Finding the impulse response of a system

[btse]

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.

 

[vela]

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.

 

[btse]

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.

 

[vela]

Show your work. I have no idea how you are managing to get a constant.

 

[DeeNos]

To find the impulse response of a system, we need to first understand what an impulse response is. An impulse response is the output of a system when an impulse is applied as the input. An impulse is a very short signal with a very high amplitude and infinitesimal duration. In mathematical terms, an impulse is represented by the Dirac delta function, δ(t).

In this problem, we are given an input signal x(t) and the corresponding output signal y(t). The input signal is a function of time, and the output signal is the integral of the input signal over a certain time period. This means that the system is a linear time-invariant (LTI) system, where the output is the convolution of the input signal with the impulse response of the system.

To find the impulse response, we can use the definition of convolution, which is given by the integral of the product of the input signal and the impulse response over all possible time delays. In this case, the input signal is δ(t-2), which means that the impulse is applied at t=2. Therefore, the integral becomes:

h(t) = integral from -infinity to t of δ(t-2)*e^-(t-tau)dtau

To solve this integral, we can use the properties of the Dirac delta function, which states that δ(t-a) = 0 for all t ≠ a, and δ(t-a) = ∞ for t = a. This means that the integral will only contribute when t = 2, and the value of the integral will be 1. Therefore, the impulse response of the system is:

h(t) = e^-(t-2)

This is the impulse response of the system, and it can be used to find the output signal for any input signal using the convolution integral. I hope this helps in solving the problem.