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In old recordings or highly compressed audio files (lossy), the higher frequencies are lost, and we hear a distinctive muffling effect. (Just like if the song was played through a wall)

Is it possible to compute the missing harmonics using any method (fourier, bilinear, etc.) and add those back to the sound/song? What is currently feasible?

*It is true that those frequencies are lost, and I am not proposing to boost the higher frequencies (as it would raise the noise floor), but I thought there would be some way to approximate the missing frequencies, since we know what all the instruments should sound like (we know their corresponding harmonics)...*

I'm well aware of the fact "garbage in, garbage out", but even so, our brain itself have very advanced processing techniques. (3D visualization using only 2D sources in vision and audition) Wound't it be possible to fool the brain in "hearing" a better quality sound, and give old recordings back their original sharpness?

Edit: To be more clear, I'm proposing to make audio "sound" better, not to recover lost information.

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    $\begingroup$ There is some literature with keyword "artificial bandwidth extension", mainly for telephony. $\endgroup$ – Olli Niemitalo May 3 '16 at 20:08
  • $\begingroup$ Very interesting... one paper describes the use of Markov Chains in order to estimate the upper band (2kHz+). However its utility in already "wideband" scenarios is to be confirmed... (worst sample rates of music files is around 20kHz) - ARTIFICIAL BANDWIDTH EXTENSION OF SPEECH SIGNALS USING MMSE ESTIMATION BASED ON A HIDDEN MARKOV MODEL $\endgroup$ – Bloc97 May 3 '16 at 22:10
  • $\begingroup$ the answer requires a book... $\endgroup$ – Fat32 May 5 '16 at 23:26
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Bandwidth expansion is actually a problem of missing audio data recovery, which is an active area of research.

Usually you assume that your signal follows some statistical model in the time-frequency domain, and the whole challenge is to infer the parameters of your model, based only on the available data.

In the case of bandwidth expansion, you would infer your model from the lower frequency part of your spectrogram, in order to recover the higher frequencies.

You can have a look at some of Paris Smaragdis' work on missing audio data and bandwidth expansion, using latent variable models. Some demos are available here.

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