本文中的例子和代码放在Github

First，什么是双二阶滤波器？wiki上是这么说的：二阶、递归、线性，含有两个极点和两个零点，“双二阶”的名字来源于它的传递函数是两个二次多项式的比值。

In signal processing, a digital biquad filter is a second order recursive linear filter, containing two poles and two zeros. "Biquad" is an abbreviation of "biquadratic", which refers to the fact that in the Z domain, its transfer function is the ratio of two quadratic functions: H(z)=(b₀+b₁z⁻¹+b₂z⁻²)/(a₀+a₁z⁻¹+a₂z⁻²) The coefficients are often normalized such that a₀ = 1: H(z)=(b₀+b₁z⁻¹+b₂z⁻²)/(1+a₁z⁻¹+a₂z⁻²) High-order IIR filters can be highly sensitive to quantization of their coefficients, and can easily become unstable.

归一化传递函数写成这样：

$$H(z)= \frac{b₀+b₁z⁻¹+b₂z⁻²}{1+a₁z⁻¹+a₂z⁻²}$$

用MATLAB的Filter Designer来设计一个：400Hz带通IIR，需要用4个Sections来实现，默认给出的滤波器结构是Direct-Form II。

在菜单栏的Analysis中选择Filter Coeffients就能看到滤波器系数了：

Numerator，分子，也就是传递函数中的b项们，从上到下依次为$b_0$，$b_1$和$b_2$。 Denominator，分母，也就是传递函数中的a项，从上到下依次为$a_0$，$a_1$和$a_2$，其中$a_0$总是为1。 Gain，增益。

用数组来存放滤波器系数：

//8-order IIR filter with 4 sections

const int sections = 4;

//nominator

const float b[4][3] = {

{ 1, -1.984454421, 1 },

{ 1, -1.999405318, 1 },

{ 1, -1.993167556, 1 },

{ 1, -1.998644244, 1 }

};

//denominator

const float a[4][3] = {

{ 1, -1.984532740, 0.9884094716 },

{ 1, -1.988571923, 0.9909378613 },

{ 1, -1.991214225, 0.9962624248 },

{ 1, -1.995917854, 0.9977478940 }

};

const float gain[4] = { 0.583805364, 0.583805364, 0.170388576, 0.170388576 };

以Direct-Form II为例来写个实现代码，计算过程是： $$y[n]=b_{0}w[n]+b_{1}w[n-1]+b_{2}w[n-2]$$ 其中$$w[n]=x[n]-a_{1}w[n-1]-a_{2}w[n-2]$$ 代码如下： 用一个数组来存放滤波器的中间状态量$w[n-1]$和$w[n-2]$：

float w[sections][2]; //filter states

在滤波计算之前，初始化滤波器：

for (int i = 0; i < sections; i++) {

w[i][0] = 0; //w[n-1]

w[i][1] = 0; //w[n-2]

}

正式开始计算：pcmIn[i]是原始的输入信号，输入section 1，section 1的输出则作为section 2的输入，以此类推，由于这个滤波器由4个sections构成，因此要循环4次。 注意输出要乘gain。

y[0] = pcmIn[i];

for (j = 0; j < sections; j++) {

tmp[j] = y[j] - a[j][1] * w[j][0] - a[j][2] * w[j][1]; //calculate w[n]

y[j+1] = tmp[j] + b[j][1] * w[j][0] + b[j][2] * w[j][1]; //calculate the j-th section filter output y[n]

w[j][1] = w[j][0]; //move w[n-1] -> w[n-2]

w[j][0] = tmp[j]; //move w[n] -> w[n-1]

y[j+1] = gain[j] * y[j+1]; //multiply with gain

}

out = y[j];

如果需要得到例如PCM16的输出，那么再对out进行限幅，that's all.