Covariant (/spl alpha/,/spl beta/), 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/sub /spl alpha/,/spl beta//. Covariant representations are developed in the (/spl alpha/,/spl beta/) 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 (/spl alpha/,/spl beta/), TF, and (a,b) domains are epitomized by the displacement function and the localization function. For conjugate operators, (/spl alpha/,/spl beta/) covariance and (a,b) covariance are equivalent.