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I want to take the Fourier transform of the following transient signal, $$f(t) = e^{-t/\tau} \cos((\omega_0 + m t)t)$$, where $m$ is some gradient parameter in units of $\rm{Hz}/s$. I thought this would be quite straight forward -- although most of my approaches have been made using Mathematica -- which struggles to provide anything useful.

I would have assumed the resultant function would have a Lorentzian-like peak profilein a similar way to if one takes the trivial Fourier transform of $$f(t) = e^{-t/\tau} \cos(\omega_0t)$$. Does anyone have any ideas on how I can approach this, or, an alternative to Fourier transforming a damped sinusoidal with a linear (or even non linear) frequency, i.e. a chirp.

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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:

  1. Evaluate the Fourier Transform of the magnitude portion and then convolve with the Fourier Transform of the LFM chirp pulse.
  2. 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.

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