This course introduces advanced time domain (statistical) and transform domain methods in signal processing. This is mainly a course on non-linear signal processing. A small portion of this course is dedicated statistical signal processing techniques like ARMA estimation, Kalman Filter and Hidden Markov Model. The main focus of this course is on inverse problems. A majority of modern signal processing problems can be recast as inverse problems - deblurring, deconvolution, denoising, reconstruction etc. to name a few. The course introduces the students to powerful optimization based non-linear inversion and filtering techniques.