Fourier transform of a damped cosine wave with a linear frequency chirp
For Fourier Transform of the LFM chirp portion - you use the Principle of Stationary Phase (POSP). The POSP essentially says the main contribution in the Fourier Integral comes from the portion of where the derivative of the phase is zero - it assumes that the integral of the oscillating components cancel themselves out.

Using the POSP - the Fourier transform of a LFM chirp is another LFM chirp in the frequency domain. The assumption is not very good in cases where you have a low time-bandwidth product.

For the complete signal, you have to options:


Evaluate the Fourier Transform of the magnitude portion and then convolve with the Fourier Transform of the LFM chirp pulse.
Keep the magnitude in while you are doing the POSP evaluation. You have to check on the assumptions the POSP make. If I recall correctly - it is usually assumes a slowly varying amplitude window.


I believe you can find the use of the POSP for LFM pulses in the signal analysis books by Papoulis, and also in "Digital Processing of Synthetic Aperture Radar Data" by Cummings and Wong.