تجزیه و تحلیل عملکرد ردیابی و توان دوم خطاها حالت پایدار الگوریتم شبه OBE

The quasi-OBE (QOBE) algorithm is a set-membership adaptive filtering algorithm based on the principles of optimal bounding ellipsoid (OBE) processing. This algorithm can provide enhanced convergence and tracking performance as well as reduced average computational complexity in comparison with the more traditional adaptive filtering algorithms such as the recursive least squares (RLS) algorithm. In this paper, we analyze the steady-state mean squared error (MSE) and tracking performance of the QOBE algorithm. For this purpose, we derive energy conservation relation of the QOBE algorithm. The analysis leads to a nonlinear equation whose solution gives the steady-state MSE of the QOBE algorithm in both stationary and nonstationary environments. We prove that there is always a unique solution for this equation. The results predicted by the analysis show good agreement with the simulation experiments.

Set-membership adaptive filtering (SMAF) algorithms are known for their superiority over the more classical estimation techniques in three aspects. First, they provide sets of acceptable estimates rather than single estimates. Second, they feature improved convergence and tracking properties generally due to the execution of some kind of set-theoretic optimization, e.g., optimization of the step size or the weighting sequence. Third, they enjoy a data-discerning update strategy, which enables them to check for the innovation in the new data at each time instant and determine whether an update is required or not. In other words, they perform an update only when it can improve the quality of estimation and obviate the expense of updating when there is no useful information in the incoming data. As a result, in addition to the enhanced convergence and tracking performance, they can also provide an appreciable reduction in the average computational complexity [1], [2] and [3]. The optimal bounding ellipsoids (OBE) algorithms are well-established SMAF algorithms that tightly outer-bound the set of feasible solutions in the associated parameter space using ellipsoids. They optimize the size of the ellipsoids in some meaningful sense. Different optimality criteria have led to different OBE algorithms. Among them, the quasi-OBE (QOBE) algorithm1[4] and [5] is particularly interesting since it shares many of the desired features of the various OBE algorithms. Furthermore, it incorporates simple but efficient innovation check and optimal weight calculation processes, which make it computationally more efficient than other OBE algorithms. The SMAF algorithms generally have high degrees of nonlinearities, which make their performance analysis complicated, especially at the transient state. The QOBE algorithm involves both data nonlinearity [6] and error nonlinearity [7]. Few works have been reported on the theoretical performance analysis of the SMAF algorithms. Steady-state mean squared error (MSE) of the set-membership normalized least mean squares (SM-NLMS) algorithm [8] is analysed in [9] and [10] and steady-state MSE of the set-membership affine projection (SM-AP) algorithm [11] is analyzed in [12], [13] and [14]. Analytical results on the steady-state MSE performance of the set-membership binormalized data-reusing LMS algorithm and the multichannel filtered-x set-membership affine projection algorithm are given in [15] and [16], respectively. The work in [4] is also dedicated to prove that, with persistent excitation [17], the QOBE algorithm exhibits central-estimator convergence while its associated hyperellipsoidal membership set cannot converge to a point set. To the best of our knowledge, no steady-state MSE or tracking performance analysis of the QOBE algorithm has been reported in the literature to date. In this paper, we present a steady-state MSE and tracking performance analysis of the QOBE algorithm. The classic approach is to analyze the transient behavior of the adaptive filter and then obtain the steady-state MSE as a limiting behavior of the transient MSE [18] and [19]. However, due to high nonlinearity of the QOBE algorithm, this approach would be very difficult. To circumvent the analysis of the transient behavior and its complications, we employ the energy conservation argument [20] and [21] and initiate our steady-state analysis from the energy conservation relation of the QOBE algorithm. For the tracking performance analysis, we consider a random walk model for time variations of the unknown system in a nonstationary environment. The outcome of the analysis is a nonlinear equation whose solution yields the theoretical steady-state MSE of the QOBE algorithm in both stationary and nonstationary environments. Simulations confirm the accuracy of the theoretical predictions for the steady-state MSE in addition to justifying the assumptions made in the derivations. In Section 2, we describe the QOBE algorithm. We analyze its steady-state MSE and tracking performance in 3 and 4, provide some simulation results in Section 5, and conclude the paper in Section 6.

Assumptions : Despite seeming unnatural, A1 is reasonably realistic and simplifies the analysis tremendously. Several similar assumptions have been made in the literature (see, e.g., [9], [18], [21] and [23]). Without utilizing A2, calculation of the expectations in (21) would be arduous. This assumption is commonly made to deal with error nonlinearities (see, e.g., [7] and [9]). It is justified for long adaptive filters via central limit arguments [24]. A3 is natural and regularly used while A4 is typical in the context of tracking performance analysis of the adaptive filters (see, e.g., [21] and [25]). A5 is necessary to make the analysis tractable. Such assumptions are frequently used in the MSE performance analysis of the adaptive filters (see, e.g., [26]). A6 is also required for tractability of the analysis. However, assuming a stationary ‖xn‖−2‖xn‖−2 is not as strict as assuming a stationary input. Therefore, our analysis is valid for any input signal as long as ‖xn‖−2‖xn‖−2 is stationary. Input signal correlation: In the numerical studies of the stationary case, we provided a comparison between the experimental and theoretical steady-state MSEs for a typical correlated input signal. Extensive investigations showed that the theoretical predictions are in good agreement with experiments for a wide range of correlated and uncorrelated input signals. Simulations of the nonstationary case: The primary objective of the provided numerical results is to corroborate our theoretical findings. That is why the simulated MISO wireless channel varies in time according to the random walk model, which is the assumed model in the analysis. Admittedly, this is not the most appropriate way to model variations of a wireless channel. However, it is well established that a Rayleigh-fading channel can be suitably modeled by a first-order autoregressive model [27]. For slow variations, the corresponding autoregressive model can be approximated by a random walk model [23]. Accuracy of the analysis : Comparison with the simulation results shows that the analytical predictions for the steady-state MSE are less accurate when View the MathML sourceγ2/ση2 is considerably small. Nonetheless, in the presented comparisons, the largest gap between the experimental and the theoretical MSEs is still only about one dB. This lesser accuracy for the low values of View the MathML sourceγ2/ση2 can mainly be attributed to the fact that the assumptions employed in the analysis, particularly A1 and A2, become less tenable in such cases. However, it is known that the practically appropriate range of values is View the MathML sourceγ2/ση2≥1 and a value of View the MathML sourceγ2/ση2<1 cannot usually be a good choice in practice [19]. Summary: Steady-state MSE and tracking performance of the QOBE algorithm was analyzed utilizing the energy conservation argument. The analysis resulted in a nonlinear equation that by solving it, the theoretical steady-state MSE of the QOBE algorithm for both stationary and nonstationary systems is calculated. Existence and uniqueness of the solution of this equation was also proved. Simulations showed that the theoretical predictions match the experimental ones well.