962 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000 CDMA Systems in Fading Channels: Admissibility, Network Capacity, and Power Control Junshan Zhang, Student Member, IEEE, and Edwin K. P. Chong, Senior Member, IEEE Abstract—We study the admissibility and network capacity of imperfect power-controlled Code-Division Multiple Access (CDMA) systems with linear receivers in fading environments. In a CDMA system, a set of users is admissible if their simulta- neous transmission does not result in violation of any of their Quality-of-Service (QoS) requirements; the network capacity is the maximum number of admissible users. We consider a single-cell imperfect power-controlled CDMA system, assuming known received power distributions. We identify the network capacities of single-class systems with matched-filter (MF) receivers for both the deterministic and random signature cases. We also characterize the network capacity of single-class systems with linear Minimum-Mean-Square-Error (MMSE) receivers for the deterministic signature case. The network capacities can be expressed uniquely in terms of the users’ signal-to-interference ratio (SIR) requirements and received power distributions. For multiple-class systems equipped with MF receivers, we find a necessary and sufficient condition on the admissibility for the random signature case, but only a sufficient condition for the deterministic signature case. We also introduce the notions of effective target SIR and effective bandwidth, which are useful in determining the admissibility and hence network capacity of an imperfect power-controlled system. Index Terms—Admissibility, CDMA, deterministic signature, fading channel, matched filter, MMSE, network capacity, power control, random signature, scale family. I. INTRODUCTION T HE last fifteen years have witnessed a tremendous growth of wireless networks. Due to the fast-growing demand for network capacity in wireless networks, it is essential to utilize efficiently the limited resources. The characterization of net- work capacity is therefore a fundamental and pressing issue in wireless network research. In this paper, we consider a model for the uplink of a single-cell symbol-synchronous Code-Divi- sion Multiple Access (CDMA) system in fading channels. The network therein consists of numerous mobile subscribers com- municating with one base station, which is typically intercon- nected to a backbone network via a wired infrastructure. Two approaches have been studied extensively to achieve efficient utilization of network resources in CDMA systems: Manuscript received September 16, 1998; revised October 12, 1999. This work was supported in part by the National Science Foundation under Grant ECS-9501652 and by the U.S. Army Research Office under Grant DAAH04-95-1-0246. The material in this paper was presented in part at the Allerton Conference, Monticello, IL, 1998 and at IEEE Infocom, New York, NY, 1999. The authors are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: {junshan} {echong}@ecn.purdue.edu). Communicated by M. L. Honig, Associate Editor for Communications. Publisher Item Identifier S 0018-9448(00)03088-1. multiuser detection and power control. Multiuser detection refers to the process of demodulating one or more user data streams from a nonorthogonal multiplex and is concerned with designing good receivers to achieve efficient interference suppression. In particular, among multiuser receivers, linear receivers have attracted a large amount of attention because they are practically appealing (see, e.g., [13]–[15], [30]). Power control, on the other hand, is implemented at the transmitter and is concerned with allocating powers to fulfill individual users’ Quality-of-Service (QoS) requirements (see, e.g., [7], [9], [28], [36]). Since both linear receivers and power control are em- ployed to suppress interference effectively and utilize network resources efficiently, it is natural to ask how linear receivers perform in power-controlled systems. In [27], Tse and Hanly characterized the network capacities for several important linear receivers via a notion of effective bandwidth, assuming users have random signatures. (See also earlier work in [9].) Viswanath, Anantharam, and Tse [31] studied the joint opti- mization problem of signature allocation and power control, and obtained simple characterizations of the network capacities of single-cell systems with linear Minimum-Mean-Square-Error (MMSE) receivers. However, both [27] and [31] assumed perfect power control in characterizing the network capacity. In a practical wireless communication system, due to delays and errors in power control and time-varying channel conditions, it is difficult to implement perfect power control, and the received powers typically fluctuate around the desired levels. Therefore, it is more appropriate to model the received powers as random. However, little work has been done on characterizing network capacity of imperfect power-controlled CDMA systems with linear receivers in fading channels. In this paper, we focus primarily on the admissibility and network capacity of imperfect power-controlled CDMA sys- tems with matched filter (MF) receivers in fading channels. Throughout this paper, we assume MF receivers unless speci- fied otherwise. Each user in the system is assigned a signature onto which the user’s information symbols are spread. Every user also has a minimum signal-to-interference (SIR) require- ment. Roughly speaking, a set of users is said to be admissible if their simultaneous transmission does not result in violation of any of their SIR requirements; the network capacity is the maximum number of admissible users. Following the approach of [27], [30], we formulate the problem in an asymptotic setting in which we allow the number of users and the degrees of freedom (length of the signatures) to grow, while keeping their ratio fixed. The results are stated in terms of this ratio of number of users per degree of freedom. A feature that distinguishes this work from [27] is that the received power for 0018–9448/00$10.00 © 2000 IEEE