Partitioned-block frequency-domain adaptive filter (PBFDAF) algorithms have become very popular, in particular, for acoustic signal processing. However, the stochastic behavior of PBFDAFs has not been extensively examined. This paper presents a comprehensive statistical analysis for a family of the overlap-save PBFDAF algorithms with 50% overlap, including the transient and steady-state performance. The frequency-domain equations of the PBFDAFs are transformed into the time-domain counterparts, which allows us to carry out the analysis completely in the time domain. By means of the independence assumption and vectorization operation, theoretical models for the mean and mean-square behavior of the PBFDAF algorithms are established without restricting the distribution of the inputs to being Gaussian. Specifically, the mean weight behavior and corresponding steady-state solution are provided. Closed-form expressions for the mean-square deviation (MSD) and mean-square error (MSE) are derived. The upper bound on the step size for the mean and mean-square stability of the PBFDAFs is specified. The theoretical model presents new insights into the convergence properties of the PBFDAFs with a sufficient number of coefficients. It was revealed that both the constrained and unconstrained PBFDAFs converge to the Wiener solution. However, the mean weight vector of the unconstrained PBFDAF algorithms cannot converge to the true solution for any inputs, while that of the constrained version can. The presented theory explains why the steady-state MSD of the unconstrained PBFDAF algorithm is larger than that of the constrained version but their minimum MSE is the same. Monte Carlo simulations provide very good support for our theory.

中文翻译：

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关于分区块频域自适应滤波器的收敛行为

分区块频域自适应滤波器 (PBFDAF) 算法已经变得非常流行，特别是用于声学信号处理。然而，PBFDAF 的随机行为尚未得到广泛研究。本文对具有 50% 重叠的一系列重叠保存 PBFDAF 算法进行了全面的统计分析，包括瞬态和稳态性能。PBFDAF 的频域方程被转换为时域方程，这使我们能够完全在时域中进行分析。通过独立性假设和向量化操作，建立了PBFDAF算法的均值和均方行为的理论模型，而没有将输入分布限制为高斯分布。具体来说，提供了平均重量行为和相应的稳态解。推导出均方偏差 (MSD) 和均方误差 (MSE) 的封闭式表达式。指定了 PBFDAF 的均值和均方稳定性的步长上限。该理论模型提供了对具有足够数量系数的 PBFDAF 收敛特性的新见解。结果表明，有约束和无约束的 PBFDAF 都收敛到 Wiener 解。然而，无约束 PBFDAF 算法的平均权向量不能收敛到任何输入的真实解，而约束版本的平均权向量可以。所提出的理论解释了为什么无约束 PBFDAF 算法的稳态 MSD 大于约束版本的稳态 MSD，但它们的最小 MSE 是相同的。