Classical Analog Prototypes
四类经典模拟滤波器：Butterworth · Chebyshev · 椭圆
IIR 设计的第一步是选一个模拟逼近族。四类经典原型在三件事上做权衡：
过渡带陡峭度、通带/阻带纹波、相位与群延迟。
想要更陡的过渡带，就得容忍纹波或更差的相位。本页只看低通原型，
把四族归一化到同一个通带边界 （在此处都衰减 dB）。
下面一个面板，三个视图：幅频响应、群延迟、零极点图，点标签切换。
Step 1 of IIR design is picking an analog approximation family. The four classics trade three things:
transition steepness, passband / stopband ripple, and phase / group delay. A sharper transition costs ripple or worse phase.
This page uses a low-pass prototype, normalized so all four reach dB at .
One panel, three views: magnitude, group delay, pole-zero map.
拖动阶数 、通带纹波 、阻带衰减 。 决定 Chebyshev I、椭圆与 Butterworth 的通带边界； 决定 Chebyshev II 与椭圆的阻带。
群延迟 衡量各频率成分被延迟多少，通带内越平坦波形失真越小。
过渡带越陡，截止附近的群延迟峰越高、越不平。
Group delay measures how much each frequency is delayed; flatter in the passband means less waveform distortion.
Curves are computed analytically from each family's exact poles, including the elliptic poles.
极点位置决定滚降快慢，有限零点决定阻带能否压出陷波。
Butterworth 极点在圆上，Chebyshev I 在椭圆上，两者都没有有限零点；
Chebyshev II 与椭圆在 轴上放有限零点制造陷波。选一族查看。
Poles set the rolloff; finite zeros create stopband notches. Pick a family.
| 族 Family | 通带 Passband | 阻带 Stopband | 同阶过渡带 Transition | 相位/群延迟 Phase | 关键参数 Params |
|---|---|---|---|---|---|
| Butterworth巴特沃斯 | 最大平坦maximally flat | 单调monotonic | 最缓gentlest | 最好（四族中）best of these | |
| Chebyshev I切比雪夫 I | 等纹波equiripple | 单调monotonic | 较陡steeper | 较差worse | |
| Chebyshev II切比雪夫 II | 单调monotonic | 等纹波equiripple | 较陡steeper | 居中moderate | |
| Elliptic椭圆 Cauer | 等纹波equiripple | 等纹波equiripple | 最陡sharpest | 最差worst |
“相位家族”的 Bessel 滤波器Bessel: the phase-approximation family
上表四族（Butterworth / Chebyshev I·II / 椭圆）都是幅度逼近：用一条 模板来定义，比的是通带/阻带纹波和过渡带陡峭度。Bessel 是个例外，它属于“相位逼近”族。
- 设计目标：通带内最大平坦群延迟，等价于最接近线性相位。分母由 Bessel 多项式给出，全极点、无有限零点；幅度只是副产品（单调、平缓）。
- 代价：滚降是五族里最差的（比 Butterworth 还缓）。本质是“用更高阶数去换更好的相位”。
- 何时用：当波形保真、时序比频率截止的陡峭更重要时。它让脉冲几乎不变形、保持信号形态（生物信号里如尽量不歪曲 QRS 波形）。
这也是它在“ 能否唯一确定原型”里成为例外的原因：其余四族由幅度定义，而 Bessel 的定义指标是群延迟（相位），偏偏 把相位信息全扔了。数学上因为它全极点，给定幅度仍能唯一恢复 ；但那抓错了它的本质。
为什么本页不画它：本页把四族摆在同一幅度坐标上对比；Bessel 的看点在相位 / 群延迟那条轴上，属于另一个比较维度。
The other four families are amplitude approximations defined by a |H|² template. Bessel is a phase approximation: it targets maximally flat group delay (most linear phase), is all-pole, and its magnitude is just a gentle, byproduct response with the poorest rolloff of the five. It is the exception to “magnitude determines the prototype” because its defining spec lives in the phase, which |H| discards (though, being all-pole, its H(s) is still mathematically recoverable from |H|²). Not plotted here because this page compares amplitude families on a common magnitude axis.
Q1: 为什么椭圆滤波器同阶最陡？Why is the elliptic filter the sharpest at a given order?
因为它在 轴上放了有限零点，在阻带制造出深陷波。Butterworth 与 Chebyshev I 只有极点、没有有限零点，阻带只能单调下滑。椭圆把这些零点紧贴在阻带边界外侧，一进阻带就把响应拉到很深，所以同阶下过渡带最陡。
It places finite zeros on the jω axis, creating stopband nulls. Butterworth and Chebyshev I have poles only (no finite zeros), so their stopband can only slide down monotonically. The elliptic packs those zeros just past the band edge, pulling the response deep right at the start of the stopband. That is the steepest possible rolloff for a given order.
Q2: 通带纹波换来了什么？What does passband ripple buy you?
Butterworth 把所有"逼近余量"集中用在 处的最大平坦，远处就没余量了，滚降慢。 等纹波（Chebyshev、椭圆）把余量在整个通带均匀分摊：每个频点都允许一点起伏，省下的余量全用来换更陡的过渡带。 所以同阶下，肯让通带有纹波，就能换到更窄的过渡带。
Butterworth spends all its approximation budget on flatness at ω=0, leaving none for the edge, so it rolls off slowly. Equiripple designs spread the budget evenly across the passband: every point is allowed a little ripple, and the saved budget buys a steeper transition. Accepting passband ripple buys a narrower transition band at the same order.
Q3: 既然椭圆最陡，为什么 Butterworth 还最常用？If elliptic is sharpest, why is Butterworth still the most used?
因为很多应用更在乎通带保真而不是极限陡峭。Butterworth 通带无纹波、相位在这四族里最好、设计简单， 而且对元件误差和量化不敏感，工程上稳健。椭圆虽然最省阶数，但两带都有纹波、相位最差、对参数最敏感。 指标宽松或看重平坦/相位时选 Butterworth；过渡带极紧又能容忍纹波时才上椭圆。
Many applications care more about passband fidelity than maximal steepness. Butterworth has no passband ripple, the best phase of these four, a simple design, and is robust to component error and quantization. Elliptic saves the most order but ripples in both bands, has the worst phase, and is the most sensitive to parameters. Pick Butterworth for loose specs or when flatness/phase matter; reach for elliptic only when the transition band is very tight and ripple is acceptable.
Q4: 同样的指标，四族所需最小阶数 怎样排序？For the same spec, how does the required minimum order rank across families?
固定 和过渡带，所需最小阶数大致为 。 越陡的族越省阶数。本页反过来固定 ，于是越省阶的族在同阶下过渡带越窄， 所以椭圆曲线总是最先掉进阻带。读数里给出了椭圆与 Chebyshev II 达到 的频率，可直接对比。
For a fixed αp, αs, and transition band, the minimum order ranks roughly as N_elliptic < N_Chebyshev < N_Butterworth. The steeper the family, the fewer poles it needs. This page fixes N instead, so the more efficient family shows a narrower transition at the same order, which is why the elliptic curve always reaches the stopband first. The readout reports where the elliptic and Chebyshev II hit αs so you can compare directly.
Butterworth（1930）：最大平坦的理想Butterworth, 1930: the maximally flat ideal
斯蒂芬·巴特沃斯（Stephen Butterworth，1885-1958），英国政府物理学家（国家物理实验室 NPL，后在海军部研究实验室），在 1930 年的论文《On the Theory of Filter Amplifiers》中提出。当时滤波器设计靠经验试凑，通带里总有纹波。他主张理想滤波器应当对通带内各频率有均匀的灵敏度，并给出在原点尽可能平坦的幅频响应：。他用真空管放大器把多个二阶节级联，做出高阶滤波器，并画出 2、4、6、8、10 极点的响应曲线。这一思想发表后约沉寂了三十年才被广泛采用，如今却是最常用的 IIR 原型。
Stephen Butterworth (1885-1958), a British government physicist (the National Physical Laboratory, later the Admiralty Research Laboratory), introduced the design in his 1930 paper "On the Theory of Filter Amplifiers." Filter design then relied on trial and error and left ripple in the passband. He argued that an ideal filter should have uniform sensitivity across the wanted band, and gave the response that is as flat as possible at the origin. He realized higher orders by cascading two-pole sections through valve amplifiers, and plotted the 2, 4, 6, 8 and 10 pole responses. The idea then sat largely unused for about thirty years, yet it is now the most common IIR prototype.
文献 / References: S. Butterworth, "On the Theory of Filter Amplifiers," Experimental Wireless & the Wireless Engineer 7 (1930) 536-541. · Wikipedia: Stephen Butterworth
Chebyshev：等纹波，来自 19 世纪的数学家Chebyshev: equiripple, from a 19th-century mathematician
切比雪夫滤波器以俄国数学家 Pafnuty Chebyshev（1821-1894）命名。他本人从未设计过电子滤波器，名字纪念的是他的多项式 与等波动（minimax）逼近理论。把逼近误差在整个频带上均匀摊开，而不是把"平坦"集中在一点，正是通带（I 型）或阻带（II 型）等纹波的来源，也让它在同阶下比 Butterworth 过渡更陡。真正的电路实现，是 1930 年代网络综合时期才完成的。
The Chebyshev filter is named after the Russian mathematician Pafnuty Chebyshev (1821-1894). He never designed an electronic filter; the name honours his polynomials and his minimax (equioscillation) approximation theory. Spreading the approximation error evenly across the band, instead of concentrating flatness at one point, is exactly what produces the equiripple passband (Type I) or stopband (Type II) and a steeper transition than Butterworth at the same order. The circuit realization came later, in the network synthesis of the 1930s.
文献 / References: Wikipedia: Chebyshev filter · Pafnuty Chebyshev
椭圆 / 考尔（1930s）：最陡，生于电话网Elliptic / Cauer, 1930s: the sharpest, born in the telephone network
椭圆滤波器又称考尔（Cauer）滤波器，以德国数学家 Wilhelm Cauer（1900-1945）命名。他在读到 Foster 1924 年的电抗定理后，开创了"网络综合"这一领域。他把椭圆有理函数用于滤波器设计，以解决德国电话工业中多路复用的信道分离问题（多路话音共用一条线且串扰极小）。允许通带与阻带都等纹波，换来同阶下所有常见族里最陡的过渡带。他的"人工网络"（Artificial Network）专利于 1934 年授权。
The elliptic filter is also called the Cauer filter, after the German mathematician Wilhelm Cauer (1900-1945). After reading Foster's 1924 reactance theorem, Cauer founded the field of network synthesis. He applied elliptic rational functions to filter design to solve channel separation for telephone multiplexing in the German telephone industry, where many voice channels share one line with minimal crosstalk. Allowing equiripple in both bands buys the sharpest transition of any common family at a given order. His "Artificial Network" patent issued in 1934.
文献 / References: Wikipedia: Wilhelm Cauer · Elliptic filter
Bessel / Thomson（1949）：平的是群延迟，不是幅频Bessel / Thomson, 1949: flat group delay, not flat magnitude
贝塞尔滤波器因所用的贝塞尔多项式，以德国数学家、天文学家 Friedrich Bessel（1784-1846）命名，由英国工程师 W. E. Thomson 于 1949 年用于滤波器设计，故又称贝塞尔-汤姆森（Bessel-Thomson）滤波器。它优化的是另一个目标：最大平坦的群延迟（即相位最线性），从而在通带内尽量保持波形不失真。代价是这几族里滚降最缓。这也解释了为什么本页四族中"群延迟最平"的只是 Butterworth：真正的平群延迟冠军并不在这四族之内。
The Bessel filter is named after Friedrich Bessel (1784-1846) for the Bessel polynomials it uses, and was applied to filter design by the British engineer W. E. Thomson in 1949, hence the name Bessel-Thomson filter. It optimizes a different target: a maximally flat group delay, that is the most linear phase, so the waveform shape is preserved through the passband. The price is the gentlest rolloff of these families. That is why, among the four shown on this page, the flattest group delay is only Butterworth: the true champion of flat delay lies outside the four.
文献 / References: Wikipedia: Bessel filter