Spatial Cues

Receiver geometry

Top-down scene — drag the talker

The wavefront reaches the near channel first. The gold segment is the extra distance it still has to travel to the far one — divide by 343 m/s and you have the ITD. Shrink the spacing and the gold segment shrinks with it. The red ghost is the talker’s mirror image across the interaural axis: it produces exactly the same arrival-time and level difference, which is why timing and level alone can never tell front from back.

The three cues

ITD is geometric — it scales directly with how far apart the two channels are, and nothing else. ILD needs an obstacle: a body wide compared to the wavelength casts an acoustic shadow, so the head reaches ~20 dB above 4 kHz while a bare 1 cm pair sits at essentially zero for the whole speech band. Spectral coloring is the only one that does not need size at all — it needs shape. Notice that the green and red curves land on top of each other in the first two charts at every angle, and separate only in the third.

Cone of confusion

Every source on a cone around the interaural axis shares one ITD and one ILD — the classic cone of confusion. In the flat 2D slice shown here that cone collapses to a single mirror pair, θ and 180° − θ. Two channels and two cues give you two numbers to solve for one unknown, and the map is still two-to-one, so the ambiguity is structural rather than a matter of precision: no amount of extra SNR fixes it. Only a receiver whose frequency response differs between the two directions breaks the tie. That is what a pinna is for, and it is exactly the job SonicSieve hands to a piece of printed resin.

Listen — two channels, two ears

The source is filtered through the model above — the exact ITD, ILD and spectral coloring for the current geometry — and the two channels are sent to your two ears. With the head selected, Orbit swings the voice clearly from ear to ear, and you can hear the timbre shift as it passes behind you — though with a generic model rather than your own measured ears, it will tend to stay inside your head rather than fully externalize. Switch to the bare 1 cm pair and it collapses into a point in the middle of your skull: 29 µs of ITD and no level difference is simply not enough information for your auditory system to externalize. Add the microstructure and the sound picks up a direction-dependent timbre — not a convincing outside-the-head image, but a difference a neural network can be trained to read.

Why not just spread the microphones out?

Estimated angular resolution

Resolution here is a Fisher-style bound: how fast each cue changes with angle, divided by how much that cue wobbles from room to room. The head lands near 3° straight ahead — the right order for the 1–2° listeners actually achieve. A bare 1 cm pair sits in the teens, and gets even that entirely on phase, since it has no level cue at all; swing the talker toward the baseline axis and it degrades past 40°, because the cue slope goes to zero there.

Now compare two ways of fixing it. Widening the bare pair all the way to 16 cm buys about 4° — but that is a receiver as wide as a head, and it is still front/back ambiguous, because spacing adds no coloring. Adding the microstructure to the 1 cm pair gets to roughly 2–4° at a sixteenth the aperture, and breaks the ambiguity. Then try widening that pair: the number gets worse, not better. Once the cue comes from shape, extra spacing contributes nothing but environmental noise — which is precisely why the design puts the two microphones as close together as it can.

How it works

• Direction is not in the signal — it is in the geometry. A pressure waveform carries no heading. What a receiver actually measures is the source convolved with a transfer function that happens to depend on arrival angle. Localization is the inverse problem: recover θ from the coloring, knowing neither the source spectrum nor the room.
• Rayleigh’s duplex theory splits the band. Below roughly 1.5 kHz the wavelength is longer than the head, so sound diffracts around it — no shadow, no ILD — but the phase difference is unambiguous, so ITD carries the low end. Above that, phase wraps and becomes ambiguous while the head starts casting a real shadow, so ILD takes over. Neither cue covers the whole band on its own.
• A classical array encodes direction in the steering vector. Write the received signal as x(f,t) = a(f,θ)·s(f,t) + u(f,t), where a(f,θ) is the set of geometric delays across the elements. Beamforming inverts that. It works — but resolving spatial ambiguity this way typically wants 4 to 8 microphones spread across an aperture, which a phone does not have.
• SonicSieve rewrites the model. A passive microstructure over one microphone makes the received signal x(f,t) = Mθ(f)·s(f,t) + u(f,t) — direction now lives in a filter, not in a delay across elements. The catch is that a single channel cannot separate Mθ(f) from the source’s own spectrum or the room’s. So a second, bare microphone goes right next to it as a reference, giving the model a before/after view; the IPD and ILD between the two channels then expose the coloring.
• Close spacing is a feature, not a compromise. Because the cue now comes from the structure rather than the aperture, spreading the mics apart buys nothing and costs robustness. Putting the in-line earphone mic within 1 cm of the phone’s top mic keeps both channels in the same acoustic environment, so the difference between them is attributable to the structure instead of to the room. The result is 5.0 dB of separation gain over a 30° region with two microphones, beating a flat five-mic array.

Cue models here are textbook approximations, not measurements: Woodworth’s spherical-head ITD, a ka-dependent shadow for ILD, and a two-notch pinna response. The microstructure filter uses the same multipath model as the SonicSieve module. Environmental-variation figures follow the separation experiment in Yuan et al., CHI ’26.