ICA Demo

While driving the other day through a mountain pass the radio kept alternating between two stations which broadcast on the same frequency. It would play one station, loud and clear and after a few seconds switch to the other station. To me that’s surprising, as I would have expected to listen to a mash of both stations rather than a clear signal, albeit toggling between two stations. How could the radio tell the two stations apart? The answer, fortunately, is simple thanks to the FM capture effect [FCE]… which has nothing to do with what this post is about.

Expensive receivers have multiple antennas and are capable of discerning between same-frequency transmissions – but how do they use those many antennas to pick out a clear signal? The search lead to Independent Component Analysis [ICA] which can reconstruct original signals from a mix of signals, provided there are multiple receivers who receive different mixes. The underlying intuition is that dispersed receivers observe different mixtures, which are linear combinations (thank you, physics!) of the original signals. The more observers there are, the better the reconstructed signals will be.

[2025.08.20] Let’s assume we have three source signals s1(t), s2(t) and s3(t) which are read by receivers r1, r2 and r3. Each receiver reads a combination of the three signals:

r₁(t) = a₁*s₁(t-φ₁) + b₁*s₂(t-θ₁) + c₁*s₃(t-ω₁)

r₂(t) = a₂*s₁(t-φ₂) + b₂*s₂(t-θ₂) + c₂*s₃(t-ω₂)

r₃(t) = a₃*s₁(t-φ₃) + b₃*s₂(t-θ₃) + c₁*s₃(t-ω₃)

where:

a_n, b_n , c_n are attenuation components between 0 and 1, where lower values mean that the reception for that signal is weak (eg. the source is far away from the receiver or there is an obstacle between them).

φ_n, θ_n , ω_n are phase offsets (in time units) which correspond for the time it takes for a signal to reach each receiver.

ICA is given r₁, r₂, r₃ and reconstructs s₁, s₂, s₃.

I wrote a simple demo [GIT] based on the fantastic ICA Python implementation from scikit-learn [SCI]. The demo consists of two experiments. The first experiment mixes together a few simple, synthetic signals, one mix per receiver, feeds ICA with the mixes and plots the results.

Graph depicting mixed signals (observations), true sources, and ICA estimated sources in three separate charts. — Comparison of mixed signals, true sources, and ICA estimated sources for synthetic signals.

The second experiment uses old radio recordings (cowbells, horse races and a radio interview). The audio clips were mixed together three times. Each time the amplitudes and phase offsets of each clip in the mix were varied which resulted in different compositions for each receiver.

Original sound clips: cowbells, horse races, interview
Mixed signals: receiver1, receiver2, receiver3
Reconstructed signals: cowbells, horse races, interview

Graph showing two sections: 'Observed signals' with overlapping waveforms in green, orange, and blue, and 'ICA estimated sources' with separated waveforms in the same colors. — Graph showing observed signals and the estimated sources using Independent Component Analysis (ICA) for real radio recordings.

The source material and the reconstructed audio can be listened to here: https://github.com/ggeorgovassilis/ICA/tree/main/audio

Parting thoughts

It’s frankly amazing that ICA exists and actually works, and fast. The experiment shows that the more complex synthetic signal (composite sine) was reconstructed nearly perfectly, the rather simple pulse was mutilated badly, but at least the base frequency and phase survived. The radio recordings were reconstructed rather well, too, considering the source material’s quality (still: thank you BBC and Library of Congress for making these recordings available!). However, silent passages in the source signal are mercilessly overwritten with whichever of the other signals screamed loudest at that time.

[2025.08.20] Wikipedia says about ICA:

The success of ICA separation of mixed signals relies on […] assumptions […]

The values in each source signal have non-Gaussian distributions.

That is a fancy way of saying that in order for signals to be recoverable, they should look like something “meaningful”, which is in parts a heuristic criterium and introduces a degree of subjectivity.

Future experiments should cover: