Problems 97 3.9 Problems 3.1 Prove that for a hexagonal geometry, the co-channel reuse ratio is given by , where N = i 2 + ij + j 2 . Hint: Use the cosine law and the hexagonal cell geometry. 3.2 If two independent voltage signals, v1(t) and v2(t), are added together to provide a new resulting signal, prove that under certain conditions the resulting signal has the same power as the sum of the individual powers. What are these conditions? What special conditions must apply for this result to be valid when the signals are uncorrelated? 3.3 Show that the frequency reuse factor for a cellular system is given by k/S, where k is the average number of channels per cell and S is the total number of channels available to the cellular service provider. 3.4 If 20 MHz of total spectrum is allocated for a duplex wireless cellular system and each sim- plex channel has 25 kHz RF bandwidth, find: (a) the number of duplex channels. (b) the total number of channels per cell site, if N = 4 cell reuse is used. 3.5 A cellular service provider decides to use a digital TDMA scheme which can tolerate a signal- to-interference ratio of 15 dB in the worst case. Find the optimal value of N for (a) omnidirectional antennas, (b) 120° sectoring, and (c) 60° sectoring. Should sectoring be used? If so, which case (60° or 120°) should be used? (Assume a path loss exponent of n =4 and consider trunking efficiency.) 3.6 You are asked to determine the signal-to-interference ratio (SIR or C/I) on the forward link of a cellular system, when the mobile is located on the fringe of its serving cell. Assume that all cells have equal radii, and that base stations have equal power and are located in the centers of each cell. Also assume that each cell transmits an independent signal, such that interfering signal powers may be added. Let us define a “tier” of cells as being the collec- tion of co-channel cells that are more-or-less the same distance away from the mobile in the serving cell. This problem explores the impact of the cluster size (i.e., frequency reuse dis- tance), the number of tiers used in the calculation of C/I and the effect of the propagation path loss exponent on C/I. (a) What is the average distance (in terms of R) between the mobile on the fringe of the serving cell and the first tier of co-channel cells? (These cells are called the “nearest neighbors.”) How many cells are located in the first tier? Solve for the case of N = 1, N = 3, N = 4, N = 7, and N = 12 cluster sizes. How does the average distance compare to the value of D = QR, where ? (b) What is the average distance (in terms of R) between the mobile on the fringe of the serving cell and the second and third tier of co-channel cells, and how many cells are in the second and third tier of co-channel cells for the cases of N = 1, N = 3, N = 4, N = 7, and N = 12 cluster sizes? (c) Determine the forward link C/I for the following frequency reuse designs: N = 1, N = 3, N = 4, N = 7, and N = 12. Assume that the propagation path loss exponent is four, and evaluate the S/I contribution due to just the first tier and then due to additional outer tiers of co-channel cells. Indicate the number of tiers at which there is a diminishing contribu- tion to the interference at the mobile. (d) Repeat part (c), except now consider a line-of-sight path loss exponent of n = 2. Notice the huge impact that the propagation path loss exponent has on C/I. What can you say Q 3 N = Q 3 N = 03_57_104_final.fm Page 97 Tuesday, December 4, 2001 2:17 PM