Sparsity based formulations for dereverberation

Acoustic signals recorded in concerts, meetings or conferences are effected by the room impulse response and noise. Estimating the clean source signals from the observations is referred as the dereverberation problem. If the room impulse responses are known, the problem is non-blind dereverberation problem. In this thesis non-blind dereverberation problem is posed using convex penalty functions, with a convex minimization procedure. The convex minimization problems are solved using iterative methods. Through the thesis sparse nature of the time frequency spectrum is referred. In order to transform the time domain signal to a time frequency spectrum Short Time Fourier Transform is used. In the thesis, to begin with, the general problem is defined in time domain. The basics of the dereverberation is proposed. Basics of the convex minimization procedure is explained. Douglas Rachford Algorithm which is used to solve complicated convex minimization problems is explained. The chapter 3 proposes a derevereberation formulation based on sparsity. The dereverberation problem, with known room impulse response, is conventionally posed as a sparsity based minimization problem, by masking Short Time Fourier Transform coefficients. The sparsity constraint can be posed using an l`1 norm type penalty function. However, in such formulations, especially if the room impulse response is longer than the windowing function of the Short Time Fourier Transform the reverberation effects can not be directly represented in the transform domain. Therefore, the minimization iterations require transform and its inverse in order to mask Short Time Fourier Transform coefficients after time domain deconvolution. Changing domains is more time consuming compared to masking STFT coefficients, in turn increases computational time dramatically. In order to get rid of the transformation requirement, the room impulse response is represented in Short Time Fourier Transform frequency bands. With the approximation filters room impulse response is denoted as a convolutive operator in each frequency band. In this chapter an algorithm proposed, that does not require Short Time Fourier Transform and its inverse, using the proposed approximates of the room impulse response. Also the room impulse response approximation and the dereverberation with the sparsity constraint are justified with experiments. Experiments show that, sparsity based solution yields musical noise. Inthechapter4,musical noise is suppressed using phase information of the coefficients in a frequency band. It can be observed that in a frequency band, time consecutive coefficients are active through a harmonic. These coefficients tend to have close magnitudes. In addition, phase shift between coefficients in harmonics can be considered as constant and phase information is unimportant outside the harmonics. It can be considered that for each harmonic, there lies a complex number that maps time consecutive coefficients together. Outside the harmonics the matching constraints tend to be 0. Therefore, a piecewise constant mask can be found that maps a frequency band to its own phase shifted version. This mask, in fact, satisfies the sparsity property, as it is mapping a sparse frequency band to another. In this chapter, a method for estimating the mask is proposed. The mask is applied on least squares estimate of the signal. The least squares estimate can be performed with known impulse response and noise properties. The dereverberation performance is justified using experiments. Through the experiments different audio signals with different input signal to ratio values are taken into consideration. Also, different weight compositions are taken into consideration. In the chapter 5, different from the chapters 3 and 4, multiple microphone case is taken into consideration. Multichannel case is often solved making use of microphone array geometry or using multichannel penalty functions. Another important struggle in dereverberation is estimating RIRs. Instead of measuring explicit RIRs for the observations, a common filter can be obtained using preliminary observations. This information can be exploited in multichannel estimation. In this chapter a minimization procedure with a multichannel penalty function is proposed. In multiple microphone case, the observations share a common information. A time frequency coefficient is expected to be active in all the observations, if the microphones are close. In order to make use of that information the estimation can be modified into multichannel estimation. Instead of estimating the source signal from observations, it is assumed that the observations are formed from different sources. The sources are defined as shortly reverberated versions of the source signal. In order to obtain these observations, relatively short room impulse response definition is required. With the definition the room impulse response can be divided into two: the common part, which is the same for all observations and the independent part, which differs with position. Thus, a formulation for mixed norm is proposed using relatively short impulse responses. However, this algorithm can be generalized. In order to relax the condition on time frequency coefficients, it can be assumed that time shift between harmonics between observations is relatively small. Thus the harmonic structure is investigated using blocked mixed norm regularization. Both algorithms for mixed norm and blocked mixed norm regularization are justified and compared using experiments on speech signals.