discrete data points (= N), at frequencies given by then the formula for calculating the discrete Fourier transform is The calculations can be carried out by means of a straightforward computer algorithm, but this is not very efficient in computer usage. Consequently, more efficient algorithms have been developed, especially Fast Fourier Transforms (FFT). Computer programs which implement the FFT are widely available in the literature and in proprietary CAD programs such as Mathcad, MATLAB, and Mathematica. In the following example a linear chirp with time bandwidth product of 25 is sampled at 128 points (i.e. N = 128). In the figure samples of the real part of the waveform are shown - note that these are samples in the time domain. The FFT process assumes the waveform is cyclic, so these 128 data points can be considered to be part of an endlessly repeating sequence in time. By calculating the N-point FFT of this data, the discrete spectrum of the sequence is obtained. The magnitude of this spectrum is shown in the attached figure, where these data points are samples in frequency. The data is cyclic so, in the plot, the zero frequency point is at n = 0 and also at n = 128 (i.e. both points are the same frequency). The point n = 64 corresponds to +fs/2 (and also to -fs/2). To display the spectrum in more detail (but not necessarily with more resolution), the time sequence can be extended by zero padding. For example, extending the 128 point time sequence with zeros to give N = 4096 results in that part of the spectrum originally presented in 16 samples, now being presented in 512 samples, as shown. Spectral spread There is very little spectral content beyond the sweep frequency range of a chirp pulse and this is especially true for waveforms where the time-bandwidth product is large. The full line on the graph of the adjacent figure shows results for linear chirps. It shows, for example, that only about 2% of the total power resides at frequencies outside the sweep range F when the time-bandwidth is 100, and it is less than 1/2% when T.F is 500. In the case of a non-linear chirp, or a linear chirp shaped by amplitude weighting, the fraction of power outside F is even lower, as is shown on the graph, where the dashed line is for spectra with Hamming