Questions: Filter Design and Specifications

Question 1

An audio amplifier manufacturer needs a low-pass filter with the top priority that all frequencies within the passband are reproduced at the same volume — no frequency should be louder or quieter than another. Which filter family is most appropriate?

• A. Chebyshev — because it achieves the steepest roll-off for a given order
• B. Elliptic — because it achieves the steepest possible roll-off and has the sharpest stopband
• C. Butterworth — because it has a maximally flat passband with no ripple
• D. Any filter family works equally well; the choice doesn't affect audio reproduction

Answer: (C)

The Butterworth filter achieves a maximally flat (monotonically decreasing) magnitude response with no ripple anywhere in the passband. For audio, uniform gain across all passband frequencies is essential to avoid coloring the sound. Chebyshev and elliptic filters achieve sharper cutoffs but introduce passband ripple — some frequencies are slightly louder or quieter — which is audible as tonal distortion. The tradeoff is that Butterworth requires higher order (more components) to achieve the same stopband attenuation as a Chebyshev or elliptic design.

Question 2

A designer cascades two second-order filter sections to create a 4th-order low-pass filter. At frequencies well above the cutoff, what roll-off rate does this filter achieve?

• A. 20 dB/decade — each section contributes equally, and they average out
• B. 40 dB/decade — only the steeper of the two sections dominates
• C. 80 dB/decade — each second-order section contributes 40 dB/decade, and they add
• D. 160 dB/decade — the sections cascade exponentially

Answer: (C)

Roll-off rate scales as n × 20 dB/decade, where n is the filter order. A 4th-order filter rolls off at 4 × 20 = 80 dB/decade. Each second-order section contributes 40 dB/decade, and cascading sections multiplies their transfer functions — their dB attenuation values add. This additive property in dB is why cascading biquad sections is the standard way to build high-order filters with steep stopband attenuation.

Question 3

A Chebyshev filter achieves a steeper roll-off than a Butterworth filter of the same order, but at the cost of ripple in the passband.

• T. True
• F. False

Answer: True

This is the fundamental Butterworth-Chebyshev tradeoff. A Chebyshev filter allows equiripple behavior in the passband — the gain oscillates between defined limits — and uses this 'budget' of allowable variation to achieve significantly steeper roll-off at the passband edge. For the same stopband attenuation requirement, a Chebyshev design needs fewer poles than a Butterworth, reducing circuit complexity. The cost is passband ripple, acceptable in communications receivers but not in audio reproduction.

Question 4

Increasing filter order typically improves most aspect of filter performance simultaneously, with no downside.

• T. True
• F. False

Answer: False

Higher filter order increases roll-off rate and achieves tighter transition bands, but it comes with real costs: more poles require more circuit elements (more biquad sections, more components), increasing size, cost, power consumption, and component sensitivity. For active filters, each additional stage adds noise and phase shift. Higher-order filters also have more complex group delay characteristics. Filter design is always a tradeoff; raising order alone does not eliminate the fundamental tensions between passband flatness, roll-off sharpness, and circuit complexity.

Question 5

Why can't a filter simultaneously have a perfectly flat passband, an infinitely steep transition, and infinite stopband attenuation?

Model answer: A filter with a perfectly flat passband, zero transition band width, and infinite stopband attenuation would require an impulse response of infinite duration (the ideal 'brick wall' filter), which is physically unrealizable. In practice, filter performance is limited by filter order: sharpness of the transition band, passband flatness, and stopband attenuation all improve with order, but any finite-order filter must trade among these properties. The Butterworth, Chebyshev, and elliptic families each represent different choices about which property to optimize given the constraint of finite order.

This is why filter design is fundamentally a tradeoff problem. Real-world filters make deliberate engineering choices about which deviations from ideal behavior are acceptable for the application at hand. The three classical families systematize these tradeoffs into well-characterized designs with predictable behavior.