Questions: Chebyshev Filters and Equiripple Response

Question 1

A designer needs a 5th-order filter to achieve 45 dB attenuation at twice the cutoff frequency, but a 5th-order Butterworth only provides 30 dB there. What should the designer consider?

• A. Use a 10th-order Butterworth for better roll-off at the same cutoff
• B. Use a Chebyshev Type I filter — by accepting equal-magnitude ripple in the passband, it achieves sharper roll-off than Butterworth at the same order
• C. Use a Chebyshev Type II filter — it achieves a perfectly flat passband and provides the needed stopband attenuation
• D. No 5th-order filter can achieve 45 dB at twice cutoff; the order must be increased regardless of filter type

Answer: (B)

A Chebyshev Type I filter of the same order achieves significantly sharper roll-off than Butterworth by distributing equal-magnitude ripple across the passband instead of insisting on maximum flatness. This equiripple trade-off is exactly why a 5th-order Chebyshev with 1 dB ripple can reach ~45–50 dB at twice cutoff while Butterworth reaches only ~30 dB. Type II is wrong here because it has equiripple in the stopband (not passband) — it achieves a flat passband but achieves the same roll-off improvement, which may or may not meet the spec depending on the passband flatness requirement.

Question 2

A Chebyshev Type I filter is redesigned with 3 dB allowed passband ripple instead of 0.5 dB. How does this change the filter's roll-off performance at the same order?

• A. The transition band becomes less steep — more ripple indicates a less optimal design
• B. The transition band becomes steeper — allowing more ripple (larger ε) pushes poles further from the imaginary axis, sharpening the roll-off
• C. The stopband ripple increases proportionally, but roll-off steepness is unchanged
• D. There is no effect; ripple magnitude and transition sharpness are independent design parameters

Answer: (B)

The ripple tolerance ε² = 10^(Rₚ/10) − 1 controls the pole positions. Larger Rₚ (more allowed ripple) means larger ε, which moves the poles further from the imaginary axis. This makes the frequency response fall off more steeply outside the passband. The fundamental trade-off is: ripple tolerance in exchange for transition sharpness. Relaxing the passband ripple spec always buys sharper stopband attenuation for a fixed filter order.

Question 3

Chebyshev Type I filters have equiripple in the passband and monotonic response in the stopband, while Type II filters have the reverse: monotonic passband and equiripple stopband.

• T. True
• F. False

Answer: True

This is the defining distinction between the two types. Type I is derived directly from Chebyshev polynomials in the passband; Type II (inverse Chebyshev) places the equiripple behavior in the stopband instead. Type I is preferred when the passband flatness can tolerate some oscillation; Type II is preferred when the passband must be maximally smooth but some residual signal in the stopband is acceptable.

Question 4

Chebyshev filters achieve sharper roll-off than Butterworth filters by using a higher filter order, not by changing the response shape.

• T. True
• F. False

Answer: False

Chebyshev filters achieve sharper roll-off at the SAME order as Butterworth, by abandoning the maximally-flat passband constraint in favor of equiripple behavior. This is the entire point of the design: distributing equal-magnitude oscillations throughout the passband is more 'efficient' (in the Chebyshev polynomial sense) than trying to minimize the maximum deviation at any single passband point. The order is a separate design lever — increasing order sharpens any filter type, but the Chebyshev achieves more at each order than Butterworth.

Question 5

What is the fundamental design trade-off that distinguishes a Chebyshev filter from a Butterworth filter? Explain why accepting ripple in the passband allows a sharper transition band.

Model answer: Butterworth achieves a maximally flat passband — it minimizes deviation from 0 dB at the cost of a gentle roll-off. Chebyshev accepts equal-magnitude oscillations (equiripple) throughout the passband, which allows the response to 'use up' its error budget more efficiently. Chebyshev polynomials have the property of oscillating between ±1 in the passband while growing faster than any other polynomial of the same degree outside it — meaning for a fixed number of poles, the stopband attenuation is maximized once ripple is permitted. The ripple amplitude (ε) is the designer's lever: more allowed ripple → larger ε → steeper roll-off for the same order.

The mathematical insight is that maximally-flat responses waste 'approximation capacity' by concentrating effort on one point (near DC). Equiripple spreads the approximation error evenly, achieving a minimax optimum. This is why Chebyshev filters are preferred when passband variation is acceptable and transition-band sharpness matters — and why pushing to equiripple in BOTH bands (elliptic/Cauer filter) achieves the theoretical maximum roll-off for any given order.