Butterworth Digital Filter Calculator
Find the minimum filter order and cutoff frequency for a Butterworth low-pass filter from your passband and stopband specifications.
📉 What is a Butterworth Digital Filter?
A Butterworth filter is one of the classic analog and digital filter families, prized for having the flattest possible passband, with no ripple at all, at the cost of a comparatively wide transition band between the passband and stopband. It was first described by British engineer Stephen Butterworth in 1930 and remains the default choice whenever passband flatness matters more than roll-off steepness.
Audio engineers reach for Butterworth low-pass and high-pass filters in crossovers and anti-aliasing stages precisely because a flat passband means no unwanted coloration of the signal that stays inside the passband. Instrumentation and measurement systems use Butterworth filters ahead of ADCs to reject out-of-band noise without distorting the in-band signal's amplitude. Communications engineers use them as channel filters whenever preserving relative signal levels across the passband is more important than an aggressively narrow transition band.
A common misconception is that "higher order always means a better filter." A higher order does mean a steeper roll-off and more stopband attenuation, but it also adds more group delay, more computational cost per sample, and more sensitivity to coefficient quantization in fixed-point implementations. The right order is the minimum one that meets the actual spec, not the largest one available.
This calculator implements the standard textbook order formula: given a passband edge frequency and ripple, and a stopband edge frequency and attenuation, it finds the minimum whole number of poles that meets both requirements, plus the resulting -3 dB cutoff frequency.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Audio Anti-Aliasing Filter
fpass = 1,000 Hz (1 dB ripple), fstop = 5,000 Hz (40 dB attenuation)
Example 2 — Tight Instrumentation Filter
fpass = 2,000 Hz (0.5 dB ripple), fstop = 8,000 Hz (60 dB attenuation)
Example 3 — Wide-Ripple Communications Filter
fpass = 500 Hz (3 dB ripple), fstop = 1,500 Hz (30 dB attenuation)
❓ Frequently Asked Questions
🔗 Related Calculators
What is a Butterworth filter?
A Butterworth filter is a filter design with the flattest possible passband response (no ripple) among the classic analog filter families, achieved at the cost of a wider transition band compared to Chebyshev or elliptic designs of the same order.
How do you calculate the order of a Butterworth filter?
Order N = log10[(10^(0.1*As) - 1) / (10^(0.1*Ap) - 1)] / (2 * log10(fstop/fpass)), where Ap is the passband ripple in dB, As is the stopband attenuation in dB, and fpass/fstop are the passband and stopband edge frequencies. Always round the result up to the next whole number.
Why round the order up instead of down?
The order formula gives the minimum fractional order that exactly meets the spec. Rounding down would fail to meet the stopband attenuation requirement, so the practical pole count always rounds up, which means the built filter actually exceeds the requested attenuation.
What is the -3 dB cutoff frequency for a Butterworth filter?
The cutoff (-3 dB point, or more precisely -Ap dB if a different passband ripple is specified) is fc = fpass / (10^(0.1*Ap) - 1)^(1/(2N)). It sits slightly above the passband edge frequency because the passband edge is defined by the ripple spec, not by the -3 dB point itself.
How is a Butterworth filter different from a Chebyshev filter?
A Butterworth filter has a perfectly flat passband but needs a higher order to reach a given stopband attenuation. A Chebyshev Type I filter allows small, equal ripples in the passband in exchange for a steeper roll-off and a lower required order for the same specification.
Does the order formula work directly for digital filters?
The formula computes the analog-prototype order, the standard first step in digital IIR design. To get the exact digital pole locations, engineers apply the bilinear transform (with frequency pre-warping) to the analog Butterworth poles, but the order itself, the number of poles needed, does not change in that step.
What happens if I request a very steep transition band?
Requesting a stopband edge very close to the passband edge (a small fstop/fpass ratio) forces the log10(ratio) denominator toward zero, driving the required order very high. Physically this means a much steeper filter needs many more poles, each adding cost, group delay, and numerical sensitivity.
Can this calculator design a high-pass or band-pass Butterworth filter?
This calculator finds the order for a low-pass prototype. High-pass, band-pass, and band-stop Butterworth filters use the same order formula on a transformed frequency variable, so the order calculation here is the correct starting point for any of those variants.
Why does the actual stopband attenuation exceed my requested value?
Because the order must be a whole number of pole pairs, rounding the exact fractional order up to the next integer always produces slightly more attenuation than requested. This calculator reports both numbers so you can see the margin your rounded design actually achieves.
What passband ripple value should I use for Butterworth?
Common practical choices are 0.5 dB, 1 dB, or 3 dB (the classical -3 dB point). Ap only defines where the passband edge sits relative to the flat 0 dB gain at DC, it does not introduce actual ripple, since Butterworth filters are monotonic by construction.
How many poles does a typical audio Butterworth low-pass need?
A modest 1 dB passband ripple with 40 dB stopband attenuation two octaves above the passband edge (a 4:1 frequency ratio) typically needs 4 to 6 poles; tighter transition bands or deeper attenuation targets push the order into the 8 to 12 pole range.