巴特沃斯滤波器，原理与Matlab实现

  巴特沃斯滤波器（英语：Butterworth Filter），也被称作最大平坦滤波器，是一种通频带之频率响应曲线平坦无纹波的信号处理滤波器，最先由英国工程师、物理学家史蒂芬·巴特沃斯提出。

传递函数求解

  巴特沃斯滤波器对不同频率下的增益进行了设计，$n$阶巴特沃斯滤波器的增益可以表示为（截止频率${\omega }_c$）：
$$\left|H_n\left(j\omega \right)\right|=\frac{1}{\sqrt{1+{\left(\frac{\omega }{{\omega }_c}\right)}^{2n}}}\iff {\left|H_n\left(j\omega \right)\right|}^2=\frac{1}{1+{\left(\frac{\omega }{{\omega }_c}\right)}^{2n}}$$
  上述表达式仅对幅频特性进行了描述，我们更希望得到传递函数形式的表达式，从而兼具幅频特性和相频特性，考虑到$\overline{H_n\left(j\omega \right)}=H_n\left(-j\omega \right)$，于是有：
{ H n ( s ) H n ( − s ) = H n ( j ω ) H n ( − j ω ) = H n ( j ω ) H n ( j ω ) ‾ = ∣ H n ( j ω ) ∣ 2 = 1 1 + ( ω ω c ) 2 n = 1 1 + ( j ω j ω c ) 2 n = 1 1 + ( s j ω c ) 2 n = 1 1 + ( − s 2 ω c 2 ) n \left\{

$$\begin{aligned} H_n\left( s \right)H_n\left( { - s} \right) &= {H_n}\left( {j\omega } \right){H_n}\left( { - j\omega } \right) = {H_n}\left( {j\omega } \right)\overline { {H_n}\left( {j\omega } \right)} = {\left| { {H_n}\left( {j\omega } \right)} \right|^2}\\ &= \frac{1}{ {1 + { {\left( {\frac{\omega }{ { {\omega _c}}}} \right)}^{2n}}}} = \frac{1}{ {1 + { {\left( {\frac{ {j\omega }}{ {j{\omega _c}}}} \right)}^{2n}}}} = \frac{1}{ {1 + { {\left( {\frac{s}{ {j{\omega _c}}}} \right)}^{2n}}}}\\ &= \frac{1}{ {1 + { {\left( { - \frac{ { {s^2}}}{ { {\omega _c}^2}}} \right)}^n}}} \end{aligned}$$ \right. ⎩ ⎨ ⎧​Hn​(s)Hn​(−s)​=Hn​(jω)Hn​(−jω)=Hn​(jω)Hn​(jω)​=∣Hn​(jω)∣2=1+(ωc​ω​)2n1​=1+(jωc​jω​)2n1​=1+(jωc​s​)2n1​=1+(−ωc​2s2​)n1​​
  进一步求解$H_n(s)$的具体表达式，假设$H_n(s)$的$n$个极点分别为$s_1,s_2,...,s_n$，则$H_n(-s)$的$n$个极点为相应的负对称点，$-s_1=s_1{e}^{j\pi },-s_2=s_2{e}^{j\pi },...,-s_n=s_n{e}^{j\pi }$。一般地，$H_n(s)$和$H_n(-s)$可以写作：
$$\{\begin{array}{rl}{H}_n\left(s\right)& =\frac{1}{\left(1-\frac{s}{s_1}\right)\left(1-\frac{s}{s_2}\right)\cdot \cdot \cdot \left(1-\frac{s}{s_n}\right)}\\ H_n\left(-s\right)& =\frac{1}{\left(1-\frac{s}{s_1{e}^{j\pi }}\right)\left(1-\frac{s}{s_2{e}^{j\pi }}\right)\cdot \cdot \cdot \left(1-\frac{s}{s_n{e}^{j\pi }}\right)}\end{array}$$
  显然，$s_1,s_2,...,s_n$和$s_1{e}^{j\pi },s_2{e}^{j\pi },...,s_n{e}^{j\pi }$也构成$H_n\left(s\right)H_n\left(-s\right)$的$2n$个极点。因此，可以通过求解$H_n\left(s\right)H_n\left(-s\right)$的全部极值点，选择其中$n$个左半平面极值点，构成$H_n\left(s\right)$。选左半平面极值点可以保证$H_n(s)$的稳定性，剩余的$n$个右半平面的极值点对应$H_n(-s)$函数。

  以下对$H_n\left(s\right)H_n\left(-s\right)$的极值点进行求解：
{ ( − s 2 ω c 2 ) n = − 1 = e j ( 2 k − 1 ) π → − s 2 ω c 2 = e j ( 2 k − 1 ) π n → s 2 ω c 2 = e j ( 2 k + n − 1 ) π n → s = ± ω c e j ( 2 k + n − 1 ) π 2 n ， k = 1 , 2 , . . . , n \left\{

$$\begin{array}{l} {\left( { - \frac{ { {s^2}}}{ { {\omega _c}^2}}} \right)^n} = - 1 = {e^{j\left( {2k - 1} \right)\pi }}\\ \to - \frac{ { {s^2}}}{ { {\omega _c}^2}} = {e^{\frac{ {j\left( {2k - 1} \right)\pi }}{n}}}\\ \to \frac{ { {s^2}}}{ { {\omega _c}^2}} = {e^{\frac{ {j\left( {2k + n - 1} \right)\pi }}{n}}}\\ \to s = \pm{\omega _c}{e^{ \frac{ {j\left( {2k + n - 1} \right)\pi }}{ {2n}}}}，k=1,2,...,n \end{array}$$ \right. ⎩ ⎨ ⎧​(−ωc​2s2​)n=−1=ej(2k−1)π→−ωc​2s2​=enj(2k−1)π​→ωc​2s2​=enj(2k+n−1)π​→s=±ωc​e2nj(2k+n−1)π​，k=1,2,...,n​

传递函数表达式

  求解1~8阶巴特沃斯滤波器的传递函数，列于下表。如有离散化需求，请参考：传递函数离散化方法。

Matlab波特图

% 设置频率范围
w = logspace(-1, 1, 1000);

% 设置截止频率
wc = 1;

% 定义各阶巴特沃斯滤波器的传递函数
H1 = 1 ./ (1 + 1j*w/wc);
H2 = 1 ./ (1 + sqrt(2)*1j*w/wc + (1j*w/wc).^2);
H3 = 1 ./ ((1 + 1j*w/wc) .* (1 + 1j*w/wc + (1j*w/wc).^2));
H4 = 1 ./ ((1 + 0.7654*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.8478*1j*w/wc + (1j*w/wc).^2));
H5 = 1 ./ ((1 + 1j*w/wc) .* (1 + 0.6180*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.6180*1j*w/wc + (1j*w/wc).^2));
H6 = 1 ./ ((1 + 0.5176*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.414*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.9318*1j*w/wc + (1j*w/wc).^2));
H7 = 1 ./ ((1 + 1j*w/wc) .* (1 + 0.4450*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.247*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.8022*1j*w/wc + (1j*w/wc).^2));
H8 = 1 ./ ((1 + 0.3986*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.111*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.6630*1j*w/wc + (1j*w/wc).^2) .* (1 + 1.9622*1j*w/wc + (1j*w/wc).^2));

% 绘制波特图
figure;
subplot(2, 1, 1);
semilogx(w, 20*log10(abs(H1)), 'DisplayName', '1st Order');
hold on;
semilogx(w, 20*log10(abs(H2)), 'DisplayName', '2nd Order', 'LineWidth', 2);
semilogx(w, 20*log10(abs(H3)), 'DisplayName', '3rd Order');
semilogx(w, 20*log10(abs(H4)), 'DisplayName', '4th Order', 'LineWidth', 2);
semilogx(w, 20*log10(abs(H5)), 'DisplayName', '5th Order');
semilogx(w, 20*log10(abs(H6)), 'DisplayName', '6th Order', 'LineWidth', 3);
semilogx(w, 20*log10(abs(H7)), 'DisplayName', '7th Order');
semilogx(w, 20*log10(abs(H8)), 'DisplayName', '8th Order', 'LineWidth', 3);
hold off;
title('Magnitude Response');
xlabel('Frequency (rad/s)');
ylabel('Magnitude (dB)');
legend;
grid on;
ylim([-160, 20]); % 设置幅值 y 轴的限制

subplot(2, 1, 2);
semilogx(w, angle(H1), 'DisplayName', '1st Order');
hold on;
semilogx(w, angle(H2), 'DisplayName', '2nd Order', 'LineWidth', 2);
semilogx(w, angle(H3), 'DisplayName', '3rd Order');
semilogx(w, angle(H4), 'DisplayName', '4th Order', 'LineWidth', 2);
semilogx(w, angle(H5), 'DisplayName', '5th Order');
semilogx(w, angle(H6), 'DisplayName', '6th Order', 'LineWidth', 3);
semilogx(w, angle(H7), 'DisplayName', '7th Order');
semilogx(w, angle(H8), 'DisplayName', '8th Order', 'LineWidth', 3);
hold off;
title('Phase Response');
xlabel('Frequency (rad/s)');
ylabel('Phase (radians)');
legend;
grid on;
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基本性质小结

1. 不同阶次巴特沃斯滤波器，在截止频率处的增益都是$-3$dB，这也是截止频率的定义；
2. 巴特沃斯滤波器的幅度特性是 $\omega$ 的单调函数，滤波器阶数越高，衰减速度越快；
3. 截止频率以上，每十倍频增益衰减$20n$ dB，$20{\log }_{10}\left|H_n\left(j\omega \right)\right|=20{\log }_{10}\frac{1}{\sqrt{1+{\left(\frac{\omega }{{\omega }_c}\right)}^{2n}}}\approx -20n{\log }_{10}\left(\frac{\omega }{{\omega }_c}\right)$；
4. 对于$n$阶巴特沃斯低通滤波器，幅频特性$\left|H_n\left(j\omega \right)\right|$，及其一阶导数、二阶导数、…、$2n-1$阶导数，在$\omega =0$时取值为零，也因此被称为最大平坦的滤波器。
5. 相比较于理想滤波器，巴特沃斯滤波器（特别是高阶型）对其幅频特性进行了很好的模拟，但忽略了相频特性。若更追求相频特性，可参考贝塞尔滤波器。

参考文献

参考1：巴特沃斯，维基百科
参考2：传递函数离散化方法
参考3：信号无失真传输，理想滤波器
参考4：巴特沃斯，CSDN
参考5：巴特沃斯，CSDN