will give acceptable results, but if both a(t) and |S()| are defined by rectangular functions, then the product T.F needs to be much greater, typically over 100. Examples Typically, in the radar case, a(t) is a constant over the duration of the signal and, for convenience, is assumed here to be unity. So the phase and amplitude characteristics, in the frequency domain, are related by There are two solutions for (), which are complex conjugates of each other. The two filters with these characteristics can be used as the transmitter and receiver filters of a radar system and are interchangeable. The group delay characteristic D(), (where D()=-d/d), is so So in the case of a rectangular time envelope, the dispersive delay characteristic is given by the integral of the square of the envelope. If the positive sign is taken, then the group delay increases with increasing frequency and vice versa. The result is only approximate, but is more accurate for large values of the time bandwidth product. Consider, as an example, the case of a spectrum that is uniform over the range -/2 to /2, then so Put D(-/2) = 0 and D(/2) = T, where T is the pulse duration, then K = T/2 and A = (2πT)/ωmax so, finally As expected, a flat topped frequency spectrum corresponds to a linear frequency sweep. The linear chirp is just one special case which, in any case, can be calculated more precisely by the methods of the earlier section. The particular usefulness of the stationary phase method lies in its ability to provide results when the frequency sweep is non linear. In such cases the spectral response can be shaped to meet some desired design criteria, for example, low side-lobes when a chirp is compressed. One such family of spectral functions that has been studied is given by It is possible to find the group delay characteristics of these functions in a similar manner to that carried out above and the results for n = 1 to 4 have been calculated. Although these cosine functions are amenable to mathematical manipulation, they are rarely chosen to define the spectral characteristics of a chirp, in practice, because when compressed they give broad main pulses with high side-lobe levels. A better characteristic (among many) is the Hamming function, given by A plot of this characteristic is shown, plotted over the range -/2 to /2.