\hat{x}(t) = x(t) + \tilde{x}(t)

= \frac{1}{2} A(t) \exp\left( i \phi(t) \right) \left[ 1 + \frac{\tilde{A}(t)}{A(t)} \exp\left( i \left( \tilde{\phi}(t) - \phi(t) \right) \right) \right]

= \frac{1}{2} A(t) \exp\left( i \phi(t) \right) + \frac{1}{2} \tilde{A}(t) \exp\left( i \tilde{\phi}(t) \right)

= \frac{1}{2} A(t) \exp\left( i \phi(t) \right) \left[ 1 + \alpha \exp\left( i \Delta \phi \right) \right]

x(t) = \frac{1}{2} A(t) \exp\left( i \phi(t) \right)