Time-frequency complexity and information

2. TIME-FREQUENCY REPRESENTATIONS Many functions have been proposed for estimating signal information content and complexity on the time-frequency plane, including moment-based measures such as the timebandwidth product and the Shannon and RCnyi entropies. When applied to a time-frequency representation from COhen’s class, the R h y i entropy conforms closely to t.he visually based notion of complexity that we use when inspecting time-frequency images. A detailed discussion reveals many of the desirable properties of the Rknyi information measure for both deterministic and random signals. We will study the complexity and information content of signals indirectly via Cohen’s class of T F b , a set of quadratic distributions that a t tempt to indicate the energy content of a signal s as a function of both time 2 and frequency f . A T F R C,(t, f) from Cohen’s class can be expressed as [I] d u d8 d r , (1) e J z m ( e u e t r f ) where the function a(@, r ) is called the kernel of the TFR. Examples of Cohen’s class T F R s include the W D (@(e, r ) = I ) , the spectrogram (‘P = ambiguity function of the timeThe term component is ubiquitous in the literature on reversed window function), and the exponential distribution joint time-frequency representations (TFRs) [l]. For example, one talks of suppression of Wigner distribution (WD) ( @ ( ‘ I T ) = cross-components, concentration and resolution of autoThe kernel completely determines the properties of its corcomponents, and the property that TFRs separate signal responding TFR. For example, a fixed-kernel T F R possesses components such as parallel chirps that overlap in both time the energy preservation property and frequency. Very often the quality of a particular T F R nents of the signal being analyzed. Intuitively, a component is a concentration of energy in the time-frequency plane, but this notion is difficult to translate into a quantitative concept. and the In fact, the concept of a signal component has never been and may never be clearly defined. In this paper, rather than address the question “what is a component?” directly, we will investigate several quantiprovided @(8,0) = ‘P(0, T ) = 1 V 8, T . (The function S(f) tative measures of signal complexity and information on the denotes the Fourier transform of the signal s(2).) We will time-frequency plane. While they do not yield direct answers assume throughout this paper that the signal energy is norregarding the locations and shapes of components, these meamalized to one, that is, sures are intimately related to the concept of a signal compoThe formulas (2) and (3) evoke an analogy between a nent, the connection being the intuitively reasonable assumpTFR and the probability density function (PDF) of a twotiOn that signals of high complexity (and therefore high infordimensional random variable. l-his has been exmation content) must be constructed from large numbers of ploited with much SUCCeSS in the past [1,2]. fact, as we will information include the time-bandwidth product and other bor’rowed directly from probability theory, H ~ ~ ~ ~ ~ ~ , there tion appropriated from probability theory by Williams et d. because of the freedom of choice of kernel function the T F R [2], and parametric techniques based on decompositions into of a given signal is nonunique, second, most co~en~s TFRs are nonpositive and, therefore, cannot be interpreted strictly as densities of signal energy.2 Nevertheless, concepts from probability theory still have considerable merit in timefrequency analysis, provided caution is exercised in their interpretation.