Applying the equations given above, the group delay characteristic which achieves this spectral shape can be obtained. It is Now because the principle of stationary phase shows that there is a direct relationship between elapsed time and the instantaneous signal delay then, for the Hamming window, t/T can be related to / by This characteristic which is time as a function of frequency is shown here. Inverting the plot gives the more usual (and more useful) plot of frequency as a function of time, which is also shown. Other spectral shapes can be investigated in the same way and the results, although approximate, are surprisingly accurate, especially when the time bandwidth product of the pulse is high. The stationary phase method does not predict or deal with Fresnell ripples, so it is unable to offer any means by which these ripples can be minimized. As an example, the figure below shows a chirp spectrum with T.F =250 obtained for a non-linear chirp aiming to match the Hamming window, using the methods described above. The figure shows that the spectral profile matches the Hamming characteristic quite well, but Fresnell ripples, not predicted by the method, are very much in evidence. Via numerical methods Sampling Whenever a Fourier integral cannot be evaluated by analytical means, an approximate solution is usually possible by numerical analysis. Such a procedure requires the function to be sampled, usually at equi-spaced intervals in time. One consequence of sampling is that the resultant spectrum is periodic in the frequency domain. In addition to the (desired) baseband spectrum, additional versions of the spectrum occur, centered on multiples of the sampling frequency. To ensure that there is no overlapping of frequency data (i.e. no aliasing) the Nyquist sampling theorem must be satisfied. In practice, a sampling rate substantially higher than that dictated by the sampling theorem is advisable Spectrum of a sampled Signal - the Fourier transform of a discrete time signal A straightforward way to approximate an integral, such as a Fourier integral, is to use the standard 'rectangle rule' for numerical integration. The method assumes the signal value taken at a sample instant remains constant for one sampling interval, until the next sample is taken. This procedure is sometimes referred to as a 'box-car generator', or a zero order sample and hold. If the time interval between samples is W, then sn = s(nW), and the desired integral