Monte Carlo Solutions for Blind Phase Noise Estimation
© Frederik Simoens et al. 2009
Received: 30 June 2008
Accepted: 7 January 2009
Published: 12 February 2009
This paper investigates the use of Monte Carlo sampling methods for phase noise estimation on additive white Gaussian noise (AWGN) channels. The main contributions of the paper are (i) the development of a Monte Carlo framework for phase noise estimation, with special attention to sequential importance sampling and Rao-Blackwellization, (ii) the interpretation of existing Monte Carlo solutions within this generic framework, and (iii) the derivation of a novel phase noise estimator. Contrary to the ad hoc phase noise estimators that have been proposed in the past, the estimators considered in this paper are derived from solid probabilistic and performance-determining arguments. Computer simulations demonstrate that, on one hand, the Monte Carlo phase noise estimators outperform the existing estimators and, on the other hand, our newly proposed solution exhibits a lower complexity than the existing Monte Carlo solutions.
Instabilities of local oscillators are an inherent impairment of coherent communication schemes [1, 2]. Such instabilities give rise to a time-varying phase difference between the oscillator at the transmitter and the receiver sides. As the phase of the transmitted symbols conveys (part of) the information of a coherent transmission, the carrier phase must be known to the receiver before the recovery of the transmitted information can take place. Estimation of the carrier phase is henceforth a crucial task of a coherent receiver.
As long as frugality with respect to the available resources is deemed important, this estimation process should occur without inserting too many training or pilot symbols into the transmitted data sequence. The presence of training symbols in the data sequence reduces the spectral efficiency and power efficiency of the transmission. Estimating the carrier phase based on the unknown information carrying data symbols is definitely more efficient in that respect.
Spurred by its great importance, the research on phase noise estimation evolved into a relatively mature state nowadays. There already exists a myriad of estimation strategies and most of them achieve a satisfactory performance—at least under the specific circumstances for which they were designed [1–5]. The existing estimators range from feed-forward techniques assuming a piecewise constant carrier phase over the duration of a predefined interval [1–3] to more advanced algorithms which track the movements of the carrier phase from symbol to symbol [4, 5]. Despite all these ad hoc efforts, no optimal solutions—from a classical estimation point of view—to the phase noise estimation problem have yet been presented. Optimal estimation of the phase noise, for example, in a maximum-likelihood or maximum a posteriori sense, without knowing the transmitted information turns out to be an extremely complicated task.
The purpose of the present paper is exactly to investigate the phase noise problem within a classical estimation context. We will define an optimal receiver strategy and explore the extent to which Monte Carlo methods can be used to obtain a practical implementation of this optimal receiver. In doing so, we will furnish a thorough overview of Monte Carlo methods and their application to phase noise estimation. It is only fair to point out that Monte Carlo methods have already been considered for phase noise estimation in the past [6, 7]. However, these solutions are limited to uncoded systems and explore only one of the possible Monte Carlo techniques. In this paper, we will lay out a more general Monte Carlo framework and integrate the existing estimators within this framework. We will also present a novel estimator and demonstrate that it bears a lower complexity than the existing techniques.
This paper is organized as follows. Section 2 describes the channel model. The objective of the paper and the connection with existing phase noise estimators is outlined in Section 3. Since it is unfair to assume that everyone working in the field of phase noise estimation is acquainted with Monte Carlo methods, we devote an entire and relatively large section of this paper to the introduction of Monte Carlo methods and sequential importance sampling in particular (Section 4). The framework presented in Section 4 is thereafter applied to the phase noise problem for uncoded and coded systems in Sections 5 and 6, respectively. Finally, Section 7 provides numerical results and Section 8 wraps up the paper.
2. Channel Model
2.1. Phase Noise Channel Model
2.2. Linearized Phase Noise Channel Model
where represents an initial estimate of the phase at instant . This approximation is valid as long as . Hence, the linearized channel model can only be invoked if is small, and an accurate phase estimate is available.
3. Problem Formulation and Prior Work
In a coherent communication scheme, the receiver needs to know the phase at each time instant before detection can take place. The traditional way to acquire this information is by estimating the carrier phase. If the carrier phase remains constant over a relatively long period, standard feed-forward estimation techniques can be applied. In the presence of severe phase noise, however, other more ingenious techniques are called upon. Before we describe our approach in that regard, let us review some of the existing solutions.
3.1. Prior Work
Existing phase noise estimators or trackers have one thing in common. Their derivation does not stem from a probabilistic analysis, but is rather driven by pragmatic (and scenario dependent) arguments. Incidentally, the use of feedback loops or phase-locked loops is common practice .
where is a positive parameter, denotes the phase estimate at instant , and denotes an estimate (soft or hard decision) of , using the phase estimate from a previous time instant and possible additional information from a decoder (see also Section 6). Obviously, there exist other estimators as well, for example, . To our knowledge, however, their application is limited to pilot symbols only. Estimators of the form (6) are based on the linear model (5) and exploit the fact that hazards an estimate of the difference between and the true value of . The impact of the phase noise and the additive (thermal) noise can be balanced by tuning the parameter . Provided the linearized model (5) is a valid approximation, the optimal values, in a minimum mean squared error sense, of follow from the extended Kalman filter equations .
For a wide range of applications, these existing estimators render a satisfactory performance, but they nevertheless lack a rock-solid theoretical foundation. In the next section, we will outline our strategy to settle this issue.
3.2. Probabilistic Solution
A closed-form expression for follows immediately from the combination of (3) and the prior distribution . Hence, the remainder of this paper will focus on the derivation of and the ensuing computation of the integral in (8). In particular, we will investigate the use of Monte Carlo methods for the computation of (8).
4. Monte Carlo Framework
The purpose of this section is to provide a succinct introduction to Monte Carlo techniques. Section 5 addresses the specific application to our phase noise problem.
4.1. Particle Representation
Representing a distribution by means of samples or particles drawn from it is an appealing alternative in case the actual distribution defies an analytical representation. The rationale behind the particle filtering approach is that as long as we generate enough samples from the distribution, further processing with this distribution can be performed using particles of the distribution rather than the actual distribution. An example will serve to illustrate this benefit.
It can be shown that converges to as the number of particles grows . Hence, as long as we are able to draw samples from , it is not necessary to solve the integral from (9) analytically. The next section elaborates the case when sampling from is not that straightforward.
4.2. Importance Sampling
These weights are normalized such that . The idea is to assign different weights to the samples to compensate for the difference between the target distribution and the importance sampling distribution . Again, it can be shown that converges to for a large number of samples and under mild conditions with respect to the choice of .
4.3. Sequential Importance Sampling
where denotes the (hidden) state variable of the Markov process and the symbol means that the right-hand side is the probability function of the variable on the left-hand side. Note that we do not impose any restriction about the nature of or , these can be discrete or continuous, scalar or vector variables.
A typical problem associated with a Markov process involves the derivation of the a posteriori state distribution or inferences thereof. The purpose of this section is to explain how to draw samples from in a recursive manner, the process called sequential importance sampling (SIS).
4.3.1. Derivation of the Algorithm
The choice of the importance sampling distribution plays an important role with respect to the performance and stability of the algorithm. The next section elaborates this issue furthermore. To conclude this section, we summarize the operation of the SIS algorithm in Algorithm 1.
4.3.2. Degeneracy of Sequential Importance Sampling
One particularly annoying problem with SIS is that the variance of the importance weights increases as becomes larger . This is an adverse property as it is intuitively clear that for a fixed number of samples, the best approximation, in terms of its ability to evaluate the expectation of a function (11), to a distribution is obtained using equal-weight samples. The increasing variance is so persevering that almost all samples bear a negligible weight after a few recursions. This implies that the distribution is represented by far less particles than the original particles. Obviously, this does not bode well for the accuracy of the approximation of the distribution and the performance of ensuing algorithms. A detriment that manifests itself especially when dealing with high-dimensional state spaces, that is, where the state variable is actually a vector. Fortunately, this problem can be resolved by taking the following measures.
(1) Choice of the Sampling Distribution
Note that the weight update (18) does not depend on the current sample . This intuitively explains the optimality of (17) since the particular choice of the samples does not alter the weights, and hence, does not affect (read: increase) their variance. Unfortunately, this design measure will only slow down the process of degeneration; it will not bring it to a standstill. Furthermore, as will become apparent through the remainder of this paper, it is often very difficult to draw samples from (17). In this case, there is no alternative than to use a suboptimal distribution. The prior importance distribution forms a good alternative as it is often easy to sample from it. The corresponding weight update function follows from (16) and is given by .
A more effective approach to avoid degeneracy is resampling. The idea is to remove samples with negligible weight from the set and to include better chosen samples (which actually contribute in a meaningful manner to the representation of the target distribution). There are several methods to implement this rule in practice. The prevailing method is simply to draw new and equal-weight samples from the old distribution (defined by the weights of the old samples). Samples associated with low importance weights are most probably eliminated by this rule [11, 12].
and where . Again, it can be shown that converges to , defined in (9), for a large number of samples. Obviously, it only makes sense to rearrange (9) into (19) if can be computed analytically, and the integration from (20) is tractable.
5. Phase Noise Estimation for Uncoded Systems
Geared with the Monte Carlo framework from the previous section, we are now ready to tackle our original phase noise problem.
5.1. Joint Phase and Symbol Sampling
In a first attempt, we cast the problem under investigation immediately into the SIS algorithm by defining . The original state space model from (1), (2) is then a special case of the general model from (13). Application of the SIS algorithm immediately results in a sampled version of the a posteriori probability function .
Benefits and Drawbacks
The benefit of this algorithm is that it renders an asymptotically optimal solution, for a high number of particles, to the phase noise problem, provided that the linearized channel model approximation is accurate.
The sample space is two-dimensional. In general, more samples are required to represent a distribution of more than one variable. Obviously, this weighs on the overall complexity.
To overcome the drawbacks encountered with the previous method, we explore the application of the Rao-Blackwellization method in this section. We distinguish two separate approaches. The first one is a symbol-based sampling method. This method is not new and has already been investigated in , albeit without establishing the link with the Rao-Blackwellization framework. For completeness, we provide a Rao-Blackwellized derivation of the algorithm in this paper.
In the second and new approach, we only draw samples of the carrier phase. As we will demonstrate, this offers significant computational advantages.
5.2.1. Symbol-Based Sampling
Denote the mean and variance of the carrier variable at instant conditioned on the observations up to instant by and , as follows: This succinct derivation captures the main idea and furnishes the key equations of the symbol-based sampling approach.
Benefits and Drawbacks
The main benefit of this approach is the reduction of the sample space to one dimension. By running a Kalman filter in parallel with the particle filter, the posterior distribution of the carrier phase can be tracked analytically.
The algorithm still relies on the linearized channel model and suffers from the disadvantages mentioned in Section 5.1.
The computational complexity remains high due to the required evaluation of (33), (34), and the Kalman filter evaluation.
5.2.2. Phase-Based Sampling
In this second method, samples are drawn of the carrier phase rather than of the data symbols. We will distinguish two different approaches within this method. In the first approach, we use the optimal importance sampling distribution, whereas in the second approach, an alternative distribution is explored. We will show that the suboptimal sampling method results in a lower overall complexity.
(a) Optimal Distribution
The second transition follows from the fact that is a Markov process, provided that the transmitted symbols are independent. The first distribution in the last line has already been derived in Section 5.1. We can simply reuse the result obtained there if we replace by in (28). The second factor in (35) is also known and given by (26). Hence, as it turns out, is a mixture of Gaussian distributions. Sampling from this, a distribution is very simple. First, draw a sample from . Then, draw a phase sample from . The weight update equation is again given by (31).
This approach is almost identical to the approach from Section 5.1. The only difference is that the samples of the data symbols are not stored. Hence, this method will not mitigate the inconveniences of the earlier described methods. Note that this approach has also been investigated in .
(b) Prior Distribution
The functions in the right-hand side of (37) follow immediately from the channel model and are known.
Benefits and Drawbacks
The apparent simplicity of the latter method raises high hopes regarding the computational complexity. The only drawback of this method is that it does not use the optimal importance sampling distribution. However, as we will show in Section 7, the slightly more samples required to surmount degeneration are more than compensated by the reduced complexity of the method.
6. Phase Noise Estimation for Coded Systems
Straightforward application of the SIS algorithm is no longer possible for two reasons. First, the code constraint prohibits to draw samples from in a recursive manner. In particular, the evaluation of the importance sampling and particle update equations is prohibitive in the presence of a code constraint on the symbols. Second, the integral in (38) cannot be evaluated using the importance sampling technique as we have no closed-form solution for . The evaluation of requires a complicated decoding step, which has to be executed for every possible sample of . Obviously, this becomes impractical for a large number of samples.
Fortunately, we can extend the algorithms described above to a coded setup by means of iterative receiver processing. As shown in [15–19], there exists a solid framework based on factor graph theory that dictates how the estimation and the decoding can be decoupled in a coded setup. It can be shown that the factor graph solution converges to the optimal solution under mild conditions. The loops that arise in the factor graph representation of the receiver should not be too short. Extending the above algorithms to a coded system boils down to replacing the prior probabilities of the symbols with the extrinsic probabilities provided by the decoder. These extrinsic probabilities are updated by the decoder and exchanged in an iterative fashion with the estimator which, on its turn, updates the phase estimates. This process repeats until convergence of the algorithm is achieved. More details on this approach can be found in [15, 16]. Section 7 illustrates the performance of the resulting iterative receiver.
7. Numerical Results
The following paragraphs tender a discussion of the obtained results.
Let us begin with an uncoded configuration. If the transmitted symbols are unknown, it is impossible to assess the true value of the carrier phase based on the received signal. For QPSK, for instance, the carrier phase can only be known up to a four-fold ambiguity. Figure 1 demonstrates this fact. It portrays a histogram of the samples from the distribution , which were obtained through the evaluation of the phase-based sampling algorithm from Section 5.2.2 (with the optimal sampling distribution). In Figure 1, only the symbols at instants are known to the receiver. Hence, the distribution for is based solely on unknown symbols. As expected, the distribution exhibits 4 local maxima (at intervals). At , however, these ambiguities have been resolved because of the known symbols inserted before . This result indicates that it is necessary to insert pilot symbols in the data stream (at regular time instants).
Figures 2 and 3 illustrate the BER performance of various algorithms for an uncoded and coded setups, respectively. We considered the transmission of blocks of QPSK symbols, with the periodic insertion of one pilot symbol per symbols ( pilot overhead). The scenarios labeled phase-based A and B correspond to the phase-based sampling algorithm from Section 5.2.2, using the optimal and prior importance sampling distributions, respectively. The symbol-based algorithm corresponds to the algorithm which was proposed in  and has also been described in Section 5.2.1. These Monte Carlo approaches have also been compared to conventional phase noise estimators. Performance curves are included for an extended Kalman filter, using either hard-symbol decisions, soft-symbol decisions, or pilot symbols only (see also Section 3.1). In a coded setup, these soft or hard symbol decisions are based on the available posteriori probabilities of the symbols (available during the specific iteration).
As we can observe from Figures 2 and 3, it definitely pays to exploit information from the unknown data symbols. The estimators that are only based on pilot symbols give rise to a significant performance degradation. On the other hand, there is no much difference between the performance of the various blind estimators in the uncoded setup. This confirms that in an uncoded setup, the conventional estimators exhibit a satisfactory performance. In the coded configuration, however, the Monte Carlo methods outperform the conventional methods. Apparently, these conventional ad hoc methods fail to operate at the lower SNR-values that can be achieved with the use of coding. We furthermore observe that the phase-based estimators exhibit the best performance. The reason that the symbol-based method performs not as good is due to the fact that at high SNRs, the importance sampling distribution is very peaky. Therefore, almost all samples drawn from the distribution will be equal to each other. Hence, it takes a lot more samples to provide an accurate representation of this latter distribution, and the algorithm will suffer from cycle-slip-like phenomena .
Finally, we will examine the computational complexity of the different Monte Carlo-based methods. First, we note that the complexity of each of the presented algorithms scales linearly with the number of samples. Hence, it suffices to determine (i) the complexity per sample and (ii) the number of samples required to achieve a satisfactory performance.
It is hard to assess the complexity of the algorithms in an analytical manner. Therefore, we compared their relative complexity per sample based on the duration of an actual implementation on a Matlab simulation platform. Table 1 displays the results. Apparently, the phase-based sampling method with the prior importance sampling distribution bears the lowest complexity. Based on the simplicity of this estimator operation (see Section 5.2.2), this result does not come as a surprise.
Comparison of the complexity per sample of the Monte Carlo methods (for QPSK signaling).
Phase-based sampling A
Phase-based sampling B
This paper explored the use of Monte Carlo methods for phase noise estimation. Starting with a short survey on Monte Carlo methods, several techniques were introduced, such as sequential importance sampling and Rao-Blackwellization, laying the foundation for the development of various phase noise estimators. It turned out that there are two feasible Monte Carlo approaches to tackle the phase noise problem. The first one boils down to drawing samples from the a posteriori distribution of the symbols and updating them in a recursive manner. The carrier phase trajectory is hereby tracked analytically. This approach has previously been examined in . The other approach entails the sequential sampling of the a posteriori carrier phase distribution. Two different importance sampling distributions can be used for this method. The use of the optimal sampling distribution has been explored in , whereas this paper also considers the use of the prior sampling distribution. Computer simulations show that the performance complexity tradeoff is optimized for the phase-based sampling method with a prior importance sampling distribution.
The first author gratefully acknowledges the support from the Research Foundation-Flanders (FWO Vlaanderen). This work is also supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless Communications NEWCOM++ (Contract no. 216715), the Turkish Scientific and Technical Research Institute (TUBITAK) under Grant no. 108E054, and the Research Fund of Istanbul University under Projects UDP-2042/23012008, UDP-1679/10102007.
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