -     2   ⁢     y   2                     (   12   )             
 
 Line l′ 3  passes R 1 (0,0) and A′(x 2 ,y 2 ). It can be expressed by equation (13)  
             y   =         y   2       x   2       ⁢   x             (   13   )             
 
 Point T 2  is at the intersection of l′ 2  and l′ 3 . Its location can be derived from the equations (14):  
             {           y   =           x   2         c   /   2     -     y   2         ⁢   x     +           c   2     /   4     -     d   2   2         c   -     2   ⁢     y   2                         y   =         y   2       x   2       ⁢   x                     (   14   )             
 
      The solution to equation (14) is shown in equations (15):  
             {             x   T2     =       (         c   2     4     -     d   2   2       )     ⁢       x   2         cy   2     -     2   ⁢     d   2   2                           y   T2     =       (         c   2     4     -     d   2   2       )     ⁢       y   2         cy   2     -     2   ⁢     d   2   2                             (   15   )             
 
 Because T 1  and T 2  are in fact the same point in the system, the relationships shown in equations (16) hold.  
             {             x   T1     =     x   T2                   y   T1     =     y   T2                     x   T1   2     +     y   T1   2       =       x   T2   2     +     y   T2   2                       (   16   )             
 
 Substituting (9) and (15) into (16) gives equations (17).  
             {                   c   2     -     d   1   2       2     ⁢       x   1         cy   1     -     d   1   2           =       (         c   2     4     -     d   2   2       )     ⁢       x   2         cy   2     -     2   ⁢     d   2   2                                 c   2     -     d   1   2       2     ⁢       y   1         cy   1     -     d   1   2           =       (         c   2     4     -     d   2   2       )     ⁢       y   2         cy   2     -     2   ⁢     d   2   2                                 (       c   2     -     d   1   2       )     2     4     ⁢       d   1   2         (       cy   1     -     d   1   2       )     2         =         (         c   2     4     -     d   2   2       )     2     ⁢       d   2   2         (       cy   2     -     2   ⁢     d   2   2         )     2                         (   17   )             
 
      Equation (18) follows from the second and third of the equations (17).  
               y   2     =         ⅆ   2       ⅆ   1       ⁢     y   1               (   18   )             
 
 Substituting equation (18) into the second of the equations (17) leads to equation (19):  
                     c   2     -     d   1   2       2     ⁢       y   1         cy   1     -     d   1   2           =       (         c   2     4     -     d   2   2       )     ⁢           ⅆ   2       ⅆ   1       ⁢     y   1           c   ⁢       ⅆ   2       ⅆ   1       ⁢     y   1       -     2   ⁢     d   2   2                     (   19   )             
 
      Equation (19) can be further simplified as equations (20) and (21).  
                     c   2     -     d   1   2       2     ⁢     1       cy   1     -     d   1   2           =       (         c   2     4     -     d   2   2       )     ⁢       d   2           cd   2     ⁢     y   1       -     2   ⁢     d   1     ⁢     d   2   2                     (   20   )                   (       c   2     -     d   a   2       )     ⁢     (         cd   2     ⁢     y   1       -     2   ⁢     d   1     ⁢     d   2   2         )       =     2   ⁢       d   2     ⁡     (         c   2     4     -     d   2   2       )       ⁢     (       cy   1     -     d   1   2       )               (   21   )             
 
 y 1  can be obtained from equation (21) as shown in equation (22):  
               y   1     =       2   ⁢       d   1     ⁡     [         d   2     ⁡     (       c   2     -     d   1   2       )       -       d   1     ⁡     (         c   2     /   4     -     d   2   2       )         ]           c   ⁡     (         c   2     /   2     -     d   1   2     +     2   ⁢     d   2   2         )                 (   22   )             
 
      Equation (23) then follows from equation (22):  
                 cy   1     -     d   1   2       =         d   1     ⁡     [       2   ⁢     c   2     ⁢     d   2       -     2   ⁢     d   1   2     ⁢     d   2       -       c   2     ⁢     d   1       +     d   1   3       ]             c   2     /   2     -     d   1   2     +     2   ⁢     d   2   2                   (   23   )             
 
      Substituting (23) into the left side of the third equation of equations (17) gives a distance, dist, between the transmitter and the receiver that can be described by equation (24).  
                   dist   =         (       c   2     -     d   1   2       )     4     ⁢       d   1   2         (       cy   1     -     d   1   2       )     2                     =           (       c   2     -     d   1   2       )     2     4     ⁢         (         c   2     /   2     -     d   1   2     +     2   ⁢     d   2   2         )     2         (       2   ⁢     c   2     ⁢     d   2       -     2   ⁢     d   1   2     ⁢     d   2       -       c   2     ⁢     d   1       +     d   1   3       )     2                       (   24   )             
 
      The distance is expressed in terms of three known values 
          c—the distance between the receiver antennas R 1  and R 3  and     d 1 —the propagation difference between the transmitter and the receiver antennas R 1  and R 3 .     d 2 —the propagation difference between the transmitter and the receiver antennas R 1  and R 2 .        

      In fact, x T  and y T  can also be calculated by using (3) and (9) as shown in equations (25)  
             {             y   T     =           c   2     -     d   1   2       2     ⁢       y   1         cy   1     -     d   1   2                         x   T     =           c   2     -     d   1   2       2     ⁢           d   1   2     -     y   1   2             cy   1     -     d   1   2                           (   25   )             
 
 When T(x T ,y T ) rotates around the axes R 1 -R 2 -R 3 , a circle is formed shown in  FIG. 7 . The points on this circle have the same distance to R 1 , R 2 , and R 3  respectively. Thus, in this situation, the distance may be determined, but the position of the transmitter T cannot be uniquely determined. 
 
       FIG. 8  is a topology diagram of an alternative exemplary embodiment for more precisely determining the location to include the position of the transmitter T.  FIG. 8  indicates the relative positions of four antenna elements R 1 , R 2 , R 3 , and R 4  according to the present invention, which are coupled to a receiver (not shown). The antennas R 1 , R 2  and R 3  are on the same line and, thus, in the same plane, as shown in  FIG. 8 . In this example, R 1  is separated from R 3  by a distance c, and R 2  which is between R 1  and R 3  is separated from both R 1  and R 3  by a distance c/2. Antenna R 4  is separated from antennas R 1  and R 3  by a distance c 2  and from antenna R 2  by a distance c 1 . The relationship shown in equation (26) may be derived from  FIG. 8 . 
   c   2   2   =c   1   2   +c   2 /4  (26)  
 A coordinate system is selected such that the receive antennas R 1 , R 2  and R 3  and the transmitter T are in the same plane shown in  FIG. 9  so that z T =0. The distance between T and R 1  may be defined by equation (27).  
                     d     T   -   R1     2     =       x   T   2     +     y   T   2     +     z   T   2                   =       x   T   2     +     y   T   2                     (   27   )               
 and distance between T and R 4  may be defined by equation (28).  
                     d     T   -   R4     2     =         (       x   T     -     x   r       )     2     +       (       y   T     -     y   r       )     2     +       (       z   T     -     z   r       )     2                   =         (       x   T     -     x   r       )     2     +       (       y   T     -     c   /   2       )     2     +     z   r   2                     (   28   )               
 The difference, Δ, between d T-R1  and d T-R4  can be derived from equations (27) and (28) as shown in equation (29).  
                   Δ   =       d     T   -   R4     2     -     d     T   -   R1     2                   =         -   2     ⁢     x   r     ⁢     x   T       +     x   r   2     -     cy   T     +       c   2     4     +     z   r   2                     (   29   )               
      In  FIG. 9 , the point R 4 ′ is the image of R 4  on the XOY plane. The relationship shown in equation (30) can be derived from the triangle formed by the points R 4 , R 4 ′ and R 2 . 
 
 x   r   2   +z   r   2   =c   1   2   (30) 
 
 Equations (31) may be derived from equations (29) and (30).  
             {           Δ   =         -   2     ⁢     x   r     ⁢     x   T       +     x   r   2     -     cy   T     +       c   2     4     +     z   r   2                       x   r   2     +     z   r   2       =     c1   2                     (   31   )             
 
 Substituting the second equation of the equations (31) into the first equation (31), results in equation (32).  
             Δ   =         -   2     ⁢     x   r     ⁢     x   T       +     c1   2     -     cy   T     +       c   2     4               (   32   )             
 
 Equations (33), describing the X and Z coordinates of the transmitter, may be derived from equation (32).  
             {             x   r     =         d     T   -   R1     2     -     d     T   -   R4     2     -     cy   T     +     cl   2     +       c   2     /   4         2   ⁢     x   T                       z   r     =     ±         c1   2     -     x   r   2                           (   33   )             
 
      The result shown in equation (33) may be expressed in another way by rotating the coordinate around the y axes, as shown in  FIG. 10  so that: {overscore (x r )}=−c 1  &amp; {overscore (z r )}=0, such that equations (34) hold.  
             ∵     {               x   r     _     =           x   r     ⁢   cos   ⁢           ⁢   θ     +       z   r     ⁢   sin   ⁢           ⁢   θ       =     -   c1                       z   r     _     =           -     x   r       ⁢   sin   ⁢           ⁢   θ     +       z   r     ⁢   cos   ⁢           ⁢   θ       =   0                       (   34   )             
 
 From the second equation (34) it can be seen that  
       θ   =     arctan   ⁢       z   r       x   r             
 
 After this rotation, the new coordinates of T may be expressed as shown in equations (35).  
             {               ⁢         x   T     _     =           x   T     ⁢   cos   ⁢           ⁢   θ     +       z   T     ⁢   sin   ⁢           ⁢   θ       =       x   T     ⁢   cos   ⁢           ⁢   θ                         ⁢         y   T     _     =     y   T                           ⁢     z   T       _     =           -     x   T       ⁢   sin   ⁢           ⁢   θ     +       z   T     ⁢   cos   ⁢           ⁢   θ       =       -     x   T       ⁢   sin   ⁢           ⁢   θ                       (   35   )             
 
 The position of the transmitter, T, relative to the antennas R 1 , R 2 , R 3 , and R 4  can be determined by equations (35) as T({overscore (x T )}, {overscore (y T )}, {overscore (z T )}). 
 
      This invention concerns a mechanism to estimate the location of at least images of transmitters, such as UWB transmitters in UWB communications systems. No line of sight propagation path is required. No transmission from the receivers is needed. In MIMO systems, the same receiver antennas can be used and very limited extra calculation is needed to provide the described location functions. This invention may be used, for example, in UWB ad-hoc networks to improve performance of hand-off management and in other location aware applications.  
      It is contemplated that one or more of the components may be implemented in software running on a general purpose computer. In this embodiment, one or more of the functions of the various components may be implemented in software that controls the general purpose computer. This software may be embodied in a computer readable carrier, for example, a magnetic or optical disk, a memory-card or an audio frequency, radio-frequency or optical carrier wave.  
      In addition, although the invention is illustrated and described herein with reference to specific embodiments, the invention is not intended to be limited to the details shown. Rather, various modifications may be made in the details within the scope and range of equivalents of the claims and without departing from the invention.