Positive-instantaneous frequency and approximation

Abstract

Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance, although the involved concept itself is paradoxical. The desire and practice of uniqueness of such frequency representation (decomposition) raise the related topics in approximation. During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations. The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies. The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values, and in particular, promotes kernel approximation for multi-variate functions. This article mainly serves as a survey. It also gives two important technical proofs of which one for a general convergence result (Theorem 3.4), and the other for necessity of multiple kernel (Lemma 3.7).

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 + iH f, where H stands for the Hilbert transformation of the context. We develop fast converging expansions of F in orthogonal terms of the form

$$F = \sum\limits_{k = 1}^\infty {{c_k}{B_k},} $$

where B_k’s are also Hardy space functions but with the additional properties

$${B_k}(t) = {\rho _k}(t){e^{i{\theta _k}(t)}},\,\,\,\,\,\,{\rho _k} \geqslant 0,\,\,\,\,\,\,\,\theta _k^\prime (t) \geqslant 0,\,\,\,\,\,\,{\rm{a}}.{\rm{e}}.$$

The original real-valued function f is accordingly expanded

$$f = \sum\limits_{k = 1}^\infty {{\rho _k}(t)\cos {\theta _k}(t)} $$

which, besides the properties of ρ_k and θ_k given above, also satisfies

$$H({\rho _k}\cos {\theta _k})(t) = {\rho _k}(t)\sin {\theta _k}(t).$$

Real-valued functions f (t)= ρ(t)cos θ(t) that satisfy the condition

$$\rho \geqslant 0,\,\,\,\,\,\,\,{\theta ^\prime }(t) \geqslant 0,\,\,\,\,\,\,\,\,H(\rho \cos \theta )(t) = \rho (t)\sin \theta (t)$$

are called mono-components. If f is a mono-component, then the phase derivative θ′(t) is defined to be instantaneous frequency of f. The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion. Mono-components are crucial to understand the concept instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the studies on the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We finally provide an account of related studies in pure and applied mathematics.