the frequency characteristic of the chirp, as shown. The method is effective because amplitude and phase distortions having functional similarity can produce similar effects when the distortion factors are small. Suggested values for these distortion regions, to give good results are: Later work proposed slightly different values, namely: but the outcome can doubtless be improved by optimizing values for each particular situation. Two plots show the effects of frequency pre-correction and can be compared to the results in the earlier sections. The ripple reduction achieved by frequency pre-correction, although significant, is seen to be less successful than that achieved by the amplitude modulation methods of the previous section. However, it has been suggested that by implementing cubic (rather than quadratic) phase pre-correction, comparable results can be achieved. Deriving a Waveform from a Target Frequency Spectrum This method uses an inverse Fourier transform in order to derive a waveform which has a spectrum with the phase characteristic of a chosen chirp but a new amplitude profile which is rectangular and ripple free. The method is very effective but, unfortunately, the waveform that is so derived has a semi-infinite time duration. If, for convenience, the newly derived waveform is truncated to a practical length, then some ripple is reintroduced onto the spectrum. As an example, a linear chirp waveform with a time bandwidth of 25 is shown together with its spectrum magnitude (shown by a full line) which, as demonstrated earlier, has a large ripple component. It is possible to find, by means of an inverse FFT, a chirp waveform which, in the frequency domain, has the same phase characteristic as before, but with the rectangular magnitude characteristic shown by the dashed line on the plot. The chirp waveform resulting from this process has a very long time duration, but when it is truncated to say, a length 2T, then the spectrum acquires some ripple once more, as shown. Applying Window Functions There are many applications in which a spectrum with a rectangular magnitude profile is not ideal. For example, when a chirp waveform is compressed by means of its matched filter, then the resultant waveform approximates to the sinc function and, consequently, has annoyingly high sidelobes. Often, to improve the characteristics of the pulse and lower the sidelobe levels, its spectrum is modified, typically to a bell-shaped profile. Similar problems arise in digital signal processing where the spectral shaping is provided