The estimator doesn’t try to combine signals optimistically. Every stage has a gate that can kill the measurement. The system is built to be skeptical of its own results-that’s what makes the surviving claims defensible.
Coherent Acoustic FMCW Range-Response Sharpening on a Stock iPhone via an Aliased Second-Harmonic Band
By using both the original signal and this hidden extra copy, the phone gets more detail than it normally should. More detail means finer separation between nearby objects, so the system can resolve structure more clearly without changing the hardware.
When an iPhone plays a high-pitched sound sweep, the speaker also creates a faint second copy at an even higher pitch. Part of that extra copy folds back into a signal the phone can still record, and it carries the same distance information as the original. By detecting and combining both signals, we effectively give the phone more usable sound bandwidth, which makes its distance measurements sharper and improves resolution.
The estimator doesn’t try to combine signals optimistically. Every stage has a gate that can kill the measurement. The system is built to be skeptical of its own results-that’s what makes the surviving claims defensible.
Compares chirp-on interval SNR against guard-interval noise floor. Then checks split-half reproducibility: if odd and even chirp subsets disagree, the alias isn’t real.
Generalized likelihood ratio test confirming that the fundamental and alias share a common time-of-flight. If the delays diverge, something is wrong-abort.
Coherent combining via β-weighted generalized cross-correlation over the full 8–20 kHz aperture. Validated with held-out odd/even interleaved split.
The held-out validation is key. Chirp cycles are interleaved odd/even, so the training and test sets see identical acoustic conditions. If the combined estimate doesn’t sharpen on both halves independently, the result is rejected. 200 chirp cycles per measurement (50 ms chirps + 25 ms guard intervals).
Early on, the matched filter for the alias band produced nothing. Zero correlation. The expected down-chirp template-a textbook ideal generated from continuous-time math-was completely wrong.
The problem: the alias isn’t a clean analytical down-chirp. It’s the result of a discrete-time squaring operation followed by bandpass filtering and ADC folding. The phase structure of the actual alias has subtle but critical differences from the continuous-time idealization.
| Template | Peak ρ | Result |
|---|---|---|
| Ideal analytic down-chirp (continuous-time) | 0.001 | × |
| Time-reversed fundamental chirp | 0.003 | × |
| Windowed analytic ± phase offset grid search | 0.012 | × |
| Discrete x²[n] → bandpass filter | 0.74 | ✓ |
The fix was conceptually simple but easy to miss. Generate the alias reference template by squaring the actual transmitted chirp samples in discrete time, then bandpass-filtering to the alias band:
Where x[n] is the transmitted chirp sampled at 48 kHz. Squaring in discrete time produces a template that matches the real alias because it is the real alias-just without the propagation delay or noise. This was the single biggest unlock.
All results are on stock iPhone hardware. Without any external jailbreak or use of any external sensors. Stock 48 kHz audio path.
| Confirmed held-out narrowing | >2× |
| Best single-reflector cases | 2.1–2.8× |
| Theoretical ceiling | 3× |
| External hardware required | None |
| Chirp cycles per measurement | 200 |
Drive a small speaker hard enough and it clips. That clipping is a memoryless nonlinearity - the output is roughly proportional to the square of the input. The trig identity is all you need:
A 16–20 kHz chirp, squared, produces a component at 32–40 kHz-the second harmonic. The iPhone mic’s anti-alias filter is designed to reject everything above 24 kHz before the ADC samples at 48 kHz. But it doesn’t fully kill the harmonic. Enough leaks through.
The 48 kHz ADC then folds anything above Nyquist back into the baseband. A signal at frequency f aliases to 48 kHz − f. So 32–40 kHz becomes 8–16 kHz-a down-chirp that mirrors the original sweep direction.
This aliased chirp carries the same time-of-flight information as the fundamental. If we can validate it, align it, and combine the two coherently, we triple our effective bandwidth.
In FMCW sensing, the width of the matched-filter mainlobe is inversely proportional to bandwidth. Double the bandwidth, halve the mainlobe. The arithmetic is simple: