Distance Metrics for Discrete Time-Frequency Representations

This paper presents a new method for addressing problems related to the sparsity of the discrete time-frequency representation. Although each real signal x[n] has only N free parameters, its representation has N² elements, along N(N + 1)=2 independent dimensions. This leads to problems in modeling, classification, and recognition tasks. An overview of our discrete time-frequency representations is presented, together with a brief discussion of previous work in continuous time-frequency representations. These discrete time-frequency representations have all of the descriptive power of conventional discrete spectral features, together with a powerful framework that describes how the spectrum evolves over time. Unfortunately, using these features directly in their natural, high-dimensional space tends to be unworkable. A geodesic distance measure is introduced, that leverages our knowledge of the set of valid time-frequency representations to reduce the apparent dimensionality of the problem. Applications of this geodesic distance are found in signal classification and nonlinear time-frequency interpolation.