Chebyshev Digital Filter Calculator
Find the minimum order and ripple factor for a Chebyshev Type I low-pass filter, and see how it compares to an equivalent Butterworth design.
📈 What is a Chebyshev Digital Filter?
A Chebyshev Type I filter is a filter design that allows small, equal-height ripples in the passband to achieve a steeper transition band than a Butterworth filter using the same number of poles. Named after Pafnuty Chebyshev, whose polynomials define the filter's magnitude response, it is the standard choice whenever pole count (cost, size, or group delay) matters more than a perfectly flat passband.
RF and communications engineers favor Chebyshev filters for channel selection, where a narrow, well-defined transition band matters more than passband flatness, since receiver AGC circuits already compensate for small in-band gain variations. Power supply and switching converter designers use Chebyshev low-pass filters to reject switching noise with fewer stages than an equivalent Butterworth design. Test equipment designers reach for Chebyshev whenever board space or component count is tightly constrained, since each avoided pole is a real saving in parts and power.
A common misconception is that Chebyshev is simply a "better" Butterworth. It is not; it is a different trade-off. Chebyshev sacrifices passband flatness (real ripple the signal will pick up) to gain a lower pole count for the same stopband requirement. Whether that trade is worthwhile depends entirely on whether the application can tolerate that ripple.
This calculator computes the minimum Chebyshev Type I order needed for a given passband ripple and stopband attenuation spec, and directly compares it against the Butterworth order the identical spec would require, so you can see the trade-off in concrete pole counts.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Compact Anti-Aliasing Filter
fpass = 1,000 Hz (1 dB ripple), fstop = 5,000 Hz (40 dB attenuation)
Example 2 — Narrow 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 Chebyshev filter?
A Chebyshev Type I filter is a filter design that allows small, equal-sized ripples in the passband in exchange for a steeper roll-off into the stopband than a Butterworth filter of the same order. It uses Chebyshev polynomials to place its poles, giving equiripple passband behavior.
How do you calculate the order of a Chebyshev filter?
Order N = acosh[sqrt((10^(0.1*As) - 1) / (10^(0.1*Ap) - 1))] / acosh(fstop/fpass), using the inverse hyperbolic cosine (acosh). As with Butterworth, always round the result up to the next whole number of poles.
What is the ripple factor epsilon?
Epsilon is epsilon = sqrt(10^(0.1*Ap) - 1), derived only from the passband ripple Ap in dB. It sets how deep the equiripple oscillations are inside the passband and appears directly in the Chebyshev polynomial that defines the filter's poles.
Why does Chebyshev need fewer poles than Butterworth for the same spec?
Chebyshev's equiripple passband uses the full ripple budget at every point in the passband, packing more selectivity into each pole than Butterworth's maximally flat response, which only just touches its ripple limit at a single frequency (DC). The trade-off is that the passband is no longer perfectly flat.
What happens if I set the passband ripple to a very small value?
A very small Ap drives epsilon toward zero, and the Chebyshev design starts to resemble a Butterworth filter, needing a similarly high order. The lower-order advantage of Chebyshev only shows up once you are willing to accept a non-trivial amount of passband ripple.
Is Chebyshev Type I the same as Chebyshev Type II?
No. Chebyshev Type I ripples in the passband and is flat (monotonic) in the stopband, the variant this calculator computes. Chebyshev Type II (inverse Chebyshev) is the opposite: flat in the passband and rippled in the stopband. Both use the same order formula, just applied on different sides of the design.
How does group delay compare between Chebyshev and Butterworth?
For the same stopband spec, Chebyshev's lower order can mean less total group delay, but its poles sit closer to the unit circle near the passband edge, giving Chebyshev filters a more non-linear phase response and worse group delay variation than a Butterworth filter of similar selectivity.
Why use acosh (inverse hyperbolic cosine) in the order formula?
The Chebyshev polynomial of order N used to define the filter's magnitude response is naturally expressed with cosh and acosh functions once the argument (the frequency ratio) exceeds 1, which is exactly the stopband region. Solving the response equation for N produces the acosh-based order formula directly.
What order does a typical Chebyshev crossover filter need?
A 1 dB passband ripple with 40 dB stopband attenuation at a 5:1 frequency ratio (the same spec as the Butterworth calculator's featured example) needs only 3 Chebyshev poles versus 4 Butterworth poles, a common margin for moderate specs.
Can I use this calculator for a high-pass Chebyshev filter?
Yes, in terms of order. The order formula only depends on the frequency ratio between the stopband and passband edges, not on whether the filter is low-pass or high-pass, so the same N applies once you transform the prototype to your desired filter type.
Does a higher-ripple Chebyshev filter sound or perform worse?
A larger passband ripple (higher Ap) means more amplitude variation within the passband, which can be audible in audio applications or introduce measurement error in instrumentation. Choosing the smallest Ap that still gives an acceptable order is the usual practical trade-off.