EXAMPLE 2.10 The Delta Function as a Limiting Form of the Gaussian Pulse Consider a Gaussian pulse of unit area, defined by (2.66 0:25 05 T- where t is a variable parameter. The Gaussian function () has two useful properties: (1) its derivatives are all continuous, and (2) it dies away more rapidly than any power of t. The delta function () is obtained by taking the limit -0. The Gaussian pulse then becomes infi- nitely narrow in duration and infinitely large in ampli- tude, and yet its area remains finite and fixed at unity. Figure 2.13a illustrates the sequence of such pulses as the parameter varies. The Gaussian pulse g(t), defined here, is the same as the normalized Gaussian pulse exp-Ar) derived in Example 2.6, except for the fact that it is now expanded in time by the factor and compressed in amplitude by the same factor. Therefore, applying the linearity and time-scaling properties of the Fourier transform to the transform pair of Eq. (2.38), we find that the Fourier transform of the Gaussian pulse g(1) defined in Eq. (2.66) is also Gaussian, as shown by G() = exp(-x") Figure 2.136 illustrates the effect of varying the pe rameter on the spectrum of the Gaussian pulse g(). Thus, putting 1 = 0, we find, as expected that the Four- ier transform of the deta function is unity. -10 05 10 GUD -10 10 FIGURE 2.13(a) Gaussian pulse of varying duration. (b) Come- sponding spectrum 025 1-1 -10 10 10 T-25 Te -2 -10 10 FIGURE 2.13 (a) Gaussian pulse of varying duration. (b) Come sponding spectrum 100 points) Regenerate Figure 2.13 a & b using Matlab/Ocatve