Overview :

For Synthetic signals, original signal is a harmonic serial, which fundamental frequency is 400Hz, has 40 partials with exponentially decreasing amplitudes, thus having energy up to 16KHz. The signal is also temporally windowed by a Gaussian window.

The learning process is conducted on a set of similar signals, having only their fundamental frequencies varying.

The results are the following:
For one note:

synthetic_original

synthetic_damaged_fc6k_destructif

synthetic_restored_fc6k_destructif

For a mix of two notes:

synthetic_2notes_original

synthetic_2notes_damaged_fc5k_destructif

synthetic_2notes_recovered_fc5k_destructif

It sounds like a buzz. Damaged signal is the same except for which we intentionally destroyed frequencies above 8KHz, and sounds like the original being heavily filtered. This damaging is destructive, which means all partials above 8KHz are really destroyed, not just lowered.

The enhancement method manages to recreate missing parts of the spectrum, by replacing atoms from the processed dictionary by corresponding full band versions of the same atoms. It can easily be heard that the reconstituted signal is very close to the original one, and that a great part of missing frequencies have been reconstructed.

For real trumpet signals, a dictionary of trumpet spectra is learned using the RWC Database. Then we chose a short segment of trumpet notes, apply a damaging filter on it, then reconstructed it with the described method.