An audio engineer needs to remove frequencies above 5 kHz from a music signal without any amplitude coloration — no peaks or ripple — near the cutoff. Which second-order filter response should she choose?
AChebyshev, because it has the steepest roll-off in the transition band
BButterworth (Q = 0.707), because it provides a maximally flat passband with no peaking near cutoff
CBessel, because it has the sharpest transition band for a given filter order
DA high-Q Sallen-Key stage with Q = 2, because higher Q always means sharper frequency selectivity
Butterworth is defined by its maximally flat passband — no ripple, no peaking anywhere in the passband. Chebyshev exchanges flatness for a steeper roll-off but introduces ripple in the passband, which would create audible coloration. High Q produces a peak near cutoff (which distorts audio), and Bessel prioritizes linear phase at the cost of a very gradual transition. For flat, uncolored frequency response, Butterworth is the correct choice.
Question 2 Multiple Choice
A Sallen-Key second-order low-pass filter is designed with Q = 2. Compared to a Butterworth design (Q = 0.707) with the same cutoff frequency ω₀, what characterizes the frequency response of the high-Q design?
AA steeper roll-off beyond ω₀ with a completely flat passband below ω₀
BA more gradual roll-off everywhere, but with better preservation of pulse shapes
CA peak in the magnitude response near ω₀, where the response exceeds the DC value before rolling off
DIdentical passband behavior to Butterworth, but with improved stopband attenuation
High Q places the poles close to the imaginary axis in the s-plane, which creates a resonant peak in the frequency response near ω₀. The filter's magnitude actually rises above the passband level before dropping, distorting any signals with energy near the cutoff frequency. This is not 'sharper selectivity' in the useful sense — it is distortion. The common misconception is that higher Q always means a better filter; in reality, Q must be chosen to match the desired response shape.
Question 3 True / False
A single second-order Sallen-Key stage can achieve arbitrarily steep roll-off by increasing its quality factor Q to very high values.
TTrue
FFalse
Answer: False
A second-order stage always produces -40 dB/decade roll-off in the stopband, regardless of Q. Increasing Q changes the shape of the response near cutoff — creating peaking — but does not increase the asymptotic roll-off rate. To achieve steeper roll-off (e.g., -80 dB/decade), you must cascade additional second-order sections (biquads) to build a 4th-order or higher filter. This is a fundamental constraint imposed by the number of poles, not by Q.
Question 4 True / False
A high-Q second-order filter corresponds to a highly underdamped system in the time domain — the frequency-domain peaking near ω₀ and the time-domain ringing after a step input are two descriptions of the same underlying pole placement.
TTrue
FFalse
Answer: True
The Q factor describes pole placement in the s-plane. High Q places poles close to the imaginary axis, which in the time domain means slow decay of natural oscillations (ringing) and in the frequency domain means a sharp resonant peak near ω₀. These are not separate phenomena — they are the same mathematical object viewed through the Laplace transform. Understanding this connection is why studying resonance in RLC circuits directly informs filter design.
Question 5 Short Answer
Explain the tradeoff you accept when choosing a Chebyshev filter over a Butterworth filter for the same order and cutoff frequency.
Think about your answer, then reveal below.
Model answer: A Chebyshev filter achieves a steeper transition band — it rolls off faster outside the passband — but at the cost of ripple within the passband. The magnitude response oscillates between its maximum and minimum passband values, rather than monotonically decreasing as in Butterworth. In exchange, signals just outside the passband are more strongly attenuated. The tradeoff is: better rejection of unwanted frequencies vs. more distortion of signals inside the passband. Butterworth is better when passband flatness is required; Chebyshev is better when steep skirts matter more than passband uniformity.
The tradeoff is formalized in the filter design parameter Q. Chebyshev designs use Q > 0.707, which places poles closer to the imaginary axis — more resonance, sharper skirts, but also more peaking. Butterworth uses Q = 0.707 exactly, balancing response shape optimally. Neither is universally superior: the right choice depends on the application's tolerance for passband ripple versus its need for stopband rejection.