最小平方误差受限的FIR滤波器设计

在设计FIR滤波器时,往往会指定过渡带大小,但过渡带的引入只是为了便于滤波器的设计,而并不是物理上的需要,所以在设计中仅需指定截至频率。这是第一个设计理念。此外,在FIR滤波器的设计中存在两种准则:一是等波纹设计准则 (即最大误差最小化或者Chebyshev准则 ),另一种是平方误差最小准则。但在现实中两种准则往往要同时兼顾,所以仅基于其中一种准则来设计不能得到最佳结果。这是第二个设计理念。基于上述两种设计理念,提出了一种新的FIR滤波器设计算法。该算法采用最陡梯度下降法来对平方误差最小化下的最佳滤波器系数进行迭代修正,得到最佳结果。

Constrained Least Squared Error Design of FIR Filters

In FIR design, transition bands are usually specified. However, the introduction of transition bands is just to simplify the design and is not the physical requirement, that is to say, only cutoff frequence is needed for the design. This is the first design notion. Furthermore, there are two criteria for FIR filters design, one is equiripple criterion (minimum peak error or Chebyshev criterion) and the other is least squared error (LSE) criteria. But in practice the two criteria are usually considered jointly, that is to say, the design is not optimum just based on the one of two criteria. This is the second design notion. A new design algorithm of FIR filters is proposed based on above two notions, which uses the steepest gradient descent (SGD) algorithm to correct the coefficients of optimum MMSE filters iteratively to achieve the optimum result.