Also known as: group delay, envelope delay, group-delay variation
Group delay is the time by which a system delays the envelope of a narrow band of frequencies, defined as the negative derivative of phase with respect to angular frequency, τ_g = −dφ/dω.1 When group delay is not constant across a signal’s occupied band, different frequency components arrive at slightly different times — a form of phase distortion that spreads each symbol into its neighbours and raises intersymbol interference.
How it works
A signal is only passed undistorted in phase if the system has linear phase — phase that changes proportionally with frequency. The derivative of a straight line is a constant, so linear phase means constant group delay: every frequency component is held up by the same amount and the waveform shape is preserved, merely shifted in time.
When phase deviates from a straight line, group delay varies with frequency and the system is dispersive. Higher-frequency components of a pulse may be delayed more (or less) than lower ones, so the reassembled pulse spreads and rings. It is the variation in group delay across the used band — the group-delay flatness — that causes trouble, not the absolute delay: a constant 10 µs delay harms nothing, but 2 µs of variation across a channel can close an eye diagram.
Sharp filters are the usual culprit. A steep IIR filter has strongly non-flat group delay near its cutoff, worst right at the band edges where the phase bends most. A symmetric FIR filter can be made exactly linear-phase — its group delay is a flat constant of half the filter length — which is a major reason FIR structures are favoured in communications where waveform fidelity matters.
In practice
Because group-delay variation and amplitude ripple both degrade a channel, systems either design for flatness or correct for it. A matched root-raised-cosine pulse-shaping pair is chosen to give an overall linear-phase, controlled-ISI response at the sampling instants. Where the channel itself adds dispersion, an adaptive equalizer in the receiver estimates and inverts the combined amplitude and group-delay distortion. Group-delay flatness is a standard line item on filter and cable datasheets, and excess variation shows up directly as a raised error vector magnitude.
Relevance to SDR
Group-delay flatness matters most for wideband and high-order-modulation signals, where a distorted phase response smears symbols and closes the eye. TETRA, P25, and DMR all rely on well-behaved, near-linear-phase transmit and receive filtering to keep their constellations clean; sharp analog IF filters in older receivers were a classic source of group-delay distortion that digital FIR filtering now avoids.
GopherTrunk performs its channel filtering and decimation digitally, and its pulse-shaping and channelizing stages use FIR structures whose linear phase gives constant group delay by construction — so the decode chain does not itself introduce phase dispersion. The concept remains useful for diagnosis: if a captured signal decodes poorly with an otherwise healthy SNR, group-delay distortion introduced before the SDR (in a cheap external filter or a long marginal feedline) is one candidate the concept helps rule in or out.
Sources
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Group delay and phase delay — Wikipedia, definitions and the linear-phase / constant-delay relationship. ↩