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Measures of complexity -- time-bandwidth (TW) product being the leading example -- used by engineers and scientists to characterize signals are often assumed to describe inherent attributes of the signal. Signals of low TW product are viewed as `simple,' and those of high TW, such as a spread spectrum signal, appease our intuition by appearing complicated. Following precedent from classical physics, this sort of complexity can also be quantified in terms of degrees of freedom or occupied dimensions. One thesis of this paper is that the conventional association of complexity with large values of a dimension measure results from the signal representations -- impulses and sine waves -- we instinctively use in time- and frequency-domain thinking. These basis functions are easy to visualize, thus putting the burden of complexity on the expansion coefficients. Conclusions reached under this paradigm do not necessarily hold up under the enlarged scope of mixed time-frequency representations, and thus we make the case that there is no inherent dimensionality to an individual signal, and that dimension properties are entirely representation dependent. Corresponding conclusions concerning sets of signals compose the second thesis of the paper, and these turn out to be somewhat different. Simple examples illustrate that under certain constraints on the basis functions, a nontrivial minimum dimension can be assigned to a signal set based on its maximally compact representation over eligible bases. The central ideas are made precise by postulating some desirable attributes for a signal dimension measure relative to a basis set, and showing that these imply a definition intimately related to both the quantum-mechanical, probabalistic interpretation of expansion coefficients and the entropy function. A logical extension from signal dimension to signal set dimension is then presented. No fully inherent dimension definition is reached. Instead we find that basis sets must exhibit some structure enabling the dimension problem to be posed and solved, and that a well-defined inherent dimension of a signal set exists only with respect to subclasses of bases. Certain highly structured basis function families that provide settings within the signal processing framework with respect to which the signal set dimension measure may prove useful are discussed.
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