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The bell-shaped Gaussian function, m) = sap H (\"'3'\")2], is one of the most widely used functions in science and technology. The parameters or and s 1) 0 are prescribed real numbers. Make a program for evaluating this function when m = D, a = 2, and a: = 1. Verify the program's rllult by comparing with hand calculations on a calculator. 3. The Gaussian function ex has many applications which in statistics, its [15] normalized form is often referred to as normal distribution with zero mean. Find the Fourier transform of the following function e-ax cos bx, and establish that Fourier transform of the Gaussian function is also a Gaussian. ( that is show that F(e "x ] = eam? Further, use the above result to find the value of the following integral identities (i) Le mxa dx (ii) Sex dx (ii) fo xe x dx (i)So xle-x dx (v) Pe z dxExample 2-4 Gaussian function s(t) = (2.27) which is the standard Gaussian function with the mean time t, The Fourier trans- form is s(w) - Jamexp -"(t-1 ) expi-jothat = exp. -zoo +jot. (2.28) The formula (2.28) usually is remembered as the Gaussian characteristic function, Eq.(2.28) shows that the Fourier transform of the Gaussian function is also Gauss- ian. The time-shift in (2.27) corresponds to the phase-shift in (2.29). Moreover, the variance of the frequency representation in (2.28) is the reciprocal of the time vari ance in (2.27), The narrower the spread in the time domain is, the wider the spread in the frequency domain, or vice versa. Unlike the sinusoidal or rectangular pulse functions discussed earlier, the Gaussian function is localized in both time and fre- quency. As we shall see later, among all possible functions, the Gaussian function is optimally concentrated in joint time and frequency domain