8.3 Acoustic filters and the room as a transfer function

Every linear element of an acoustic system is a filter — characterised by its impulse response $h(t)$ in the time domain, or by its transfer function $\tilde h(\omega)$ in the frequency domain. The two are related by the Fourier transform; when a signal passes through the filter, the spectra multiply (convolution theorem). This lesson catalogues the acoustic systems that act as filters and what their transfer functions look like.

What acts as a filter

In an acoustic system, everything between the source and the listener that is linear and time-invariant contributes to the overall transfer function:

• The source itself. A loudspeaker has its own frequency response — typically rolling off at the low end (woofers can’t move enough air at long wavelengths) and at the high end (treble drivers get directional and physically small).
• The propagation medium. Atmospheric absorption (Sound chapter 10) is a frequency-dependent loss — high frequencies attenuate more.
• The room. A complete impulse response: direct sound + early reflections + reverberant tail. Each spatial position has its own room transfer function.
• The listener’s body. The head, torso, and pinna act as a frequency- and direction-dependent filter (the head-related transfer function, HRTF) before sound reaches the eardrum.
• The ear canal and middle ear. A short open-closed tube with its own resonances. Roughly a band-pass with a peak around 3 kHz from the ear-canal mode.
• The cochlea. A bank of overlapping band-pass filters, one per place on the basilar membrane.

The end-to-end transfer function from a sound source out in the world to a neural spike in the auditory nerve is the product of all these filter functions in the frequency domain. Each lesson in the Hearing book picks one of them apart.

A few canonical filter shapes

Acoustic filters are usually built from combinations of four canonical types:

• Low-pass: $|H(\omega)|$ near 1 below a cutoff $\omega_c$, falling at higher frequencies. Removes high-frequency content. A blanket draped over a speaker is a passive low-pass.
• High-pass: rising from 0 below $\omega_c$ to 1 above. Removes DC and low-frequency content. The roll-off of small loudspeakers below their resonance is a high-pass.
• Band-pass: peaks around a centre frequency $\omega_0$, falling on both sides. Each cochlear place is a band-pass. So is a Helmholtz resonator at the bottle’s neck mode.
• Notch: flat with a narrow zero at $\omega_0$. Used to suppress mains hum, feedback frequencies, or specific resonant peaks.

Most real acoustic systems combine several of these. A room’s transfer function viewed from one point looks like a band-pass (with the speaker’s roll-off cutting low and the atmospheric absorption cutting high) plus many narrow peaks at the room modes.

The simplest non-trivial filter: a first-order lowpass

Consider the filter

• $|H(0)| = 1$ — no attenuation at DC.
• $|H(\omega_c)| = 1/\sqrt{2}$ — the $-3$ dB cutoff.
• For $\omega \gg \omega_c$, $|H| \propto 1/\omega$ — falling at $-6$ dB per octave.
• $\arg H(\omega) = -\arctan(\omega/\omega_c)$ — phase lags 45° at the cutoff, approaching $-90°$ at high frequencies.

In the time domain, the impulse response is $h(t) = \omega_c\, e^{-\omega_c t}$ for $t > 0$. A short impulse at the input dies away exponentially with time constant $1/\omega_c$. Same filter; two domains.

This filter — equivalent to a passive RC circuit or a single bin of damped diffusion — is the simplest non-trivial LTI filter and the building block for many more complex designs (Butterworth, Chebyshev, Bessel, elliptic). The choice between them is the engineering tradeoff between magnitude sharpness and phase fidelity.

The room as a filter

A room is a very complex LTI filter. Its impulse response — the reverberation pattern discussed in chapter 7 — has three components: direct sound (a sharp peak), early reflections (a series of discrete peaks at times $\tau_i = r_i/c$), and a reverberant tail (an exponentially-decaying noisy signal).

Fourier-transforming gives the room’s transfer function $H_\text{room}(\omega)$ for that source-listener pair. Its features:

• A roughly flat baseline modulated by:
• Sharp peaks at each room mode below the Schroeder frequency (~50–500 Hz for typical rooms).
• A dense, statistically-flat pattern above the Schroeder frequency, with the envelope set by the room’s absorption properties.
• An overall high-frequency roll-off from atmospheric absorption (for very large rooms or open spaces).

The convolution theorem says: the sound you hear is the source signal multiplied by this transfer function in the frequency domain. Different listening positions in the room have different $H_\text{room}$ — the “sweet spot” for a stereo speaker setup is the point where $H_\text{room}$ for the two speakers is most similar.

Convolution reverbs

Modern audio software uses this idea explicitly. Convolution reverb plug-ins record the impulse response of a real space (a concert hall, a cathedral, a stairwell, a guitar amplifier, a spring reverb tank) and store it as $h_\text{space}(t)$. To make dry audio sound as if it were performed in that space, the plug-in convolves the audio with $h_\text{space}$. Equivalently — and how it’s actually implemented in software — it Fourier-transforms both, multiplies the spectra, and inverse-transforms.

You can listen to your dry guitar recording “in Carnegie Hall” because the convolution of your signal with Carnegie Hall’s impulse response is mathematically what your guitar would have sounded like there.

Cascading filters

If a signal passes through filter $H_1$ then filter $H_2$, the overall transfer function is $H_\text{total}(\omega) = H_2(\omega) H_1(\omega)$. Filters in cascade multiply in the frequency domain; filters in parallel add. Combine cascading and parallel composition, and you can build essentially any LTI system from simple building blocks.

This is how mixing consoles work, how synthesizers work, and how the auditory pathway works.

Looking ahead

The next and last lesson of this chapter returns to resonance from chapter 2 and views it in the frequency domain. The quality factor $Q$ that we introduced as a ratio of stored to dissipated energy reappears as the bandwidth of the resonator’s transfer function — the same number, two pictures.