Optimal sequences for CDMA with decision-feedback receivers

We consider a symbol-synchronous code-division multiple-access (CDMA) system that is equipped with a multiuser decision-feedback receiver and for which power control is available. The users are each assigned a quality-of-service (QoS) threshold to be guaranteed by the system, and to cover scenarios for which there are multiple classes of users, these are not required to be equal to each other. For an ideal decision-feedback receiver, it is known that with enough power the system can always meet the users' QoS thresholds, so we instead minimize the sum of the users' received powers over system designs (i.e., signature sequences, power-control policy, and decision-feedback receiver) which guarantee the QoS requirements. It is found that the optimal design produces two classes of users, those whose sequences and powers satisfy with equality the generalized Welch bound inequality and those oversized users that are mutually orthogonal to each other and the rest of the users. In terms of power and bandwidth savings, the optimal sequences for the decision-feedback receiver are found to compare very favorably to optimal designs for linear receivers and to random sequences for the decision-feedback receiver.     

User-capacity maximization in synchronous CDMA subject to RMS-bandlimited signature waveforms

We consider the symbol-synchronous code-division multiple-access (CDMA) channel equipped with either a multiuser linear receiver or a multiuser decision-feedback receiver. The network, or user, capacity is defined to be the number of users that can be supported with available resources, such that every user achieves a certain quality of service (QoS). In this paper, the QoS threshold is given as a signal-to-interference ratio and the bandwidth is given as the root mean squared bandwidth of the received power spectral density of the users' transmitted waveforms. Given the QoS threshold and constraints on bandwidth and the sum of the users' received powers, we maximize user capacity for both the linear and decision-feedback receivers by optimally and jointly designing the users' signature waveforms and power-control polices.     

Linear multiuser detectors for incompletely known symmetric signals in CDMA systems

In this paper, we consider a synchronous code-division multiple-access (CDMA) system with a multiuser receiver. All users are assumed to have symmetric signature sequences, but the presence of a subset of the users is unknown to the receiver. We first calculate the signal-to-interference ratio (SIR) in this environment for the matched-filter receiver, the decorrelating receiver, and the linear minimum mean-square error (MMSE) detector. We then identify the user capacity for a single-class system, and the effective bandwidth for a multiple-class system. The result is compared to the case of random sequences and of optimum sequences. For symmetric sequences, the effective bandwidth cannot be expressed by a scalar as in , because two constraints have to be satisfied simultaneously to satisfy the SIR requirement. We introduce a two-dimensional (2-D) vector notion of effective bandwidth with and without unknown users. For both the decorrelator and the MMSE detector, the user capacity is 1 when all users are known to the receivers and is reduced to (1-N/L) when N users are unknown (with L the processing gain). The performance of these three linear detectors, with and without unknown users, is compared.     

Effective interference and effective bandwidth of linear multiuserreceivers in asynchronous CDMA systems

The performance of linear multiuser receivers in terms of the signal-to-interference ratio (SIR) achieved by the users has been analyzed in a synchronous CDMA system under random spreading sequences. In this paper, we extend these results to a symbol-asynchronous but chip-synchronous system and characterize the SIR for linear receivers-the matched-filter receiver the minimum mean-square error (MMSE) receiver and the decorrelator. For each of the receivers, we characterize the limiting SIR achieved when the processing gain is large and also derive lower bounds on the SIR using the notion of effective interference. Applying the results to a power controlled system, we derive effective bandwidths of the users for these linear receivers and characterize the user capacity region: a set of users is supportable by a system if the sum of the effective bandwidths is less than the processing gain of the system. We show that while the effective bandwidth of the decorrelator and the MMSE receiver is higher in an asynchronous system than that in a synchronous system, it progressively decreases with the increase in the length of the observation window and is asymptotic to that of the synchronous system, when the observation window extends infinitely on both sides of the symbol of interest. Moreover, the performance gap between the MMSE receiver and the decorrelator is significantly wider in the asynchronous setting as compared to the synchronous case     

Linear multiuser receivers: effective interference, effectivebandwidth and user capacity

Multiuser receivers improve the performance of spread-spectrum and antenna-array systems by exploiting the structure of the multiaccess interference when demodulating the signal of a user. Much of the previous work on the performance analysis of multiuser receivers has focused on their ability to reject worst case interference. Their performance in a power-controlled network and the resulting user capacity are less well-understood. We show that in a large system with each user using random spreading sequences, the limiting interference effects under several linear multiuser receivers can be decoupled, such that each interferer can be ascribed a level of effective interference that it provides to the user to be demodulated. Applying these results to the uplink of a single power-controlled cell, we derive an effective bandwidth characterization of the user capacity: the signal-to-interference requirements of all the users can be met if and only if the sum of the effective bandwidths of the users is less than the total number of degrees of freedom in the system. The effective bandwidth of a user depends only on its own SIR requirement, and simple expressions are derived for three linear receivers: the conventional matched filter, the decorrelator, and the MMSE receiver. The effective bandwidths under the three receivers serve as a basis for performance comparison     

Resource pooling and effective bandwidths in CDMA networks withmultiuser receivers and spatial diversity

Much of the performance analysis on multiuser receivers for direct-sequence code-division multiple-access (CDMA) systems is focused on worst case near-far scenarios. The user capacity of power-controlled networks with multiuser receivers are less well-understood. Tse and Hanly (see ibid., vol.45, p.541-657, 1999) have shown that under some conditions, the user capacity of an uplink power-controlled CDMA cell for several important linear receivers can be very simply characterized via a notion of effective bandwidth. We show that these results extend to the case of antenna arrays. We consider a CDMA system consisting of users transmitting to an antenna array with a multiuser receiver, and obtain the limiting signal-to-interference (SIR) performance in a large system using random spreading sequences. Using this result, we show that the SIR requirements of all the users can be met if and only if the sum of the effective bandwidths of the users is less than the total number of degrees of freedom in the system. The effective bandwidth of a user depends only on its own requirement. Our results show that the total number of degrees of freedom of the whole system is the product of the spreading gain and the number of antennas. In the case when the fading distributions to the antennas are identical, we show that a curious phenomenon of “resource pooling” arises: the multiantenna system behaves like a system with only one antenna but with the processing gain the product of the processing gain of the original system and the number of antennas, and the received power of each user the sum of the received powers at the individual antennas     

Ergodic sum capacity maximization for CDMA:Optimum resource allocation

We solve for the optimum signature sequence and power allocation policies that maximize the information-theoretic ergodic sum capacity of a code-division multiple-access (CDMA) system subject to fading. We show that at most N users may transmit at any given channel state, where N is the processing gain; and those users who are transmitting should be assigned orthogonal signature sequences. We also show that the power allocation policy that maximizes the capacity together with the choice of these signature sequences is single-user water-filling over sets of channel states that are favorable to each user. That is, the capacity maximizing signaling scheme is shown to dictate that the users allocate their powers and signature sequences in such a way that they always avoid interference from each other.     

CDMA systems in fading channels: admissibility, network capacity,and power control

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 simultaneous 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     

A Unified Approach to Power control in Large Energy-Constrained CDMS Systems

A unified approach to power control is proposed for maximizing utility in terms of energy efficiency in code-division multiple access (CDMA) networks. The approach is applicable to a large family of multiuser receivers including the matched filter, the decorrelator, the linear minimum mean-square error (MMSE) receiver, and the (nonlinear) optimal detectors. It exploits the linear relationship between the transmit power and the output signal-to-interference-plus-noise ratio (SIR) for each user in the large-system limit. Suppose that each user seeks to selfishly maximize its own energy efficiency, a unique Nash equilibrium is shown to exist and be SIR-balanced, thus extending a previous result on linear receivers. A unified power control algorithm for reaching the Nash equilibrium is proposed, which adjusts transmit powers iteratively by computing the large-system multiuser efficiency, which is independent of instantaneous spreading sequences. The convergence of the algorithm is proved for linear receivers, and is demonstrated via simulation for the multiuser maximum likelihood detector. Moreover, the performance of the algorithm in finite-size systems is studied and compared with that of a conventional power control scheme, in which user powers depend on the instantaneous spreading sequences.     

Bandwidth-efficient multiple access (BEMA): a new strategy based onsignal design under quality-of-service constraints forsuccessive-decoding-type multiuser receivers

This paper considers the design of signature waveforms for successive-decoding-type multiuser receivers (including the optimum successive decoder (OSD)) in a correlated-waveform multiple-access channel. The problem is to obtain signature waveforms that require as little bandwidth as possible while allowing the receiver to meet a given set of quality-of-service (QoS) objectives. The QoS objectives are specified for each user in terms of capacity, or equivalently, the signal-to-interference ratio. A (generally unachievable) lower bound is obtained on the minimum bandwidth required to achieve these QoS constraints. Moreover, a simple algorithm is proposed for obtaining signal sets that meet the QoS constraints when used with the OSD, and which, while not optimal, require a bandwidth that can be very close to the minimum required bandwidth. It is also shown that such signal sets allow for a significantly more efficient use of bandwidth than do orthogonal signals used in time- or frequency-division multiple access (TDMA/FDMA). Based on our signal design approach, we propose a new multiple-access strategy that we refer to as bandwidth-efficient multiple access (BEMA). While BEMA is more bandwidth efficient than TDMA or FDMA, it retains their desirable feature of needing only single-user coding (and decoding) for each user     

Multiuser Constrained Water-Pouring for Continuous-Time Overloaded Gaussian Multiple-Access Channels

In this paper, signature waveforms and signal powers are jointly optimized for multiuser communications over strictly bandlimited, continuous-time, overloaded channels corrupted by additive white Gaussian noise (AWGN). The total signal power is minimized subject to general asymmetric signal-to-interference-plus-noise ratio (SINR) constraints at the output of linear minimum mean-squared error (LMMSE) receivers. Using vectorized Fourier transform (VFT) technique, the optimal solutions as well as a necessary and sufficient condition for the existence of a feasible solution are derived in the frequency domain. It turns out that every optimal solution performs the same as the optimal frequency-division multiple-access (FDMA) system. A geometric procedure called multiuser constrained water-pouring is developed to construct every possible optimal solution, of which profiles of signal power and equivalent bandwidth are, respectively, the same as those of signal power and physical bandwidth of the optimal FDMA system. It is shown that orthogonal signature waveforms are assigned to oversized users and continuous-time equivalents of generalized Welch bound equality (CTE-GWBE) sequences are assigned to non-oversized users.     

Ensuring convergence of the MMSE iteration for interference avoidance to the global optimum

Viswanath and Anantharam (1999) characterize the sum capacity of multiaccess vector channels. For a given number of users, received powers, spreading gain, and noise covariance matrix in a code-division multiple-access (CDMA) system, Viswanath and Anantharam present a combinatorial algorithm to generate a set of signature sequences that achieves the maximum sum capacity. These sets also minimize a performance measure called generalized total square correlation (TSC/sub g/). Ulukus and Yates (2001) propose an iterative algorithm suitable for distributed implementation: at each step, one signature sequence is replaced by its linear minimum mean-square error (MMSE) filter. This algorithm results in a decrease of TSC/sub g/ at each step. The MMSE iteration has fixed points not only at the optimal configurations which attain the global minimum TSC/sub g/ but also at other configurations which are suboptimal. The authors of claim that simulations show that when starting with random sequences, the algorithm converges to optimum sets of sequences, but they give no formal proof. We show that the TSC/sub g/ function has no local minima, in the sense that given any suboptimal set of sequences, there exist arbitrarily close sets with lower TSC/sub g/. Therefore, only the optimal sets are stable fixed points of the MMSE iteration. We define a noisy version of the MMSE iteration as follows: after replacing all the signature sequences, one at a time, by their linear MMSE filter, we add a bounded random noise to all the sequences. Using our observation about the TSC/sub g/ function, we can prove that if we choose the bound on the noise adequately, making it decrease to zero, the noisy MMSE iteration converges to the set of optimal configurations with probability one for any initial set of sequences.     

Blind adaptive multiuser detection

The decorrelating detector and the linear minimum mean-square error (MMSE) detector are known to be effective strategies to counter the presence of multiuser interference in code-division multiple-access channels; in particular, those multiuser detectors provide optimum near-far resistance. When training data sequences are available, the MMSE multiuser detector can be implemented adaptively without knowledge of signature waveforms or received amplitudes. This paper introduces an adaptive multiuser detector which converges (for any initialization) to the MMSE detector without requiring training sequences. This blind multiuser detector requires no more knowledge than does the conventional single-user receiver: the desired user's signature waveform and its timing. The proposed blind multiuser detector is made robust with respect to imprecise knowledge of the received signature waveform of the user of interest     

Large system performance of linear multiuser receivers in multipathfading channels

A linear multiuser receiver for a particular user in a code-division multiple-access (CDMA) network gains potential benefits from knowledge of the channels of all users in the system. In fast multipath fading environments we cannot assume that the channel estimates are perfect and the inevitable channel estimation errors will limit this potential gain. We study the impact of channel estimation errors on the performance of linear multiuser receivers, as well as the channel estimation problem itself. Of particular interest are the scalability properties of the channel and data estimation algorithms: what happens to the performance as the system bandwidth and the number of users (and hence channels to estimate) grows? Our main results involve asymptotic expressions for the signal-to-interference ratio of linear multiuser receivers in the limit of large processing gain, with the number of users divided by the processing gain held constant. We employ a random model for the spreading sequences and the limiting signal-to-interference ratio expressions are independent of the actual signature sequences, depending only on the system loading and the channel statistics: background noise power, energy profile of resolvable multipaths, and channel coherence time. The effect of channel uncertainty on the performance of multiuser receivers is succinctly captured by the notion of effective interference     

Optimum power control for CDMA with deterministic sequences in fading channels

We specify the capacity region for a power-controlled, fading code-division multiple-access (CDMA) channel. We investigate the properties of the optimum power allocation policy that maximizes the information-theoretic ergodic sum capacity of a CDMA system where the users are assigned arbitrary signature sequences in a frequency flat-fading environment. We provide an iterative waterfilling algorithm to obtain the powers of all users at all channel fade levels, and prove its convergence. Under certain mild conditions on the signature sequences, the optimum power allocation dictates that more than one user transmit simultaneously in some nonzero probability region of the space of all channel states. We identify these conditions, and provide an upper bound on the maximum number of users that can transmit simultaneously at any given time. Using these properties of the sum capacity maximizing power control policy, we also show that the capacity region of the fading CDMA channel is not in general strictly convex.     

Iterative construction of optimum signature sequence sets insynchronous CDMA systems

Optimum signature sequence sets that maximize the capacity of single-cell synchronous code division multiple access (CDMA) systems have been identified. Optimum signature sequences minimize the total squared correlation (TSC); they form a set of orthogonal sequences, if the number of users is less than or equal to the processing gain, and a set of Welch (1994) bound equality (WBE) sequences, otherwise. We present an algorithm where users update their transmitter signature sequences sequentially, in a distributed fashion, by using available receiver measurements. We show that each update decreases the TSC of the set, and produces better signature sequence sets progressively. We prove that the algorithm converges to a set of orthogonal signature sequences when the number of users is less than or equal to the processing gain. We observe and conjecture that the algorithm converges to a WBE set when the number of users is greater than the processing gain. At each step, the algorithm replaces one signature sequence from the set with the normalized minimum mean squared error (MMSE) receiver corresponding to that signature sequence. Since the MMSE filter can be obtained by a distributed algorithm for each user, the proposed algorithm is amenable to distributed implementation     

Asymptotically optimal water-filling in vector multiple-accesschannels

Dynamic resource allocation is an important means to increase the sum capacity of fading multiple-access channels (MACs). In this paper, we consider vector multi-access channels (channels where each user has multiple degrees of freedom) and study the effect of power allocation as a function of the channel state on the sum capacity (or spectral efficiency) defined as the maximum sum of rates of users per unit degree of freedom at which the users can jointly transmit reliably, in an information-theoretic sense, assuming random directions of received signal. Direct-sequence code-division multiple-access (DS-CDMA) channels and MACs with multiple antennas at the receiver are two systems that fall under the model. Our main result is the identification of a simple dynamic power-allocation scheme that is optimal in a large system, i.e., with a large number of users and a correspondingly large number of degrees of freedom. A key feature of this policy is that, for any user, it depends on the instantaneous amplitude of channel state of that user alone and the structure of the policy is “water-filling.” In the contest of DS-CDMA and in the special case of no fading, the asymptotically optimal power policy of water-filling simplifies to constant power allocation over all realizations of signature sequences; this result verifies the conjecture made in Verdu and Shamai (1999). We study the behavior of the asymptotically optimal water-filling policy in various regimes of number of users per unit degree of freedom and signal-to-noise ratio (SNR). We also generalize this result to multiple classes, i.e., the situation when users in different classes have different average power constraints     

Optimal sequences and sum capacity of symbol asynchronous CDMA systems

The optimal signature sequences that maximize the sum capacity of a direct sequence code-division multiple-access (CDMA) system are characterized in the general case of symbol delay profile and user power constraints. It is shown that the optimal sum capacity of the symbol asynchronous system equals that of the symbol synchronous system with the same user power constraints. With the optimal signature sequence set, the maximum sum capacity is achieved with white Gaussian input signals. The existence of the optimal signature sequence set is proved by the proposal of an explicit construction method for arbitrary user delay profiles and power constraints.     

CDMA with power control and sequence design: the capacity region with and without multidimensional signaling

We consider the symbol-synchronous code-division multiple-access (CDMA) channel in which every user is assigned a rate at which arbitrarily reliable transmission in the Shannon sense is to be guaranteed. For an overloaded system in which the number of active users exceeds the available processing gain, we optimally design the users' signature sequences and a power-control policy to minimize the required sum-power (i.e., sum of the users' powers) while meeting the rate-tuple constraint with a (joint) maximum-likelihood receiver. This result is extended to find the power-constrained capacity region of the system; this is the set of all achievable rate-tuples over all signature sequences and power-control policies whose sum-power is constrained. Furthermore, it is shown that this capacity region may be substantially and maximally expanded in those regions where there are oversized users whose rate requirements are relatively large compared to those of the other users; this is accomplished by allowing for the flexibility of multidimensional signaling in the sense of a user simultaneously transmitting several different scalar symbols, each modulated by its own signature sequence. From the viewpoint of resource efficiency, this means that a multicarrier approach is essential in systems that support multiple classes of users. Finally, we also address the dual problem of determining the region of valid power-control policies subject to a sum-capacity constraint on the system.     

Large-system performance analysis of blind and group-blindmultiuser receivers

We present a large-system performance analysis of blind and group-blind multiuser detection methods. In these methods, the receivers are estimated based on the received signal samples. In particular, we assume binary random spreading, and let the spreading gain N, the number of users K, and the number of received signal samples M all go to infinity, while keeping the ratios K/N and M/N fixed. We characterize the asymptotic performance of the direct-matrix inversion (DMI) blind linear minimum mean-square error (MMSE) receiver, the subspace blind linear MMSE receiver, and the group-blind linear hybrid receiver. We first derive the asymptotic average output signal-to-interference-plus-noise ratio (SINR) for each of these receivers. Our results reveal an interesting "saturation" phenomenon: The output SINR of each of these receivers converges to a finite limit as the signal-to-noise ratio (SNR) of the desired user increases, which is in stark contrast to the fact that the output SINR achieved by the exact linear MMSE receiver can get arbitrarily large. This indicates that the capacity of a wireless system with blind or group-blind multiuser receivers is not only interference-limited, but also estimation-error limited. We then show that for both the blind and group-blind receivers, the output residual interference has an asymptotic Gaussian distribution, independent of the realizations of the spreading sequences. The Gaussianity indicates that in a large system, the bit-error rate (BER) is related to the SINR simply through the Q function