Asynchronous CDMA Systems with Random Spreading-Part II: Design Criteria

TL;DR

This paper explores the design of low-complexity linear multiuser detectors for asynchronous CDMA systems, analyzing their performance and effects of system parameters, with a focus on large-system limits and practical implementation considerations.

Contribution

It introduces design criteria for multistage linear detectors in asynchronous CDMA, including recursive weight design and performance analysis in large systems.

Findings

Performance of linear detectors is characterized in large-system limits.

Oversampling improves discrete-time statistics and system performance.

Design criteria enable low-complexity implementation with performance close to optimal.

Abstract

Totally asynchronous code-division multiple-access (CDMA) systems are addressed. In Part I, the fundamental limits of asynchronous CDMA systems are analyzed in terms of spectral efficiency and SINR at the output of the optimum linear detector. The focus of Part II is the design of low-complexity implementations of linear multiuser detectors in systems with many users that admit a multistage representation, e.g. reduced rank multistage Wiener filters, polynomial expansion detectors, weighted linear parallel interference cancellers. The effects of excess bandwidth, chip-pulse shaping, and time delay distribution on CDMA with suboptimum linear receiver structures are investigated. Recursive expressions for universal weight design are given. The performance in terms of SINR is derived in the large-system limit and the performance improvement over synchronous systems is quantified. The considerations distinguish between two ways of forming discrete-time statistics: chip-matched filtering and oversampling.

Figures

Equations

$${\mathchoice{\mbox{\boldmath$\displaystyle C$}}{\mbox{\boldmath$\textstyle C$}}{\mbox{\boldmath$\scriptstyle C$}}{\mbox{\boldmath$\scriptscriptstyle C$}}}=\left(\begin{array}[]{cccc}{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{0}&{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{1}&\cdots&{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{N-1}\\ {\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{N-1}&{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{0}&\cdots&{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{N-2}\\ \vdots&\vdots&&\vdots\\ {\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{1}&{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{2}&\cdots&{\mathchoice{\mbox{\boldmath$\displaystyle B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}}_{0}\\ \end{array}\right)$$

$$c_s(\Omega )=\frac{1}{\sqrt{2\pi }}\sum _{k=0}^{N-1}{c}_{sk}{\mathrm{e}}^{-j\Omega k}s=1,\dots ,r,$$

$${\mathchoice{\mbox{\boldmath$\displaystyle C$}}{\mbox{\boldmath$\textstyle C$}}{\mbox{\boldmath$\scriptstyle C$}}{\mbox{\boldmath$\scriptscriptstyle C$}}}=\mathfrak{F}\{{\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}}(\Omega)\}.$$

$$G(\omega )=\{\begin{array}{ll}\frac{1}{\sqrt{E_{\psi }}}& |\omega |\le \frac{\pi r}{T_c}\\ 0& |\omega |>\frac{\pi r}{T_c}.\end{array}$$

$$\varphi (\Omega ,\tau )\stackrel{△}{=}\frac{1}{T_c}\sum _{s=-\infty }^{+\infty }{\mathrm{e}}^{j\frac{\tau }{T_c}(\Omega +2\pi s)}{\Phi }^{\ast }\left(\frac{j(\Omega +2\pi s)}{T_c}\right).$$

$$\mathcal{Y}=\mathcal{H}\mathcal{B}+\mathcal{W}$$

$${\boldsymbol{\mathcal{H}}}=\left[\begin{array}[]{ccccccc}\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ \ldots&\mathbf{0}&{{\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}}_{d}^{(m-1)}&{{\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}}_{u}^{(m)}&\mathbf{0}&\ldots&\ldots\\ \ldots&\ldots&\mathbf{0}&{{\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}}_{d}^{(m)}&{{\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}}_{u}^{(m+1)}&\mathbf{0}&\ldots\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ \end{array}\right].$$

$${{\mathchoice{\mbox{\boldmath$\displaystyle H$}}{\mbox{\boldmath$\textstyle H$}}{\mbox{\boldmath$\scriptstyle H$}}{\mbox{\boldmath$\scriptscriptstyle H$}}}}^{(m)}={{\mathchoice{\mbox{\boldmath$\displaystyle S$}}{\mbox{\boldmath$\textstyle S$}}{\mbox{\boldmath$\scriptstyle S$}}{\mbox{\boldmath$\scriptscriptstyle S$}}}}^{(m)}{{\mathchoice{\mbox{\boldmath$\displaystyle A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}}}$$

$${{\mathchoice{\mbox{\boldmath$\displaystyle S$}}{\mbox{\boldmath$\textstyle S$}}{\mbox{\boldmath$\scriptstyle S$}}{\mbox{\boldmath$\scriptscriptstyle S$}}}}^{(m)}=\left({\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}_{1}{{\mathchoice{\mbox{\boldmath$\displaystyle s$}}{\mbox{\boldmath$\textstyle s$}}{\mbox{\boldmath$\scriptstyle s$}}{\mbox{\boldmath$\scriptscriptstyle s$}}}}_{1}^{(m)},{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}_{2}{{\mathchoice{\mbox{\boldmath$\displaystyle s$}}{\mbox{\boldmath$\textstyle s$}}{\mbox{\boldmath$\scriptstyle s$}}{\mbox{\boldmath$\scriptscriptstyle s$}}}}_{2}^{(m)},\ldots{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}_{K}{{\mathchoice{\mbox{\boldmath$\displaystyle s$}}{\mbox{\boldmath$\textstyle s$}}{\mbox{\boldmath$\scriptstyle s$}}{\mbox{\boldmath$\scriptscriptstyle s$}}}}_{K}^{(m)}\right)$$

$${\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}_{k}=\left[\begin{array}[]{c}{\mathchoice{\mbox{\boldmath$\displaystyle 0$}}{\mbox{\boldmath$\textstyle 0$}}{\mbox{\boldmath$\scriptstyle 0$}}{\mbox{\boldmath$\scriptscriptstyle 0$}}}_{\overline{\tau}_{k}}\\ \widetilde{{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}}_{k}\\ {\mathchoice{\mbox{\boldmath$\displaystyle 0$}}{\mbox{\boldmath$\textstyle 0$}}{\mbox{\boldmath$\scriptstyle 0$}}{\mbox{\boldmath$\scriptscriptstyle 0$}}}_{N-\overline{\tau}_{k}}\end{array}\right]$$

$$\widetilde{{\mathchoice{\mbox{\boldmath$\displaystyle\Phi$}}{\mbox{\boldmath$\textstyle\Phi$}}{\mbox{\boldmath$\scriptstyle\Phi$}}{\mbox{\boldmath$\scriptscriptstyle\Phi$}}}}_{k}=\mathfrak{F}({\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}}(\widetilde{\tau}_{k})),$$

$${\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}}(\widetilde{\tau}_{k})=\left[{\phi}(\Omega,\widetilde{\tau}_{k}){\phi}(\Omega,\widetilde{\tau}_{k}-\tfrac{T_{c}}{r}),\ldots,{\phi}(\Omega,\widetilde{\tau}_{k}-\tfrac{(r-1)T_{c}}{r})\right].$$

$${\chi }_{L,k}^{(m)}(\mathcal{H})=\mathrm{span}({\mathcal{T}}^{\ell }{\mathcal{U}}_k^{(m)}){|}_{\ell =0}^{L-1}.$$

$$\widehat{b}_{k}=\sum_{\ell=0}^{L-1}({{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}_{k}^{(m)})_{\ell}{\boldsymbol{\mathcal{U}}}_{k}^{(m)H}{\boldsymbol{\mathcal{T}}}^{\ell}{\boldsymbol{\mathcal{Y}}}$$

$${\boldsymbol{\mathcal{M}}}_{k}^{(m)}=\sum_{\ell=0}^{L-1}({{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}_{k}^{(m)})_{\ell-1}{\boldsymbol{\mathcal{U}}}_{k}^{(m)H}{\boldsymbol{\mathcal{T}}}^{\ell}.$$

$$\displaystyle{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}_{k}^{(m)}$$

$$\displaystyle=\underset{\overline{{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}}_{k}^{(m)}}{\text{argmin}}\mathrm{E}\left\{\left\|\overline{{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}}_{k}^{(m)H}{{\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}}}_{k}^{(m)}-b_{k}^{(m)}\right\|^{2}\right\}$$

$${{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}_{k}^{(m)}=(\boldsymbol{\Xi}_{k}^{(m)})^{-1}\boldsymbol{\xi}_{k}^{(m)}$$

$${\mathbf{\Xi }}_k^{(m)}$$

$${\mathbf{\xi }}_k^{(m)}$$

$$\mathrm{SINR}_{k}=\frac{\overline{{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}}_{k}^{(m)H}\boldsymbol{{\xi}}_{k}^{(m)}\boldsymbol{{\xi}}_{k}^{(m)T}\overline{{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}}_{k}^{(m)}}{\overline{{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}}_{k}^{(m)H}(\boldsymbol{{\Xi}}_{k}^{(m)}-\boldsymbol{{\xi}}_{k}^{(m)}\boldsymbol{{\xi}}_{k}^{(m)T})\overline{{{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}}_{k^{(m)H}}}.$$

$${\mathrm{SINR}}_k=\frac{{\mathbf{\xi }}_k^{(m)T}{\mathbf{\Xi }}_k^{(m)-1}{\mathbf{\xi }}_k^{(m)}}{1-{\mathbf{\xi }}_k^{(m)T}{\mathbf{\Xi }}_k^{(m)-1}{\mathbf{\xi }}_k^{(m)}}.$$

$${{\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}}}_{k}^{\infty}=\lim_{K=\beta N\rightarrow\infty}\boldsymbol{\Xi}_{k}^{(m)\;-1}\boldsymbol{\xi}_{k}^{(m)}.$$

$$\underset{K=\beta N\to \infty }{\lim }({\mathcal{R}}^{\ell }{)}_{k,m}=\underset{K=\beta N\to \infty }{\lim }{\mathcal{U}}_k^{(m)H}\mathcal{T}{\mathcal{U}}_k^{(m)}\stackrel{a.s.}{=}{R}_{\ell }(|a_k{|}^2,{\tilde{\tau }}_k)$$

$${R}_{\ell}(\lambda,\tau)=\sum_{s=0}^{\ell-1}g({{{\mathchoice{\mbox{\boldmath$\displaystyle T$}}{\mbox{\boldmath$\textstyle T$}}{\mbox{\boldmath$\scriptstyle T$}}{\mbox{\boldmath$\scriptscriptstyle T$}}}}}_{\ell-s-1},\lambda,\tau){R}_{s}(\lambda,\tau)$$

$$\displaystyle{{{\mathchoice{\mbox{\boldmath$\displaystyle T$}}{\mbox{\boldmath$\textstyle T$}}{\mbox{\boldmath$\scriptstyle T$}}{\mbox{\boldmath$\scriptscriptstyle T$}}}}}_{\ell}(\Omega)$$

$$\mathbf{f}(R_{\ell },\Omega )$$

$$\displaystyle g({{{\mathchoice{\mbox{\boldmath$\displaystyle T$}}{\mbox{\boldmath$\textstyle T$}}{\mbox{\boldmath$\scriptstyle T$}}{\mbox{\boldmath$\scriptscriptstyle T$}}}}}_{\ell},\lambda,\tau)$$

$${\mathchoice{\mbox{\boldmath$\displaystyle\Delta$}}{\mbox{\boldmath$\textstyle\Delta$}}{\mbox{\boldmath$\scriptstyle\Delta$}}{\mbox{\boldmath$\scriptscriptstyle\Delta$}}}_{\phi,r}(\Omega,{\tau})=\left(\begin{array}[]{c}{\phi}(\Omega,\tau)\\ {\phi}(\Omega,\tau-\frac{T_{c}}{r})\\ \vdots\\ {\phi}(\Omega,\tau-\frac{T_{c}(r-1)}{r})\\ \end{array}\right).$$

$$m_{\mathcal{R}}^{(\ell )}=\mathrm{E}\{R_{\ell }(\lambda ,\tau )\}$$

$$\underset{K=\beta N\to \infty }{\lim }({\mathcal{R}}^{\ell }{)}_{k,m}=\underset{K=\beta N\to \infty }{\lim }{\mathcal{U}}_k^{(m)H}{\mathcal{T}}^{\ell -1}{\mathcal{U}}_k^{(m)}\stackrel{a.s.}{=}{R}_{\ell }(|a_k{|}^2)$$

$$R_{\ell }(\lambda )=\sum _{s=0}^{\ell -1}\lambda R_s(\lambda ){\nu }_{\ell -s-1}$$

Full text

The effects of excess bandwidth, chip-pulse shaping, and time delay distribution on CDMA with suboptimum linear receiver structures are investigated. Recursive expressions for universal weight design are given. The performance in terms of SINR is derived in the large-system limit and the performance improvement over synchronous systems is quantified. The considerations distinguish between two ways of forming discrete-time statistics: chip-matched filtering and oversampling.

Index Terms - Asynchronous code-division multiple-access (CDMA), channel capacity, effective interference, minimum mean-square error (MMSE) detector, multistage detector, multiuser detection, random matrix theory, random spreading sequences.

I Introduction

In Part I of this paper [1], we analyzed asynchronous CDMA systems with random spreading sequences in terms of spectral efficiency constrained to a given chip pulse waveform and in terms of SINR at the output of an optimum linear multiuser detector. The analysis showed that under realistic conditions, chip-asynchronous CDMA systems significantly outperform chip-synchronous CDMA systems. In order to utilize the benefits from chip-asynchronous111As already shown in Part I of this paper [1], asynchronism is beneficial when the relative delays between users are not integer multiples of a chip interval. To emphasize this requirement we use the term chip-asynchronism instead of asynchronism. CDMA, we need efficient algorithms to cope with multiuser detection for chip-asynchronous users. Therefore, in part II of this work, we focus on the generalization of known design rules for low-complexity multiuser detectors to chip-asynchronous CDMA.

A unified framework for the design and analysis of multiuser detectors that admit a multistage representation for synchronous users was given in [2]. The class of multiuser detectors that admit a multistage representation is large and includes popular linear multiuser detectors like linear MMSE detectors (e.g. [3]), reduced rank multistage Wiener filters [4, 5], polynomial expansion detectors [6] or conjugate gradient methods (e.g. [7]), linear parallel interference cancellers (PIC, e.g. [8, 9]), eventually weighted (e.g. [10]), and the single-user matched filters. Multistage detectors are constructed around the matched filter concept. They consist of a projection of the signal into a subspace of the whole signal space by successive matched filtering and re-spreading followed by a linear filter in the subspace.

Multistage detectors based on universal weights have been proposed in [11, 12] for CDMA systems in AWGN channels and extended to more realistic scenarios in [13, 14, 2]. These references make use of the self-averaging properties of large random matrices to find universal weighting coefficients for the linear filter in the subspace. More specifically, the universal weights are obtained by approximating the precise weights designed according to some optimality criterion with asymptotically optimum weights, i.e. the optimum weights for a CDMA system whose number of users and spreading factor tend to infinity with constant ratio. Thanks to the properties of random matrices, asymptotically, these weights become independent of the users’ spreading sequences and depend only on few macroscopic system parameters, as the system load or number of transmitted symbols per chip, the variance of the noise, and the distribution of the fading. In this way, the weight design for long-code CDMA simplifies considerably, its complexity becomes independent of both the number of users in the system and the spreading factor. Moreover, the weights need updating only when the macroscopic system parameters change.

The fact that users are not received in a time-synchronized manner at the receiver causes two main problems from a signal processing perspective: (i) the need for an infinite observation window to implement a linear MMSE detector and (ii) the potential need for oversampling to form sufficient discrete-time statistics. The need for an infinite observation window is primarily related to asynchronism on the symbol-level, not the chip-level. This aspect was addressed in [15, 16] where it was found that multistage detectors need not have infinite observation windows and can be efficiently implemented without windowing at all. A detailed overview of the state of art about statistics, sufficient or not, for multiuser CDMA systems and how to form them was addressed in Part I of this paper [1]. In part I we presented general results with the only constraint that the sampled noise at the output of the front-end was white. For the sake of clarity and to get insights into systems of practical interests, in this part II we focus on two groups of statistics implementable in practical systems:

1. (A)

Sufficient statistics obtained by filtering the received signal by a lowpass filter with bandwidth $B_{\mathrm{LOW}}$ larger than the chip-pulse bandwidth and subsequent sampling at rate $2B_{\mathrm{LOW}}$. 2. (B)

Statistics obtained by sampling the output of a filter matched to the chip waveform at the chip rate (chip rate sampling). In this case, the sampling instants need to be synchronized with the time delay of each user of interest. Thus, different statistics for each user are required. Additionally, the chip pulses at the output of matched filter need to satisfy the Nyquist criterion. In the following we refer to them as root Nyquist chip-pulse waveforms.

General results for the design of linear multistage detectors with both kind of statistics are provided in this work. The chip pulse waveforms are assumed to be identical for all users.

For asynchronous CDMA, low-complexity detectors with universal weights are conveniently designed for statistics (A). In fact, these observables enable a joint processing of all users without loss of information. Multistage detectors with universal weights and statistics (A) have a complexity order per bit equal to $\mathcal{O}(rK)$ if the sampling rate is $\frac{r}{T_{c}}.$ On the contrary, discretization scheme (B) provides different observables for each user and does not allow for simultaneous joint detection of all users. An implementation of multistage detectors with universal weights using such statistics implies a complexity order per bit equal to $\mathcal{O}(K^{2}).$ This approach is still interesting from a complexity point of view if detection of a single user is required. However, it suffers from a performance degradation due to the sub-optimality of the statistics.

This work is organized in six additional sections. Section II and III introduce the notation and the system model for asynchronous CDMA, respectively. In Section IV, multistage detectors for asynchronous CDMA are reviewed and a implementation which does not suffer from truncation effects is given. The design of universal weighting is addressed in Section 25. Finally, the analytical results are applied to gain further insight into the system in Section VI where methods for pulse-shaping, forming sufficient statistics and synchronization are compared. Conclusions are summed up in Section VII.

II Notation and Some Useful Definitions

Throughout Part II we adopt the same notation and definitions already introduced in Part I of this work [1]. In order to make Part II self-contained we repeat here definitions useful in this part. Upper and lower boldface symbols are used respectively for matrices and vectors corresponding to signals spanning a specific symbol interval $m.$ Matrices and vectors describing signals spanning more than a symbol interval are denoted by upper boldface calligraphic letters.

In the following, we utilize unitary Fourier transforms both in the continuous time and in the discrete time domain. The unitary Fourier transform of a function $f(t)$ in the continuous time domain is given by $F(\omega)=\frac{1}{\sqrt{2\pi}}\int f(t)\mathrm{e}^{-j\omega t}\mathrm{d}t$. The unitary Fourier transform of a sequence $\{\ldots,c_{-1},c_{0},c_{1},\ldots\}$ in the discrete time domain is given by $c(\Omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{+\infty}c_{n}\mathrm{e}^{-j\Omega n}$. We will refer to them shortly as Fourier transform. We denote the argument of a Fourier transform of a continuous function by $\omega$ and the argument of a Fourier transform of a sequence by $\Omega.$ They are the angular frequency and the normalized angular frequency, respectively. A function in $\Omega$ is periodic with respect to integer multiples of $2\pi$.

For further studies it is convenient to define the concept of $r$-block-wise circulant matrices of order $N$.

Definition 1

Let $r$ and $N$ be positive integers. An $r$-block-wise circulant matrix of order $N$ is an $rN\times N$ matrix of the form

[TABLE]

with {\mathchoice{\mbox{\boldmath\displaystyle B}}{\mbox{\boldmath\textstyle B}}{\mbox{\boldmath\scriptstyle B}}{\mbox{\boldmath\scriptscriptstyle B}}}_{i}=(c_{1,i},c_{2,i},\ldots,c_{r,i})^{T}.

In the matrix $\textstyle C$ an $r\times N$ block row is obtained by circularly right shift of the previous block. Since the matrix $\textstyle C$ is univocally defined by the unitary Fourier transforms of the sequences $\{c_{s,0},c_{s,1},\ldots c_{s,N-1}\}$, for $s=1...r,$

[TABLE]

there exists a bijection $\mathfrak{F}$ from the frequency dependent vector {\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}}(\Omega)=[c_{1}(\Omega),c_{2}(\Omega),\dots,c_{r}(\Omega)] to $\textstyle C$. Thus,

[TABLE]

Furthermore, the superscripts $\cdot^{T},$ $\cdot^{H},$ and $\cdot^{*},$ denote the transpose, the conjugate transpose, and the conjugate of the matrix argument, respectively. {\mathchoice{\mbox{\boldmath\displaystyle I}}{\mbox{\boldmath\textstyle I}}{\mbox{\boldmath\scriptstyle I}}{\mbox{\boldmath\scriptscriptstyle I}}}_{n} is the identity matrix of size $n\times n$ and $\mathbb{C}$, $\mathbb{Z}$, $\mathbb{Z}^{+},$ $\mathbb{N},$ and $\mathbb{R}$ are the fields of complex, integer, nonnegative integers, natural, and real numbers, respectively. $\mathrm{tr}(\cdot)$ is the trace of the matrix argument and \mathrm{span}({\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}}_{1},{\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}}_{2},\ldots,{\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}}_{s}) denotes the vector space spanned by the $s$ vectors {\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}}_{1},{\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}}_{2},\ldots{\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}}_{s}. $\mathrm{diag}(\ldots):\mathbb{C}^{n}\rightarrow\mathbb{C}^{n\times n}$ transforms an $n$-dimensional vector $\textstyle v$ into a diagonal matrix of size $n$ having as diagonal elements the components of $\textstyle v$ in the same order. $\mathrm{E}\{\cdot\}$ and $\mathrm{Pr}\{\cdot\}$ are the expectation and probability operators, respectively. $\delta_{ij}$ is the Kronecker symbol and $\delta(\lambda)$ is the Dirac’s delta function. $\mathrm{mod}$ denotes the modulus and $\lfloor\cdot\rfloor$ is the operator that yields the maximum integer not greater than its argument.

III System Model

In this section we recall briefly the system model for asynchronous CDMA introduced in Section IV and VII of Part I of this work [1]. The reader interested in the details of the derivation can refer to [1].

Let us consider an asynchronous CDMA system with $K$ active users in the uplink channel with spreading factor $N$. Each user and the base station are equipped with a single antenna. The channel is flat fading and impaired by additive white Gaussian noise with power spectral density $N_{0}$. The symbol interval is denoted with $T_{s}$ and $T_{c}=\frac{T_{s}}{N}$ is the chip interval. The modulation of all users is based on the same chip pulse waveform $\psi(t)$ bandlimited with bandwidth $B,$ unitary Fourier transform $\Psi(\omega),$ and energy $E_{\psi}=\int_{-\infty}^{\infty}|\psi(t)|^{2}\mathrm{d}t.$

The time delays of the $K$ users are denoted with $\tau_{k},$ $k=1,\ldots,K.$ Without loss of generality we can assume (i) user 1 as reference user so that $\tau_{1}=0$, (ii) the users ordered according to increasing time delay with respect to the reference user, i.e. $\tau_{1}\leq\tau_{2}\leq\ldots\leq\tau_{K}$; (iii) the time delay to be, at most, one symbol interval so that $\tau_{k}\in[0,T_{s})$.222For a thorough discussion on this assumption the reader can refer to [3].

As for the results presented in Part I, the mathematical results presented in this second part hold for any front-end that keeps the sampled noise white at its output. However, in order to get better insights into the physical system we focus on two front-ends of practical and theoretical interest. Both of them satisfy the more general assumption underlying the results in Part I. We refer to them as Front-end Type A and Front-end Type B333For the sake of compactness of some of the results, we adopt a different normalization from the one in Part I. Here, the signal energy at the output of the front-end is equal to one. In Part I, the energy of the analog filter’s impulse response is normalized to unity. The variance of the sampled noise at the front-end output changes accordingly..

Front-end Type A consists of

• •

An ideal lowpass filter with cut-off frequency $\omega=\frac{\pi r}{Tc}$ where $r\in\mathbb{Z}^{+}$ satisfies the constraint $B\leq\frac{r}{2T_{c}}$ such that the sampling theorem applies. The filter is normalized to obtain a unit overall amplification factor, i.e. the transfer function is

[TABLE]

• •

A subsequent continuous-discrete time conversion by sampling at rate $\frac{r}{T_{c}}.$

This front-end satisfies the conditions of the sampling theorem and, thus, provides sufficient discrete-time statistics. For convenience, the sampling rate is an integer multiple of the chip rate. Additionally, the discrete-time noise process is white with zero mean and variance $\sigma^{2}=\frac{N_{0}r}{E_{\psi}T_{c}}.$

Front-end Type B consists of

• •

A filter $G(\omega)$ matched to the chip pulse and normalized to the chip pulse energy, i.e. $G(\omega)={\Psi^{*}(\omega)}{{E_{\psi}^{-\frac{1}{2}}}};$

• •

Subsequent sampling at the chip rate.

When used with root Nyquist chip pulses, the discrete time noise process $\{w[p]\}$ is white with variance $\frac{N_{0}}{E_{\psi}T_{c}}$. For a synchronous systems with square root Nyquist chip pulses, this front end provides sufficient statistics whereas the observables are not sufficient if the system is asynchronous.

The chip waveform at the filter output is denoted by $\phi(t)$ and its unitary Fourier transform by $\Phi(\omega).$ The well-known relations $\phi(t)=\psi(t)\ast g(t)$ and $\Phi(\omega)=\Psi(\omega)G(\omega)$ hold. The unitary Fourier transform of the chip pulse waveform $\phi(t)$ sampled at rate $\frac{1}{T_{c}}$ and delay $\tau$ is given by

[TABLE]

Sufficient statistics for asynchronous CDMA require an infinite observation window. In the following, we introduce a matrix system model corresponding to an infinite observation window.

Let us denote with {{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}}}^{(m)} and {{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}}}^{(m)} the vectors of transmitted and received signals at time instants $m\in\mathbb{Z}.$ The baseband discrete-time asynchronous system is given by

[TABLE]

where {\boldsymbol{\mathcal{Y}}}=[\ldots,{{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}}}^{(m-1)T},{{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}}}^{(m)T},{{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}}}^{(m+1)T}\ldots]^{T} and {\boldsymbol{\mathcal{B}}}=[\ldots,{{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}}}^{(m-1)T},{{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}}}^{(m)T},{{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}}}^{(m+1)T}\ldots]^{T} are infinite-dimensional vectors of received and transmitted symbols respectively; ${\boldsymbol{\mathcal{W}}}$ is an infinite-dimensional noise vector; and ${\boldsymbol{\mathcal{H}}}$ is a bi-diagonal block matrix of infinite size given by

[TABLE]

Here, {{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{u}^{(m)} and {{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{d}^{(m)} are matrices of size $rN\times K$ obtained by the decomposition of the $2rN\times K$ matrix {{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}^{(m)} into two parts such that {{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}^{(m)}=[{{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{u}^{(m)T},{{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{d}^{(m)T}]^{T}. For {{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}^{(m)} the relation

[TABLE]

holds where $\textstyle A$ is the $K\times K$ diagonal matrix of the received amplitudes $a_{k}$ and {{\mathchoice{\mbox{\boldmath\displaystyle S}}{\mbox{\boldmath\textstyle S}}{\mbox{\boldmath\scriptstyle S}}{\mbox{\boldmath\scriptscriptstyle S}}}}^{(m)} is the $2rN\times K$ matrix whose $k$-th column accounts for the spreading of the symbol transmitted by user $k$ in the symbol interval $m$ and due to the actual spreading sequence, the channel delay, and filtering and sampling at the front-end. We refer to it as the matrix of virtual spreading. More specifically, the matrix of virtual spreading is given by

[TABLE]

where {{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k}^{(m)} is the $N$-dimensional column vector of the spreading sequence of user $k$ for the transmitted symbol $m$ and {\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{k} is the $2rN\times N$ matrix taking into account the effects of the chip pulse shape and the time delay $\tau_{k}$ of user $k.$ Let us decompose $\tau_{k}$ in $\overline{\tau}_{k}=\left\lfloor\frac{\tau_{k}}{T_{c}}\right\rfloor$ and $\widetilde{\tau}_{k}=\tau_{k}-T_{c}\overline{\tau}_{k}=\tau_{k}\mod T_{c},$ the integer number of chips the signal is delayed and its delay within a chip, respectively. The matrix {\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{k} is of the form

[TABLE]

where {\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}}_{\overline{\tau}_{k}} and {\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}}_{N-\overline{\tau}_{k}} are zero matrices of dimensions ${\overline{\tau}_{k}}\times N$ and $\left(N-\overline{\tau}_{k}\right)\times N$, respectively; \widetilde{{\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}}_{k} is an $r$-block-wise circulant matrix of order $N$ as in (2)

[TABLE]

with

[TABLE]

Thus, the virtual spreading sequences are the samples of the delayed continuous-time spreading waveforms at sampling rate $r/T_{c}$.

Throughout this work we assume that the transmitted symbols are uncorrelated and identically distributed random variables with unitary variance and zero mean, i.e. $\mathrm{E}({\boldsymbol{\mathcal{B}}})=\boldsymbol{\mathcal{O}}$ and $\mathrm{E}({\boldsymbol{\mathcal{B}}}{\boldsymbol{\mathcal{B}}}^{H})=\boldsymbol{\mathcal{I}}$ being $\boldsymbol{\mathcal{O}}$ and $\boldsymbol{\mathcal{I}}$ the unlimited zero vector and the unlimited identity matrix, respectively. The elements of the spreading sequences {{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k}^{(m)} are assumed to be zero mean i.i.d. Gaussian random variables over all the users, chips, and symbols with \mathrm{E}\{{{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k}^{(m)}{{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k}^{(m)H}\}=\frac{1}{N}{\mathchoice{\mbox{\boldmath\displaystyle I}}{\mbox{\boldmath\textstyle I}}{\mbox{\boldmath\scriptstyle I}}{\mbox{\boldmath\scriptscriptstyle I}}}_{N}. Finally, ${\boldsymbol{\mathcal{U}}}_{k}^{(m)}$ denotes that column of the matrix ${\boldsymbol{\mathcal{H}}}$ containing the $k^{\mathrm{th}}$ column of the matrix {\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}^{(m)}. We define the correlation matrices ${\boldsymbol{\mathcal{T}}}={\boldsymbol{\mathcal{H}}}{\boldsymbol{\mathcal{H}}}^{H}$ and ${\boldsymbol{\mathcal{R}}}={\boldsymbol{\mathcal{H}}}^{H}{\boldsymbol{\mathcal{H}}}.$ The system load $\beta=\frac{K}{N}$ is the number of transmitted symbols per chip.

IV Multistage Structures for Asynchronous CDMA

We consider the large class of linear multistage detectors for asynchronous CDMA. Let $\chi_{L,k}^{(m)}({\boldsymbol{\mathcal{H}}})$ be the Krylov subspace [17] of rank $L\in\mathbb{Z}^{+}$ given by

[TABLE]

A multistage detector of rank $L\in\mathbb{Z}^{+}$ for user $k$ is given by

[TABLE]

where {{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}_{k}^{(m)} is the $L$-dimensional vector of weight coefficients.

It has been shown in [16] that, given the weight vector {{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}_{k}^{(m)} the detection of the symbol $b_{k}^{(m)}$ by the multistage detector of rank $L$ in (12) can be performed with finite delay $L$ using the implementation scheme in Figure 1. Although infinite length vectors and infinite dimension matrices appear in (12), the multistage detector in Figure 1 implements exactly (12) and does not suffer from truncation effects. Equivalently, the multistage detector in Figure 1 can be considered as a multistage detector processing data over an observation window of size $2L.$ The projection of the received vector ${\boldsymbol{\mathcal{Y}}}$ onto the subspaces $\chi_{L,k}^{(m)}({\boldsymbol{\mathcal{H}}})$, for $k=1\ldots K,$ is performed jointly for all users and requires only multiplications between vectors and matrices. The size of those vectors and matrices does not depend on the observation window. For further details the interested reader is referred to [16], [18].

The class of multistage detectors includes many popular multiuser detectors:

• •

the single-user matched filter for $L=1$,

• •

the linear parallel interference canceller (PIC) [19, 20] for weight coefficients chosen irrespective of the properties of the transfer matrix ${\boldsymbol{\mathcal{H}}}$,

• •

the polynomial expansion detector [6] and the conjugate gradient method [7], if the weight coefficients are identical for all users and chosen to minimize the mean square error,

• •

the (reduced rank) multistage Wiener filter [5] if the weight coefficients are chosen to minimize the mean square error, but are allowed to differ from user to user.

Throughout this work we refer to detectors that minimize the MSE in the projection subspace of the user of interest as optimum detectors in the MSE sense. More specifically this class of multistage detectors includes the linear MMSE detector and the multistage Wiener filter but not the polynomial expansion detector.

In the following we focus on the design of multistage Wiener filters implemented as in Figure 1. This reduces the problem to the design of the filter coefficients {{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}_{k}^{(m)}. The multistage Wiener filter for the detection of the symbol $m$ transmitted by user $k$ reads

[TABLE]

The weight vector {{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}_{k}^{(m)} that minimizes the MSE $\mathrm{E}\{\|{\boldsymbol{\mathcal{M}}}_{k}^{(m)}{\boldsymbol{\mathcal{Y}}}-b_{k}^{(m)}\|^{2}\}$ is given by

[TABLE]

where {{\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}}}_{k}^{(m)} is an $L$-dimensional vector with $j^{\text{th}}$ element ({{\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}}}_{k}^{(m)})_{j}={\boldsymbol{\mathcal{U}}}_{k}^{(m)H}{\boldsymbol{\mathcal{T}}}^{j-1}{\boldsymbol{\mathcal{Y}}}. This optimization problem is solved by the Wiener-Hopf theorem [21] and {{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}_{k}^{(m)} is given by

[TABLE]

where \boldsymbol{\Xi}_{k}^{(m)}=\mathrm{E}\{{{\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}}}_{k}^{(m)}{{\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}}}_{k}^{(m)H}\} and \boldsymbol{\xi}=\mathrm{E}\{b_{k}^{(m)*}{{\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}}}_{k}^{(m)}\}. It is straightforward to verify that in this case

[TABLE]

where ({\boldsymbol{\mathcal{R}}}^{s})_{k,m}={{\mathchoice{\mbox{\boldmath\displaystyle h}}{\mbox{\boldmath\textstyle h}}{\mbox{\boldmath\scriptstyle h}}{\mbox{\boldmath\scriptscriptstyle h}}}}^{(m)H}_{k}{\boldsymbol{\mathcal{T}}}^{s-1}{{\mathchoice{\mbox{\boldmath\displaystyle h}}{\mbox{\boldmath\textstyle h}}{\mbox{\boldmath\scriptstyle h}}{\mbox{\boldmath\scriptscriptstyle h}}}}_{k}^{(m)} is the diagonal element of the matrix ${\boldsymbol{\mathcal{R}}}^{s}$ corresponding to the $m^{\text{th}}$ symbol transmitted by user $k.$

V Universal Weight Design

Consider the SINR of any linear detector that admits a multistage representation. Let \overline{{{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}}_{k,m} be the weight vector for the detection of the $m^{\mathrm{th}}$ symbol transmitted by user $k.$ Then, the SINR at the output of the multistage detector is given by

[TABLE]

The performance of multistage Wiener filters simplifies to

[TABLE]

From (16), (23), and (24) it is apparent that the diagonal elements of the matrix ${\boldsymbol{\mathcal{R}}}^{s}$ play a fundamental role in the design and analysis of multistage detectors.

It has been shown in [2] that, if the spreading sequences are random and the CDMA system is synchronous, the diagonal elements of the matrix ${\boldsymbol{\mathcal{R}}}^{s},$ $s\in\mathbb{Z}^{+},$ converge to deterministic values as $K,N\rightarrow\infty$ with constant ratio. This asymptotic convergence holds for some classes of random matrices and is a stronger property than the convergence of the eigenvalue distribution. The Stieltjes transform of the asymptotic eigenvalue distribution of ${\boldsymbol{\mathcal{R}}}$ is related to the SINR at the output of the linear MMSE detector, as pointed out first in [22] for synchronous CDMA systems. The asymptotic eigenvalue moments of ${\boldsymbol{\mathcal{R}}}$ enable the asymptotic performance analysis of reduced rank multistage Wiener filters [23] and the design of multistage detectors with quadratic complexity order per bit [14], [13]. The convergence of the diagonal elements of ${\boldsymbol{\mathcal{R}}}^{s}$ has been utilized in [2] for the design of multistage detectors with linear complexity order per bit in synchronous CDMA systems and for the asymptotic analysis of any multistage detector not necessarily optimum in a MSE sense. In the following we extend the results in [2] to the case of asynchronous CDMA systems making use of the asymptotic properties of the random matrix ${\boldsymbol{\mathcal{R}}}$ for asynchronous CDMA systems.

The design of low complexity multistage detectors is based on the approximation of the weight vectors {{\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}}}_{k}^{(m)} by their asymptotic limit when $K,N\rightarrow\infty$ with constant ratio $\beta$

[TABLE]

Thanks to the fact that the diagonal elements of ${\boldsymbol{\mathcal{R}}}^{s}$ can be computed by a polynomial in few macroscopic system parameters, the computation of the weight vectors becomes independent of the size of ${\boldsymbol{\mathcal{R}}}$ and independent of $m$. Thus, the effort for the computation of the weights becomes negligible and the complexity of the detector is dominated by the joint projection of the received signal ${\boldsymbol{\mathcal{Y}}}$ onto the subspaces $\chi_{k}^{(m)}({\boldsymbol{\mathcal{H}}}),$ $k=1\ldots K$ and $m\in\mathbb{Z}.$ This projection has linear complexity per bit if the multistage detector in Figure 1 is utilized.

The convergence of the diagonal elements of ${\boldsymbol{\mathcal{R}}}^{\ell}$ to deterministic values is established in the following theorem. The definitions and the assumptions in the statement of Theorem 1 summarize and formalize the characteristics of system model (5) for ${\tau}_{k}\in[0,T_{s}].$

Theorem 1

Let $K,N\in\mathbb{N}$ and {{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}\in\mathbb{C}^{K\times K} be a diagonal matrix with $k^{\text{th}}$ diagonal element $a_{k}\in\mathbb{C}.$ $T_{s}$ and $T_{c}$ are positive reals with $T_{s}=NT_{c}$. Given $\{\tau_{1},\tau_{2},\ldots\tau_{K}\}$ a set of delays in $[0,T_{s}),$ we introduce the sets of delays in $[0,T_{c})$ defined as $\{\widetilde{\tau}_{k}:\widetilde{\tau}_{k}=\tau_{k}\mathrm{mod}T_{c},\,k=1,\ldots K\}$ and the set of normalized delays $\left\{\overline{\tau}_{k}:\overline{\tau}_{k}=\left\lfloor\frac{\tau_{k}}{T_{c}}\right\rfloor\right\}.$ Given a function $\Phi(\omega):\mathbb{R}\rightarrow\mathbb{C}$, let $\phi(\Omega,\tau)$ be as in (4). Given a positive integer $r,$ let {\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{k},\,k=1,\ldots K, be $r$-block-wise circulant matrices of order $N$ defined in (10) and {{\mathchoice{\mbox{\boldmath\displaystyle S}}{\mbox{\boldmath\textstyle S}}{\mbox{\boldmath\scriptstyle S}}{\mbox{\boldmath\scriptscriptstyle S}}}}^{(m)}=\left({\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{1}{{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{1}^{(m)},{\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{2}{{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{2}^{(m)},\ldots{\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{K}{{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{K}^{(m)},\right) with {{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k}^{(m)} $N$-dimensional random column vector. Let {{{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}}=({{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{u}^{(m)T},{{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{d}^{(m)T})^{T}={{{\mathchoice{\mbox{\boldmath\displaystyle S}}{\mbox{\boldmath\textstyle S}}{\mbox{\boldmath\scriptstyle S}}{\mbox{\boldmath\scriptscriptstyle S}}}}}{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}} with {{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{u}^{(m)},{{\mathchoice{\mbox{\boldmath\displaystyle H}}{\mbox{\boldmath\textstyle H}}{\mbox{\boldmath\scriptstyle H}}{\mbox{\boldmath\scriptscriptstyle H}}}}_{d}^{(m)}\in\mathbb{C}^{rN\times K} and ${\boldsymbol{\mathcal{H}}}$ the infinite block row and block column matrix of the same form as in (6), ${\boldsymbol{\mathcal{T}}}={\boldsymbol{\mathcal{H}}}{\boldsymbol{\mathcal{H}}}^{H},$ ${\boldsymbol{\mathcal{R}}}={\boldsymbol{\mathcal{H}}}^{H}{\boldsymbol{\mathcal{H}}},$ and ${\boldsymbol{\mathcal{U}}}_{k}^{(m)}$ the column of ${\boldsymbol{\mathcal{H}}}$ corresponding to {\mathchoice{\mbox{\boldmath\displaystyle\Phi}}{\mbox{\boldmath\textstyle\Phi}}{\mbox{\boldmath\scriptstyle\Phi}}{\mbox{\boldmath\scriptscriptstyle\Phi}}}_{k}{{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k}^{(m)}.

We assume that the function $\Phi(\omega)$ is upper bounded and has finite support. The receive filter is such that the sampled discrete time noise process is white. The vectors {{\mathchoice{\mbox{\boldmath\displaystyle s}}{\mbox{\boldmath\textstyle s}}{\mbox{\boldmath\scriptstyle s}}{\mbox{\boldmath\scriptscriptstyle s}}}}_{k} are independent with i.i.d. zero-mean circularly symmetric Gaussian elements with variance $\mathrm{E}\{|s_{ij}|^{2}\}=N^{-1}$. Furthermore, the elements $a_{k}$ of the matrix $\textstyle A$ are uniformly bounded for any $K.$ The sequence of the empirical joint distributions F_{|{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}|^{2},\widetilde{T}}^{(K)}(\lambda,\widetilde{\tau})=\frac{1}{K}\sum_{k=1}^{K}1(\lambda-|a_{k}|^{2})1(\widetilde{\tau}-\widetilde{\tau}_{k}) converges almost surely, as $K\rightarrow\infty$, to a non-random distribution function F_{|{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}|^{2},\widetilde{T}}(\lambda,\widetilde{\tau}).

Then, conditioned on $(|a_{k}|^{2},\widetilde{\tau}_{k})$, the corresponding diagonal elements of the matrices ${\boldsymbol{\mathcal{R}}}^{\ell}$ converge almost surely to the deterministic value

[TABLE]

with ${R}_{\ell}(|a_{k}|^{2},\widetilde{\tau}_{k})$ determined by the following recursion

[TABLE]

and

[TABLE]

with

[TABLE]

The recursion is initialized by setting {{{\mathchoice{\mbox{\boldmath\displaystyle T}}{\mbox{\boldmath\textstyle T}}{\mbox{\boldmath\scriptstyle T}}{\mbox{\boldmath\scriptscriptstyle T}}}}}_{0}(\Omega)={\mathchoice{\mbox{\boldmath\displaystyle I}}{\mbox{\boldmath\textstyle I}}{\mbox{\boldmath\scriptstyle I}}{\mbox{\boldmath\scriptscriptstyle I}}}_{r} and ${R}_{0}(\lambda,\tau)=1.$

Theorem 1 is proven in Appendix A.

Note that the asymptotic diagonal elements of ${\boldsymbol{\mathcal{R}}}^{\ell}$ depend on the delay $\tau_{k}$ only via the delay of a chip pulse waveform within a chip, i.e. via $\widetilde{\tau}_{k},$ while any delay multiple of $T_{c}$ leaves the diagonal elements unchanged.

From Theorem 1 we can obtain $m_{{\boldsymbol{\mathcal{R}}}}^{(\ell)}$, the asymptotic eigenvalue moment of the matrix ${\boldsymbol{\mathcal{R}}}$ of order $\ell$ by using the relation

[TABLE]

where the expectation is taken over the limit distribution F_{|{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}|^{2},\widetilde{T}}(\lambda,\widetilde{\tau}). For $r=1$ and F_{|{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}|^{2},\widetilde{T}}(\lambda,\widetilde{\tau})=F_{|{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}|^{2}}(\lambda)\delta(\widetilde{\tau}), i.e. for synchronous systems sampled at the chip rate, and $\Phi(\omega)$ satisfying the Nyquist criterion the recursive equations (28), (29), and (30) reduce to the recursion in [2] Theorem 1.

This theorem is very general and holds for all chip pulses of practical interest. Furthermore, no constraint is imposed on the time delay distribution. The choice of the front end in this work is restricted only by the applicability of (23) or (24), which imply white noise at the front end. Then, since both Front-end A and Front -end B keep the sampled noise white, Theorem 1 applies to both of them.

Now, we specialize Theorem 1 to a case of theoretical and practical interest, where sufficient statistics are utilized in the detection, the chip pulse waveform $\phi(t)$ is band-limited, and the sequence of the empirical distribution functions of the time delays converges to a uniform distribution function as $K\rightarrow+\infty.$ The constraint to use sufficient statistics restricts the class of front-ends. The following results apply to Front-end A but, in general, not to Front-end B.

Corollary 1

Let us adopt the same definitions as in Theorem 1 and let the same assumptions of Theorem 1 be satisfied. Additionally, assume that the random variables $\lambda$ and $\widetilde{\tau}$ in F_{|{{\mathchoice{\mbox{\boldmath\displaystyle A}}{\mbox{\boldmath\textstyle A}}{\mbox{\boldmath\scriptstyle A}}{\mbox{\boldmath\scriptscriptstyle A}}}}|^{2},\widetilde{T}}(\lambda,\widetilde{\tau}) are statistically independent and the random variable $\widetilde{\tau}$ is uniformly distributed. Furthermore, $\Phi(\Omega)$ is bounded in absolute value, and bandlimited with bandwidth $B\leq\frac{r}{2T_{c}}.$ Then, given $(|a_{k}|^{2},\widetilde{\tau}_{k})$ and $m\in\mathbb{Z}$, the corresponding diagonal element of the matrix ${\boldsymbol{\mathcal{R}}}^{\ell}$ converges almost surely to a deterministic value, conditionally on $|a_{k}|^{2}$,

[TABLE]

with ${R}_{\ell}(\lambda)|_{\lambda=|a_{k}|^{2}}$ determined by the following recursion:

[TABLE]

and

[TABLE]

The recursion is initialized by setting ${T}_{0}(\omega)=1$ and ${R}_{0}(\lambda)=1.$

Corollary 1 is derived in Appendix B.

The eigenvalue moments of ${\boldsymbol{\mathcal{R}}}$ can be expressed in terms of the auxiliary quantities $f({R}_{s})$ and $\nu_{s}$ in the recursion of Corollary 1 by the following expression:

[TABLE]

Applying Corollary 1 we obtain the following algorithm to compute the asymptotic limits of the diagonal elements of ${\boldsymbol{\mathcal{R}}}^{\ell}$