I have a set of songs for which I extracted the STFT (Short-Time Fourier Transform) and used the magnitude spectrum |S| to calculate the mel spectrogram by using a mel filterbank matrix M, so X=log(M×|S|). I want to know is there any method to reverse this process, i.e. convert from the mel spectrogram back to the spectrogram. I performed some dimensionality reduction on the mel spectrogram, and reconstructed the mel spectrogram from lower dimensions. Now I want to regenerate the audio signal from the reconstructed mel spectrogram, so I guess first reconstruct the spectrogram and then the audio signal.
The problem is that the mel filter bank matrix is not a square matrix, since we the reduce the no of frequency bins, so inverse of M cant be used like this : ˆS=M−1exp(X). So is there any way to generate the inverse mapping, like some inverse transfer function that can convert from X to S?
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