What makes a Butterworth filter 'maximally flat'?

The Butterworth approximation sets all derivatives of the magnitude-squared function to zero at omega = 0 (DC). For an Nth-order filter, the first 2N minus 1 derivatives of |H(jw)|^2 are zero at DC, meaning the response is as flat as mathematically possible near DC for a given filter order. The result is a monotonically decreasing magnitude response with no ripple in the passband. At the cutoff frequency omega_c, the magnitude is exactly minus 3 dB (1/sqrt(2) in voltage, 1/2 in power). The poles of the Butterworth filter are equally spaced on the left half of the unit circle in the s-plane, with angles theta_k = pi(2k plus N minus 1)/(2N) for k = 1 to N. This equal spacing produces the smoothest possible transition from passband to stopband for a given order.

How do you design a Butterworth filter for a specific attenuation?

First, determine the required filter order N from the specification. Given the stopband frequency ratio (omega_s/omega_c) and required attenuation A_s in dB: N = ceil(log(10^(A_s/10) minus 1) / (2 times log(omega_s/omega_c))). For example, 40 dB attenuation at twice the cutoff frequency: N = ceil(log(10^4 minus 1)/(2 times log(2))) = ceil(13.29/0.602) = ceil(6.64) = 7. A 7th-order Butterworth provides 40 dB at 2 times omega_c with 140 dB per decade ultimate roll-off. Next, use the Butterworth g-value formula: g_k = 2 sin(pi(2k minus 1)/(2N)) for k = 1 to N, with g_0 = g_(N+1) = 1 for equal termination. These normalized values are frequency- and impedance-scaled to obtain actual L and C values for the target cutoff frequency and system impedance.

When should you use Butterworth versus Chebyshev or elliptic filters?

Butterworth provides the flattest passband with no ripple, best group delay flatness of the three, and simplest design equations. It is the best choice when passband flatness and linear phase (good group delay) are priorities, such as IF filters in receivers, anti-aliasing filters before ADCs, and wideband amplifier interstage matching. Chebyshev Type I allows equiripple passband variation (0.01 to 3 dB configurable) in exchange for sharper transition from passband to stopband: a 5th-order Chebyshev with 0.5 dB ripple provides the same stopband rejection as a 7th-order Butterworth, saving 2 reactive elements. Elliptic (Cauer) filters allow ripple in both passband and stopband, achieving the sharpest transition for a given order but with the worst group delay variation. For narrowband RF bandpass filters where sharp skirts are critical (duplexer, channel filters), Chebyshev or elliptic designs dominate.

Filter Design

Butterworth Filter

/but-er-werth fil-ter/ — Maximally Flat

An analog filter with maximally flat magnitude response in the passband. Transfer function: |H(jω)|² = 1/(1 + (ω/ω_c)^2N). Roll-off: 20N dB/decade (6N dB/octave). Zero passband ripple with all derivatives zero at DC. Poles equally spaced on the s-plane unit circle. Prototype g-values: g_k = 2 sin(π(2k-1)/(2N)) define LC ladder element values. Used for anti-aliasing, IF filtering, and interstage matching where flatness and group delay are priorities.

Ripple: 0 dB (flat)

Roll-off: 20N dB/dec

@ ω_c: -3 dB

Understanding Butterworth Filters

Stephen Butterworth published his filter design in 1930, seeking the flattest possible frequency response. The Butterworth approximation achieves this by forcing all derivatives of the magnitude-squared function to zero at DC. This "maximally flat" property means the filter responds as uniformly as possible to signals within the passband, introducing no amplitude ripple that could distort wideband signals. The tradeoff is a more gradual transition from passband to stopband compared to Chebyshev or elliptic designs of the same order.

In RF engineering, Butterworth filters are the starting point for filter synthesis. The normalized prototype g-values provide a universal set of element values that can be scaled to any cutoff frequency and impedance level. For a low-pass to bandpass transformation, the g-values determine the coupling coefficients and resonator Q requirements. The Butterworth response is also the foundation for distributed filter design: coupled-line, hairpin, and interdigital bandpass filters can all be synthesized starting from Butterworth (or Chebyshev) prototypes using impedance inverter theory.

Butterworth Design Equations

Magnitude response:
|H(jω)|² = 1 / (1 + (ω/ω_c)^2N)
@ ω = ω_c: |H| = -3 dB
@ ω = 2ω_c: |H| = -10 log(1+2^2N) dB

Order selection:
N = ⌈log(10^A_s/10 − 1) / (2 log(ω_s/ω_c))⌉
A_s = required stopband attenuation (dB)

Prototype g-values:
g_k = 2 sin(π(2k−1)/(2N))
g₀ = g_N+1 = 1
N=3: g = {1, 1, 2, 1, 1}
N=5: g = {1, 0.618, 1.618, 2, 1.618, 0.618, 1}

Impedance/frequency scaling:
L_k = g_k × Z₀ / (2πf_c)
C_k = g_k / (Z₀ × 2πf_c)

Filter Approximation Comparison

Filter Type	Passband	Transition	Group Delay	Order for 40 dB @ 2×f_c
Butterworth	Maximally flat	Gradual	Good	N = 7
Chebyshev I (0.5 dB)	Equiripple	Sharper	Moderate	N = 5
Chebyshev II	Flat	Sharp	Moderate	N = 5
Elliptic (Cauer)	Equiripple	Sharpest	Poor	N = 4
Bessel	Non-flat	Very gradual	Best (linear)	N = 10+

Common Questions

Frequently Asked Questions

What makes Butterworth "maximally flat"?

The first 2N-1 derivatives of |H(jω)|² are zero at DC, making the response as flat as mathematically possible for order N. Poles are equally spaced on the left half of the s-plane unit circle. No passband ripple, monotonically decreasing magnitude. At ω_c: exactly -3 dB. The smoothest transition from passband to stopband for any given order.

How do you determine filter order?

From attenuation specification: N = ceil(log(10^(A_s/10) - 1) / (2 log(ω_s/ω_c))). Example: 40 dB at 2×f_c requires N = 7. Then g-values g_k = 2 sin(π(2k-1)/(2N)) define the LC ladder network. Scale to target impedance and frequency: L = g×Z_0/(2πf_c), C = g/(Z_0×2πf_c).

When to use Butterworth vs. Chebyshev?

Butterworth: flattest passband, best group delay, simplest design. Use for anti-aliasing, IF filters, wideband matching where flatness matters. Chebyshev I: allows passband ripple (0.01-3 dB) for sharper transition, saves 2+ elements for same stopband rejection. Elliptic: sharpest cutoff but worst group delay. For narrowband RF bandpass (duplexers, channel filters), Chebyshev or elliptic dominate.

Related Terms

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