Page 270 - Fundamentals of Radar Signal Processing
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4.6.2   The Principle of Stationary Phase
               The Fourier transform of Eq. (4.84) is a relatively complicated result involving
               the  sine  integral  Si(F)  (Rihaczek,  1996).  A  very  useful  and  much  simpler
               approximation can be derived using the principle of stationary phase (PSP), an
               advanced  technique  in  Fourier  analysis.  The  PSP  is  useful  for  approximate
               evaluation  of  integrals  with  highly  oscillatory  integrands;  thus,  it  applies

               particularly well to Fourier transforms. Write x(t) in amplitude and phase form,
               x(t) = A(t) exp[jθ(t)], and consider its Fourier transform






                                                                                                       (4.86)

               Define the phase ϕ(t, Ω) of the Fourier integral as the combination of the signal
               phase and the Fourier kernel phase









                                                                                                       (4.87)


               Of  course,  the  exact  Fourier  transform  is  known  for  many  signals  having
               relatively simple phase functions τ(t). The PSP is most useful when the signal
               phase  function  and  thus  the  total  integral  phase ϕ(t,  Ω)  is  continuous  but
               nonlinear or otherwise complicated.
                     Define a stationary point of the integrand as a value of t = t  such that the
                                                                                              0
               first  time  derivative  of  the  integral  phase ϕ′(t ,  Ω)  =  0.  Then  the  PSP
                                                                            0
               approximation to the spectrum is (Born and Wolf, 1959; Papoulis and Pillai,
               2002; Raney, 1992)







                                                                                                       (4.88)

               where ϕ′(t , Ω) is the second time derivative of ϕ(t, Ω) evaluated at t = t . If
                           0
                                                                                                         0
               there are multiple stationary points the spectrum is the sum of such terms for
               each stationary point. Equation (4.88) states that the magnitude of the spectrum
               at a given frequency Ω is proportional to the amplitude of the signal envelope at
               the  time  that  the  stationary  point  occurs  and,  more  importantly,  is  inversely
               proportional to the square root of the rate of change of the frequency ϕ′(t , Ω) at
                                                                                                     0
               that  time.  The  PSP  also  implies  that  only  the  stationary  points  significantly
               influence X(Ω).
                     The PSP can be applied to estimate the spectrum of the LFM waveform.
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