Questions: Filter Classification and Design Basics

Question 1

An engineer must filter a signal to meet strict stopband attenuation requirements within a narrow transition band. Phase distortion is not a concern. Which filter approximation minimizes the required filter order?

• A. Butterworth — its maximally flat passband guarantees minimum order for any specification
• B. Bessel — it has the most linear phase response, which also produces efficient stopband rejection
• C. Chebyshev Type I — its equiripple passband achieves steeper rolloff than Butterworth at the same order
• D. Elliptic (Cauer) — it places equiripple in both passband and stopband, achieving minimum order for given passband ripple, stopband ripple, and transition width

Answer: (D)

The elliptic filter is optimal by the criterion of minimum order for specified passband ripple, stopband attenuation, and transition bandwidth. By placing equiripple oscillations in both the passband and the stopband, it uses every degree of freedom to minimize the transition width — achieving steeper rolloff at a given order than either Butterworth (no passband ripple) or Chebyshev Type I (no stopband equiripple). The cost is the most non-linear phase response of the three. Since phase distortion is explicitly stated to be irrelevant here, elliptic is the correct choice. Butterworth is actually the worst choice by this criterion — its 'wasted' flatness requires higher order for the same rolloff.

Question 2

A filter for a data communications system must preserve signal timing precisely — group delay must be constant across the passband. Which property should guide filter selection?

• A. Maximally flat magnitude (Butterworth), because flat magnitude implies flat group delay
• B. Equiripple magnitude (Chebyshev), because ripple in frequency domain corresponds to uniformity in time domain
• C. Maximally flat group delay (Bessel), because constant group delay means all frequency components are delayed by the same amount, preserving signal shape
• D. Minimum order (Elliptic), because fewer poles means less phase shift overall

Answer: (C)

Group delay is the negative derivative of phase with respect to frequency: τ_g(ω) = −dφ/dω. Constant group delay means all frequency components in a signal arrive at the output with the same time delay — the signal shape is preserved. The Bessel filter is specifically designed to maximize group delay flatness (not magnitude flatness). Butterworth is designed for magnitude flatness, which does not imply phase linearity — Butterworth actually has better phase than Chebyshev or Elliptic, but Bessel sacrifices magnitude rolloff entirely to optimize phase. For timing-sensitive applications, Bessel (or a post-equalized elliptic) is the appropriate choice.

Question 3

A Chebyshev Type I filter achieves steeper rolloff than a Butterworth filter of the same order by tolerating equiripple oscillations within the passband.

• T. True
• F. False

Answer: True

This is the fundamental Butterworth-Chebyshev trade-off. Butterworth concentrates all its degrees of freedom on making the passband as flat as possible (all derivatives of |H|² zero at ω = 0). Chebyshev redirects those degrees of freedom: instead of zero ripple in the passband, it allows a specified ripple (e.g., 0.5 dB), using the resulting oscillation to push more attenuation into the stopband. The Chebyshev polynomial, which oscillates exactly N times between ±1 on the passband interval, is what generates this equiripple property. Result: for the same order N and same cutoff, Chebyshev reaches a given stopband attenuation at a lower stopband frequency than Butterworth.

Question 4

A higher-order Butterworth filter generally achieves steeper stopband rolloff than a lower-order Chebyshev filter, making order the primary relevant metric for comparing filter types.

• T. True
• F. False

Answer: False

Order is not the only relevant metric — the approximation type determines how efficiently each order is used. A Chebyshev filter of order N consistently achieves steeper stopband rolloff than a Butterworth filter of the same order N, because Chebyshev trades passband flatness for rolloff steepness. Similarly, an Elliptic filter of order N outperforms both for minimum transition bandwidth. A 4th-order Chebyshev will outperform a 4th-order Butterworth in stopband attenuation. The relevant comparison depends on the design specification: how much passband ripple is acceptable, how steep the transition must be, and whether phase linearity matters — there is no universally superior approximation.

Question 5

Why is there no universally optimal filter approximation, and what are the key trade-offs a designer must weigh when choosing among Butterworth, Chebyshev, and Elliptic designs?

Model answer: Each approximation optimizes a different objective. Butterworth maximizes passband flatness (zero ripple), using its degrees of freedom for smoothness rather than rolloff steepness — good when passband gain uniformity is critical. Chebyshev Type I accepts a specified passband ripple to achieve steeper rolloff at the same order — good when rolloff is important and small passband variation is tolerable. Elliptic minimizes filter order for given passband ripple, stopband attenuation, and transition width by optimizing both regions simultaneously — good when order (complexity or component count) must be minimized. Running orthogonal to all three is phase linearity: Butterworth has the most linear phase, Elliptic the least. Applications requiring signal shape preservation (communications, control) may need to prioritize phase, potentially choosing Bessel or adding an all-pass equalizer.

The key insight is that filter design is a multi-objective problem with no single dominant solution. The choice depends on which specification — passband flatness, transition steepness, stopband attenuation, phase linearity, or filter order — is the binding constraint. Understanding what each approximation sacrifices is more important than memorizing which is 'best,' because the answer is always: best at what?