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Depending on the context, the use of the complex form could be for mathematical convenience or for a no-kidding need for both real and imaginary parts. When you factor the expression, you get $$u(t) = e^{{\eta}t^2}e^{j{\beta}t^2}$$ Where the first exponential is a generic magnitude envelope, in this case Gaussian. The second exponential is the chirp itself ...
Instead of phase delay $td(f)$, use group delay $\tau(f)$: $$\tau(f) = -\frac{d}{d\omega}\phi(f),\tag{1}$$ calculated as the negative of the derivative of the phase $\phi(f)$ with respect to the angular frequency $\omega$, which is defined by: $$\omega = 2\pi f\quad\Leftrightarrow\quad f = \frac{\omega}{2\pi}.\tag{2}$$ Using your phase shift $\phi(f)$: $$\... 2 The book is not wrong, but it does present the concepts on LFM in a clunky manner and can be misleading. The book presents the analytical expression for the LFM spectrum, which is an approximation. It also plays with the plot views and most likely unwraps the phase angles, which is usually required to see the phases you expect. Usually when you're ... 2 i is the symbol for \sqrt{-1} There is a very important formula called Euler's Equation.$$ e^{i\theta}=\cos(\theta) + i \sin(\theta) = (e^i)^\theta "$e^i$" is a point on the unit circle one radian along the circumference. Any point on the unit circle raised to a power will stay on the unit circle and its distance along the circumference ...