Cancelling out known signal with sampling offset

Question

I would like to cancel out a known audio signal at my receiver.

Specifically at my transmitter I am playing a song, and at my receiver I am recording it. Both transmit and receive sampling rate is 44.1kHz.

At the receiver I basically want to null out the song so that the only thing that is left is the ambient noise in the room.

When I lined up the received and transmitted signal, I noticed that there is sampling offset so I cannot simply subtract out the transmitted signal.

Are there tools in Matlab that are designed for this task? And what are the general principles I can use to solve this?

Answer 1

What you are describing is basically an acoustic echo cancellator, as used in e.g. any car with a «hands free» phone solution. Use the (known) dry song track as a reference, and use a stochastic gradient search algorithm (eg NLMS) to online identify the FIR filter which relate your recorded signal to the known reference. As long as the channel (room acoustics) remains relatively stationary (LTI) and the ambient noise is uncorrelated to the reverberated music, you should be able to adopt standard methods to your application.

Answer 2

So bad news first: you can't just subtract your transmission.

Any room has more than one path that sound waves can take. Hence, there will be multiple different "copies" of your transmit signal, and they will arrive at the receiver with different delays, according to the path lengths. There, they will add up. You know this as echo / reverberation.

This has a lot of interesting effects, such as frequency selectiveness of your audio channel. Long story short: you can't just assume the signal you receive is transmit signal + noise. That's not how the basic physics below this works, at all.

Then: Good news is that there's ways to "reverse" the effect of a room. We call these methods equalizers; you'll need one that perfectly matches the room (including all obstacles in it). There's ways of "learning" about the room impulse response (which fully describes the described multipath environment, should it not change over time and be linear).

Especially, when you have a reference signal you know has been sent: You can use the same song as kind of a matched filter (if you're from a wireless comms background), or pulse compression (same thing, but if you're from a RADAR background) by correlating with it in the receiver. By doing so, you'd get an estimate for the overall system's impulse response, from DAC to speaker to room to microphone to ADC.

Problem is that it's just an estimate, and the less easy your system model is, the more complex this computation would have to look. Essentially, the assumption that a cross-correlation is sufficient to get the channel impulse response is that your system is linear and time-invariant.

While time-invariance might be given for a static room with no movement, linearity is usually simply not fully true for audio problems. Your speaker does not cause twice the amplitude in air pressure when actuated with twice the signal, for example.

One can, these days, to a high degree calibrate all these disturbing effects out. But I think you might have been underestimating the complexity of your problem there.