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The impulse function can also be written as the derivative of the unit step function: dðtÞ¼ d dt uðtÞðA:1-5Þ The impulse function can be obtained by limiting operations on a number of functions whose integral has the value 1. Some examples are given below. dðtÞ¼ lim a!1 ½ae atuðtÞ lim a!1 1 2 e ajtj lim a!0þ 1. What is Impulse Response? - Impulse Response of a system is the result that is presented with a brief input signals. - Linear, Time-Invariant Systems are characterized by the response of their impulses. - It is easy to analyze the systems that implements transfer functions. This is done by Laplace transform of the impulse response function. Find the Impulse Response d2y dt2 + 13 dy dt + 12y = f(t) 1. Find the General Solution with f(t) = 1 Complimentary function is y = Ae−12t+ Be−t Particular integral is y =1 12 General solution is y =1 12+ Ae −12t+ Be−t 2. Set boundary conditions y(0) = ˙y(0) = 0 to get the step response. 1 12+ A + B = 0 −12A − B = 0 ⇒ A =1 132and B = − 1 11.