Butterworth Filter Interview Questions

Q1. What is a Butterworth filter and what makes it unique among filter types?

A Butterworth filter is designed to have the flattest possible passband frequency response — no ripple in the passband — at the cost of a moderate roll-off rate compared to Chebyshev or elliptic filters. A 4th-order Butterworth LPF with fc = 10kHz maintains gain within ±0.01dB from DC to 10kHz before rolling off at 80dB/decade. This maximally flat magnitude characteristic makes it the first choice when signal integrity in the passband is more important than sharp transition band cutoff.

Follow-up: What filter type would you choose if you need a sharper cutoff but can tolerate some passband ripple?

Q2. What is the magnitude response equation for an nth-order Butterworth filter?

|H(jω)|² = 1/(1 + (ω/ωc)^(2n)), where n is the order and ωc is the cutoff radian frequency. For n=1 at ω = ωc, this gives |H| = 1/√2 ≈ 0.707, confirming −3dB at cutoff for all Butterworth orders. This equation shows that increasing n steepens the transition without moving the −3dB frequency, which is why order selection is the primary design parameter.

Follow-up: How does the Butterworth magnitude response compare to the ideal brick-wall filter response?

Q3. Where are the poles of an nth-order Butterworth filter located in the s-plane?

Butterworth poles lie on a circle of radius ωc in the left-half s-plane, equally spaced in angle at intervals of 180°/n, starting at angle 90°+90°/n from the positive real axis. For a 3rd-order Butterworth with ωc = 1 rad/s, the poles are at s = −1, and s = −0.5 ± j0.866. Only left-half plane poles are used to ensure stability; the right-half plane mirror poles are discarded.

Follow-up: Why must all poles of a stable filter be in the left-half s-plane?

Q4. How do you determine the order of a Butterworth filter required for a given specification?

The minimum order n is found using n ≥ log[(10^(As/10)−1)/(10^(Ap/10)−1)] / (2×log(Ωs/Ωp)), where As is the stopband attenuation, Ap is the passband ripple, and Ωs/Ωp is the frequency ratio. For a filter needing 40dB attenuation at twice the cutoff frequency with 3dB passband ripple, n ≥ log(9999/1)/(2×log(2)) ≈ 6.6, so n = 7 is chosen. Always round up to the next integer to meet the specification.

Follow-up: How does the required order change if you tighten the stopband attenuation requirement from 40dB to 60dB?

Q5. What is a normalized Butterworth filter and how do you denormalize it?

A normalized Butterworth filter is designed with cutoff frequency ωc = 1 rad/s; actual designs are obtained by frequency scaling all capacitor values by 1/(2πfc) or resistor values accordingly. For a normalized 2nd-order Butterworth with R=1Ω and C=1.414F, scaling to fc=10kHz and R=10kΩ gives C=1.414/(2π×10000×10000) = 2.25nF. Denormalization allows engineers to use standard pole-zero tables and simply scale to the target frequency.

Follow-up: How do you also impedance-scale a normalized filter design to use practical component values?

Q6. What is the transfer function of a 2nd-order Butterworth low-pass filter?

H(s) = ωc² / (s² + √2×ωc×s + ωc²), which has Q = 1/√2 = 0.707 and a natural frequency ωn = ωc. For ωc = 2π×1000 rad/s (fc=1kHz), the denominator becomes s² + 8886s + (6283)². The √2 coefficient in the s term represents the damping, and Q=0.707 produces the maximally flat Butterworth response without any peak near cutoff.

Follow-up: What would the transfer function look like if Q was increased to 1? What effect would that have on the frequency response?

Q7. How is a Butterworth filter implemented using op-amps in practice?

A Butterworth LPF is implemented using Sallen-Key stages, each realizing a second-order section, with component values chosen from Butterworth polynomial tables. A 4th-order Butterworth at 1kHz uses two Sallen-Key stages: Stage 1 with Q=0.541 and Stage 2 with Q=1.307, with RC values adjusted accordingly. Each stage uses a standard op-amp like the TL072, and the stages are cascaded with buffers if needed to prevent loading.

Follow-up: Why are Butterworth filters usually implemented as cascaded 2nd-order sections rather than one high-order section?

Q8. How does a Butterworth filter compare to a Chebyshev filter?

A Butterworth filter has a maximally flat passband with no ripple but a gentler roll-off transition, while a Chebyshev filter allows equiripple in the passband (or stopband for Type II) to achieve a steeper roll-off for the same order. A 5th-order Chebyshev Type I with 1dB ripple achieves 40dB more attenuation at twice the cutoff compared to a 5th-order Butterworth. However, Chebyshev filters have worse group delay variation, causing signal distortion in wideband applications.

Follow-up: What is group delay and why does group delay flatness matter in data communication filters?

Q9. What is the −3dB frequency relationship to the cutoff frequency for different Butterworth orders?

For all orders of Butterworth filter, the −3dB frequency equals the design cutoff frequency ωc exactly — this is the defining property. This is because at ω = ωc, the magnitude formula gives 1/√(1+1) = 1/√2 regardless of n. This makes Butterworth filters predictable: specifying the cutoff is unambiguous, unlike Chebyshev filters where cutoff and ripple edge must both be specified.

Follow-up: How does the −3dB definition of cutoff differ between Butterworth and Chebyshev filters?

Q10. What is the group delay of a Butterworth filter and is it constant?

The group delay of a Butterworth filter is not constant — it varies with frequency, particularly near the cutoff frequency. A 4th-order Butterworth shows group delay peaking near fc, meaning frequency components near cutoff are delayed more than those in the flat passband. For audio and communications applications where signal fidelity is critical, a Bessel filter with maximally flat group delay is preferred over Butterworth.

Follow-up: In what application would you prefer a Bessel filter over a Butterworth filter, and why?

Q11. How do you design a Butterworth high-pass filter from a Butterworth low-pass prototype?

A Butterworth HPF is obtained by applying the LP-to-HP frequency transformation s → ωc²/s to the normalized LPF prototype, which swaps the roles of resistors and capacitors in the circuit realization. A 2nd-order Butterworth LPF with R=10kΩ and C=11.25nF (fc=1.414kHz) converts to an HPF by replacing each R with a C of the same impedance at fc and each C with an R. The HPF Sallen-Key topology swaps R and C positions compared to the LPF.

Follow-up: How would you design a Butterworth band-pass filter starting from a low-pass prototype?

Q12. What is the significance of the Butterworth polynomial?

Butterworth polynomials are the denominators of Butterworth filter transfer functions, giving the specific pole locations that produce maximally flat magnitude. The 3rd-order Butterworth polynomial is (s+1)(s²+s+1), producing poles at s=−1 and s=−0.5±j0.866 for a normalized filter. These polynomials are tabulated and allow engineers to directly write transfer functions and design circuits without deriving poles from scratch.

Follow-up: How do you use Butterworth polynomial tables to design a 5th-order Butterworth LPF?

Q13. What is the effect of component tolerances on a Butterworth filter's frequency response?

Component tolerances shift the pole locations, causing the response to deviate from the ideal Butterworth flatness — the passband may develop small ripple or the cutoff frequency may shift. A 5% tolerance on capacitors in a 4th-order Butterworth can introduce up to 0.5dB ripple in the passband. For precision designs, 1% resistors and C0G capacitors are used; Monte Carlo simulation in SPICE is used to evaluate the yield of a design across component variation.

Follow-up: How would you use sensitivity analysis to identify which component in a Butterworth filter most affects the cutoff frequency?

Q14. How is a Butterworth filter used in a DAC reconstruction (smoothing) application?

After a DAC converts digital data to a staircase waveform, a Butterworth LPF is used as a reconstruction filter to smooth out the quantization steps and remove imaging frequencies above Nyquist. For a 44.1kHz audio DAC like the PCM1792, a 4th-order Butterworth LPF with fc = 20kHz removes the 24.1kHz image while preserving audio bandwidth. The flat passband of the Butterworth ensures that the audio frequencies are not pre-distorted or amplitude-modulated.

Follow-up: What is an imaging frequency in a DAC output and why must it be filtered?

Q15. Can a Butterworth filter be implemented digitally? Explain how.

Yes, a digital Butterworth filter is designed using the bilinear transform, which maps the analog s-domain prototype to the z-domain by substituting s = (2/T)×(z−1)/(z+1), where T is the sampling period. A 2nd-order Butterworth LPF with fc = 1kHz sampled at 44.1kHz maps to a IIR digital filter with coefficients computed from the bilinear transform. The bilinear transform warps the frequency axis, so the analog prototype cutoff must be pre-warped before transformation to get the correct digital cutoff.

Follow-up: What is frequency warping in the bilinear transform and how do you pre-warp the cutoff frequency?