# Audio and Room Acoustics *The minimum-phase / allpass decomposition is what separates the parts of a loudspeaker or a room that can be equalised from the parts that cannot — a direct application of Kramers–Kronig in the audio domain.* This chapter uses the **engineering convention** `e^{+jωt}` (standard in DSP and audio), with causal responses analytic for Re s ≥ 0 (Chapter 16). We'll mostly work in discrete time on the unit circle, using the discrete K-K / Hilbert pair from Chapter 19. ## Loudspeaker transfer functions An ideal moving-coil driver, modelled electromechanically with a single resonant mode and viscous damping, has a minimum-phase transfer function from voltage to cone acceleration: all its zeros of H(z) lie inside the unit disk. For such a driver, the phase response is completely determined by the magnitude response via the discrete K-K pair ``` ∠H(e^{jω}) = −(1/2π) P ∫_{−π}^{π} ln|H(e^{jω′})| · cot((ω′ − ω)/2) dω′ ``` and inverse filtering with a minimum-phase IIR equaliser simultaneously corrects magnitude **and** phase. One EQ, both fixed — this is the fundamental reason driver-level EQ is so effective. Departures from minimum-phase arise when mechanical compliance and acoustic loading introduce additional resonances whose damping trajectories cross the unit circle in the z-plane. Woofers approaching their excursion limit and tweeters with break-up modes above their pass-band both exhibit measurable non-minimum-phase behaviour. ## Crossovers: where magnitude and phase part company A multi-way loudspeaker crossover sums the outputs of two or more drivers, each with its own frequency response. Standard analogue crossover topologies (Butterworth, Linkwitz–Riley) are *not* minimum- phase when considered as the system from input voltage to combined acoustic pressure at the listening point: the summation introduces allpass behaviour where the two drivers overlap. A Linkwitz–Riley 4th-order crossover, for example, has **flat summed magnitude** but **non-flat phase**, with a 360° phase rotation through the crossover region. That phase rotation is an allpass contribution and cannot be removed by any causal minimum-phase EQ; removing it requires an FIR filter with sufficient latency to realise a non-causal inverse. ## Room impulse responses A room impulse response (RIR) from a source to a microphone in a reverberant space is generally **non-minimum-phase**. The physical reason is that modal interference and multi-path reflections scatter zeros of H(z) outside the unit disk: a reflection arriving at the microphone later than the direct sound with greater amplitude puts a zero at |z| > 1. Any causal stable H(z) can be factored (Chapter 19) as ``` H(z) = H_min(z) · H_ap(z) ``` where H_min is minimum-phase and carries all the magnitude response, and H_ap is allpass with |H_ap(e^{jω})| = 1. For a RIR: - The **minimum-phase part** H_min is amenable to K-K reconstruction from magnitude alone, and is invertible with a stable causal filter. Room-EQ schemes that correct only the minimum-phase component give perceptually large improvements at modest latency. - The **allpass part** H_ap contains the directional and reflective information; inverting it would require a non-causal filter or a delay. Any practical room correction that tries to flatten the full RIR must introduce latency proportional to the excess group delay of H_ap, typically tens of milliseconds. This is the origin of the "listening-window vs power response" debate in loudspeaker design — K-K reconstruction works for the axial minimum- phase component, but the diffuse-field response and the reflections are allpass-like and cannot be EQ-corrected universally. ## Psychoacoustics and group delay The ear is remarkably tolerant of group-delay distortion. Blauert & Laws (1978) established audibility thresholds for group delay: roughly 2 ms within a critical band, rising to tens of ms across bands. Above about 1 kHz the threshold is essentially constant at a few ms; below 200 Hz it rises to 20–30 ms because the critical bands are wider and the ear effectively averages across them. Practically, this means a minimum-phase EQ for a non-minimum-phase system — which fixes magnitude but leaves an allpass residual — is **often inaudible** as a timing error even when the measured group- delay ripple is 10 ms or more at bass frequencies. It is one reason common room-EQ schemes (Dirac Live, Audyssey, ARC) succeed without needing to tackle the allpass factor directly. ## Analytic signal and Hilbert envelope The Hilbert transform, sibling of the K-K relation, is the workhorse of audio envelope extraction. Given a real audio signal x(t), its analytic signal is ``` x_a(t) = x(t) + j · ℋ[x](t) ``` The instantaneous amplitude |x_a(t)| is the Hilbert envelope, and arg x_a(t) is the instantaneous phase. Modulation-domain analysis, ring-modulator detection, single-sideband synthesis, and pitch-shifting via complex demodulation all rely on a Hilbert transform (Chapter 28 covers the numerical implementation). ## Warped and Kautz filters for room modelling Room modes cluster unevenly — densely at low frequencies, sparsely at high — so uniform-frequency IIR modelling is inefficient. **Warped filters** precompose the z-transform with an allpass warping map, concentrating design resolution where the modes are. **Kautz filters** generalise this to arbitrary pole sets matched to the modal structure of the room: a Kautz filter with poles placed at measured mode frequencies gives a compact, perceptually-adequate RIR model. Both schemes preserve causality by construction — the warping or Kautz basis is a causal allpass — and K-K enforcement on the derived modal magnitudes reconstructs the corresponding minimum-phase part. The **common-pole method** applies when multiple RIRs (different source or receiver positions) share the same room modes: one set of poles, many sets of residues. ## Summary for the practitioner - If a driver or sub-system is minimum-phase, magnitude EQ fixes phase for free. Measure |H|, K-K-reconstruct phase, invert. - If it is non-minimum-phase, decompose into H_min · H_ap. EQ H_min with IIR. Accept or compensate H_ap with FIR (and latency). - Rooms are non-minimum-phase; EQ the minimum-phase part only, and lean on psychoacoustic tolerance of the allpass residual. - Envelope extraction and SSB synthesis use the Hilbert transform — the same mathematical object as K-K, applied to audio signals rather than response functions. ## See also - [19_minimum_phase_DSP.md](19_minimum_phase_DSP.md) - [26_communications_channels.md](26_communications_channels.md) - [28_discrete_hilbert_transform.md](28_discrete_hilbert_transform.md) ## References - A. V. Oppenheim and R. W. Schafer, *Discrete-Time Signal Processing*, 3rd ed., Pearson, 2010. - H. Kuttruff, *Room Acoustics*, 6th ed., CRC Press, 2016. - J. Blauert and P. Laws, "Group delay distortions in electroacoustical systems", *Journal of the Acoustical Society of America*, vol. 63, pp. 1478–1483, 1978. - M. Karjalainen, P. A. A. Esquef, P. Antsalo, A. Mäkivirta, V. Välimäki, "Frequency-zooming ARMA modeling of resonant and reverberant systems", *JAES*, 2002.