The short-time Fourier transform and the Wigner distribution are the time-frequency representations that have received the most attention. The Wigner distribution has a number of desirable properties, but it introduces nonlinearities called cross-terms that make it difficult to interpret when applied to real multi-component signals. The short-time Fourier transform has achieved widespread use in applications, but it often has poor resolution of signal components and can bias the estimate of signal parameters. A need exists for a time-frequency representation without the shortcomings of the current techniques. This dissertation develops a data-adaptive time-frequency representation that overcomes the often poor resolution of the traditional short-time Fourier transform, while avoiding the nonlinearities that make the Wigner distribution and other bilinear representations difficult to interpret and use. The new method uses an adaptive Gaussian basis, with the basis parameters varying at different time-frequency locations to maximize the local signal concentration in time-frequency. Two methods for selecting the Gaussian parameters are presented: a method that maximizes a measure of local signal concentration, and a parameter estimation approach. The new representation provides much better performance than any of the currently known techniques in the analysis of multi-modal dispersive waveforms.