In speech synthesis, the Griffin-Lim algorithm is employed to estimate the phase information of a speech signal when only the magnitude spectrogram is available. The magnitude spectrogram represents the magnitude or intensity of different frequency components in the speech signal but lacks phase information, which is crucial for naturalness and intelligibility. The Griffin-Lim algorithm starts with an initial estimate of the complex-valued spectrogram using random phase values. It then performs an iterative process that alternates between estimating the time-domain waveform and updating the complex spectrogram. The steps involved in the Griffin-Lim algorithm are as follows: 1. Initialization: Begin with an initial estimate of the complex spectrogram, which includes the magnitude from the given magnitude spectrogram and random phase values. 2. Reconstruction: Convert the complex spectrogram back into the time domain using the inverse short-time Fourier transform (ISTFT). This step generates an initial time-domain estimate of the speech signal. 3. New Phase: Apply the short-time Fourier transform (STFT) to the reconstructed time-domain signal to obtain a new complex spectrogram. Extract the phase information from this spectrogram. 4. Phase Update: Update the phase of the complex spectrogram by replacing it with the phase from the magnitude spectrogram obtained in the previous step. This step discards the random phase values used initially. 5. Iteration: Repeat steps 2-4 for a certain number of iterations or until convergence is achieved. By iteratively refining the time-domain waveform and updating the phase information based on the magnitude spectrogram, the Griffin-Lim algorithm attempts to reconstruct the missing phase information. The iterative process helps improve the phase estimation and eventually leads to a synthesized speech signal that sounds more natural and coherent. ![[griffinlim_4_0.png]] While the Griffin-Lim algorithm is a widely used technique for phase reconstruction in speech synthesis, it has some limitations. It can only provide a coarse reconstruction of the phase, assuming that the phase information is uniformly distributed, which may not always hold true. This results in some artifacts and imperfections in the synthesized speech. Neural models trained on pairs of audio and the corresponding spectrogram can provide higher-quality results. One of the first systems to achieve this was the [[WaveNet vocoder]] ```python import numpy as np import librosa def griffinlim(mag_spec, n_iters=50): angles = np.exp(2j * np.pi * np.random.rand(*mag_spec.shape)) for i in range(n_iters) spectrogram = mag_spec.astype(np.complex) * angles inverse = librosa.istft(spectrogram) rebuilt = librosa.stft(inverse) angle = np.exp(1j * np.angle(rebuilt)) spectrogram = mag_spec.astype(np.complex) * angles inverse = librosa.istft(spectrogram) return inverse ``` ## Reference D. Griffin and Jae Lim, "Signal estimation from modified short-time Fourier transform," in _IEEE Transactions on Acoustics, Speech, and Signal Processing_, vol. 32, no. 2, pp. 236-243, April 1984, doi: 10.1109/TASSP.1984.1164317. Speech Processing Book, [5.9. The Griffin-Lim algorithm: Signal estimation from modified short-time Fourier transform](https://speechprocessingbook.aalto.fi/Modelling/griffinlim.html)