A Theory on Non-Constant Frequency Decompositions and Applications

Positive time-varying frequency representation of transient signals has been a hearty desire of signal analysts due to its theoretical and practical importance. During approximately the last two decades there has been formulated a signal decomposition and reconstruction method rooting in harmonic and complex analysis and giving rise to the desired signal representation. The method decomposes a signal into a few basic signals that possess positive-instantaneous frequencies. The theory has profound relations with classical mathematics and can be generalized to signals defined in higher dimensions with vector or matrix values. Such representations, in particular, promote rational approximations in higher dimensions. This article mainly serves as a survey. It also gives a new proof of a general convergence result, as well as a proof of a result concerning multiple selections of the parameters. Expositorily, for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f. Such function F has the form F = f + iHf, where H stands for the Hilbert transformation of the context. We develop fast converging expansions of F in orthogonal terms of the form F = ∑ k = 1 ∞ c k B k , $$\displaystyle \begin{array}{@{}rcl@{}}{} F=\sum _{k=1}^\infty c_k B_k,\end{array} $$ where B k’s are also Hardy space functions but with the additional properties B k ( t ) = ρ k ( t ) e i θ k ( t ) , ρ k ≥ 0 , θ k ( t ) ≥ 0 , a . e . $$\displaystyle \begin{array}{@{}rcl@{}}{} B_k(t)=\rho _k(t)e^{i\theta _k(t)},\quad \rho _k\ge 0, \quad \theta _k'(t)\ge 0, \quad \mathrm {a.e.}\end{array} $$ The original real-valued function f is accordingly expanded f = ∑ k = 1 ∞ ρ k ( t ) cos θ k ( t ) $$\displaystyle f=\sum _{k=1}^\infty \rho _k(t) \cos \theta _k(t) $$ which, besides the properties of ρ k and θ k given above, also satisfies the relation H ( ρ k cos θ k ) ( t ) = ρ k ( t ) sin θ k ( t ) . $$\displaystyle H( \rho _k\cos \theta _k)(t)= \rho _k(t) \sin \theta _k(t). $$ Real-valued functions f ( t ) = ρ ( t ) cos θ ( t ) $$f(t)=\rho (t)\cos \theta (t)$$ that satisfy the condition ρ ≥ 0 , θ ′ ( t ) ≥ 0 , H ( ρ cos θ ) ( t ) = ρ ( t ) sin θ ( t ) $$\displaystyle \rho \ge 0, \quad \theta '(t)\ge 0, \quad H(\rho \cos \theta )(t)= \rho (t) \sin \theta (t) $$ are called mono-components. Phase derivative in the above definition will be interpreted in a wider sense. If f is a mono-component, then the phase derivative θ′(t) is defined to be instantaneous frequency of f. The above defined positive-instantaneous frequency expansion is a generalization of the Fourier series expansion. Mono-components are crucial to understand the concept of instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into their principal or intrinsic mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the study of the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We provide an account of the related studies in pure and applied mathematics, and in signal analysis, as well as applications of the developed theory.