The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform. The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics. The simplest way to derive the spectrum of a chirp, now that computers are widely available, is to sample the time-domain waveform at a frequency well above the Nyquist limit and call up an FFT algorithm to obtain the desired result. As this approach was not an option for the early designers, they resorted to analytic analysis, where possible, or to graphical or approximation methods, otherwise. These early methods still remain helpful, however, as they give additional insight into the behavior and properties of chirps. Chirp pulse A general expression for an oscillatory waveform, centered on frequency is where and (t) give the amplitude and phase variations of the waveform , with time. The frequency spectrum of this waveform is obtained by calculating the Fourier Transform of , i.e. so In a few special cases, the integral can be solved to give an analytical expression, but often the characteristics of and (t) are such that the integral can only be evaluated by an approximation algorithm or by numerical integration. Linear chirp In the special case where s(t) is constrained to be a down-chirp, flat topped pulse with its instantaneous frequency varying as a linear function of time, then an analytical solution is possible. For convenience, the pulse is considered to have unit amplitude and be of duration T, with the amplitude and phase defined over the time interval -T/2 to +T/2. The total frequency sweep is F, varying in a linear manner from -F/2 to +F/2 in the defined time interval. When the frequency is a linear function of time, the phase is a quadratic function, and s(t) can be written The spectrum of this linear FM signal is By completing the square and using the Fresnel integrals C(X) and S(X), defined by the expression can be evaluated to give: where and are given by The linear