Griffin-Lim Algorithm

The Griffin-Lim Algorithm: A Method for Spectrogram Phase Reconstruction

If you have ever listened to digital music or spoken with someone on a video call, you have benefited from the Fourier transform, a mathematical technique that helps convert time domain signals into frequency domain signals. One specific application of the Fourier transform is the short-time Fourier transform (STFT), which allows us to analyze signals over time by breaking them into small, overlapping segments.

While the STFT provides valuable information about a signal's frequency content, it does not contain information about its phase. In the context of audio, this means that we can have an accurate spectrogram (a visual representation of a signal's frequency content) without being able to reproduce the original sound.

What is the Griffin-Lim Algorithm?

The Griffin-Lim Algorithm (GLA) is a phase reconstruction method that promotes consistency within spectrograms by iteratively projecting them onto two sets: the set of consistent spectrograms and the set of spectrograms with the same amplitude as the original signal. By doing so, the algorithm aims to recover a complex-valued spectrogram that is consistent and maintains the given amplitude.

The GLA is composed of two projections applied in alternating fashion, as shown in the following equation:

Here, $\mathbf{X}$ is the complex-valued spectrogram that is updated at each iteration, $P_{\mathcal{S}}$ is a metric projection onto a set $\mathcal{S}$, and $m$ is the iteration index. The set $\mathcal{C}$ is the set of consistent spectrograms, while the set $\mathcal{A}$ is the set of spectrograms with the same amplitude as the original signal.

The metric projection onto the set of consistent spectrograms is given by:

where $\mathcal{G}$ represents the STFT, and $\mathcal{G}^{\dagger}$ is the pseudo-inverse of the STFT (iSTFT). The metric projection onto the set of spectrograms with the same amplitude as the original signal is given by:

Here, $\odot$ and $\oslash$ represent element-wise multiplication and division, respectively, and division by zero is replaced by zero.

Optimization Problem

The GLA is obtained as an algorithm that solves the following optimization problem:

where $ || \cdot ||_{Fro}$ represents the Frobenius norm. This equation minimizes the energy of the inconsistent components under the constraint that the amplitude of the reconstructed spectrogram must be equal to the amplitude of the original signal.

While the GLA has been widely utilized due to its simplicity, it often involves many iterations before converging to a certain spectrogram, resulting in low reconstruction quality. This is because the cost function only requires consistency and does not take any prior knowledge of the target signal into account.

The Griffin-Lim Algorithm is a commonly used phase reconstruction method in audio signal processing. By promoting consistency within spectrograms through iterative projections onto sets of consistent spectrograms and spectrograms with the same amplitude as the original signal, the algorithm aims to recover the phase of the signal's frequency components. While the GLA is simple and effective, it may require many iterations to converge to a certain spectrogram and may result in low reconstruction quality.

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