Abstract:
A method and system for generating three-dimensional antenna patterns from two-dimensional cross sections. The method involves an estimate ( 1006 ), on a given vertical plane, obtained by rotating a gain value ( 1010 ) from the front of the vertical pattern using the horizontal pattern ( 1004 ) as a weight; a second estimate, which could be on a separate vertical plane, obtained by rotating a gain value ( 1014 ) from the back of the vertical pattern, and a final estimate ( 1206 ) obtained by connecting the first two estimates across their respective planes. The invention yields smooth reasonable surfaces ( 1704 ) that satisfy the vertical and horizontal boundary conditions, exhibits no mathematical artifacts, and improves the accuracy of propagation calculations of radio frequency signals. The method is implemented in a software system ( 1812 ) that provides interactive analysis and visualization capabilities for antenna patterns in three dimensions.

Description:
TECHNICAL FIELD OF THE INVENTION 
     The present invention relates to a method for generating three-dimensional antenna patterns for use in predicting radio frequency signals in wireless communication networks. More specifically, the present invention relates to the extraction of three-dimensional patterns from cross sectional two-dimensional data. 
     BACKGROUND OF THE INVENTION 
     The planning and optimization of wireless communications networks requires accurate propagation models. Propagation predictions are used to estimate quantities such as coverage, serving areas, interference, etc. These quantities, in turn, are used to arrive at equipment settings, such as channel assignments, power levels, antenna orientations, and heights. The goal is to optimize these settings to extract the most capacity and coverage without sacrificing the quality of the network. Thus, it is extremely important to employ a propagation model that is as accurate and reliable as possible. Naturally, the accuracy of the predictions also depends on the quality of the geographical data used as input. 
     There is another important factor that affects the quality of propagation predictions: the accuracy of the antenna radiation pattern used to estimate the spatial distribution of the transmitted RF power. Accurate pattern information is readily available from antenna manufacturers. Unfortunately, such data are usually available only for cross sections at the vertical and horizontal planes. Since a typical calculation involves arbitrary orientations, a full three-dimensional pattern needs to be generated. It is this step that can introduce considerable error as the spatial distribution of the power radiating from an antenna is generated from only two cross sections. Clearly, the generated pattern will not be unique—the only piece of information that we have is that this surface has to match the two patterns when it intersects the vertical and horizontal planes. In fact, there are an infinite number of 3D surfaces that can be shaped so that they agree with the values available for the two cross sections, and therefore, there is no “correct” generated surface. The best one can hope for is a reasonable estimate and the problem then focuses on finding the algorithm that produces the best estimate. 
     From a practical point of view, the idea of only using vertical and horizontal data is very attractive, in spite of the uniqueness problem. For example, a pattern stored at one degree increments would require 360×2=720 measured antenna gains. A full 3D surface at the same resolution would require 360×180=64,800 measurements, a number almost two orders of magnitude larger. To our knowledge, no antenna vendor routinely provides this kind of detail. In a limited number of cases, antenna pattern values are available for a few cross sections in addition to the vertical and horizontal. For those cases, one can use the additional information to validate proposed algorithms for 3D surface generation. For most antenna patterns, however, one would still need to rely on some sort of approximation. 
     As an example of the problem addressed by this invention, a wireless communications link is schematically illustrated in  FIG. 1 . A typical link includes a transmitting base station  104  and a receiving base station  110  located at some distance from each other. These stations have respective antennas  102  and  108  mounted at some height above the local terrain  114 . The quantity of interest is the power that reaches the receiving station  110 . This power is given by (in dB units)
 
 P   r   =P   t   +G   t   −L+G   r ,  (1)
 
where
         P t =transmitter power,   G t =transmitter antenna gain,   L=propagation path loss,   G r =receiver antenna gain.       

     A popular view is that once the transmitted power and the two antennas  102 ,  108  are selected, the propagation problem reduces to evaluating the propagation path loss. The path loss is regarded as the difficult part of the calculation and a considerable amount of effort has focused on improving its predictive accuracy. It is interesting to note that even though the literature is full of papers on how to calculate the path loss, not much work on how to apply the antenna patterns has been reported. However, as can be seen in the equation above, errors in the antenna gain terms can be as important as errors in the path loss, especially if the antennas are directional. 
       FIG. 1  also displays the shape of the antenna patterns  106  and  112  superimposed on the vertical plane defined by the two stations. As can be seen there, these patterns can have a very complex structure with numerous nulls and side lobes. It is also clear that the side lobes can be so sharp that a small error in angle can lead from a peak  116  to a deep null  118 , and vice versa. 
     An antenna pattern is the spatial distribution of the electromagnetic power radiating from an antenna. Typically, the size of the antenna (a couple of meters) is much smaller than the transmitter-receiver distance (a few kilometers) and the antenna can be regarded as a point source. Therefore, it is convenient to analyze a 3D radiation pattern in spherical coordinates, ρ, θ, and φ. In practice, it is desirable to have the θ coordinate defined with respect to the horizontal plane, and therefore, the modified spherical coordinate system shown in  FIG. 2  will be used. Here the origin  202  represents the antenna and the point  204  represents some arbitrary location of interest. The radial coordinate  206  represents the antenna gain G, φ is the standard azimuth coordinate and θ represents the angular elevation relative to the X-Y plane. The antenna is mounted on some vertical physical structure oriented along the Z axis. An advantage of this coordinate system is that θ can also be used to describe the amount of electrical tilt applied to the antenna main lobe, which by default will be oriented along the positive X-axis. 
     A note about terminology: Since the coordinate system used here is similar to the geocentric coordinate system used to describe locations on the surface of the earth and since the unit sphere will be used throughout this paper, it will be convenient to use geographical terminology to describe zones and lines on the surface of the unit sphere. Thus, the equator is defined as the circle, on the X-Y plane, that divides the sphere into northern and southern hemispheres. All points having the same θ form a line called a parallel and all points of the same φ form a meridian line. The prime (φ=0) meridian divides the sphere into east and west hemispheres. Finally, the north and south poles are the points where θ=π/2 and θ=−π/2, respectively. Using this terminology, the horizontal pattern will lie on the equatorial plane and the vertical pattern will lie on the plane defined by the prime meridian. 
     Notice that the patterns supplied by the antenna manufacturer may not conform to this coordinate system, and indeed, the vertical pattern will not conform and the appropriate coordinate transformation will need to be applied. The reason for this is that the patterns are provided as simple tabulated arrays of gain values. Thus, the vertical array will contain values for vertical angles that usually range from 0 to 2π, while the θ coordinate of  FIG. 2  only ranges from −π/2 to π/2. In addition, the direction, clockwise or counterclockwise, will need to be specified. 
     It is assumed that a set of measured vertical and horizontal patterns, g v (θ′) and g h (φ′) respectively, are available, and that they are normalized to unit maximum gain. These two patterns come from measurements tabulated as functions of vertical and horizontal angles θ′ and φ′, respectively. A schematic (circular) representation of a vertical and horizontal pattern pair is shown in  FIGS. 3A and 3B , respectively. These patterns need to be transformed into the coordinate system of  FIG. 2 . The horizontal pattern  304  is placed on the equatorial plane, while the vertical  302  is placed on the plane of the prime meridian. In order to properly use these values in the coordinate system of  FIG. 2 , a mapping of the front half of the vertical pattern to the φ=0 meridian and the back half to the φ=π or meridian must be constructed. In other words, the transformation 
                     θ   ′     =     {           θ   ,             0   ≤   θ   ≤     π   /   2       ,           φ   =   0                 π   -   θ     ,             0   ≤   θ   ≤     π   /   2       ,           φ   =   π                 π   -   θ     ,                 -   π     /   2     ≤   θ   &lt;   0     ,           φ   =   π                   2   ⁢   π     +   θ     ,                 -   π     /   2     ≤   θ   &lt;   0     ,           φ   =   0                     Eq   .           ⁢     (   2   )                 
is applied when accessing the vertical pattern array. The mapping for the azimuthal coordinate is trivial because φ′ and φ are equivalent, so we write φ=−φ′ or φ=φ′, depending on whether the pattern is tabulated in the clockwise or counterclockwise direction. This coordinate mapping allows the display of the vertical and horizontal pattern pair in 3D, as will be shown below.
 
     There are some important points to make before an antenna radiation pattern in 3D space can be generated. In addition to the uniqueness problem, there is a possible ambiguity about the meaning of the horizontal pattern provided by antenna vendors when the vertical pattern is tilted. To illustrate this, consider the measured vertical  404  and horizontal  406  patterns of  FIGS. 4A and 4B . These two patterns correspond to antenna model 1309.17.0007 manufactured by Huber and Suhner. This antenna has an electrical down-tilt of 13 degrees. If the horizontal and vertical patterns are interpreted as cross sections of the 3D pattern surface, the two patterns are expected to coincide at the places where their two planes intersect. In other words,
 
 g   v (θ′=0)= g   h (φ′=0)  Eq. (3)
 
and
 
 g   v (θ′=π)= g   h (φ′=π)  Eq. (4)
 
     It can clearly be seen that in an instance such as  FIG. 5 , since the main lobe of the vertical pattern  502  lies below the horizontal pattern  504  on the X-Y plane, the maximum vertical gain cannot geometrically match the maximum horizontal gain. Thus, at the outset, inconsistencies for electrically down-tilted antennas are observed. This suggests that the horizontal pattern be shifted up or down until it matches the maximum value of the vertical front lobe. In other words, the horizontal pattern for this case might be considered a shaping function and not a cross section of the 3D surface. 
     Finally, it is important to recognize that many of the antenna patterns supplied by antenna vendors will not satisfy the above requirements, especially Eq. (4), even when they are not electrically tilted. The inconsistencies may be due to uncertainties in the measured values, or to gaps in the array of measurements. So, in practice, a technique for generating the 3D surface must be robust enough to tolerate inconsistencies at these two points and not produce shape artifacts. 
     DESCRIPTION OF THE RELATED ART 
     Previous work on this problem consists of two basic approaches: Rotation and interpolation. The first approach, well known to those skilled in the art and mentioned in S. R. Saunders, “ Antennas and Propagation for Wireless Communication Systems, ” Wiley, N.Y., 1999, pp. 65-66, assumes that the pattern is separable into the product of the vertical and horizontal cross sections. In effect, this method is equivalent to taking one of the cross sections and rotating it while using the second cross section as a weight to modulate it, hence the rotation name. The rotation method has been also discussed by Araujo-Lopes, et. al., “Generation of 3D Radiation Patterns: A Geometrical Approach,”  Proceedings of the IEEE Vehicular Technology Conference , May, 2002. It is important to note that when implementing this technique, one must choose either the front or the back lobe for rotation. The front lobe of the vertical pattern is usually used for rotation and the horizontal pattern is used as a weight. 
     A 3D antenna pattern generated with this method is shown in  FIGS. 6A ,  6 B and  6 C, wherein the 3D gain values are represented as a shaded surface  604 , in respective side, top, and bottom views. This shaded surface, and all those that follow below, have been generated by using a grid spacing of one degree in the θ and six degrees in the φ direction. The fact that the horizontal pattern is used as a weight is an important consideration. It means that the horizontal pattern is not attached to the equator, but rather, it is moved up and down the various parallels as needed. This technique yields realistic looking surfaces and is very easy to calculate. Since the horizontal pattern is only used as a modulating weight, this method easily handles a down-tilted main lobe  606  and correctly reproduces the vertical gain on the equatorial plane. Note, however, that it has the disadvantage that only one half of the vertical pattern can be rotated, usually the front lobe, and the other half has to be discarded. This means that the 3D back lobe may or may not match the original vertical cross section. In some cases this represents no problem, but in others it will predict very inaccurate back patterns, especially when the front and the back parts of the pattern have a different number of lobes. As an example,  FIG. 6A  shows dark areas  610 ,  612 , and  614  that represent the parts of the vertical back lobe that are not properly accounted for by the simple rotation of the front lobe. In fact, those several back lobes are entirely missing from the 3D surface. Even though the actual gains involved are weak, the difference of over 12 dB observed between the measured and predicted back lobe can have a profound effect on interference calculations, especially for air-to-ground and ground-to-air links. 
     The second approach involves linear interpolation, discussed by Gil, et. al., in the paper “A 3D Interpolation Method for Base-Station-Antenna Radiation Patterns,”  IEEE Antennas and Propagation Magazine , Vol. 43, April 2001, pp. 132-137. A simpler variation of this method is also briefly discussed by P. J. Marshall, U. S. Pat. No. 6,834,180, entitled “Radio Propagation Model Calibration Software” issued Dec. 21, 2004. There are two antenna software tools that use interpolation, as described in a marketing brochure entitled “Wavezebra 3D Antenna Visualization and Field Analysis,” by Wavecall S. A., of Lausanne, Switzerland, and in the document entitled “AMan Graphical Editor for Antenna—User Reference,” Copyright ©2000, Antennas, Wavepropagation and Magnetics (AWE) of Gartringen, Germany. 
     Interpolation methods estimate the antenna gain values at some arbitrary point in 3D by linearly interpolating between the two cross sections. This method requires that the horizontal pattern be fixed at the equator during the interpolation process, a requirement that, as discussed above, leads to incorrect predictions for electrically down-tilted antennas. This effect is clearly seen in  FIGS. 7A ,  7 B and  7 C, which show the 3D gain values as a shaded surface  704 . This illustration shows that the back lobes  710  and  712  calculated with this method are consistent with the vertical pattern used to generate them. The front lobe, on the other hand, is clearly incorrect as this method attempts to compensate for inconsistent requirements. What is obtained is an artificial lobe  708  that incorrectly predicts higher gain, about 8.6 dB for the θ=0 angle at the front lobe. This error can have a very significant effect when predicting coverage since the down-tilt is effectively cancelled by this artifact. A further artifact introduced by the inconsistent requirements imposed by electrically down-tilted antennas is that the artificial lobe suddenly collapses at φ=0 and creates a notch  708  clearly visible from the top view shown in  FIG. 7B . 
     In addition, there is another serious problem with linear interpolation using polar coordinates—it introduces “heart” shape artifacts, especially when only a few points are available. These shapes,  714 , and  716  are clearly visible in the top and bottom views shown in  FIGS. 7A ,  7 B and  7 C. In this case, if a one degree resolution is requested,  358  interpolated values would be generated from only two measured points. Consider interpolation along a parallel as an example. The values are generated by interpolating between two points on the vertical pattern, one at the front lobe, φ=0, and the other at the back lobe, is φ=π. For illustrative purposes one can assume the following sequence of vertical gain pairs, (1, 0), (1, ¼), (1, ½), (1, ¾) and interpolate. In this notation for a pair (f, b), f represents the antenna gain at the front lobe, and b represents the gain at the back lobe. 
     The interpolation results are summarized in  FIG. 8  which shows the linear interpolation weights  802 ,  804 ,  806 , and  808  used and in  FIG. 9 , which shows the corresponding interpolated values  902 ,  904 ,  906 , and  908 . In the trivial case where both points are identical, a circle would be obtained. However, when the two points differ, the interpolation formula attempts to go from one circle radius to the other, resulting in the shapes shown in the figure. The effect will be more pronounced the larger the difference between the front and back vertical pattern values. Any method to generate 3D patterns by interpolation will be subjected to this kind of undesirable artifact. 
     SUMMARY OF THE INVENTION 
     Accordingly, a need exists for a method and system for generating three-dimensional antenna patterns for use in accurately predicting radio frequency signals. There is also a need for a software system that manages all the related pattern information and displays both the 2D and the 3D patterns. 
     An object of the present invention is to provide a simple, robust, self-consistent method and/or a corresponding system for estimating three-dimensional antenna radiation surfaces from cross-sectional slices. 
     The method should provide smooth, reasonable surfaces that satisfy the vertical and horizontal plane boundary conditions and exhibit no mathematical artifacts. 
     According to a preferred implementation of the invention, a 3D surface estimate of an antenna radiation pattern is generated using a hybrid approach—elements of a rotation technique and elements of an interpolation technique, are combined in a way that is designed to mitigate their disadvantages. 
     More particularly, the method starts with antenna gain values such as those taken from a vertical plane pattern and a horizontal plane pattern. The method then continues by obtaining a first estimate by rotating a gain value from a front portion of the vertical pattern, and then obtaining a second estimate by rotating a gain value from a back portion of the horizontal pattern. A final estimate is then obtained by interpolating between the first and second estimates. 
     As an optional step, the resulting grid points from the final estimate may be used to estimate the 3D surface. 
     According to one preferred embodiment, the method for generating 3D antenna surfaces is implemented as a software system that provides interactive analysis and visualization capabilities. Such a system may optionally provide a database to contain the 2D antenna pattern information and the ability to edit the 2D antenna gains used in the calculation. In addition, such a system may also provide detailed output views of the generated 3D surfaces. 
     According to another embodiment, the method for generating 3D antenna surfaces is implemented as an executable software library that can be invoked by wireless network planning tools, or for that matter any software program that employs wireless propagation calculations. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and other objects, features and advantages of the invention, which are not meant to be limiting to the invention, will be apparent from the following more particular description of preferred embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. 
         FIG. 1  is a schematic representation of a wireless link. 
         FIG. 2  illustrates the coordinate system used to represent 3D antenna radiation patterns. 
         FIGS. 3A and 3B  illustrates the coordinate system used to represent the 2D vertical and horizontal patterns in 3D space. 
         FIGS. 4A and 4B  show vertical and horizontal radiation patterns for a specific antenna, model Huber and Suhner 1309.17.00007. 
         FIG. 5  shows vertical and horizontal cross sections of the radiation pattern of antenna model Huber and Suhner 1309.17.00007 in 3D space. 
         FIGS. 6A ,  6 B, and  6 C show side, top, and bottom views, respectively, of the 3D surfaces for the Huber and Suhner 1309.17.00007 antenna generated by the prior art rotation method. 
         FIGS. 7A ,  7 B, and  7 C show side, top, and bottom views, respectively, of the 3D surfaces for the Huber and Suhner 1309.17.00007 antenna generated by the prior art interpolation method. 
         FIG. 8  shows interpolation weights for linear interpolation along a parallel using two known points with values at azimuths of 0 and 180 degrees. 
         FIG. 9  shows linear interpolated curves calculated for several point pairs using the weights of  FIG. 8 . 
         FIG. 10  shows a schematic representation of the new algorithm for generating 3D radiation patterns. 
         FIG. 11  is a flow chart of a method for generating a 3D antenna surface. 
         FIG. 12  shows a schematic representation of the transition between the two estimated results of  FIG. 10 . 
         FIG. 13  shows a top view of the transition between the two estimated results of  FIG. 12 . 
         FIG. 14  shows interpolation weights for cubic interpolation along a parallel using two known points with values at azimuths of 0 and 180 degrees. 
         FIG. 15  shows cubic interpolated curves calculated for several point pairs using the weights of  FIG. 14 . 
         FIG. 16  is a flow chart of a method for generating a single-point 3D gain. 
         FIGS. 17A ,  17 B, and  17 C show side, top, and bottom views, respectively, of the 3D surfaces for the Huber and Suhner 1309.17.00007 antenna generated as by the method of the present invention. 
         FIG. 18  shows an exemplary software system for the 2D and 3D analysis of antenna patterns. 
         FIGS. 19A ,  19 B,  19 C, and  19 D show topography and geographical ground-to-ground coverage maps according to the prior art methods and the method of the present invention. 
         FIGS. 20A ,  20 B, and  20 C show geographical ground-to-air coverage maps according to the prior art methods and the method of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       FIG. 2  shows the preferred geocentric coordinate system used with the present invention. computational grid is prepared by specifying a θ and a φ spacing and a radius value of unity. Initially, the grid is simply the unit sphere, i.e., an isotropic antenna. As the calculation proceeds, the various radial distances on this grid will be replaced by the gain values calculated and the result will be a surface that describes the shape of the antenna pattern. For convenience, the angular spacing is usually uniform, but it can be totally arbitrary. 
     Although a geocentric coordinate system is preferred because it simplifies the mathematical derivation, any 3D coordinate system may be employed.  FIG. 10  shows, as an example, and without loss of generality, a horizontal pattern  1004  shaped like an ellipse, placed on the X-Y plane, and an arbitrarily shaped vertical pattern  1002 , placed on the Y-Z plane. The vertical pattern is shown as a vertically hatched shape. 
       FIG. 11  shows a flow chart that outlines the 3D surface generation method  1102 . The method begins with the construction, in step  1104 , of a grid that will contain the antenna gain values such as the grid shows in  FIG. 2 . From this grid, a vertical angle, θ P , is selected in step  1106 . This vertical angle is mapped into the θ′ coordinate system. This leads to two known gain values: One value  1012  at φ=0, on the front lobe, and another  1016  at φ= 90  , on the back lobe. The point on the back lobe is actually tabulated at π−θ P  if θ P  is positive, or 2π+θ P  if θ P  is negative. If the tabulated vertical pattern does not provide a gain at the selected value θ P , it is simply estimated by interpolating between neighboring entries. For simplicity, it is assumed that θ P  is positive in the discussion that follows. 
     In step  1108  a curve  1006  with a set of estimates on the horizontal plane is constructed by scaling and translating the horizontal pattern  1004  along the Z axis in such a way that its φ=0 gain matches g v , (φ P ), which is shown as point  1010  in  FIG. 10 . Equivalently, this step can be viewed as a rotation of the vertical gain  1010  using the scaled horizontal pattern as a weight. This results in an estimate of the gains for this plane as the φ coordinate is swept around. In this step the horizontal pattern  1004  is regarded as a template whose shape is to be replicated at every horizontal plane. This is a key aspect of the present approach, because the horizontal pattern is no longer viewed as a set of points that must lie on the X-Y plane, the equator. Instead, they can lie on any plane defined by any parallel. This point of view is essential when dealing with vertical patterns with electrical down-tilt. 
     Mathematically, the scaling and translation of step  1108  is expressed as
 
 G   R (θ P , φ)=[ g   h (φ)/ g   h (0)] g   v (θ P ),  Eq. (5)
 
where G R (θ P , φ) is the intermediate result of the rotation, and the term in brackets represents the shape of the horizontal pattern normalized to the gain at boresight. For the rare case where the bore sight horizontal gain is close to zero, the horizontal pattern can be normalized with respect to the maximum gain found in the horizontal pattern array, and use the following equation,
 
 G   R (θ P , φ)=[ g   h (φ)/ g   h   max   ]g   v (θ P ),  Eq. (6)
 
where g h   max  represents the maximum the maximum horizontal gain.
 
     In step  1110  a second horizontal plane is constructed at the point defined by the vertical gain  1014  at the back lobe. Point  1014  is found by examining the array of vertical gains and locating the one that corresponds to angle θ P  on the back lobe. In general, this gain will not match the gain previously obtained from the front lobe, which means it will lie on a separate plane. Again, a scaled version of the horizontal pattern  1008  is placed on this plane, but this time the scaling is done so that the φ=π horizontal gain matches the vertical gain on the back lobe. 
     In step  1112  a transfer function that smoothly goes from one plane to the other is constructed so that as the vertical gain on the front lobe is rotated, it smoothly makes a transition to the second plane on the back lobe. This transition is schematically illustrated by shape  1206  in  FIG. 12 , which shows the two estimates on the two planes and the final estimate that bridges these two planes. The transition shape  1206  is actually a distorted version of the horizontal pattern. It has been distorted so that it agrees with the corresponding vertical gains at both the front  1208  and back  1212  lobes. In general, this transition shape will need to go from one horizontal plane to another. A possible technique for arriving at this transfer function is to calculate what Eq. 5 predicts at the back lobe and compare with the actual value g v (π−θ P ) from the vertical pattern. This is accomplished with the aid of  FIG. 13 , which represents a view of  FIG. 12  from the top, looking down the Z axis toward the origin. In this illustration, the pattern generated by rotating the front-lobe gain  1308  is shown as curve  1302 . Similarly, the pattern obtained by rotating the back lobe gain  1310  is shown as curve  1304 . The objective here is to smoothly transition from curve  1302  to a curve such as  1306  that agrees with both the front and back gain values  1308  and  1310 . To obtain curve  1306 , the shape of estimate  1302  needs to be corrected by the amount  1312  shown as a shaded area in  FIG. 13 . The gain at the back lobe is corrected first, resulting in 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     This expression only applies to the φ=π point. To generalize this correction to all φ we need to attenuate this correction as we go from φ=π back to φ=0. One possible way to do this is to use a linear attenuation function, or given that linear interpolation is subject to “heart” shape artifacts, one could use a higher order interpolation function with a smoother transition. In either case, using the notation W φ , to denote this transition function, we obtain
 
Δ(θ P , φ)= W   φ   {g   v (π−θ P )−[ g   h (π)/ g   h (0)] g   v (θ P )}  Eq. (8)
 
     This correction is shown as the diagonally shaded area in  FIG. 13 . 
     Finally, we add this correction to G R (θ P , φ) to arrive at the hybrid result
 
 G   New (θ P , φ)= G   R (θ P , φ)+ W   φ   {g   v (π−θ P )−{ g   h (π)/ g   h (0)} g   v (θ P )}  Eq. (9)
 
Or, equivalently,
 
 G   New (θ P , φ)=[ g   h (φ)/ g   h (0)− W   φ   g   h (π)] g   v (θ P )+ W   φ   g   v (π−θ P )   Eq. (10)
 
     The formula of Eq. (9) can be viewed as a rotation of the front lobe with a correction to provide the correct value as we approach the back lobe. The alternate formula of Eq. (10) can be viewed as an interpolation between the front and back vertical gains, using a new set of interpolation weights that correctly make the transition from the front to the back lobe. Note that the new interpolation weights incorporate the horizontal pattern itself—a new result. 
     This derivation applies to positive θ P  angles, i.e., for the northern hemisphere. For the southern hemisphere the gains are calculated according to 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                   . 
                   
                       
                   
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                     ( 
                     11 
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     As indicated in step  1114  this process is repeated for all other θ P  angle values in the grid. Each θ P  angle leads to a modified version  1306  of the horizontal pattern. When done with all vertical angles, the final step  1116  is to connect the grid points as triangular or quad surface elements to form a surface for graphical display. 
     The choice of the transition function W φ is arbitrary, the only restriction being that it has to have a value of unity at φ=π or and zero at φ=0, and monotonically go from one to the other. Since this function is being applied to a correction, which in many instances has a small value, a simple linear function works well, with few, if any, “heart” shape artifacts. Thus, in one aspect of the invention, the linear transition function 
                     W   φ   linear     =     {             φ   π     ,           0   ≤   φ   ≤   π                     2   ⁢   π     -   φ     π     ,           π   &lt;   φ   ≤     2   ⁢   π                       Eq   .           ⁢     (   12   )                 
is employed.
 
     Another type of transition function can be obtained if the further condition that the slope vanish at φ=π and φ=0 is required. The advantage of this requirement is that it softens the sharp discontinuities  810  and  812  of  FIG. 8 . Natural choices would be a function such as (1+cos(φ))/2 or a simpler cubic function. The latter is selected here as an example. Thus, in another aspect of the invention, the cubic function
 
 W   φ   cubic =3( W   φ   linear ) 2 −2( W   φ   linear ) 3    Eq. (13)
 
     can be selected to model the transition. Here W φ   linear  represents the linear function of Eq.(12). 
       FIGS. 14 and 15  display the cubic weights and their corresponding effect on interpolating along the φ coordinate. As can be seen there, the transition is much smoother, and the “heart” artifacts are greatly diminished, except for the extreme case of interpolating between unity gain and zero. 
     According to another aspect of the invention, a fast, single-point 3D antenna gain calculation is streamlined for direct use in wireless propagation applications. A typical scenario is illustrated in  FIG. 1 , where antenna gain values are required along the line of sight between the two antennas. The flow chart of  FIG. 16  outlines the calculation process. In step  1604  some calling program specifies the (θ, φ) direction along which the antenna gain is to be calculated. In step  1606  the gain from the front lobe at angle θ is used to estimate the gain at the back lobe by rotating the front lobe gain by π degrees. Then, in step  1608  we compare with the actual value on the back lobe and generate a correction. This correction on the back lobe is then used in a transition formula that generalizes the correction to arbitrary φ angles to generate a gain estimate as shown in step  1610 . Thus, for specific arbitrary orientations θ, φ the 3D gain is calculated according to 
     
       
         
           
             
               
                 
                   
                     
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                     ( 
                     14 
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     Notice that since the gain is basically given in terms of a rotated pattern plus a simple correction, the extra computational effort is minimal when applying this technique. 
     The present invention also works when the two slices are not orthogonal. In this case, one of the slices, the one that would play the role of the horizontal pattern, would be placed on the equatorial plane and the other one, which plays the role of the vertical pattern, would be placed on a plane at the appropriate angle with respect to the horizontal. Even though generation of 3D antenna surfaces works best when the two slices are orthogonal one can still apply the method described here, except that the rotation axis is no longer the Z axis, but the axis of the slanted vertical pattern. Surface construction information is lost as a fiction of deviations from orthogonality, with no surface possible when the two slices become parallel. The present method, however, would attempt to construct the best estimate it can with the available information for moderate deviations from orthogonality. 
     For the case of more than two slices, the present method would be applied sequentially. Thus, if two orthogonal vertical cross sections are available, instead of treating points at φ=0 and φ=π, the method would be applied to the 0 to π/2 range first, then to the π/2 to π range, and so on. 
     A further advantage of the present method is that the vertical and horizontal cross sections can be swapped and the same results are obtained, rotated by 90 degrees. This is certainly not the case for the methods of the previous art. 
       FIG. 17  shows three views of the results produced by the present invention.  FIG. 17A  presents a side view of the 3D gains,  1704 , as a shaded surface. It can be clearly seen that all the features of the vertical pattern used to generate this surface are faithfully reproduced. The back lobes  1710  and  1712  are clearly visible, the main front lobe shows the correct down-tilt  1706 , land the gain,  1708 , on the horizontal plane displays the correct value. Similarly, the top and bottom views, shown in  FIGS. 17B  and  FIG. 17C , respectively, are consistent with the shape of the horizontal pattern. 
     As an example of a possible application,  FIG. 18  shows a sample screen from a software tool  1812  that implements the full surface methods of this invention. It contains a graphical user interface  1802 - 1812  that allows a user to access an antenna database  1804 , browse the specifications of the antennas in the database  1806 , tabulate the raw antenna data  1808 , and display both the input vertical and horizontal patterns,  1810 , and the calculated 3D surface  1812 . This is only one example of an antenna analysis system that can be built around the methods of the present invention. 
     As an example of the application of the single-point 3D gain calculation,  FIG. 19  shows the results of a wireless ground-to-ground propagation coverage calculation using the present invention along with the two prior-art methods for comparison. The calculation involves propagation calculations using realistic terrain and land cover for a test base station. In this example, areas where the received signal strength is greater than or equal to −95 dBm are considered covered.  FIG. 19A  shows the transmitting base station  1904 , which has the Huber and Suhner antenna previously mentioned mounted at a height of 25 meters and pointed along the East direction. The receiving antenna is assumed to be isotropic and mounted at a height of 1.5 meters, a typical mobile wireless user. This illustration also shows the terrain features in the surrounding area  1902 , as well as some roads  1906  and some neighboring base stations  1908 . For clarity, the terrain features will be omitted from the display of the coverage maps of  FIGS. 19B-19D . All calculations that follow use the same propagation model and input data, they only differ in the method used to calculate the antenna gain. 
       FIG. 19B  shows the results,  1910 , in dark gray shading, of the geographical coverage calculated using the method of the present invention to calculate the antenna gain. The coverage area is calculated to be 25.6 km 2 . For comparison, results  1912 , obtained through the use of the rotation method are shown in  FIG. 19C . As expected from the small vertical angles involved and the similarities between their corresponding shapes at those angles, the rotation results, with coverage area of 27.1 km 2 , are very similar to those of the present invention. Next,  FIG. 19D  shows the results obtained through a prior art simple interpolation method,  1914 , which clearly displays a much larger coverage area, 47.9 km 2 . The reason for this is that the interpolation artifact,  708  of  FIG. 7A , effectively cancels out the built-in electrical down-tilt and predicts much larger gains along the horizontal plane. This kind of error is intolerable in network planning calculations. 
     Thus, it seems like the rotation and the new methods provide similar results, and that is indeed the case for this particular antenna pattern. However, as pointed out in the discussion of  FIG. 6A , the missing back lobes can lead to erroneous predictions in the back direction. As a final example,  FIG. 20  compares the coverage results for the three methods when a receiving antenna has a height of 1,730 meters, i.e., for an aircraft. This ground-to-air scenario and particular height has been selected to simultaneously probe one of the back lobes,  1710 , and the secondary front lobe  1714 .  FIG. 20A  shows the geographical coverage area,  2002  calculated with the method of the present invention. The coverage area is calculated to be 30.7 km 2 . The coverage area predicted by the rotation method, shown in  2004  of  FIG. 20B  looks similar to  2002  in the direction of the front lobe, but underestimates the coverage,  2006 , in the back lobe. The coverage area for this case is 26.9 km 2 . Finally, in  FIG. 20C , the interpolation results,  2008 , clearly overestimate the coverage, predicting a coverage area of 43.0 km 2 . The main reason for this behavior is the blunt shape  714  predicted by the interpolation method as shown in  FIG. 7A . The interpolation method fails to reproduce the much sharper secondary lobe  416  of the input vertical pattern shown in  FIG. 4 . 
     While this invention has been particularly shown and described with references to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims.