# Performance of TD-CDMA systems during crossed slots

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3 Let be the CIR that can be reached by all active mobiles during a time slot n, hence: Γ i,j,n i S j,n, j Π, (5) where Γ i,j,n is the estimated CIR of mobile i. A. Downlink-cells The path gain between two mobiles i and k is denoted G k,i. From equation (4), the power transmitted by mobile k of cell l is C B,l,n /G k,l. The CIR of a mobile i in a downlink-cell j is given by: Γ i,j,n = α i,ng i,j P T,j,n (6) I d,i,n where I d,i,n is the total interference power received by mobile i during time slot n: I d,i,n = β d G i,j (1 α i,n ) P T,j,n + G i,l P T,l,n l Π IntraI d,n {j} InterID + C B,l,n, (7) l Π u,n G k,i G k,l InterIU where IntraI is the intracell-interference power, InterID is the intercell-interference power received from neighboring base stations active in downlink and InterIU is the intercell-interference power received from all the neighboring mobiles active in the uplink. By substituting the value of Γ i,j,n from (6) in (5) and using (3), we obtain: α i,n P T,j,n β d (1 α i,n ) P T,j,n + Z i,l P T,l,n l Π d,n {j} + G k,i C B,l,n (8) G i,j G k,l l Π u,n By making the sum over all mobiles of cell j and adding P T,j,n to both sides of the last inequality, we obtain: 1+ P T,j,n P T,j,n + I d,j,n (9) where I d,j,n is given by: I d,j,n = β d (N j,n 1) P T,j,n (1) + Z i,l P T,l,n l Π d,n {j} i S j,n + G k,i C B,l,n. l Π u,n i S j,n G i,l G k,l where N j,n is the number of active mobiles in cell j during slot n. If the intracell and intercell CIR balanced power control is used, the set of relations (9) in all cells gives the same achievable CIR as in [1] which is the maximum achievable CIR, because there is only one solution for (5) [9]. B. Uplink-cells The CIR of a mobile i in an uplink-cell j is: Γ i,j,n = C B,j,n, (11) I u,j,n where I u,j,n is the total interference power corresponding to mobile i during time slot n. This interference power is thesameforallmobilesofcellj: I u,j,n = β u (N j,n 1) C B,j,n + G j,l P T,l,n l Π IntraI d,n InterID + Z k,j C B,l,n. (12) l Π u,n {j} InterIU By substituting the value of Γ i,j,n from (11) in (5) and adding C B,j,n to both sides of the last inequality we obtain: 1+ C B,j,n C B,j,n + I u,j,n (13) C. Generic model By combining equations (9) for all downlink-cells and (13) for all uplink-cells, we obtain a linear system which can be represented by the following matrix inequality: Z (n) R n 1+ R n (14) where R n =[R j,n ] is the M 1 vector with elements: { PT,j,n j Π R j,n = d,n (15) C B,j,n j Π u,n Z (n) is an M M non negative matrix with elements: i S j,n Z i,l j, l Π d,n,j l 1+β d (N j,n 1) j, l Π d,n,j = l Z (n) G k,i j,l = i S j,n G i,j G k,l j Π d,n,l Π u,n G j,l j Π u,n,l Π d,n 1+β u (N j,n 1) j, l Π u,n,j = l Z k,j j, l Π u,n,j l (16) It must be noted that vector R n includes the total base station transmitted powers in downlink-cells and the base station received powers in uplink-cells. From the Perron Frobenius theorem [11], the maximum value γn of that can be reached during a time slot n with positive powers is obtained when the matrix inequality (14) became an equality: Z (n) R n= 1+γ n R γn n (17) and the value of γn is given by the following equation: γ n = 1 λ n 1 (18)

4 Slot capacity (mobiles/slot) where λ n is the largest real eigenvalue of the matrix Z (n) and R n is the corresponding eigen vector. 4 Slot capacity when the CIR threshold=-1 db IV. SYSTEM PERFORMANCE AND SIMULATIONS RESULTS The maximum achievable CIR is estimated in crossed and s using equation (18). The impact of the cell load and the impact of the asymmetry rate are evaluated for 1 samples. Simulations results have shown that uplink slots have always higher average maximum achievable CIR than s for all mobile distributions and for all asymmetry rates. In figure 4, the capacity is plotted as a function of the asymmetry rate for the three types of mobile distribution. The horizontal lines represent the capacity, which are independent from the asymmetry rate. The slot capacity C n is the total maximum number of active mobiles that can be served during slot n with a CIR threshold γ th : C n =max card (S j,n ) / jɛπ Γ i,j,n γ th iɛ (19) S j,n jɛπ where card (S) is the cardinality of the set S. As it is expected, the slot capacity is at its highest level when the second distribution is used while it is at its lowest level when the third distribution is used; In the third distribution, the presence of a high number of active mobiles in uplink in some cells and in downlink in other cells at the cell borders increases the probability of high inter-mobile interference. Moreover, mobiles at the cell borders need more power than other mobiles to achieve γ th. Therefore, the third distribution has the lowest achievable CIR while the second distribution has the highest one. Moreover, the capacity is at its lowest level when the asymmetry rate is.5. At this rate, the number of active mobiles in uplink and in downlink is at its maximum, and thus the probability that high inter-mobiles and inter-base stations interference affect the system, is at its maximum. The capacity is higher when a r =1/1 than when a r =9/1, because β u > β d, and thus the intracell interference impact is higher in uplink than in downlink. Thus, when the number of uplink cells is higher than the number of downlink cells, the intracell interference impact is increased. In figure 5, the probability density function (pdf) is plotted for the three distributions in uplink and crossed slots. The plotted figures correspond to a cell load of 6 codes/cell/slot and an asymmetry rate of 1/1. As in figure 4, the average maximum achievable CIR is higher when the uplink is used and when mobiles are distributed using the second distribution. Furthermore, the range of pdfisverylargeincrossedslotswhileitisverynarrow in s. This result is confirmed also when the confidence interval is evaluated; the range of confidence distribution 2 distribution 1 distribution Rate of asymmetry Fig. 4. The slot capacity for γ th = 1 db interval is higher in s than in s. The high variation in the maximum achievable CIR in s is due to the high number of parameters affecting the path gain matrix. In s, only path gains between mobiles and base stations has an impact on the path gain matrix. In s, new type of path gains (i.e. path gains between mobiles and path gains between base stations) appear. The path gains between mobiles are highly dependent on the mobile positions. Hence, the path gain matrix in s may change drastically dependent on the mobile positions. Though the mean of the maximum achievable CIR is higher in uplink than in s, the maximum achievable CIR in s is higher than the value in s for some mobile samples, especially when the second distribution is used. It must be noted that the maximum achievable CIR can reach db in s when the second distribution is used while the maximum value in is -3.5 db for the same distribution. The average difference between the maximum achievable CIR in crossed and s is plotted in figure 6 with the confidence interval for three values of the asymmetry rate using the first distribution. The results show that the performance in s become closer to the performance of s for very high or very low values of a r in particular when a r is at its maximum where the slot become almost a pure downlink slot; for very high value of a r (resp. very low value of a r ), the number of uplink (resp. downlink) cells is very low and the probability of having high inter-mobile or inter-base station interference is very low. Due to the high range of confidence interval, we can note also that the maximum achievable CIR in crossed slots can be higher than its value in s in some cases (e.g. the difference can reach 5 db). However, the

5 pdf mean difference of the maximum achievable CIR (db).45 distribution 2.45 distribution 1.45 distribution Fig. 5. The pdf of the maximum achievable CIR for uplink and s ar=1/1 ar=9/1 ar=1/2 depending on the path gain distributions. Therefore, a dramatic degradation may appear when high mobile-mobile interference are present though that some enhacements may appear in other cases. Therefore, must be allocated very carefully for a low number of mobiles and intelligent slot allocation technique must be used to conserve the flexibility of the TD-CDMA technique without increasing the outage probability. This trade-off may be achieved by using s for cells where the mobile distribution is concentrated near to base stations and using some techniques to avoid high interference Cell load (codes/cell/slot) Fig. 6. The mean difference between the maximum achievable CIR in uplink and s performance in uplink is better in most cases and the enhancement in the uplink case can reaches 2 db for high loads. The high range of confidence interval is due to the high variation of the maximum achievable CIR in s. V. CONCLUSIONS In this paper, we have studied the performance of TD- CDMA systems during s by using a simplified optimum power control and by considering the as a reference. The impact of different mobile distributions inside cells, the cell load and the rate of asymmetry on the system performance during s were investigated. Simulation results has shown that in some distributions where mobiles are concentrated near to base stations, the system performance may be better during s than during s. However, the system performance in s varies in a large range REFERENCES [1] D. G. Jeong and W. S. Jeon, Time slot allocation in cdma/tdd systems for mobile multimedia services, IEEE Communications Letters, vol. 4, no. 2, pp , 2. [2] H. Holma, S. Heikkinen, O.-A. Lehtinen, and A. Toskala, Interference considerations for the time division duplex mode of the UMTS terrestrial radio access, IEEE Journal on Selected Areas in communications, vol. 18, no. 8, 2. [3] J. Nasreddine and X. Lagrange, Power control and slot allocation in a TD-CDMA system, in Proc.of the 55th IEEE Vehicular Technology Conference, 22, 22. [4] J. Nasreddine and X. Lagrange, Comparison of different slot allocation schemes in a TD-CDMA TDD system, in Proc.of the second IEEE Workshop on Applications and Services in Wireless Networks, 22. [5] S. Verdu, Multiuser detection. Cambridge University Press, [6] Siemens AG, TDD UE-UE interference simulations, TSG-RAN Working Group 4 (Radio meeting n26), February 23. [7] J. Nasreddine and X. Lagrange, Time slot allocation based on path gain division scheme for TD-CDMA TDD systems, in Proc.of the 57th IEEE Semiannual Vehicular Technology Conference, 23, vol. 2, pp , 23. [8] J. M. Aein, Power balancing in systems employing frequency reuse, COMSAT tech. rev., vol. 3, no. 2, [9] J. Zander, Performance of optimum transmitter power control in cellular radio systems, IEEE Transactions on Vehicular Technology, vol.41,no.1,pp.57 62,1992. [1] Q. Wu, Performance of optimum transmitter power control in CDMA cellular mobile systems, IEEE Transactions on Vehicular Technology, vol. 48, no. 2, pp , [11] F. R. Gantmacher, The theroy of matrices, vol. 2. New York: Chelsea publishing company, 1971.

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