A new generic adaptive time-frequency transform based on the Wigner distribution is proposed for amplitude estimation of transient signals with any nonlinear non-polynomial variation of the instantaneous frequency. It is for use in situations where independent synchronous measurements of the instantaneous frequency are available. It shows that the new transform tracks the instantaneous frequency and estimates signal amplitude without errors along the curve of the instantaneous frequency. The paper also applies the new transform to signals with the non-linear sinusoidal and exponential variations in the instantaneous phase and determines formulae for transform in these cases. The paper compares the new transform with the Wigner distribution in several cases and demonstrates that the new transform is more effective at amplitude estimation of signals with nonlinear variation of the instantaneous frequency. New analytic formulae are obtained for the new transform and the Wigner distribution in both the sinusoidal and the exponential cases. An analytic formula is obtained which relates the new transform to the Wigner distribution in the sinusoidal case. This formula is inverted to obtain a previously unknown formula for the Wigner distribution of any signal. It is shown that the new transform could be used for adaptive processing of transient signals with instantaneous frequency variations. The paper studies the performance of the new transform with no adaptation, partial adaptation and complete adaptation.