Butterworth vs Chebyshev: Flat Passband or Steeper Skirts
At N = 3 and twice the cutoff, a Butterworth low-pass gives 18.1 dB of attenuation and a 0.5 dB Chebyshev gives 19.2 dB, about 1.1 dB more. The price is equal ripple in the passband and a cutoff defined at the ripple edge, not the 3 dB point.
For practitioners
Every low-pass prototype trades three things against each other: the number of sections N, the shape of the passband, and how quickly loss rises above cutoff. Butterworth and Chebyshev are the two classic answers. Butterworth makes the passband as flat as possible. Chebyshev accepts a controlled ripple and gets more attenuation in return. This page puts numbers on that trade at N = 3.
The two power loss ratios
Both responses are defined by the power loss ratio P_LR, the ratio of available source power to the power delivered to the load. Insertion loss in dB is 10·log10(P_LR).
The Butterworth (maximally flat) response is:
At ω = 0 the first 2N − 1 derivatives of P_LR are zero. That is what "maximally flat" means. At ω = ωc, P_LR = 2, which is 3.01 dB. So for a Butterworth, ωc is the 3 dB frequency. The loss rises monotonically from 0 dB at DC and never dips back.
The Chebyshev (equal-ripple) response is:
T_N is the Chebyshev polynomial of the first kind. The first few are T_1(x) = x, T_2(x) = 2x² − 1 and T_3(x) = 4x³ − 3x. Higher orders follow from the recursion:
For |x| ≤ 1, T_N swings between −1 and +1, so P_LR swings between 1 and 1 + k². That sets the passband ripple:
For |x| > 1, T_N grows like 2^(N−1)·x^N, faster than the x^N term in the Butterworth. That growth is the source of the steeper skirt.
What ωc means in each case
This is the most common source of confusion when comparing the two. In a Butterworth, ωc is the 3.01 dB point. In a Chebyshev, ωc is the edge of the equal-ripple band. At that frequency the loss equals the ripple value, 0.5 dB in the example below, not 3 dB.
So a "1 GHz Chebyshev" and a "1 GHz Butterworth" are not specified the same way. The Chebyshev still has low loss at its ωc. Its 3 dB point lies somewhat higher. Comparing the two at the same ω/ωc, as done here, is the usual textbook comparison. If you instead lined them up at their 3 dB frequencies, the Chebyshev's advantage at a given offset would be larger than the figure below shows. Before comparing curves from different sources, check which definition of cutoff each one uses.
Worked example: N = 3, 0.5 dB ripple, at ω/ωc = 2
Ripple constant. For 0.5 dB ripple:
k² = 10^(0.5/10) − 1 = 10^0.05 − 1 = 0.122
Butterworth at ω/ωc = 2. With N = 3, 2N = 6:
P_LR = 1 + 2^6 = 65, so the attenuation is 10·log10(65) = 18.1 dB.
Chebyshev at ω/ωc = 2. First the polynomial:
T_3(2) = 4·2³ − 3·2 = 32 − 6 = 26, so T_3² = 676.
P_LR = 1 + 0.122·676 = 1 + 82.5 = 83.5, so the attenuation is 10·log10(83.5) = 19.2 dB.
Difference. The Chebyshev gives about 1.1 dB more attenuation at twice the cutoff, with the same number of reactive elements.
The figure shows both N = 3 curves as |S21| in dB. Below ω/ωc = 1 the Butterworth stays close to 0 dB and then bends down to −3.01 dB at the cutoff. The Chebyshev ripples between 0 and −0.5 dB and is at −0.5 dB at the band edge. At ω/ωc = 2 the curves are at −18.1 dB and −19.2 dB, 1.1 dB apart.
Where the ripples sit, and how the gap grows
For N = 3, T_3(x) = 4x³ − 3x is zero at x = 0 and x = ±0.866, and |T_3| = 1 at x = 0.5 and x = 1. Across the passband, the loss is therefore 0 dB at DC, 0.5 dB at ω/ωc = 0.5, back to 0 dB at 0.866, and 0.5 dB again at the band edge. Every peak is the same height. T_N has N zeros in −ωc ≤ ω ≤ ωc, so a higher order packs in more ripples without making them taller.
Far above cutoff, both responses approach straight lines on a log-frequency axis, with the same slope of 20N dB per decade. The Chebyshev line is offset by:
For N = 3 and k² = 0.122 this is 10·log10(0.122·16) ≈ 2.9 dB. At ω/ωc = 2 the curves have not yet reached their asymptotes, which is why the gap there is only 1.1 dB. A larger ripple raises k² and widens the gap. A smaller ripple shrinks it, and as k² goes to zero the Chebyshev advantage disappears.
Choosing between flat and steep
The decision comes down to what the passband costs you and what the stopband earns you.
- Choose Butterworth when the passband must be monotonic and has no gain ripple to calibrate out, or when the filter sits in a chain where ripple from several stages would add up. Its response is also less sensitive to small loss variations near the band center.
- Choose Chebyshev when a rejection number at a fixed offset drives the design and every section counts. A 1 dB gain at 2ωc may not save a whole section on its own. But the asymptotic advantage, and the extra margin from a slightly larger ripple, often does.
Return loss is covered in Return Loss vs Insertion Loss: What Each Number Tells You, and how both responses look on a measured S21 trace is covered in What S21 and S11 of a Low-Pass Filter Look Like.
In plain words · for students and beginners
Butterworth vs Chebyshev in plain words
An everyday analogy
Think of two roads that both end at a cliff. The Butterworth road is perfectly smooth up to the edge, then curves gently downhill. The Chebyshev road has small, even bumps all the way along, but after the edge it drops away more steeply. If you want a smooth ride, take the first road. If you want to be as far down as possible a short walk past the edge, take the second.
Where the analogy breaks down: a real filter is not about a ride. The "bumps" are small changes in signal loss across the frequencies you want to keep (the passband). They also change how much signal bounces back to the source. And the "edge" is defined differently for the two roads, which the analogy hides.
The same example, step by step
Both filters have N = 3, meaning three reactive parts (coils or capacitors). The cutoff frequency, ωc, is where the passband ends. We look at twice that frequency.
- Butterworth: compute 1 + 2⁶ = 65. Convert to decibels (a log scale for power ratios): 10·log10(65) = 18.1 dB of loss.
- Chebyshev with 0.5 dB ripple: first find the ripple constant k² = 10^0.05 − 1 = 0.122.
- Evaluate the Chebyshev polynomial (a fixed formula, here 4x³ − 3x) at x = 2: 32 − 6 = 26. Square it: 676.
- Compute 1 + 0.122 × 676 = 83.5. In decibels: 19.2 dB.
- Compare: 19.2 − 18.1 = 1.1 dB more blocking from the Chebyshev.
The table puts it side by side. Butterworth: flat, no ripple, ωc is the 3 dB point, 18.1 dB at twice cutoff. Chebyshev: ripple between 0 and 0.5 dB, k² = 0.122, ωc is the end of the ripple band, 19.2 dB at twice cutoff.
Mistakes beginners make
- Treating ωc as the same thing in both. For Butterworth it is where loss reaches 3 dB. For Chebyshev it is where the ripple band ends, at only 0.5 dB loss here.
- Expecting a huge gain. At twice cutoff and N = 3, the extra rejection is about 1.1 dB. Far above cutoff it grows, toward about 2.9 dB for this ripple.
- Ignoring reflections. Each ripple peak means some signal is reflected back. Measuring only S21, the forward transmission, misses this.
- Thinking more ripples means bigger ripples. A higher order adds more ripples, but they all keep the same height.
Remember
- Butterworth: flat and smooth passband, 3 dB loss at ωc.
- Chebyshev: equal ripple in the passband and a steeper skirt; at N = 3 with 0.5 dB ripple, 1.1 dB more loss at 2ωc.
- Always check how each datasheet or textbook defines cutoff before comparing curves.
Related pages on RF View
Frequently asked questions
Is a Chebyshev filter always steeper than a Butterworth of the same order?
Yes, for any non-zero ripple the Chebyshev gives more attenuation beyond the passband edge at the same order. At N = 3 and ω/ωc = 2, a 0.5 dB Chebyshev gives 19.2 dB against 18.1 dB for the Butterworth.
What does ωc mean in a Chebyshev filter?
In a Chebyshev design, ωc is the edge of the equal-ripple passband, where the loss equals the ripple value. It is not the 3 dB frequency, as it is for a Butterworth.
How do I convert Chebyshev ripple in dB to the ripple constant k²?
Use k² = 10^(ripple/10) − 1. For 0.5 dB ripple, k² = 10^0.05 − 1 = 0.122.
When should I choose a Butterworth over a Chebyshev?
Choose Butterworth when the passband has to be monotonic and as flat as possible near the center, and a little less skirt attenuation is acceptable. Choose Chebyshev when you need more rejection from the same number of sections and can tolerate a known ripple.