Finding the impulse response of a system

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Discussion Overview

The discussion revolves around finding the impulse response h(t) of a system given the relationship y(t) = integral from -infinity to t of e^-(t-tau)*x(tau-2)dtau. Participants are exploring the application of the Dirac delta function in this context, particularly how to manipulate the integral to derive h(t).

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant suggests substituting delta(t) for x(t) to start solving the problem, leading to the equation y(t) = integral from -infinity to t of e^-(t-tau)*delta(tau-2)dtau.
  • Another participant provides a basic relationship for the Dirac delta function, indicating that it picks out the value of a function at a specific point.
  • A participant expresses confusion about manipulating the integral, noting that their attempts yield h(t) as a constant rather than a function of t.
  • One participant requests to see the work done by another to understand how a constant was obtained from the integral.

Areas of Agreement / Disagreement

The discussion remains unresolved, with participants expressing different levels of understanding and confusion regarding the manipulation of the integral involving the delta function.

Contextual Notes

Participants have not fully articulated their assumptions regarding the properties of the delta function or the specific steps taken in their calculations, leading to potential gaps in understanding.

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Homework Statement


So the problem asks to find the impulse response h(t) provided y(t) = integral from -infinity to t of e^-(t-tau)*x(tau-2)dtau

Homework Equations


none

The Attempt at a Solution


I understand that the way to begin this problem is to substitute delta(t) for x(t). Therefore the equation becomes y(t) = integral from -infinity to t of e^-(t-tau)*delta(tau-2)dtau

However, at this point I am unsure of how to begin solving the integral.

Thanks for any help.
 
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The basic relationship for the Dirac delta function is

[tex]\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx = f(x_0)[/tex]

It picks out the value of the function f(x) when the argument of delta function is 0.
 
I understand the concept of the delta function. However, I am having trouble manipulating the integral. Each time I try something I achieve h(t) equal to a constant which is not correct at all. h(t) should be in terms of t.
 
Show your work. I have no idea how you are managing to get a constant.
 

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