We define the complex signal \(s(t)\) as $$ s(t) = A(t) e^{i \left(2 \pi f t + \phi(t)\right)} \label{s}$$ where \(i = \sqrt{-1}\), \(A(t)\) we call the amplitude as a function of time, \(t\), the constant \(f\) we call the carrier frequency, and \(\phi(t)\) we call the applied phase shift as a function of \(t\); and \(A(t)\), \(f\), \(t\), and \(\phi(t)\) are all real numbers. We get \(x(t)\), the "real" signal, by taking the real part of \(s(t)\), \(x(t) = \Re\left[s(t)\right]\). The choose of the minus sign in the exponent is somewhat arbitrary we could have picked \(s(t) = A(t) e^{- i \left(2 \pi f t + \phi(t)\right)}\), but it turns out that convention dictates we use equation \ref{s}. We can complete this complex signal model by saying $$ s(t) = x(t) + i y(t) \label{s_xy}$$ where $$ y(t) = A(t) \sin(2 \pi f t + \phi(t)) \label{y} . $$ Note, we are just defining \(y(t)\) for convenience.
Now lets remove the carrier wave from this complex signal, \ref{s_xy}, like we did before in chapter IQ_variables with just the real signal. We define the change of variables \(s(t) \to S(t)\) with $$s(t) = S(t) e^{i 2 \pi f t} \label{S_s}$$ or conversely \(S(t) \to s(t)\), by multiplying \ref{S_s} by \(e^{ - i 2 \pi f t}\) we can get $$S(t) = s(t) e^{ - i 2 \pi f t}$$ and that with \ref{s} gives $$S(t) = A(t) e^{i \phi(t)} \label{S} . $$ Equation \ref{S} shows that complex signal, \(S(t)\), is independent of the carrier signal frequency, \(f\). And so, we can think of \(S(t)\) as the complex signal with the carrier removed.