# Communications — Group Delay, Phase Noise, Channel Response *In communications every physically realisable channel is causal, so Kramers–Kronig constrains what a designer can and cannot achieve: magnitude and phase are not independent, group delay is pinned by attenuation, and a flat passband with brick-wall rolloff and zero group delay is a mathematical impossibility.* This chapter uses the **engineering convention** `e^{+jωt}`, with H(jω) = |H(jω)| e^{jφ(ω)} and causal h(t) = 0 for t < 0. The corresponding analyticity domain is Re s ≥ 0 (Chapter 16). Physics-form K-K has the opposite sign on the right-hand side. ## K-K on log H: the gain–phase link The central tool for communications is Bode's gain–phase theorem (Chapter 20): for a stable, minimum-phase causal transfer function H(s), log H(s) is itself analytic in Re s ≥ 0, so the Hilbert-transform pair ``` φ(ω) = (1/π) P ∫_{−∞}^{∞} log|H(jω′)| / (ω′ − ω) dω′ ``` holds (up to a constant allpass factor). Attenuation α(ω) = −log|H(jω)| and phase β(ω) = −φ(ω) along a minimum-phase channel are a Hilbert pair. Everything in this chapter is a consequence. ## Group delay cannot be designed independently of magnitude The group delay of a channel is ``` τ_g(ω) = −dφ(ω)/dω ``` For a minimum-phase channel, differentiating Bode's integral under the sign gives τ_g directly in terms of d(log|H|)/dω. Two consequences: 1. **No flat-group-delay-with-rolloff filter exists.** If |H(ω)| rolls off at band edges (any real filter does), then dφ/dω is forced to have a matching bump. A causal realisation *cannot* have both flat gain **and** flat group delay while having a finite passband. 2. **Minimum phase gives minimum group delay** for a prescribed |H(ω)|. Any allpass factor adds extra phase that integrates to extra positive group delay — it delays without changing magnitude. These are why Bessel filters (maximally flat group delay) give up magnitude sharpness, and why Butterworth/Chebyshev filters have phase ripple where the magnitude is sharp. ## Pre-equalisation and the zero-location trap A digital comms receiver often pre-equalises or post-equalises the channel to flatten the effective response. Inverting the channel H_eq(z) = 1/H_ch(z) is stable *only if the channel is minimum-phase* — i.e. all its zeros lie strictly inside the unit circle. In that case, the inverse has all-poles-inside and is causal and stable. Non-minimum-phase channels — any channel with a zero on or outside the unit circle — have no stable causal inverse. Equalisation then requires either: - an allpass factor to reflect the exterior zeros inward, accepting the extra group delay, or - a non-causal inverse implemented with a **delay**: the receiver looks forward in time by buffering samples, so that future inputs are available when computing current output. This is the origin of the "decision delay" in DFE and MLSE receivers and the implementation latency of FIR equalisers in optical coherent receivers. ## Phase noise and Leeson's model In oscillators, short-term amplitude and phase fluctuations a(t) and φ(t) produce sidebands around the carrier. The single-sideband phase noise L(f_m) at offset f_m from the carrier is related to the near- carrier amplitude noise through a K-K-like structure for random processes: causality of the resonator response forces the phase-noise spectrum to share structure with the magnitude response of the resonator. Leeson's model (1966) writes ``` L(f_m) = 10 log₁₀ [ (FkT/2P_s) · (1 + (f_0 / 2Q f_m)²) · (1 + f_c/f_m) ] ``` where f_0/(2Q) is the resonator half-bandwidth, f_c is the 1/f-noise corner, F is the noise figure, P_s the signal power. The (1 + (f_0/2Q f_m)²) factor is precisely the Lorentzian roll-off of the minimum-phase resonator: the phase noise is tied to the magnitude response by the same K-K logic. Seeing an oscillator's L(f_m) is, in effect, seeing the imaginary part of a K-K-compliant resonator transfer function. ## The Paley–Wiener criterion A closely related causality criterion: a complex-valued H(ω) on the real line is the Fourier transform of a causal square-integrable h(t) if and only if ``` ∫_{−∞}^{∞} |log|H(ω)|| / (1 + ω²) dω < ∞ (Paley–Wiener, 1934) ``` The condition says |H(ω)| cannot vanish on a set of positive Lebesgue measure — in particular, **no realisable filter has |H(ω)| = 0 on any finite interval**. A brick-wall filter with |H| strictly zero on a stopband is not causal. Practical filters have small but non-zero stopband gain, with the roll-off set by the filter order. Paley–Wiener can be viewed as the non-zero-ness envelope of K-K: the integrand `log|H|/(1 + ω²)` is the kernel that appears when one applies the Hilbert transform to log H in the half-plane sense, and integrability of that kernel is the precise causality constraint. A proof via the K-K inversion of log H is in Papoulis. ## Channel impulse response and equaliser design The discrete-time channel h[n] in a baseband communications system is causal by construction. Its DTFT H(e^{jω}) satisfies the discrete Hilbert-transform pair on the unit circle (Chapter 19) when minimum- phase; when not, a minimum-phase + allpass decomposition separates the K-K-determined part from the free allpass degrees of freedom. Matched- filter / whitened-matched-filter theory rests on this decomposition: the whitened matched filter realises the minimum-phase equivalent channel, whose phase response is determined by magnitude alone. ## A designer's heuristic - If a specification demands flat magnitude **and** flat group delay over a band, either the band is finite and the phase rolls off at the edges, or a delay is allowed (FIR with latency). There is no third option. - If a channel's magnitude is measured but not its phase, reconstruct phase via K-K on log|H| and *declare* the channel minimum-phase; use that as the starting model and add allpass factors only when evidence demands. - Oscillator phase noise shape near the carrier is the Lorentzian wing of the resonator: if the close-in L(f_m) doesn't follow that shape, the resonator model is wrong, not the K-K. ## See also - [19_minimum_phase_DSP.md](19_minimum_phase_DSP.md) - [20_bode_gain_phase.md](20_bode_gain_phase.md) - [25_vna_measurements.md](25_vna_measurements.md) - [27_audio_room_acoustics.md](27_audio_room_acoustics.md) ## References - J. G. Proakis and M. Salehi, *Digital Communications*, 5th ed., McGraw-Hill, 2008. - S. Haykin, *Communication Systems*, 5th ed., Wiley, 2009. - D. B. Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum", *Proceedings of the IEEE*, vol. 54, no. 2, pp. 329–330, 1966. - R. E. A. C. Paley and N. Wiener, *Fourier Transforms in the Complex Domain*, AMS Colloquium Publications, 1934.