A sequential Monte Carlo technique for blind synchronization and detection in frequency-flat Rayleigh fading wireless channels
Introduction
Narrow band mobile communication links are generally modeled as frequency-flat Rayleigh fading channels. Recently, a lot of research work has been focused on the detection of signals over such channels [5], [15], [22]. However, most of these contributions assume a perfect knowledge of the so-called synchronization parameters. It is broadly recognized that many practical communication channels present a high degree of structure and they can be accurately characterized through a set of reference parameters with a clear physical meaning. Since the observed signals collected by the receiver are affected by these parameters, they should be estimated, and compensated for, prior to data detection in order to achieve optimal or close-to-optimal performance. The generalized synchronization problem consists of the recovery of a set of such physical parameters as the symbol timing, phase offset and carrier frequency error. Unfortunately, optimal estimators of the parameters of interest cannot be derived in closed form and practical methods found in the literature [14], [19] are either heuristic or based on approximate maximum likelihood (ML) arguments.
Sequential Monte Carlo (SMC) techniques [8], [10] (also referred to as particle filtering methods) are powerful tools for Bayesian estimation that employ discrete measures with random support for representing posterior probability distributions of unknowns of interest. Recently, particle filtering has been successfully applied in digital communication problems, including applications such as channel estimation, equalization or space–time decoding (see [7] for a recent review of the subject). The SMC approach is also potentially useful for joint symbol detection and synchronization because it provides a way to numerically compute optimal Bayesian estimators when exact solutions cannot be derived analytically.
In order to apply common SMC algorithms, e.g., sequential importance sampling (SIS) [10], the only (and mild) requirement is that the observed signals can be written as a dynamic system in state-space form, which is usually simple to achieve with most communication signals. Existing SMC-based schemes which involve timing recovery and phase offset correction can be found in [2], [3], [12], [16], [17], [18]. In [2], [3] a pilot-data aided particle filtering algorithm is used for the estimation of the delay and the channel complex amplitude (which includes the phase offset) in a system with direct sequence spread spectrum (DSSS) modulation. A similar problem is addressed in [12], where the code-delay and the complex channel is estimated using a suboptimal (but complexity-restrained) combination of SMC methods and extended Kalman filtering, together with linear data detection. The problem of joint timing recovery, phase correction and data detection in DSSS modulated systems is addressed in [16], [17], [18] using particle filtering tools. In particular, a SMC algorithm based on deterministic sampling is presented and numerically evaluated in [18].
In this paper, we consider the problem of joint detection and non-data-aided generalized synchronization in general linearly modulated transmission systems using particle filtering. Specifically, we propose a new method for jointly and adaptively estimating the physical channel parameters (symbol timing, phase offset, channel amplitude and frequency error) and the transmitted data sequence, without the aid of any pilot symbols or training sequences. The SMC algorithm at the core of the new receiver is derived by considering an extended dynamic system where the symbol delay and the frequency error are modeled as first-order autoregressive (AR) stochastic processes [11] and the fading process of the complex channel (amplitude and phase) is modeled as a second-order AR process driven by complex white Gaussian noise [15]. The transmitted symbols are assumed independent and identically distributed (i.i.d.) random variables with a discrete uniform distribution. Within this framework we suggest two possible configurations to build an adequate receiver architecture that allows for complete removal of inter-symbol interference (ISI) and attains close-to-optimal symbol error rate (SER):
An open-loop structure consisting of two branches, where the first one processes the received signal in order to compute frequency error, delay and complex channel estimates which are used in the second branch to sample the received signal with corrected timing epochs and adequately rotate the resulting observation to compensate the phase and frequency offsets.
A closed-loop scheme that exploits the sequential structure of the proposed SMC algorithm to adaptively adjust the receiver timing, phase offset and frequency error estimates.
The proposed receiver performance is assessed both analytically, by deriving the posterior Cramér–Rao bound (PCRB) for timing estimation, and through computer simulations. The latter allow to illustrate both the comparison of the delay estimates with the PCRB and the overall performance of the receiver in terms of its SER.
The remaining of the paper is organized as follows. Section 2 describes the signal model. The proposed SMC algorithm and the suggested receiver architectures are introduced in Sections 3 and 4, respectively. In Section 5, we proceed with an analytical study of the PCRB for timing estimation. Computer simulation results are presented in Section 6 and, finally, brief concluding remarks are made in Section 7.
Section snippets
Received signal
Let us consider a digital communication system where symbols from a discrete alphabet, , are transmitted in frames of length M. The baseband-equivalent received signal has the formwhere is the complex multiplicative noise introduced by the frequency-flat Rayleigh fading channel, is the carrier frequency error, is the mth transmitted symbol, is a squared-root raised-cosine pulse waveform, T is the symbol period, is the
A sequential Monte Carlo method for joint synchronization and data detection
In this section, we propose an efficient particle filtering algorithm for joint reference parameters estimation and data detection. We focus on the joint estimation of the symbols, , the delays, , and the frequency error, , from the available observations, . The complex fading process is handled as a nuisance process which is analytically integrated out. Nevertheless, estimates of can be easily
Receiver architecture
Although the transmitted data with their timing can be estimated together using the SIS algorithm described above, it is important to notice that the proposed method does not remove the ISI. In other words, although the relative symbol delays, , are estimated, the sampling instants, , are not corrected to attain a better timing and avoid ISI. As a consequence, the SER that can be attained by detecting according to (54) is lower bounded by the SER of the maximum likelihood sequence
Posterior Cramér–Rao bound
Posterior distribution estimates based on SIS algorithms converge asymptotically to the true posterior distribution as the number particles, N, approaches infinity. In practice, however, a finite number particles is used to estimate parameters of interest. As a result of this approximations a certain degradation in performance of the estimation is expected. In order to study the efficiency of the proposed estimation method, it is of great interest to compute the variance bounds on the
Computer simulations
Finally, we present computer simulations that illustrate the validity of our approach. We have considered a differentially encoded BPSK modulation (symbol alphabet ) with symbol period of and a flat fading channel with fading rate (AR parameters , and selected accordingly). The delay has been modeled as a first-order AR process with parameter and noise variance . Similarly, the frequency error AR process is assigned parameters and noise
Conclusions
We have presented a novel algorithm for blind generalized synchronization and data detection in frequency-flat fast-fading wireless channels based on a Bayesian estimation approach and the SMC methodology. The proposed SMC technique allows to obtain asymptotically optimal estimates of the symbol time varying delays, the also time-varying frequency error and the fast fading complex channel process (which includes both amplitude attenuation and phase offset). The design of the resulting blind
Acknowledgements
Joaquín Míguez ackowledges the support of Ministerio de Ciencia y Tecnología of Spain and Xunta de Galicia (project TIC2001-0751-C04-01). Tadesse Ghirmai, Mónica F. Bugallo and Petar M. Djurić acknowledge the support of the National Science Foundation (Award CCR-0082607).
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