Butterworth, Chebyshev & Elliptic Designs | DSP-101

The Classical Filter Families

With the IIR structure in hand — a rational transfer function H(z) = B(z)/A(z) with poles that create sharp frequency selectivity — the next question is: where exactly should those poles go? Classical analog filter theory answers this with three famous families: Butterworth, Chebyshev, and Elliptic. Each makes a different trade-off between flatness, sharpness, and phase behaviour, and each is the "best" design for a specific combination of priorities.

These families were developed for analog circuits, but because the bilinear transform (next lesson) maps analog poles and zeros directly to digital ones, mastering the analog prototypes is the standard route to designing digital IIR filters. Engineers rarely design digital IIR filters from scratch — they design analog prototypes and then transform them.

Butterworth: Maximally Flat

The Butterworth filter is the gentlest of the three. Its defining property is a maximally flat magnitude response in the passband — no ripple whatsoever. The magnitude squared of an N-th order Butterworth low-pass filter is:

Butterworth Magnitude Response

|H(j\Omega)|^2 = \frac{1}{1 + \left(\dfrac{\Omega}{\Omega_c}\right)^{2N}}

N is the filter order and Ω_c is the cutoff frequency (−3 dB point). The response is perfectly flat at Ω = 0 and rolls off at −20N dB/decade in the stopband. All derivatives of the magnitude at Ω = 0 are zero — hence "maximally flat."

The Butterworth poles lie on a circle of radius Ω_c in the s-plane, equally spaced in angle. For a stable filter, only the left-half-plane poles are kept. The result is a smooth, monotonic roll-off — predictable and well-behaved, but not the sharpest transition for a given order.

When to Choose Butterworth

Choose Butterworth when flatness in the passband is more important than transition sharpness. Typical uses: anti-aliasing filters for audio ADCs, smoothing filters in control systems, and any application where phase and amplitude distortion within the band must be minimised.

Monotonic in both passband and stopband — no ripple anywhere

Chebyshev: Ripple for Sharpness

The Chebyshev filter trades the smoothness of Butterworth for a sharper transition. It allows equiripple — a controlled, equal-amplitude oscillation — in one band, concentrating the filter's "budget" on making the transition as steep as possible.

Chebyshev Type I places the equiripple in the passband and has a monotone stopband. Its magnitude is defined using Chebyshev polynomials T_N:

Chebyshev Type I Magnitude

|H(j\Omega)|^2 = \frac{1}{1 + \varepsilon^2\, T_N^2\!\left(\dfrac{\Omega}{\Omega_c}\right)}

ε controls the passband ripple amplitude (ripple = 10·log₁₀(1 + ε²) dB). T_N(Ω/Ω_c) is the N-th order Chebyshev polynomial, which oscillates between ±1 for |Ω| ≤ Ω_c and grows monotonically outside.

Chebyshev Type II (inverse Chebyshev) flips this: it has a flat passband and equiripple in the stopband. The stopband ripple is at a fixed minimum attenuation level, and the passband is monotone. Type II is useful when you want Butterworth-like passband behaviour with a sharper stopband than Butterworth can provide for the same order.

Type I

Equiripple Passband

Ripple in the passband; monotone stopband. Sharper transition than Butterworth of the same order. The ripple amplitude is a design parameter (e.g., 0.5 dB, 1 dB, 3 dB).

Type II

Equiripple Stopband

Flat passband; ripple in the stopband. Finite zeros in the stopband create notches. Behaves like Butterworth in the band but with better stopband attenuation.

Elliptic (Cauer): Sharpest of All

The elliptic filter — also called the Cauer filter — is the most powerful of the three. It places equiripple in both the passband and the stopband simultaneously. By using both the passband and stopband ripple budgets together, it achieves the steepest possible transition for a given filter order. No other filter type can match a given set of passband/stopband specifications with a lower order.

Elliptic Filter Order Formula

N \geq \frac{K(k)\, K\!\left(\sqrt{1-k_1^2}\right)}{K(k_1)\, K\!\left(\sqrt{1-k^2}\right)}, \quad k = \frac{\Omega_p}{\Omega_s},\; k_1 = \frac{\varepsilon_p}{\varepsilon_s}

K(·) is the complete elliptic integral of the first kind. ε_p and ε_s are the passband and stopband ripple parameters. This formula gives the minimum integer order N required to meet both specs simultaneously — always less than or equal to the required Butterworth or Chebyshev order.

The elliptic filter places its zeros at finite frequencies in the stopband, creating sharp notches that push the stopband response down quickly. The poles and zeros together are positioned using the mathematics of elliptic functions — complex but well-tabulated and handled automatically by design tools like MATLAB's ellip() function.

The Efficiency Champion

For any given passband ripple, stopband attenuation, and transition ratio, the elliptic filter requires the lowest possible order. The trade-off is the most nonlinear phase of the three families and the highest sensitivity to coefficient quantization.

Equiripple passband + equiripple stopband = minimum order for any spec

Order Comparison

The practical impact of these design philosophies is dramatic when you look at the filter orders required to meet the same specification:

Specification	Butterworth Order	Chebyshev I Order	Elliptic Order
1 dB passband ripple, 40 dB stopband att., 2× freq ratio	8	5	4
0.5 dB ripple, 60 dB att., 1.5× ratio	20	9	6
3 dB ripple, 80 dB att., 1.2× ratio (tight transition)	51	16	9

The saving is not a fixed factor — it grows as the transition tightens, which is exactly where it matters. Against Butterworth, the elliptic design in the table above needs half the poles at a 2:1 transition ratio (8 → 4), under a third at 1.5:1 (20 → 6), and under a fifth at 1.2:1 (51 → 9) — ratios of 2.0×, 3.3× and 5.7×. For resource-constrained DSP implementations, this is a decisive advantage.

Phase and Group Delay Comparison

Sharpness and phase linearity are fundamentally at odds in IIR design. As a filter's transition becomes sharper, its group delay — how long different frequencies are delayed — becomes more nonuniform:

Best phase

Butterworth

Most nearly linear phase of the three classical IIR families. Group delay variation is relatively mild, especially away from the cutoff. The gentlest trade-off.

Moderate

Chebyshev I & II

More phase distortion than Butterworth, especially near the cutoff. Type II tends to have better phase behaviour than Type I for the same order.

Worst phase

Elliptic

Most severe group delay variation, particularly in the transition band. The price of minimum-order efficiency. Unacceptable for phase-sensitive applications.

Alternative

Bessel / Thomson

A fourth family not covered here: designed for maximally flat group delay (linear phase approximation). Very poor magnitude selectivity — only use when phase is the only goal.

Design Trade-off Summary

No single filter family wins on all fronts. The right choice depends on which properties your application can tolerate versus which it requires:

Family	Passband	Stopband	Phase	Order Efficiency
Butterworth	Flat	Monotone	Best	Lowest
Chebyshev I	Equiripple	Monotone	Moderate	Good
Chebyshev II	Flat	Equiripple	Moderate	Good
Elliptic	Equiripple	Equiripple	Worst	Highest

Key Takeaways

Butterworth is maximally flat — no ripple in passband or stopband — with the smoothest phase, but requires the highest order for a given spec.
Chebyshev Type I allows equiripple in the passband to achieve a sharper transition; Type II puts the ripple in the stopband for a flat passband with better stopband rejection.
Elliptic (Cauer) filters have equiripple in both bands and achieve the steepest possible transition for a given order — the minimum-order solution for any spec.
The cost of sharpness is phase distortion: Butterworth has the most nearly linear phase, elliptic has the most nonlinear group delay.
Elliptic filters need roughly 2× fewer poles than Butterworth at a loose transition ratio and nearly 6× fewer at a tight one — the saving grows with the difficulty of the specification, so it is largest exactly where cost bites: low-power and real-time DSP.
In practice, engineers use design tools (MATLAB's butter(), cheby1(), cheby2(), ellip()) to generate the coefficients, then verify performance by examining the pole-zero plot and frequency response.
The next step is the bilinear transform — which maps these analog prototype filters into equivalent digital IIR filters.