Questions: All-Pass Filters for Phase Shaping

Question 1

A filter design achieves excellent magnitude response but has strongly nonlinear group delay. A colleague proposes cascading an all-pass filter to fix the group delay. A second engineer objects: 'Adding more poles and zeros will ruin the magnitude response we just designed.' Who is correct?

• A. The second engineer is correct; any additional poles or zeros will alter the magnitude response
• B. The colleague is correct; all-pass filters add poles and zeros that cancel exactly in magnitude while contributing phase — they leave the existing magnitude response untouched
• C. Both are partially correct; the all-pass filter will introduce small magnitude ripple but acceptable phase correction
• D. Neither; group delay cannot be corrected after the original filter design is finalized

Answer: (B)

This is the key insight of all-pass filters. Their poles and zeros are placed as mirror images across the imaginary axis — for every zero at s = +a, there is a pole at s = -a. At any frequency on the imaginary axis (s = jω), the distance to the zero equals the distance to the pole, so the magnitude contributions cancel: |H(jω)| = 1. The angles (phases) from the zero and pole do not cancel — they add constructively to produce a frequency-dependent phase shift. This elegant pole-zero structure gives all-pass filters independent control of phase without touching magnitude, allowing cascade addition without disturbing the already-designed magnitude response.

Question 2

In a first-order all-pass filter H(s) = (s − a)/(s + a) with a > 0, why does |H(jω)| = 1 at every frequency?

• A. The numerator and denominator have the same leading coefficient, so magnitudes must be equal by definition
• B. The zero at s = +a and the pole at s = −a are mirror images across the imaginary axis, so they are equidistant from every point jω, and their magnitude contributions cancel exactly
• C. The filter is lossless because it contains no resistive elements in its circuit implementation
• D. Unity gain is a design choice enforced by the normalization of the transfer function

Answer: (B)

For any point jω on the imaginary axis, the distance to the zero at +a is |jω − a| = √(ω² + a²), and the distance to the pole at −a is |jω + a| = √(ω² + a²). These distances are identical, so |jω − a|/|jω + a| = 1 for all ω. The angles, however, are different: the zero contributes +arctan(ω/a) and the pole contributes −arctan(ω/a), giving a total phase of −2 arctan(ω/a) that varies from 0° at DC to −180° at high frequency. The geometric symmetry of the pole-zero pair across the imaginary axis is what simultaneously enforces unity magnitude and enables nonzero phase.

Question 3

An all-pass filter can be cascaded with an existing filter to adjust phase response (and thus group delay) without changing the magnitude response already designed.

• T. True
• F. False

Answer: True

This is the primary application of all-pass filters in filter design. Once a filter is designed to meet magnitude specifications (Butterworth, Chebyshev, etc.), group delay equalization is accomplished by cascading one or more all-pass sections designed to flatten the total group delay. Because |H_AP(jω)| = 1 at every frequency, the cascade product |H_total(jω)| = |H_original(jω)| × 1 = |H_original(jω)|. The all-pass sections add purely to the phase response. This separation of magnitude and phase design is possible only because all-pass filters provide independent phase control.

Question 4

A non-minimum-phase system with right-half-plane zeros can be corrected to minimum-phase behavior by cascading a stable most-pass filter that cancels the right-half-plane zeros.

• T. True
• F. False

Answer: False

This is a fundamental limitation. Cancelling a right-half-plane zero requires placing a pole at the same location — but right-half-plane poles correspond to unstable system modes. Any stable, causal filter cannot cancel RHP zeros without introducing instability. The all-pass factor in a non-minimum-phase system represents irreducible phase lag: it cannot be removed by cascading any stable causal system. This limitation directly constrains achievable control bandwidth — a control loop with a non-minimum-phase plant (RHP zero) cannot be stabilized at frequencies above approximately the inverse of the RHP zero location, because the all-pass phase lag makes the loop margin disappear.

Question 5

What is group delay equalization, and why would you cascade an all-pass filter to achieve it rather than simply redesigning the original filter?

Model answer: Group delay τ_g(ω) = −dφ/dω measures how much time delay each frequency component experiences through a filter. When group delay varies with frequency, different spectral components of a signal arrive at the filter output at different times, smearing edges and distorting pulse shapes — a phenomenon called phase distortion. Group delay equalization means adding a phase response that, combined with the existing filter's phase, produces a total group delay that is approximately constant across the passband. An all-pass filter is used because it contributes only phase (no magnitude change), providing extra degrees of freedom: the original filter's poles and zeros were spent to meet the magnitude specification, and the all-pass adds new poles and zeros that cancel in magnitude but flatten the phase.

Redesigning the original filter to have both the desired magnitude response and flat group delay is typically impossible within the same filter order and structure — the two requirements conflict. Butterworth and Chebyshev filters are optimal for magnitude flatness or equiripple, respectively, but they achieve this at the cost of nonlinear phase. By separating the design into two cascaded stages (magnitude design, then phase correction), the engineer gains independent control. The Bessel filter is an alternative that starts with maximally flat group delay but sacrifices magnitude response — illustrating that magnitude and phase cannot both be optimized simultaneously with a single filter design.