Abstract:
A system and method are presented for reducing intermodulation products in a cellular radio system. The cellular radio system constructs and stores modified channel sets that have a reduced measure of homogeneity. The cellular radio system selects a channel number from the appropriate modified channel set in order to support a call. This invention reduces radio interference, thus improving the quality of service and increasing the call capacity of the cellular radio system.

Description:
FIELD OF THE INVENTION 
     This invention relates to cellular radio systems, and in particular relates to reducing intermodulation products by choosing radio channels for use by a specific antenna sector of a base station. 
     BACKGROUND OF THE INVENTION 
     Cellular radio service is expanding at an explosive rate and will be ubiquitous in the near future. Thus, it is important that the radio spectrum be used to provide service for more customers with little or no extra cost to the service provider. In cellular radio service, a predetermined radio frequency is allocated to carry the communication between a user&#39;s cellular telephone and the service provider&#39;s base station (the gateway into the cellular switching center). The spectrum is divided into frequency channels, commonly referred to as “channel numbers,” and are reused by base stations within a service provider&#39;s area. The greater the reuse of frequency channels, the greater the number of cellular radio subscribers that can be simultaneously served. However, one frequency channel cannot be used by two adjacent base stations because each will interfere with one another. While reusing frequency channels more often increases the frequency spectrum efficiency, it also increases the resulting interference. Thus, one skilled in the art balances each factor against the other in order to achieve a compromise. 
     Radio technology has long recognized the problem of intermodulation (IM) products in radio communications systems (including cellular radio systems). The mixing of two sinusoidal signals having different frequencies in a nonlinear system generates IM products that may interfere with other frequency channels, thus degrading signal quality. IM products correspond to the sum and difference frequency components that are attributed to the “heterodyning process.” The heterodyning process is discussed in Carson, Ralph S.,  Radio Communications Concepts,  John Wiley and Sons, 1990, pp. 94-99. Heterodyning does not occur in a completely linear system because no new frequency components can be created. A linear system is a system that has the property of superposition. Superposition means that the output signal of the system resulting from a plurality of input signals can be determined by adding the individual output signals corresponding to each of the plurality of input signals. If the system is not completely linear, new frequency components are created whenever two or more original frequency components exist. 
     To illustrate the hetrodyning problem, assume that the original frequency components are f.sub. 1  and f.sub. 2 . Third-order nonlinear characteristics generate third-order IM products having frequency components of 2f.sub. 1 −f.sub. 2 , 2.sub. 2 −fsub. 1 , f.sub. 1 +2f.sub. 2 , 2f.sub. 1 +f.sub. 2 , 3f.sub. 1 , and 3f.sub. 2 . The IM products corresponding to differences are of greater concern because these are more difficult to filter than those corresponding to sums. As an example, let f.sub. 1  equal 871.920 MHz and f.sub. 2  equal 872.550 MHz. Third-order IM products corresponding to differences are generated at 871.290 MHz (2f.sub. 1 −f.sub. 2 ) and at 873.180 MHz (2f.sub. 2 −f.sub.  1 ). Third-order IM products corresponding to sums are generated at 2617.020 MHz (f.sub. 1 +2f.sub. 2 ), 2616.390 MHz (2f.sub. 1 +f.sub. 2 ), 2615.760 MHz (3f.sub. 1 ), and 2617.650 MHz (3f.sub. 2 ). Higher-order nonlinear characteristics generate higher-order IM products such as the fifth-order and seventh-order IM products. The nth-order IM products have frequency components of pf.sub.  1 −qf.sub. 2  and pf.sub. 2 −qf.sub. 1 , where p+q equals n and p is greater than q. Higher-order IM products have a lesser effect than the third-order IM products because the corresponding signal levels have less amplitude. Even-order IM products are generally ignored because the corresponding frequency components can be filtered. (In the above example, the second-order IM product has a frequency component of f.sub.  1 +f.sub. 2 , which equals 1744.470 MHz. This frequency is sufficiently removed from the spectrum centered around 850 MHz and thus can be easily filtered.) Third-order IM products are typically responsible for the most adverse effects on other IM products. 
     The discussion heretofore specified only two frequency components. If more than two frequency components exist, then each possible pair of all frequency components (channel numbers) must be considered, where the collection of channel numbers is commonly called a “channel set” in the art of cellular radio. If the frequency of an IM product is coincident with a channel number of the channel set, then a “hit” occurs. The total effect is determined by adding the individual effects of each pair. For example, the case in which each of two frequency pairs generate a hit on a given frequency will result in more severe effects than the case in which only one frequency pair generates a hit at the given frequency. Moreover, third-order IM products are also generated by the mixing of three signal components (triplets) having frequencies of f.sub. 1 , f.sub. 2 , and f.sub. 3 , respectively. In such cases, third-order IM products having frequency components of −f.sub. 1 +f.sub. 2 +f.sub. 3 , f.sub. 1 −f.sub. 2 +f.sub. 3 , and f.sub. 1 +f.sub. 2 −f.sub. 3  are the dominant components. Thus, the total effect of third-order IM products is exacerbated by the presence of these components. The subsequent quantitative assessment of third-order IM products includes only the effects of frequency pairs and not frequency triplets. 
     In a cellular radio system, full duplex operation is supported so that communication from the serving base station to the mobile subscriber unit (commonly associated with the downlink) and from the mobile subscriber unit to the base station (commonly associated with the uplink) can occur concurrently. The frequency of the downlink (base station to mobile subscriber unit) is spaced 45 MHz from the frequency of the uplink (mobile subscriber unit to base station). For a given call, the serving base station allocates a transmitter and a receiver. Similarly, the mobile subscriber unit tunes its transmitter and receiver to the frequencies associated with the allocated base station equipment. A channel number is associated with both a transmitting frequency and a receiving frequency. For example, the channel number  22  in the B band of the AMPS spectrum is 870.660 MHz for the base station&#39;s transmit frequency (downlink) and is 825.660 MHz for the base station&#39;s receive frequency (uplink). These frequency assignments are the mobile subscriber unit&#39;s receive frequency (downlink) and transmit frequency (uplink), respectively. 
     IM products are generated if nonlinear characteristics exist at the transmitter, receiver, or structures between the mobile subscriber unit and base station. At the base station, multiple transmitted signals, each having a corresponding frequency value, are combined by an RF combiner or power amplifier so that a common antenna can be utilized. Any nonlinear characteristics of the RF combiner, power amplifier, couplers, filters, duplexers, and cables will also cause signals corresponding to IM products to be transmitted by the antenna. These IM products are detrimental to a call if the frequency of one or more of the IM products is the same as a frequency associated with the call. Even if the RF combiner or power amplifier were completely linear, the receiver of the mobile subscriber unit is exposed to multiple signals having different frequencies. One of the signals corresponds to the frequency associated with the call while the other signals are associated with interference (i.e. calls intended for other mobile subscriber units). If the receiver of the mobile subscriber unit has nonlinear characteristics, IM products are generated. The nonlinear characteristics of the receiver are reflected in the third-order intercept point of the receiver. (Carson, Ralph S.,  Radio Communications Concepts,  John Wiley and Sons, 1990, 94-99.) 
     Furthermore, the generation of EV products can be associated with factors external to the base station and mobile subscriber unit. In fact, any nonlinear junction or device encountered by a signal in its path of propagation can generate IM products. (Boucher, Neil J.,  The Cellular Radio Handbook,  Quantum Publishing, 1995, p. 100.) Typical nonlinear junctions or devices include bolted tower joints, antenna clamps, tower guy wires, metal fences, chains, and light bulbs. Such factors may be very difficult to identify and to eliminate. 
     The discussion heretofore addresses the generation of IM products for the downlink (base station to mobile subscriber unit). However, an analogous discussion can be presented for the uplink (mobile subscriber unit to base station). IM products are detected at the base station&#39;s receiver if nonlinear characteristics exist somewhere in the uplink path or at the base station&#39;s receiver and if a plurality of mobile subscriber units are simultaneously transmitting at different frequencies. Signals associated with these different frequencies can mix at points having nonlinear characteristics to generate IM products. 
     There are several known approaches to diminish the effects of IM products in a radio system. The first approach is to reduce the nonlinear characteristics of the electronic components such as RF power devices. However, this approach may not be technically possible or economically feasible. A second approach is to cancel IM products by inducing a signal that is inverted with respect to the distortions caused by the nonlinear characteristics of the electronic circuitry. Such an approach is suggested by U.S. Pat. No. 5,606,286 that issued to Burns. This approach requires additional complexity in electronic circuitry. The third approach is to utilize only frequencies, which reduce the occurrences of IM products by a frequency planning procedure. U.S. Pat. No. 5,295,138, issued to Greenberg, et al., and assigned to Northwest Starcon Limited Partnership uses this approach. The &#39;138 patent reduces the effects of IM products within a common communication channel bandwidth by determining carrier frequencies, which are contained in this bandwidth, so that the IM products are reduced. The &#39;138 patent adjusts carrier frequencies, performs measurements of the resulting IM products, and readjusts the carrier frequencies based upon the measurement results. The &#39;138 patent addresses a problem encountered in satellite radio systems in which the assigned frequencies are randomly selected and moved in an iterative process to reduce the effects of IM products. This approach, however, does not address situations in which frequencies are reused as with cellular radio systems. 
     Moreover, current literature in the field of cellular radio leads away from the present invention. For example, one reference states that the effects of IM products “will not affect channels within the transmitted band design” when considering the RF combiner with respect to the relatively low amplitudes of the IM products that are generated. (Lee, William C.Y.,  Mobile Cellular Telecommunications Systems,  McGraw-Hill Book Company, 1989, pp. 231-232.) 
     Even though interference in a cellular radio system is a well-recognized problem (any increase of interference will degrade the call capacity of the cellular radio system), the sources of this interference may not be well defined. The current art of cellular radio recognizes co-channel and adjacent channel interference as being two sources of interference. Both co-channel and adjacent interference are addressed by the frequency planning practice detailed in a number of references, e.g. Boucher, Neil, J.,  The Cellular Radio Handbook,  Quantum Publishing, 1995, pp. 116-118. The current state of the art propounds that regularly spaced channel sets, each channel set containing a plurality of channel numbers, be assigned in a repeated fashion to cell sites within the coverage area of the cellular radio system. This approach to frequency planning for a cellular radio system fails to recognize the effects of IM products as a source of interference. 
     SUMMARY OF THE INVENTION 
     The problem of generating intermodulation (IM) products in a cellular radio system is solved and a technical advance is achieved by carefully selecting channels for a channel set assigned to a base station. The invention comprises determining a channel set for each antenna sector of each base station and assigning an appropriate channel number of the channel set to the associated base station equipment and mobile subscriber unit in order to support a cellular call. Channels are chosen so that the measure of homogeneity is reduced, thus reducing the interference associated with IM products. If the call requires a handoff from the serving base station to another base station or from one antenna sector to another antenna sector of the serving base station, the cellular radio system assigns another channel number, as determined by the invention, so that the call can continue by being served by the second base station or by the second antenna sector. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 illustrates a cellular radio system supporting a mobile subscriber unit during a call as supported by prior art; 
     FIG. 2 illustrates a cell cluster for a cellular radio system having a three-sector antenna configuration and a reuse factor of seven as supported by prior art; 
     FIG. 3 a  is a flow diagram showing the logic for implementing the invention; 
     FIG. 3 b  is a block diagram illustrating an apparatus embodiment in accordance with the present invention; 
     FIG. 4 a  illustrates partitioning channel sets into groups so that the modified zig-zag algorithm can be applied; 
     FIG. 4 b  illustrates an example of applying the modified zig-zag algorithm in order to construct a channel set that reduces IM products; 
     FIG. 4 c  shows a flow diagram for the modified zig-zag algorithm; 
     FIG. 5 illustrates a flow diagram of the zig-zag algorithm in order to construct a channel set that reduces IM products; and 
     FIG. 6 illustrates a flow diagram of the randomized algorithm in order to construct a channel set that reduces IM products. 
    
    
     DETAILED DESCRIPTION 
     FIG. 1 illustrates a cellular radio system supporting a mobile subscriber unit  102  during a call. Mobile subscriber unit  102  is located within cell  100  and is served by base station  101 . Base station  101  is approximately located at the center of cell  100 . Base station  101  communicates with mobile subscriber unit  102  over a radio channel  103 , associated with the base station&#39;s transmit frequency (mobile station unit&#39;s receive frequency) and with the mobile subscriber unit&#39;s transmit frequency (base station&#39;s receive frequency). Mobile subscriber unit  102  may move outside cell  100  to either cell  104  or cell  105 . In such a case, mobile subscriber unit  102  will be served by base station  106  or base station  107 , respectively. Base stations  101 ,  106 , and  107  are controlled by mobile switching center  108 . In addition, mobile switching center provides a telephony connection between base stations  101 ,  106 , and  107  in order to complete calls between mobile subscriber unit  102  and public switching telephone network (PSTN)  109 . 
     FIG. 2 illustrates a cell cluster  201  within a cellular radio system  200  having a three-sector antenna configuration and a frequency reuse factor of seven. The service area of cellular radio system  200  is partitioned into cells such as cell  202  and cell  203 . Cell  202  is served by base station  204 , and cell  203  is served by base station  208 . Cell  202  is configured with three antenna sectors: alpha antenna sector  205 , beta antenna sector  206 , and gamma antenna sector  207 . Each cell cluster consists of seven cells in such a configuration. Because the frequency reuse factor is seven, the frequency spectrum is reused by every cell cluster  201 . Each cell requires three channel sets; therefore, a total of twenty-one channel sets are required by cell cluster  201 . Other frequency reuse factors can be employed, but a frequency reuse factor is typically seven for a three-sector antenna configuration. For an omnidirectional configuration, a frequency reuse factor of thirteen is typical. In such a configuration, the radio spectrum is repeated every thirteen cells. Because each cell requires one channel set, a total of thirteen channel sets are required by a cell cluster. The cell clusters, or a portion thereof, are repeatedly deployed in order to expand cellular coverage as needed. 
     The usable frequency spectrum is partitioned into channel sets. Each channel set contains a plurality of channel numbers, each channel number corresponds to both a transmit frequency and a receive frequency. Tables 1, 2, and 3 illustrate an example of the partitioning of a 850 MHz cellular frequency spectrum into channel sets for the A band (limited spectrum). (Boucher, Neil, J.,  The Cellular Radio Handbook,  Quantum Publishing, 1995, p. 117.) The frequency assignments, as shown in Tables 1, 2, and 3, are followed in current practice. 
     
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Purely Homogeneous Frequency Sets for Alpha Sector 
               
             
          
           
               
                 Cell 1 
                 Cell 2 
                 Cell 3 
                 Cell 4 
                 Cell 5 
                 Cell 6 
                 Cell 7 
               
               
                   
               
             
          
           
               
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
               
               
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
               
               
                 64 
                 65 
                 66 
                 67 
                 68 
                 69 
                 70 
               
               
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
               
               
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
               
               
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
               
               
                 148 
                 149 
                 150 
                 151 
                 152 
                 153 
                 154 
               
               
                 169 
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
               
               
                 190 
                 191 
                 192 
                 193 
                 194 
                 195 
                 196 
               
               
                 211 
                 212 
                 213 
                 214 
                 215 
                 216 
                 217 
               
               
                 232 
                 233 
                 234 
                 235 
                 236 
                 237 
                 238 
               
               
                 253 
                 254 
                 255 
                 256 
                 257 
                 258 
                 259 
               
               
                 274 
                 275 
                 276 
                 277 
                 278 
                 279 
                 280 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Purely Homogeneous Frequency Sets for Beta Sector 
               
             
          
           
               
                 Cell 1 
                 Cell 2 
                 Cell 3 
                 Cell 4 
                 Cell 5 
                 Cell 6 
                 Cell 7 
               
               
                   
               
             
          
           
               
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
               
               
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
                 35 
               
               
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
               
               
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
               
               
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
               
               
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
                 119 
               
               
                 134 
                 135 
                 136 
                 137 
                 138 
                 139 
                 140 
               
               
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
               
               
                 176 
                 177 
                 178 
                 179 
                 180 
                 181 
                 182 
               
               
                 197 
                 198 
                 199 
                 200 
                 201 
                 202 
                 203 
               
               
                 218 
                 219 
                 220 
                 221 
                 222 
                 223 
                 224 
               
               
                 239 
                 240 
                 241 
                 242 
                 243 
                 244 
                 245 
               
               
                 260 
                 261 
                 262 
                 263 
                 264 
                 265 
                 266 
               
               
                 281 
                 282 
                 283 
                 284 
                 285 
                 286 
                 287 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Purely Homogeneous Frequency Sets for Gamma Sector 
               
             
          
           
               
                 Cell 1 
                 Cell 2 
                 Cell 3 
                 Cell 4 
                 Cell 5 
                 Cell 6 
                 Cell 7 
               
               
                   
               
             
          
           
               
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
               
               
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
               
               
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
               
               
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                 99 
                 100 
                 101 
                 102 
                 103 
                 104 
                 105 
               
               
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
               
               
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
               
               
                 162 
                 163 
                 164 
                 165 
                 165 
                 166 
                 167 
               
               
                 183 
                 184 
                 185 
                 186 
                 187 
                 188 
                 189 
               
               
                 204 
                 205 
                 206 
                 207 
                 208 
                 209 
                 210 
               
               
                 225 
                 226 
                 227 
                 228 
                 229 
                 230 
                 231 
               
               
                 246 
                 247 
                 248 
                 249 
                 250 
                 251 
                 252 
               
               
                 267 
                 268 
                 269 
                 270 
                 271 
                 272 
                 273 
               
               
                 288 
                 289 
                 290 
                 291 
                 292 
                 293 
                 294 
               
               
                   
               
             
          
         
       
     
     Tables 1, 2 and 3 illustrate channel sets that are assigned to each cell in a cell corresponding to the alpha, beta, and gamma sectors, respectively for a three-sector antenna configuration and a frequency reuse factor of seven. Each of the channel sets in Tables 1, 2, and 3 has a uniform frequency separation equal to 21 (which is equivalent to 21*30 KHz or 630 KHz) between each adjacent channel numbers within a channel set. Each channel set has a uniform distribution in that each adjacent channel number is separated by 21 or multiples of 21. Each channel such set is referred as a “purely homogeneous channel set.” In the context of this discussion, a channel set having this uniform property is referred as a “purely homogeneous channel set” because every entry in the channel set is separated by 21 or multiples of 21. Moreover, several purely homogeneous channel sets can be combined to form another channel set. As an example, consider the B frequency band of the 850 MHz cellular spectrum. The B frequency band is further partitioned into the basic B band (880 MHz to 890 MHz for the base station&#39;s transmit frequency and 835 MHz to 845 MHz for the base station&#39;s receive frequency) and the B&#39; band (891.5 MHz to 894 MHz for the base station&#39;s transmit frequency and 846.5 MHz to 849 MHz for the base station&#39;s receive frequency), each of which corresponds to a purely homogeneous subset. 
     Channels in a homogeneous channel set generates IM products that fall on other channels in the channel set. As known in the art, in each purely homogeneous channel set of Tables 1, 2, and 3 each channel number experiences at least 6 occurrences (“hits”) of third-order IM products (of the form 2f.sub. 1 −f.sub. 2 ) having the same frequency. As an example, there are 6 hits corresponding to channel number  22  in the first channel set of Table 1 (i.e. 2*43-64, 2*64-106, 2*85-148, 2*106-190, 2*127-232, and 2*148-274). Each hit corresponds to an IM product having an associated transmit frequency and receive frequency. Also, other third-order IM products, such as those attributable to frequency triplets, and higher-order IM products are generated. Each occurrence of an IM product produces interference to a given channel number, thus degrading the quality of communication between the base station and mobile subscriber unit. 
     FIG. 3 a  is a flow diagram showing the logic for selecting channels for a channel set according to this invention. In step  300 , one constructs a collection of channel sets that reduces the measure of homogeneity. Several algorithms that provide a reduction of homogeneity will be described. Channel sets that have a reduced measure of homogeneity are referred to herein as “modified channel sets.” In step  301 , modified channel sets are associated with the base stations according to the antenna configuration and the frequency reuse factor. A representation of the calculated modified channel sets is stored into a memory device for later retrieval. The memory device is located at an entity of the cellular radio system such as the base station or a mobile switching center that is associated with the base station. In step  302 , the cellular radio system determines whether a mobile unit is requesting a call or that there is an incoming call for a mobile subscriber unit. In step  303 , the representation of an appropriate modified channel set is retrieved from memory and a channel number is chosen from the modified channel set. The modified channel set is associated with the antenna sector in which the mobile subscriber unit is located when the call is setup. The channel number is allocated to the call. If, during the call, the cellular radio system determines that a handoff is necessary in step  304 , the cellular radio system retrieves the representation of an appropriate modified channel set and assigns a new channel number from this modified channel set in step  305 , which corresponds to the cell and antenna sector that the mobile subscriber unit has moved to. In such a case, the old channel number is relinquished so that the old channel number can be allocated to a different call. In step  306 , the cellular radio systems determines if the call is released, either by the mobile subscriber unit or by the party that was connected to the given mobile subscriber unit during the call. If the call is released, the previously allocated channel number is relinquished in step  307  so that the channel number can be allocated for a different call. 
     In step  300  in FIG. 3 a,  an algorithm transforms purely homogeneous sets, such as shown in Tables 1, 2, and 3, into modified channel sets. The purpose of the algorithm, according to this exemplary embodiment of this invention, is to decrease the degree of homogeneity. In general, the less the homogeneity, the greater the reduction of the number of resulting IM products. 
     FIG. 3 b  shows a block diagram illustrating an apparatus embodiment in accordance with the present invention. Channel processor  350  constructs modified channel sets from purely homogeneous channel sets in accordance with the call flow shown in FIG. 3 a.  Channel processor  350  can be located at the mobile switching center or base station. Alternatively, channel processor  350  can be separate from the cellular radio system so that calculations are determined at a remote location. The results, as determined by channel processor  350 , are stored in memory  351 . Memory  351  can be situated at either the mobile switching center or the base station. Assignment processor  352  appropriately selects a channel number from memory  351  in order to set up a call or to handoff a call. Assignment processor  352  instructs transmitter/receiver  353  to tune to the associated radio frequency. Assignment processor  352  can be situated either at the mobile switching center or at the base station. Transmitter/receiver  353  is located at the base station. 
     FIGS. 4 a,    4   b,  and  4   c  illustrate one possible algorithm for constructing channel sets in a cellular radio system, which is called the “modified zig-zag” algorithm. This algorithm, combined with the logic shown in FIG. 3 a,  presents an exemplary embodiment of the invention. In step  400  of FIG. 4 a,  it is determined if the number of channel numbers remaining in each of the purely homogeneous channel sets is at least  12 . (Initially, the number of remaining channel numbers is equal to all of the channel numbers. Also, each channel set has the same number of channel numbers.) If so, step  402  partitions 12 channel numbers of each purely homogeneous channel set into a group for the corresponding purely homogeneous channel set. These channel numbers are removed from subsequent processing if step  400  is repeated. If step  400  determines that there are less than 12 channel numbers remaining in each purely homogeneous channel set, step  401  appends the necessary number of unused channel numbers so that groups can be formed in step  402 . An equivalent alternative to appending is to truncate partitioning. Step  403  transforms the groups of the purely homogeneous channel sets into corresponding groups of the modified channel set. Step  403  corresponds to FIG. 4 b,  which is described shortly. In step  404 , if any channel number remains in each purely homogeneous channel set, step  400  is repeated. Otherwise, the process is completed in step  405 . 
     In FIG. 4 b,  groups  421 ,  422 , and  423  represent groups of three purely homogeneous channel sets. Each of the purely homogeneous sets is ordered so that channel numbers have progressively larger values. As an example, one can choose preferably three adjacent purely homogeneous channel sets shown in Tables 1, 2, and 3. The modified channel set is derived by processing the three purely homogeneous sets with the modified zig-zag algorithm. FIG. 4 b  shows a modified group  420 , a first group  421 , a second group  422 , and a third group  423  associated with a modified channel set, a first purely homogeneous channel set, a second purely homogeneous channel set and a third purely homogeneous channel set, respectively. First channel number  436  of modified group  420  is the same as first channel number  424  of first group  421 . Second channel number  437  of modified group  420  is equal to second channel number  425  of second group  422 . Third channel number  438  of modified group  420  is equal to third channel number  426  of first group  421 . Fourth channel number  439  of modified group  420  is equal to fourth channel number  427  of second group  422 . Fifth channel number  440  of modified group  420  is equal to fifth channel number  428  of second group  422 . Sixth channel number  441  of modified group  420  is equal to sixth channel number  429  of third group  423 . Seventh channel number  442  of modified group  420  is equal to seventh channel number  430  of second group  422 . Eighth channel number  443  of modified group  420  is equal to eighth channel number  431  of third group  423 . Ninth channel number  444  of modified group  420  is equal to ninth channel number  432  of third group  423 . Tenth channel number  445  of modified group  420  is equal to tenth channel number  433  of first group  421 . Eleventh channel number  446  of modified group  420  is equal to eleventh channel number  434  of third group  423 . Twelfth channel number  447  of modified group  420  is equal to twelfth channel number  435  of first group  421 . This process is repeated for the remaining groups of the purely homogeneous channel sets. 
     Applying the modified zig-zag algorithm to the first three purely homogeneous channel sets in Table 1, the number of IM products is greatly reduced. In this example, the first purely homogenous channel set contains channels  1 ,  22 ,  43 ,  64 ,  85 ,  106 ,  127   148 ,  169 ,  190 ,  211 ,  232 ,  253 , and  274 ; the second purely homogeneous channel set contains channels  2 ,  23 ,  44 ,  65 ,  86 ,  107 ,  128 ,  149 ,  170 ,  191 ,  212 ,  233 ,  254 , and  275 ; and the third purely homogeneous channel set contains channels  3 ,  24 ,  45 ,  66 ,  87 ,  108 ,  129 ,  150 ,  171 ,  192 ,  213 ,  234 ,  255 , and  276 . Applying the modified zig-zag algorithm to these purely homogeneous channel sets, the modified channel set contains channels  1 ,  23 ,  43 ,  65 ,  86 ,  108 ,  128 ,  150 ,  171 ,  190 ,  213 ,  232 ,  253 , and  275 . An analysis indicates that the number of occurrences of third-order IM products (having the form 2f.sub. 1 −f.sub. 2 ) that correspond to each channel number of the modified channel set is 0.26 on average. This can be compared to the case of purely homogeneous channel sets in Tables 1, 2 and 3, in which the number of occurrences is at least 6. When constructing modified channel sets, a number of factors must be considered such as a minimum spacing between channel numbers within the modified channel set, adjacent channel interference, and alternate channel interference. These factors are well known to one skilled in the art. A second and third modified channel sets are constructed in a similar manner by assigning the remaining channels numbers of channel groups  421 ,  422  and  423  to these modified channel sets. The second purely homogenous channel set is associated with first group  421 , the third purely homogeneous channel set is associated with second group  422 , and the first purely homogenous channel set is associated with third group  423  when constructing the second modified channel set. Similarly, when constructing the third modified channel set, the third purely homogenous channel set is associated with first group  421 , the first purely homogeneous channel set is associated with second group  422 , and the second purely homogenous channel set is associated with third group  423 . 
     FIG. 4 c  shows a flow diagram of the modified zig-zag algorithm and provides an alternative presentation of FIG. 4 b.  Steps  460 ,  461 ,  462 ,  463 ,  464 ,  465 ,  466 ,  467 ,  468 ,  469 ,  470 , and  471  presents the determination of channel numbers  436 ,  437 ,  438 ,  439 ,  440 ,  441 ,  442 ,  443 ,  444 ,  445 ,  446 , and  447 , respectively as shown in FIG. 4 b.  When all  12  channel numbers of the modified group are determined, the routine in FIG. 4 c  is exited in step  472 . The process in FIG. 4 c  corresponds to step  403  in FIG. 4 a.    
     The modified zig-zag algorithm is applied to the other purely homogeneous channel sets of Tables 1, 2, and 3 in order to construct a complete collection of modified channel sets. To continue the example, purely homogeneous channel sets Cell 4, Cell 5, and Cell 6 in Table 1 are selected. The process shown in FIG. 4 a  and  4   b  are applied in order to construct two additional modified channel sets. Next, purely homogeneous channels sets Cell 7 (Table 1), Cell 1 (Table 2), and Cell 2 (Table 2) are selected. This process is continued until all purely homogeneous channel sets of Tables 1, 2, and 3 are selected. 
     Utilizing three purely homogeneous channel sets at a time, the modified zig-zag algorithm is amenable to configurations in which the total number of purely homogeneous channel sets is divisible by three. This algorithm can be applied to other configurations in which the total number of purely homogenous channel sets is not divisible by three by utilizing a different number of purely homogeneous channel sets. One skilled in the art will recognize how to apply this algorithm in such cases. 
     FIG. 5 illustrates a flow diagram of an alternative algorithm called the “zig-zag” algorithm. This algorithm can be used when there is an even number of homogeneous channel sets. This algorithm, combined with the logic shown in FIG. 3 a,  presents a second exemplary embodiment of the invention. The zig-zag algorithm is conceptually simpler than the modified zig-zag algorithm; however, the reduction of IM products is not as great as with the modified zig-zag algorithm. A modified channel set is constructed by processing two purely homogeneous channel sets with the zig-zag algorithm. Partitioning into groups is not required as with the modified zig-zag algorithm. As with the modified zig-zag algorithm, the purely homogeneous channel sets are ordered in progressively larger magnitude. The zig-zag algorithm selects channel numbers from first and second purely homogeneous channel sets by progressively incrementing index pointer i. In step  500 , index pointer is set to 1, corresponding to the first channel number. If no channel numbers remain in step  501 , the routine is exited in step  502 . If a channel number remains in the purely homogeneous channel sets as determined by step  501 , step  503  determines that the ith channel number of the modified channel set is equal to the ith channel number of the first purely homogeneous channel set. If step  503  is executed, index pointer i is incremented in step  504 . In step  505 , the ith channel number of the modified channel set is equal to the ith channel number of the second purely homogeneous channel set. Index pointer i is then incremented in step  506 . Step  501  is repeated. Once the first modified channel set is constructed, a second modified channel is constructed by exchanging the roles of the first purely homogeneous channel and the second purely homogenous channel set in steps  503  and  505 . The routine in FIG. 5 is executed a second time. 
     Both the modified zig-zag algorithm and the zig-zag algorithm are examples of deterministic approaches in constructing modified channel sets. Such approaches construct a modified channel set in a fixed fashion. FIG. 6 illustrates the “randomized” algorithm, which reduces the occurrence of IM products by reducing the degree of homogeneity of the modified channel sets as compared to that of purely homogeneous channel sets. This algorithm, in conjunction with the logic shown in FIG. 3 a,  presents a third exemplary embodiment of the invention. In FIG. 6, three modified channel sets are constructed from three purely homogeneous channel sets. The randomized algorithm selects channel numbers from first, second, and third purely homogeneous channel sets by progressively incrementing index pointer i. In step  600 , index pointer is set to 1, corresponding to the first channel number. If step  601  determines that no channel numbers remain in the purely homogeneous channel sets, then the routine is exited in step  602 . If a channel number remains in the homogeneous channel sets as determined by step  601 , step  603  randomly chooses an integer n from 1, 2, and 3. In step  604 , the ith channel number of the first modified channel set is equal to the ith channel number of the nth purely homogeneous channel set. In step  605 , an integer m is randomly chosen from an integer not chosen in step  603 . In the notation shown in step  605 , the integer following the “:” indicates that the specified integer is not considered when randomly choosing an integer. In step  606 , the ith channel number of the second modified channel set is equal to the ith channel number of the mth purely homogenous channel set. In step  607 , an integer p is equal to the integer not chosen in steps  603  and  605 . In step  608 , the ith channel number of the third modified channel set is equal to the ith channel number of the pth purely homogeneous channel set. Index pointer i is incremented in step  609 , and step  601  is then repeated. When this routine has completed, three modified channels have been constructed. In order to construct the next three modify channel sets, this routine is repeated with the next three homogeneous channel sets and so forth. 
     The quantitative assessment of third-order IM products heretofore includes only the effects of frequency pairs and not frequency triplets. However, the benefits of the present invention apply both to frequency triplets and to frequency pairs. Such benefits can be appreciated by one skilled in the art. 
     If a channel set contains a plurality of purely homogenous channel sets, such as the B frequency band of the 850 MHz cellular spectrum, the algorithms discussed heretofore can be applied separately to each purely homogeneous channel set. 
     Cellular radio service is currently applicable to several frequency spectra centered around 850 MHz and around 1.8 GHz and to different radio technologies such as Advanced Mobile Phone Service (AMPS), Time Division Multiple Access (TDMA), Groupe Special Mobile (GSM). One skilled in the art will recognize how to apply these algorithms to different radio spectra and radio technologies. Moreover, these algorithms are applicable to future allocated radio spectra in which the frequency reuse factor is greater than 1. 
     Other algorithms that reduce the degree of homogeneity can be devised by one skilled in the art. Moreover, the approaches described heretofore are applicable to the 850 MHz cellular frequency spectrum, the PCS frequency spectrum, the GSM frequency spectrum, ETACS/TACS frequency spectrum, the PDC frequency spectrum, and other frequency spectra that may be allocated in the future for cellular radio systems. Moreover, current art assigns channel numbers to channel sets strictly using purely homogeneous channel sets. Any deviation from this practice would fall under the claims of this invention. 
     It is to be understood that the above-described embodiment is merely an illustrative principle of the invention and that many variations may be devised by those skilled in the art without departing from the scope of the invention. It is, therefore, intended that such variations be included with the scope of the claims.