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I have an audio recording of a conversation which is obscured by a movie playing on a television.

I have an audio recording of that same movie playing on that same television with no conversation in the background. The recording device is identical as is its placement in relation to the speakers. The volume is close to identical.

How do I remove the audio from the television so that I can unobscure the conversation?

Obviously ambient noise is different and system/thermal noise in the recording device will be different. The phase of the two audio recordings will be different as there is some time error bounded by $1/F_s$. ($F_s = 44100\textrm{ Hz}$)

I was unsuccessful in attempting to find a time offset between the two signals to perfectly align the audio. I loaded the files in MATLAB and took the sum of the absolute difference between the two samples for multiple offsets and then selected the one with the lowest error.

The reconstructed signal being the difference between the shifted reference audio and the conversation.

 reconstructed_signal = conversation - reference_audio(offset:(length(conversation)-offset+1));

This time domain approach has not worked. I am considering that I should do a difference of the frequency data of the two samples with a sliding window.

Do you think that this would be a good approach?

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From the Description, I think Adaptive filters will work out for this application. As you have the reference signal("I have an audio recording of that same movie playing on that same television with no conversation in the background") and the signal from which you need to remove this reference.

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