Understanding Generalized Linear Phase in Discrete-Time Filters

April 11, 2026 · Luciano Muratore

Expanding our study of filter design, we move beyond magnitude response to examine the critical role of phase. In Digital Signal Processing, the phase response of a filter determines how much temporal distortion is introduced to the signal.

1. Motivation: Phase and Distortion

When analyzing filters in the frequency domain, the frequency response is expressed as: $H(e^{j\omega}) = |H(e^{j\omega})|e^{j\varphi(\omega)}$

While the magnitude ($|H(e^{j\omega})|$) determines how the filter shapes the spectrum, the phase ($\varphi(\omega)$) determines temporal distortion. Filters with well-structured phase responses are essential because they preserve the original waveform shape.

2. Phase and Group Delay

To quantify how a filter affects the timing of a signal, we define two key terms:

• Phase Response: $\varphi(\omega) = \arg H(e^{j\omega})$.
• Group Delay: $\tau_{g}(\omega) = -\frac{d}{d\omega}\varphi(\omega)$.

The group delay measures how much each frequency component is delayed as it passes through the system. If $\tau_{g}(\omega)$ varies with frequency, different parts of the signal arrive at different times, leading to phase distortion.

3. Strict Linear Phase

A filter is said to have strict linear phase if its phase response follows the form: $\varphi(\omega) = -\omega d$ for some constant delay $d$. This results in a frequency response of: $H(e^{j\omega}) = |H(e^{j\omega})|e^{-j\omega d}$

In this case, the group delay is constant ($\tau_{g}(\omega) = d$), meaning all frequencies are delayed by the exact same amount, resulting in no phase distortion.

4. Generalized Linear Phase

Strict linear phase is a specific case of a broader class. Generalized linear phase adds a constant phase offset $\alpha$: $\varphi(\omega) = -\omega d + \alpha$

Properties:

• Constant Group Delay: Despite the offset, the derivative remains $\tau_{g}(\omega) = d$.
• No Waveform Distortion: The signal experiences a uniform delay and a global phase rotation ($e^{j\alpha}$), but its shape remains intact.
• Constant Phase Rotation: The term $e^{j\alpha}$ multiplies the entire output by a constant complex phase but does not affect the magnitude response or the relative alignment of frequency components.

5. Generalized Linear Phase in FIR Filters

Real-coefficient FIR filters can achieve generalized linear phase if their impulse responses exhibit specific symmetry properties. There are four canonical types of linear-phase FIR filters, categorized by:

• Symmetry: Symmetric or antisymmetric impulse response.
• Length: Even or odd number of taps.
• Phase Term: $\alpha = 0$ or $\alpha = \frac{\pi}{2}$.

Regardless of the type, all four possess a constant group delay, making them highly desirable for practical DSP applications where signal integrity is paramount.

Summary