398 IEEE COMMUNICATIONS LETTERS, VOL. 4, NO. 12, DECEMBER 2000 Interference Bounds in Power Controlled Systems David Muñoz-Rodriguez, Oscar Uribe-Arambula, Cesar Vargas, and Hector Maturino Abstract—Capacity in a code-division multiple access (CDMA) system is often referred to in terms of the maximum number of users that can be supported within a carrier to interference ratio above a specified quality threshold. This paper presents a simple analysis showing that the total mean interference in the reverse channel, in a multi-tier scenario of a CDMA system with power control, is finite regardless of the number of surrounding tiers. Rie- mann–Hurwitz analysis is introduced in order to obtain upper and lower interference bounds. I. INTRODUCTION W HEN an interference analysis in a cellular system is con- ducted, the number of tiers to be considered is heuristi- cally determined. For a small system, the number of interfering cells is small though they tend to cause a strong interference. When the number of tiers grows, the cells in those tiers are fur- ther apart from the cell of interest causing a weaker interference. However, the number of interfering cells is much larger and the overall interference contribution needs to be assessed. This letter presents upper and lower interference bounds enabling calcula- tion of the maximum number of subscribers per cell in a scenario with infinite number of tiers. Although interference and capacity calculations have been considered as engineering issues for wireless systems in the lit- erature, see [1], [5], [6], [9], and [8], the present approach con- siders extreme cases scenarios in order to obtain the bounds mentioned. We use mathematical analysis that allows us to write the interference bounds in terms of the Riemann–Hurwitz func- tion, see [3] and [4]. II. MODEL DESCRIPTION Capacity in CDMA systems is determined by the amount of interference generated within the same cell as well as in cells within surrounding tiers. Thus, interference ( ) becomes an in- creasing monotonic function of the number of tiers. In order to achieve maximum capacity while keeping interference levels to a minimum and to ensure an even performance for all customers, power control techniques become an integral part of CDMA sys- tems. In this paper we initially consider a power controlled ho- mogeneous scenario where subscribers are evenly distributed. That is, the mean number of subscribers per unit area is uni- form along the whole coverage service area. Manuscript received May 11, 2000. The associate editor coordinating the re- view of this letter and approving it for publication was Dr. N. Mandayam. D. Muñoz-Rodriguez, O. Uribe-Arambula, and C. Vargas are with the Center for Electronics and Telecommunications, ITESM-Monterrey, 64849 Monterrey, N.L., Mexico. H. Maturino is with Nortel Networks, Richardson, TX 75082-4399 USA. Publisher Item Identifier S 1089-7798(00)11517-0. Simplified cell tier scenarios departing from the regular hexagonal grid are often considered to ease analysis, [7]. In this paper, we assume circular concentric tiers around , the cell of interest with radius . Tiers wide containing cells of radius (see Fig. 1) surround this center cell, hence, the area of the th tier becomes (1) where is the area of cell . From the argument above, the mean number ( ) of subscribers in the th tier is times the number of active subscribers in cell , i.e., . Let be a cell in tier . A mobile in cell is considered to exhibit a transmitted power such that the reception level at base station equals a value defined by the power con- trol process. Taking the mobile location as an interception point [2], the mobile transmitted power, , can conveniently be ex- pressed as where is the distance between the subscriber and the base station while stands for the propagation law. Then, the interference the mobile produces at is equal to , where is the distance between the mobile and the , (Fig. 1). This model assumes a homogeneous propagation environment, even in close prox- imity to base stations. Consider a mobile a distance from its base station lo- cated in tier , Fig. 2. If the mobile is located on arc ( ), the maximum interference produced by the mobile at , , will occur when the subscriber is located at the line joining and (this is, when the mobile is placed at ). Hence, we obtain (2) which is an increasing monotonic function of . From Fig. 2, if the mobile is on arc ( ) the subscriber will interfere the least when placed farthest on the line joining and (this is, when the mobile is placed at ). In this case the interference becomes (3) which is also an increasing monotonic function of . 1089–7798/00$10.00 © 2000 IEEE