An audio engineer designs a crossover filter and observes that its group delay varies significantly across the audible band. What is the consequence for audio signals passing through this filter?
ACertain frequency bands will be attenuated more than others, altering the spectral balance of the signal
BDifferent frequency components will be delayed by different amounts, potentially smearing the time alignment of transients and distorting the waveform shape
CThe filter will become unstable and produce oscillations at the frequencies where group delay is highest
DThe phase of the output signal will be shifted by a constant amount across all frequencies, introducing a fixed time offset
Group delay τ_g(ω) = −dφ/dω gives the time delay experienced by each frequency component. When group delay is not constant, components at different frequencies emerge from the filter at different times. The output waveform is a superposition of these time-shifted components — and if they are misaligned relative to each other, the reconstructed shape differs from the input: waveform distortion. This is distinct from magnitude distortion (option A), which would alter frequency balance without necessarily changing the waveform shape. Constant group delay, by contrast, means all components are delayed equally, preserving the waveform shape with only a fixed time offset.
Question 2 Multiple Choice
An FIR filter with symmetric coefficients is preferred over an IIR Butterworth filter in applications requiring waveform fidelity primarily because:
AFIR filters achieve sharper rolloff for the same filter order, preserving more signal energy in the passband
BFIR filters have lower computational cost per sample, reducing the end-to-end processing delay
CThe symmetry of FIR coefficients mathematically guarantees a linear phase response and hence constant group delay across all frequencies
DFIR filters have no poles, which means their group delay is identically zero and introduces no delay at all
For a causal FIR filter with N taps, symmetric coefficients (h[n] = h[N−1−n]) guarantee that the phase response is exactly linear: φ(ω) = −ω(N−1)/2 + constant. Since group delay is τ_g(ω) = −dφ/dω = (N−1)/2, it is constant across all frequencies — all components are delayed by the same amount (half the filter length in samples). IIR filters like Butterworth have poles that distort the phase response non-linearly, producing frequency-dependent group delay. Note that 'zero group delay' (option D) would require non-causal processing; symmetric FIR filters have constant but non-zero group delay.
Question 3 True / False
A system whose phase response is exactly φ(ω) = −5ω introduces a constant delay of 5 seconds to all frequency components, preserving the shape of any input waveform.
TTrue
FFalse
Answer: True
Linear phase φ(ω) = −ωτ₀ is the ideal case for waveform preservation. Group delay τ_g(ω) = −dφ/dω = τ₀ is constant — every frequency component is delayed by exactly τ₀ seconds. The output is therefore a perfect time-shifted replica of the input: y(t) = x(t − τ₀). No component arrives early or late relative to others, so the superposition reconstructs the original shape. The only change is a temporal offset. This is why linear phase (constant group delay) is the gold standard in audio, image processing, and data communications applications where waveform fidelity matters.
Question 4 True / False
Group delay is calculated as the phase shift φ(ω) divided by frequency ω — that is, τ_g(ω) = φ(ω)/ω.
TTrue
FFalse
Answer: False
Group delay is τ_g(ω) = −dφ/dω, the negative derivative (slope) of the phase response with respect to frequency — not φ/ω. The quantity φ/ω is called 'phase delay,' a different measure. For a linear phase system φ(ω) = −ωτ₀, both are equal: group delay = phase delay = τ₀. But for non-linear phase responses they diverge significantly. The distinction matters because group delay (the derivative) captures how the delay changes across frequencies — which is what determines whether different-frequency components drift apart in time. Phase delay describes the total phase shift at each frequency without revealing the frequency-dependence of the delay.
Question 5 Short Answer
Explain why non-constant group delay causes waveform distortion, and describe a real-world system where this distortion has serious practical consequences.
Think about your answer, then reveal below.
Model answer: A real signal is a superposition of many frequency components. Each component at frequency ω experiences a delay of τ_g(ω) seconds when passing through the system. If τ_g(ω) is constant, all components emerge with the same relative timing and the waveform shape is preserved. If τ_g(ω) varies with frequency, some components are delayed more than others — they arrive at the output at different times relative to each other. When these time-misaligned components are recombined, the reconstructed waveform has a different shape than the input: waveform distortion. In fiber-optic communications, chromatic dispersion causes different wavelengths of light to travel at different group velocities, broadening data pulses as they propagate. As pulses spread, they overlap into adjacent time slots — intersymbol interference — which corrupts the received bits and limits the achievable data rate of the fiber link.
The practical consequence is that group delay variation sets a fundamental limit on system bandwidth in dispersive channels. Dispersion compensation (using specially designed fibers or all-pass equalizer networks) is a multi-billion-dollar field in telecom engineering, entirely motivated by the need to maintain constant group delay across signal bandwidths.