Griffin-Lim Spectrogram Inversion

A short-time Fourier transform represents a signal with complex values at successive time frames. Each value has a magnitude and phase. Many systems predict or store only magnitude information, which is insufficient for exact reconstruction because phase also affects how frequency components combine in time. Griffin-Lim addresses this missing-phase problem by estimating a phase pattern that is compatible with a target magnitude spectrogram. The algorithm starts with an initial phase estimate, often random or otherwise initialized, and combines it with the target magnitudes to form a complex spectrogram. It then applies an inverse STFT to create a waveform, computes that waveform's STFT, and replaces the resulting magnitudes with the desired target magnitudes while keeping the newly estimated phase. Repeating this projection between waveform-consistent STFTs and the target magnitude constraint can improve consistency. Because magnitude spectrograms may not correspond exactly to any waveform under the chosen STFT settings, the projections can only seek a compatible approximation. Initialization, window size, hop length, window function, centering, and iteration count affect results. More iterations may improve some forms of consistency but do not guarantee perceptually better audio in every case. The estimate does not guarantee recovery of the original phase, and the output can contain roughness or musical noise. Griffin-Lim is valuable as a transparent baseline and diagnostic tool. It can turn a predicted spectrogram into audio without training a separate synthesis model, but quality may be limited for natural speech or music. Neural vocoders learn a mapping from acoustic representations to waveforms and are common in modern neural TTS pipelines, though they require trained models and introduce their own assumptions and artifacts. When comparing reconstructions, keep the STFT definition matched to the one that produced the magnitudes. Listen to output and inspect task-relevant quality, since a spectrogram or scalar metric cannot capture every audible defect. For speech systems, evaluate intelligibility and naturalness separately from numerical spectral consistency.