Wigner-Ville distribution (WVD) is recognized as being a powerful tool and a nucleus in time-frequency representation (TFR) which gives an excellent time-frequency concentration, and more importantly, has many desirable properties. A major shortcoming of WVD is the inherent cross-term (CT) interference. Although solutions to this problem from the bulk of contributions to the literature concerning TFR are currently available, none has been able to completely eliminate the CT’s in WVD. It is therefore a common belief that if there exists an auxiliary time-frequency distribution (TFD) which has the same auto-terms (AT’s) as that in WVD, but has CT’s with the opposite sign, then, by adding the auxiliary TFD to WVD, an ideal TFD, which preserves the concentration of WVD while annihilating the CT’s, is readily obtained. However, we prove that the auxiliary TFD does not exist. Moreover, it is found that in general, CT free joint distributions with their concentrations close to that of WVD do not exist either. Wigner-Ville distribution (WVD) is recognized as being a power tool and a nucleus in time-frequency representation (TFR) which gives an excellent time-frequency concentration, and more importantly, has many desirable properties. A major shortcoming of WVD is the inherent solutions to this problem from the bulk of contributions to the literature concerning TFR are currently available, none has been able to completely eliminate the CT’s in WVD. It is therefore a common belief that if there exists an auxiliary time-frequency distribution (TFD) which has the same auto-terms (AT’s) as that in WVD, but has CT’s with the opposite sign, then, by adding the auxiliary TFD to WVD, an ideal TFD, which preserves the concentration However, we prove that the auxiliary TFD does not exist. Moreover, it is found that in general, CT free joint distributions with their concentrations close to that of WVD do not exist either.