First of all it is necessary to invert the correlogram in order to get a reconstruction of the cochleagram and then from this, by another inversion, we obtain a sound wave. The correlogram is a short time autocorrelation made on all the outputs of the cochleagram. From the autocorrelation of a signal it is possible to extract the spectral power of the same signal, in fact the Fourier transform of its autocorrelation is equal to the square of its Fourier transform magnitude, that is: where is the autocorrelation of x(t). In the same way the magnitude ot the STFT can be calculated from its STA. Therefore, by simple operations, we can obtain the magnitude of the short time Fourier transforms of all the output sequences of the cochleagram. The main problem rely on the fact that we have to reconstruct signals from the magnitude of their STFTs, that is we have no information about their phases. To achieve this operation Slaney et al. suggest to use the iterative algorithm of Griffin and Lim [5]. This algorithm, at each iteration, reconstructs the phase of the signal in order to decrease the square error between the STFT magnitude of the reconstructed signal and the STFT magnitude a priori known. At each iteration the new signal is calcolated using a procedure similar to the overlap-add method. The sequences to overlap and add are obtained with the inverse Fourier Transform of the STFT composed by the known magnitude, and by the phase of the STFT of the reconstruction of the previous iteration :

where w(n) is the analysis window and S is the window shift.

The next step is to obtain a signal coherent with the obtained inverted cochleagram. In order to do that, all the parts of the auditory model (Filter bank, HWR and AGC) have to be inverted in reversed order. The inversion of the AGC stages is relatively simple as the samples of the signal have to be divided for a value that is computable from the output values of the previous samples.

Subsequently the negative parts of the signals, that have been cut off by the HWR stages, have to be reconstructed. Slaney et al. propose the use of the convex projections tecnique [7] to invert the HWR stage. In this case two projections are made: the first in the time domain and the second in the frequency domain. The first projection is made assigning to the signal the known positive part, while the second is made filtering the signal with a bandpass filter. In our implementation the same filters of the auditory model have been used. These operations are made iteratively for each channel, and it has been empirically seen that the error stabilizes after few iterations (5-10).

Finally the filter bank inversion have to be be executed, that is we have to reconstruct a signal from the output of the filters. This has been made with the tecnique of analysis-resynthesis using the same filter bank used for the analysis [4].

The example shown in the figure refers to the signal corresponding to the Italian word /'skOtta/ (it is hot), sampled at 16 kHz. The STAs have been calculated using the FFT executed on a number of samples that was twice the number of the analysis window length. A modified Hamming window has been used, which has the property to reduce the amount of computations in this algorithm. The window length is 256 samples while the shift between them is 64 samples. Sounds resynthesised are of good quality and the perceptual differences from the original signals are almost insignificant.

original (mp3)            resinthesised (mp3)

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