Questions: Chebyshev Type I Filter Design

Question 1

A signal processing engineer needs a filter that attenuates a frequency component at twice the cutoff frequency by at least 60 dB. A 5th-order Butterworth achieves only 40 dB at that frequency. What is the most efficient solution?

• A. Increase the Butterworth to 8th order to achieve the required attenuation while maintaining a flat passband
• B. Switch to a 5th-order Chebyshev Type I with 3 dB ripple, which achieves steeper rolloff and can meet the 60 dB spec at the same order
• C. Use two cascaded 5th-order Butterworth filters to double the attenuation
• D. Redesign the Butterworth with a lower cutoff frequency to shift the 40 dB point inward

Answer: (B)

A 5th-order Chebyshev Type I with 3 dB ripple can achieve approximately 60+ dB at twice the cutoff frequency — meeting the spec without increasing filter order. The Chebyshev design achieves this by allowing controlled equiripple in the passband to buy steeper rolloff beyond it. The Butterworth's maximally-flat passband comes at the cost of less steep transition; for the same order, Chebyshev Type I offers significantly better stopband attenuation. If passband flatness is not critical but stopband performance is, switching to Chebyshev is more efficient than increasing Butterworth order.

Question 2

Why do Chebyshev Type I filter poles lie on an ellipse rather than the circle where Butterworth poles lie?

• A. The ellipse places poles farther from the jω axis, producing a flatter passband than the Butterworth circle
• B. The ellipse places poles closer to the jω axis than the Butterworth circle, creating resonances that produce the passband ripple and the steeper rolloff
• C. Chebyshev polynomials require complex pole locations that cannot be expressed on a simple geometric shape
• D. The elliptical placement results from frequency pre-warping applied to account for analog-to-digital conversion

Answer: (B)

From transfer function theory, poles near the jω axis create peaks in the magnitude response and contribute to steeper rolloff beyond those peaks. Chebyshev Type I poles lie on an ellipse with its major axis along the imaginary jω axis — meaning they are closer to the jω axis (smaller real part) than Butterworth poles at the same frequency. This proximity creates the ripple oscillations in the passband (each pole contributes a resonant peak) and produces the steep rolloff beyond the passband edge. The Butterworth circle places poles farther from the jω axis for a monotonically flat, ripple-free response.

Question 3

Allowing more passband ripple in a Chebyshev Type I design results in steeper rolloff for the same filter order.

• T. True
• F. False

Answer: True

This is the fundamental Chebyshev tradeoff. The ripple parameter ε controls both the passband variation and the rolloff steepness: larger ε means the passband oscillates between wider bounds, but the Chebyshev polynomial grows faster outside that interval, producing steeper attenuation in the stopband. This is not a free lunch — more ripple degrades signal quality in the passband and increases phase nonlinearity — but it directly buys steeper rolloff. The designer chooses how much passband variation is acceptable in exchange for a given stopband specification.

Question 4

Chebyshev Type I filters are preferred over Butterworth filters in applications where linear phase response through the passband is the primary design requirement.

• T. True
• F. False

Answer: False

Chebyshev Type I filters have more nonlinear phase response than Butterworth — a direct consequence of placing poles closer to the jω axis. Nonlinear phase causes different frequency components to experience different time delays (group delay variation), which distorts signal waveforms. For applications sensitive to pulse shape or waveform fidelity — such as data communications and certain audio processing — this phase distortion is unacceptable. Butterworth filters have better (more linear) phase response than Chebyshev. Chebyshev Type I is preferred when stopband attenuation is the priority and passband ripple with phase distortion can be tolerated.

Question 5

Explain the equiripple property of Chebyshev Type I filters, why it is considered optimal, and what practical cost is paid in exchange for the steeper rolloff it provides.

Model answer: The equiripple property means the magnitude response oscillates between exactly two fixed bounds throughout the passband — every oscillation reaches the same maximum and minimum, so no part of the allowable passband deviation is wasted. This is optimal in the minimax sense: among all Nth-order filters meeting the passband ripple constraint, the Chebyshev Type I achieves the fastest possible rolloff outside the passband. The mathematical basis is the Chebyshev polynomial, which oscillates within ±1 on [-1,1] while growing faster than any other monic polynomial outside that interval. The practical costs are twofold: passband ripple that can degrade signal quality, and nonlinear phase response that distorts waveforms. These costs are acceptable in many RF, audio, and anti-aliasing applications where stopband rejection matters more than flatness or phase linearity.

The equiripple principle also applies to Chebyshev Type II filters (equiripple in the stopband, monotonic passband) and elliptic filters (equiripple in both passband and stopband). Each represents a different point on the tradeoff surface between passband flatness, stopband attenuation, filter order, and phase linearity. Understanding why equiripple is optimal within a given domain is the conceptual key to the entire family of classical analog filter designs.