The Wigner distribution of a linear signal space

We introduce a time-frequency representation of linear signal spaces which we call the Wigner distribution (WO) ofa linear signal space. Similar to the WD of a signal, the WD of a linear signal space describes the space’s energy distribution over the time-frequency plane. The WD of a signal space can be defined both in a deterministic and in a stochastic framework, and it can be expressed in a simple way in terms of the space’s projection operator and bases. It is shown to satisfy many interesting properties which are often analogous to corresponding properties of the WD of a signal. The results obtained for some specific signal spaces are found to be intuitively satisfactory. Further topics discussed are the cross-WD of two signal spaces, a discrete-time WD version, and the extension of the WD definition to arbitrary quadratic signal representations.