# How do I generate the approximate Fourier Transform of the signal using Principle of Stationary Phase?

The following excerpt is taken from the textbook "Digital Processing of Synthetic Aperture Radar" by Ian G. Cumming

The Principle of Stationary Phase (POSP) can be briefly explained as follows. The phase of the signal is "stationary" at some time, ts, when the derivative of the phase is zero. In this case, the stationary point occurs at $$t = 0$$. Around the neighbourhood of this point, the phase is slowly varying, as are the amplitudes of the real and imaginary parts. At other times, the phase varies quite rapidly.

I wish to approximate the Fourier Transform of the signal using the principle of stationary phase.

$$s(\tau) = \exp(2\pi f_ct_d+\pi Kt_d^2 -2\pi Kt_d\tau)\tag1$$ Based on POSP, the derivative of the phase is

$$\frac{\mathrm d \phi(t)}{\mathrm dt} = -2\pi Kt_d.\tag2$$

However, there is no frequency relationship associated to time

$$f=-Kt_d.\tag3$$

How do I go about finding relationship of frequency to time and then subsequently getting the approximate Fourier transform?

• Do let me know if I need to elaborate or rephrase the question in anyway Aug 2 at 8:03
• Welcome to SE.SP! Please check (1). I suspect that you are missing a $j$ in the argument of the $\exp$ function because, as it is, the function is purely real and has no phase.
– Peter K.
Aug 2 at 12:00