Optimal conditions for signal reconstruction based on STFT Magnitude spectrum Raja Abdelmalek Signal, Image and Technology of Information Laboratory Ecole Nationale d’Ingenieurs de Tunis University Tunis El-Manar, Tunisia rajaabdelmalek32@gmail.com Zied Mnasri Signal, Image and Technology of Information Laboratory Ecole Nationale d’Ingenieurs de Tunis University Tunis El-Manar, Tunisia zied.mnasri@enit.rnu.tn Faouzi Benzarti Signal, Image and Technology of Information Laboratory Ecole Nationale d’Ingenieurs de Tunis University Tunis El-Manar, Tunisia benzartif@yahoo.fr Abstract— Signal reconstruction based on magnitude spectrum of short term Fourier Transform (STFT) only has been a challenging topic since many years. Whereas theory proves that’s completely possible to reconstruct a signal from its magnitude spectrum and minimal phase data, practical implementations are still not able to do that quite accurately. Therefore, this study aims to set the optimal initial conditions required to reconstruct a signal from the STFT magnitude spectrum with an arbitrary or zero initial phase. Results prove that an optimal selection of the frame length, the shift rate and the iterations number allows enhancing the quality of the reconstructed signals. Keywords—Signal reconstruction, Short time Fourier transform (STFT), magnitude spectrum, and minimal phase. I. INTRODUCTION Audio signal (speech and music) reconstruction algorithms in the Fourier domain are usually performed with the magnitude spectrum, whatever the phase spectrum is available or not. It is commonly assumed that the phase spectrum is perceptually less important [2]. Recently in [1], the relative importance of the Short Time Fourier Transform (STFT) magnitude and phase spectrum to reconstruct a good quality signal regarding intelligibility and naturalness has been investigated. Related to this, the magnitude spectrum and its time sequences are used in many signal processing applications, where the phase information is very poor or unavailable. The magnitude spectrum of a discrete-time signal x(n) is given from the STFT defined as ,  = ∑ .  − . −   (1) Where  is the index of the frame,  is the analysis step size,  is the Fourier angular frequency,  is the temporal-domain signal and  is the analysis window. It’s proven in [8] that the time domain signal  can be reconstructed from the magnitude spectrum |, | or from a modified version  ′ , . Furthermore, many research works show that the estimation of the STFT phase spectrum from the STFT magnitude spectrum was investigated to contribute to significant intelligibility under certain conditions, such as the choice of the window’s type and length, as discussed in details in [6], and a set of specific constraints for STFT spectrograms and window function, as developed in [10]. Whereas Signal reconstruction from magnitude spectrum only provides an acceptable intelligibility, still the problem is how to choose the optimal conditions that provide the best possible quality for the reconstructed signal, which is the issue discussed in this article. This problem has been studied since 1980, when the theorems of Hayes and Van Hove were unveiled. These two theorems have been the background of many signal reconstruction methods during the last three decades. These methods can be classified into two main categories: iterative method and non- iterative ones. The first iterative method was implemented in Griffin and Lim (G&L) algorithm, developed in 1984 [4]. To be appropriate for real time application, Beauregard & al. tried to ameliorate the performance of the G&L algorithm in order to minimize the iterations number. Therefore, the real time spectrogram inversion algorithm (RTISI) was proposed in 2005 [9] and then the RTISI Look-Ahead (RTISI-LA) algorithm in 2006 [9]. As far as the non-iterative methods are concerned, the recent single pass spectrogram inversion algorithm (SPSI) was published in 2015 by Beauregard & al. [11]. The SPSI algorithm provides an excellent phase estimation that improves the performance of the iterative spectrogram inversion techniques.