A tutorial review of several mixed time-frequency representations of signals and systems is presented. Emphasis is placed on the background, extensions, inter-relationships, and applications of the Gabor logon, the Short-time Fourier transform, the Ambiguity function and the Wigner distribution (WD). Their relationship to the instantaneous frequency, group delay, running instantaneous spectrum, Rihaczek's complex spectrum, and the signal energy flow rate as well as a comparison of their properties are also discussed. An efficient analysis algorithm is developed that can be used to approximate the WD of continuous time signals and discrete-time signals whose sampling rate satisfies the Nyquist criterion. Through the use of implicit interpolation techniques, the proposed algorithm increases the effective sampling rate without increasing the number of signal samples and thus not only reduces the effects of aliasing errors, but also reduces computation and memory storage requirements when compared with conventional WD analysis algorithms. The thesis also includes a WD synthesis algorithm that finds a discrete-time signal whose WD best approximates, in a least-squares sense, a given time-frequency function. The algorithm is used to perform a variety of time-varying signal processing operations including signal separation and time-varying filtering. Examples of window and filter design applications are also given.