later section). Non-linear chirps When the Frequency-Time characteristic is non-linear, the Fourier integral is difficult to evaluate. In such cases, it is possible to resort to an approximation method such as the stationary phase approximation, or to use numerical methods. Via stationary phase method Often (as in radar applications) a(t) is a slowly varying function of time and the phase (t) is oscillatory and varies rapidly, over the range of integration. With such waveforms, the stationary phase approximation can be used to investigate the spectrum. The method relies on the fact that the major contributions to the Fourier integral come from the region where the rate of change of phase is minimal, i.e. when Unless (t) is a constant, the point in time ts at which the phase is stationary will vary according to the instantaneous frequency . Expressing the difference between (-).t and (t) as a Taylor series about the time ts, but discarding all but the first three terms (of which the second term is zero, here), the Fourier integral can be written, approximately, as In this equation ts represents a constant time point, so terms depending on ts alone may be taken outside the integral. The expression simplifies to so where is used to indicate the dependence of the frequency variable on t. This is a very useful expression linking, as it does, the spectrum profile to the amplitude and phase characteristics of the chirp. To carry out the inverse process, i.e. to find the time domain function s(t) given frequency domain data, the inverse Fourier transform is derived. where (x) is the phase function of the spectrum. The stationary phase points for this integrand are located at and the corollary relationship, equivalent to that derived for the spectrum, can be obtained by the stationary phase method, and is In effect, stationary phase analysis gives the following (approximate) Fourier pair relationships: and Consequently, approximate expressions for a(t) and (t) can be obtained when the spectrum, including its phase function () is given and, similarly, approximate expressions for |S(| and () can be obtained when the signal characteristics are given. Several examples of the procedure are given in the literature Although the relationships are only approximate, their accuracy improves as the time-bandwidth product increases. In cases where the signal envelope and spectrum modulus are defined by smoothly varying Gaussian function then a T.F product as low as 15