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Unformatted text preview: ECE 2704 Quiz 2 Name: _______________________________________ Consider the linear timeinvariant system that satisfies the firstorder ODE , The input of the system is and output is . The impulse response of a system is the output of the . The impulse response for this system is . system when the input is the Dirac delta Compute the output of the system when the input is the complex exponential where is an arbitrary complex number. SOLUTION Since for , Since when or equivalently when (1) However, we need to be careful at this point. If turns out that for . To see this, suppose , then in order for the integral to evaluate to something other than zero, we need both that and . However, if , then the integration is from zero to a negative number. Thus the variable of integration is always nonpositive and . Thus we require and the integral in (1) should be written Now we solve the integral REMARKS Although not apparent at first glance, the solution to this problem has a special form. In the next few weeks, you will learn that Thus , then the term decays exponentially to zero. Thus the steadystate output of the Further, if system, meaning the output after transient response to initial conditions has died away, is In other words, if the input to a linear system is a complex exponential, then the output of the linear system is also a complex exponential, although scaled by a complex rational function ( ). This fact is a rather remarkable property of linear systems, and suggests why the Laplace transform and the Fourier transform are expressed in terms of complex exponentials. is the transfer function associated with the differential equation . We usually use the notation ...
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- Spring '08