Questions: Linear Phase Response and Signal Distortion

Question 1

A communications engineer applies an elliptic IIR lowpass filter to a digital audio signal. The filter has excellent magnitude performance: all passband frequencies have equal gain, and the stopband is heavily attenuated. Yet the output still sounds distorted relative to the input. What is the most likely cause?

• A. Finite-precision arithmetic in the IIR implementation is introducing quantization noise
• B. The filter's non-linear phase response delays different frequency components by different amounts, smearing transients and destroying waveform shape even though no frequencies are lost
• C. The stopband attenuation is insufficient — some high-frequency energy is leaking through
• D. The sampling rate is too low for the signal bandwidth, causing aliasing distortion

Answer: (B)

This is the classic phase distortion scenario. An elliptic IIR filter achieves very sharp magnitude transitions but has highly non-linear phase — group delay varies strongly with frequency. Even when all passband frequencies pass with equal gain, they arrive at the output at different times. For audio signals with sharp transients (drum attacks, consonants in speech), this timing smear destroys temporal detail. The distortion is purely in the phase domain, invisible in the magnitude response. This is why applications requiring waveform fidelity use linear-phase FIR filters despite their higher computational cost.

Question 2

A symmetric FIR filter with N = 51 coefficients (h[n] = h[50−n]) is designed as a lowpass filter. What is the expected group delay of this filter?

• A. 0 samples — linear phase means the filter introduces no delay whatsoever
• B. 25 samples — a constant group delay equal to (N−1)/2, independent of frequency
• C. Variable, from 0 to 50 samples depending on the frequency band
• D. Infinite at the passband edge, where the transition from passband to stopband occurs

Answer: (B)

A symmetric FIR filter of length N has a linear phase response: ∠H(ω) = −ω(N−1)/2. The group delay τ_g = −d∠H/dω = (N−1)/2 = 25 samples, constant across all frequencies. This constant group delay means every frequency component is delayed by the same 25 samples — the signal is shifted in time but not distorted in shape. The common misconception is that 'linear phase' means 'zero phase'; it means phase proportional to frequency, which corresponds to a fixed time delay. Zero phase would require a non-causal filter.

Question 3

A symmetric FIR filter (h[n] = h[N−1−n]) automatically achieves linear phase regardless of what magnitude response it is designed to produce.

• T. True
• F. False

Answer: True

This is the key result that makes symmetric FIR filters so useful. The frequency response of a symmetric FIR filter can always be written as a real-valued amplitude function multiplied by a linear phase factor e^{−jω(N−1)/2}. The symmetry condition forces the phase to be exactly linear — the magnitude design (windowing, Parks-McClellan, etc.) controls the real amplitude function, but the phase linearity is guaranteed by symmetry regardless of what magnitude shape results. This decoupling of magnitude and phase design is the defining advantage of symmetric FIR filters.

Question 4

IIR filters are generally preferred over FIR filters in applications that require exact linear phase, because IIR filters achieve sharper magnitude cutoffs with fewer coefficients.

• T. True
• F. False

Answer: False

IIR filters cannot achieve exact linear phase — their recursive structure prevents the coefficient symmetry required for linear phase. An IIR filter's denominator (feedback) terms create poles in the z-plane, and poles are inherently associated with non-linear phase. IIR filters can approximate linear phase over limited frequency ranges using all-pass equalizers, but this is an approximation with added complexity, not exact linear phase. When exact linear phase is required (medical imaging, digital communications, high-fidelity audio), FIR filters with symmetric coefficients are the only option — the computational cost is the price paid for phase integrity.

Question 5

Why is constant group delay equivalent to linear phase? And why does non-constant group delay cause waveform distortion even when the magnitude response is perfectly flat?

Model answer: Group delay τ_g(ω) = −d∠H(ω)/dω measures how much a narrowband signal centered at frequency ω is delayed by the filter. If group delay is constant (τ_g = k for all ω), integrating gives ∠H(ω) = −kω, which is exactly the definition of linear phase. A linear phase filter delays every frequency component by the same k seconds, so all components of a signal arrive simultaneously — the waveform shape is preserved, just shifted in time by k. If group delay varies with frequency, different components arrive at different times. Even if all pass with equal gain, their misalignment at the output changes the waveform shape: fast transients (which rely on many harmonics adding up in phase at the right moment) are smeared or distorted because the harmonics arrive asynchronously.

An intuitive analogy: a symphony orchestra sounds right only if all sections play in time. Magnitude response determines which instruments are audible; phase response determines whether they play together. A flat magnitude with non-linear phase is like an orchestra where every instrument plays the right notes at the right loudness but the percussion arrives late, the brass early, and the strings on time — the music is unrecognizable even though no notes were dropped.