Covariant (α,β), time-frequency, and (a,b) representations

We present a general theory of bilinear/quadratic signal representations that are covariant to time-frequency (TF) displacement operators D_α,β. Covariant representations are developed in the (α,β) domain and in the TF domain. In the conjugate case, covariant representations can also be formulated in an eigenfunction parameter domain ((a,b) domain). The relations connecting the (α,β), TF, and (a,b) domains are epitomized by the displacement function and the localization function. For conjugate operators, (α,β) covariance and (a,b) covariance are equivalent