用重叠相加和重叠保留技术实现数字FIR滤波器的频域实现
On Frequency-Domain Implementation of Digital FIR Filters Using Overlap-Add and Overlap-Save Techniques
摘要：
在本文中，新的见解，在频域实现的数字有限长度的脉冲响应滤波（线性卷积），使用数字相加和数字保存技术提供。它示出，在实际的有限字长实现，整个系统对应于一个随时间变化的系统，可以表示在本质上两种不同的方式。单向式是用失真函数和混叠函数来表示系统，本文中的失真函数和混叠函数是从多速率滤波器组表示导出的。另一种方法是使用周期性时变脉冲响应表示，或者等效地，使用一组时不变脉冲响应和相应的频率响应。本文提供了系统的推导和分析，这些表示随着滤波器的脉冲响应特性和设计实例。当分析系数量化的效果以及使用比理论上所需的更短的DFT长度时，这些表示特别有用。还提供了一个全面的计算复杂性分析，并推导出精确的公式，用于估计给定的滤波器长度的最佳DFT长度。使用最佳的DFT长度，它示出的频域实现具有较低的计算复杂度（乘法率）比相应的时域实现的滤波器的长度比那些在文献中较早报道的更短。特别是，对于一般（非对称）滤波器，频域实现被证明是更有效的所有滤波器长度。这为比较不同过滤器实现的复杂性提供了新的考虑因素。
英文摘要：
In this paper, new insights in frequency-domain implementations of digital finite-length impulse response filtering (linear convolution) using overlap-add and overlap-save techniques are provided. It is shown that, in practical finite-wordlength implementations, the overall system corresponds to a time-varying system that can be represented in essentially two different ways. One way is to represent the system with a distortion function and aliasing functions, which in this paper is derived from multirate filter bank representations. The other way is to use a periodically time-varying impulse-response representation or, equivalently, a set of time-invariant impulse responses and the corresponding frequency responses. The paper provides systematic derivations and analyses of these representations along with filter impulse response properties and design examples. The representations are particularly useful when analyzing the effect of coefficient quantizations as well as the use of shorter DFT lengths than theoretically required. A comprehensive computational-complexity analysis is also provided, and accurate formulas for estimating the optimal DFT lengths for given filter lengths are derived. Using optimal DFT lengths, it is shown that the frequency-domain implementations have lower computational complexities (multiplication rates) than the corresponding time-domain implementations for filter lengths that are shorter than those reported earlier in the literature. In particular, for general (unsymmetric) filters, the frequency-domain implementations are shown to be more efficient for all filter lengths. This opens up for new considerations when comparing complexities of different filter implementations.
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