njit-etd2003-081 - New Jersey Institute of Technology

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distributions [24]. Time-frequency distributions in the hyperbolic class are covariant to
scales and hyperbolic time shifts. The hyperbolic class intersects with the affine class
but not with Cohen's class. Baraniuk [30] has recently presented a fundamental result
concerning the existence of quadratic time-frequency distributions covariant to arbitrary
operators. One troublesome aspect of quadratic distributions is that they always contain
"cross terms". The cross terms are undesirable for two reasons. First, a true distribution
function should be non-negative and cross terms can have negative values. Second, the
cross terms do not usually represent any energy present in the signal.
It is often desired that a time-frequency distribution be positive valued, and
Cohen has created a general class of time-frequency distributions that are always
positive valued [21]. Since this class is very general, it intersects with every class
(quadratic and non-quadratic) of time-frequency distributions. The main difficulty with
this class is that it is not clear how to construct time-frequency distributions in the class.
Loughlin et. al. [32] and also Sang et. al. [33] have created positive time-frequency
distributions by iterating quadratic time-frequency distributions. There has also been
recent work in constructing distributions of quantities other than time and frequency [28-
32]. The most common distributions outside of time and frequency are those of time
and scale. The concept of scale is closely related to the concept of frequency and some
time-frequency distributions have analogous time-scale distributions. A summary of the
above methods for computing the distribution of a signal is presented in Figure 3.1