Page 270 - Fundamentals of Radar Signal Processing
P. 270
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.