The classic methods of time-frequency analysis, such as spectrogram, wavelet analysis, Wigner distribution and etc can be regarded as derivatives of Fourier analysis, the essence of which is to approximate signals by using time-frequency atoms with linear phases. Unfortunately,  most of real signals are transient signals having time-varying frequency,namely, instantaneous frequency (IF). The time-dependence of IF requires us to represent a signal as superposition of time-frequency atoms with nonlinear phases. Although the introducing of the notion of analytic signal and the algorithm named Emperical Mode Decomposition(EMD)  offer some  breakthrough to this issue, the corresponding mathematical theories are still open problems. Recent study shows that the method of nonlinear Fourier atoms offers a new approach to this issue. Here, nonlinear Fourier atom means the boundary-value on the unit circle of any inner function consisting of Blaschke products and singular functions. This project devotes to a systematic investigation of time-frequency analysis based on nonlinear Fourier atoms, which covers  integral transformations with nonlinear-phase kernels, representation theory of group beyond Heisenberg, construction of bases (frames) with nonlinear phase,adaptive approximation and algorithms. The goal of this project is to found the theory of time-frequency analysis suitable for processing of transient signals and offer fast algorithms for analysis and processing of real data.