【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

时间:2021-02-05 16:52:13

作者:桂。

时间:2017-06-06 13:25:58

链接:http://www.cnblogs.com/xingshansi/p/6943833.html

论文原文:http://pan.baidu.com/s/1hsuuQYK


前言

上一篇GSC是基于delay的框架进行处理,这是在无混响的情况下一种简单近似处理。许多更为复杂的应用场景,如存在的混响较严重Rt=450ms,则基于delay的模型是不合适的,有学者就考虑直接利用系统的响应函数,也就是传递函数(Transfer function, TF)进行处理,这也就是本文要梳理的思路。

一、理论模型

模型:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

其中

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

对应的频域变换

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

M为麦克风的数量。

问题知道了,剩下就是建模求解。仍然可以是Frost's algorithmGSC两种框架。与原始问题基本的不同点就是一个:原始的利用delay的延时补偿;此处是基于传递函数TF。用文中的话就是:

In  Frost's algorithm, a beamforming algorithm was proposed under the assumption that the TF from the desired signal source to each sensor includes only gain and delay values. In this article, we consider the general case of arbitrary TFs.

二、理论求解

  A-Frost's algorithm

滤波器频域系数

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

滤波后的信号

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

滤波后输出信号的均方值,也就是目标函数:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

Frost's algorithm基于MVDR的思路,保证目标信号不失真(限制条件),最小化输出功率(目标函数),这就是一个LCMV(linear constrained minimum variance)问题.约束条件:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

总结一下求解的理论模型

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

同样的思路,利用拉格朗日乘子法得出滤波器最优解

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

同之前的问题一样,在实际的工程应用中:

This closed-form solution is difficult to implement and does not have the ability to track changes in the environment. Therefore, an adaptive solution should be more useful.

因此需要借助梯度下降的思路,利用自适应滤波(LMS、NLMS等)进行工程化的落地。

迭代思路

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

其中

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

W的迭代也可以进一步简化:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech可以看出实现中只要A已知,整个的框架就搭建完成,问题就转换成:A如何求解?GSC框架也存在同样的问题。如果A已知,此处的TF-Frost's algorithm和TF-GSC都可以工程化落地,但实际应用中A通常未知,所以论文也只是交代了思路,只有TF-GSC利用了一种参数转化的思想进行参数估计,实现工程落地,而TF-Frost's algorithm并没有落地,因此论文的实验部分也仅仅是TF-GSC与基于delay的D-GSC对比,提前补充说明一下(个人理解,如果有误还请帮忙指出)。

回顾一下基于delay的Frost's algorithm框架:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

  B-TF-GSC

GSC的思想与Frost's algorithm的不同在于,除了利用约束条件

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

假设矩阵N个*度,约束条件利用了n个(Frost's algorithm),GSC还利用了剩下的N-n个*度:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

更直观的说,回顾基本GSC框架(本文方法的无t延迟操作)

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

上支的作用就是Frost's algorithm,下支的Block就是剩下的*度,所以上支的W与下支的B需要正交,否则冗余。通常为了简便,W与B直接给定,只优化Block之后的参数矩阵,这样一来W就不是最优了,这也是为什么说GSC是Frost's algorithm的扩展不够严谨,如果仅仅从理论框架来讲,GSC就是Frost's algorithm的扩展。

TF-GSC框架的输出

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech其中上支

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech其中

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

F的定义

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

之所以可以W0等于F,是因为

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

满足向量空间

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

下支

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech因为是利用N-n的*度,所以【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech存在于向量空间

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

剩下就是自适应滤波求取G了。

进一步讨论之前,再将GSC框架梳理一下(TF-GSC也属于该框架)

1)上支:也就是fixed beamformer,构建带噪的增强信号;例如本文的

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

2)下支:也就是Blcok matrix,构建噪声参考信号;例如本文的

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

3)自适应部分:就是针对上支信号,利用下支得到的噪声对其进行自适应处理,例如本文的

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

处理结束的信号,就是整个GSC框架增强的信号。

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

说说G的求解过程。

重新写出自适应的准则函数

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

其中

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

利用最小二乘的思路

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

考虑到工程的适应性,通常用梯度下降的思想,借助最小二乘实现

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

也可以normalized一下,即NLMS

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

其中Pest的更新方式为

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

假设G求解完毕,如何工程化实现呢?一种设计思路是借助FIR

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

理论上G求解之后,利用最佳逼近,便可得出FIR的设计,但为了防止尖峰值sharp value,考虑如下思路(其实就是为了防止脉冲的削波操作)

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

细说就是三步走:1)将估计的【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech时域变换;2)利用削波将时域的g控制在幅值【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech范围内;3)将处理后的信号变换到频域,认为是利用FIR设计的G,也就是G的最终估计。

三步走的操作可以让迭代步骤更加Robust.

至此,理论分析已经结束。

  C-参数转化

然而直接求解/估计传递函数【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech是困难的,这一点很容易理解,但绝对转化为相对,求解就容易了。

1-上支

定义

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

W0重写为

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

如果H归一化,W0可简写

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

2-下支

一种构造方式(满足正交空间即可):

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech同样可以由【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech得到。

3-自适应

这一步按上面的思路工程上完全可以实现。

至此完成了原始问题的参数转化,也就是将绝对求解转化为相对求解,剩下就是【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech的参数估计了。

  D-参数估计

考虑到

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

从而

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

进一步得出

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

实际操作中,基于平稳遍历的假设,近似估计:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

利用K帧数据进行处理

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

从而实现H的估计:

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

<.>代表均值的操作

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

至此TF-GSC完成了理论分析→理论在工程的落地→实际应用的参数估计,反过来看就是整个工程实现的思路搭建完毕。

总结一下算法流程

【论文:麦克风阵列增强】Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech

参考

  • Gannot, Sharon, David Burshtein, and Ehud Weinstein. "Signal enhancement using beamforming and nonstationarity with applications to speech." IEEE Transactions on Signal Processing 49.8 (2001): 1614-1626.