Impulse Response Function Problem

[abcz4113]

I attempted by inputting u(τ-t)u(t-τ) into the second part of the integral. Since I want to change the second part of the integral to go from [+∞, -∞]. And as for first part of the integral I added a u(t-τ) term after the x(τ) to change the integral from [t,-∞] into [+∞, -∞]. I am not sure if I did them right. And after factoring out the common term x(τ)u(t-τ) (I assume this is my input) from the two integrals I get that my IRF h(t,τ) = 1 + e^(t-τ) * u(τ-t). I am not sure about the answer and I did not know how to solve for the Unit Step Response of this system. Please Help. Thank you.

 

[jfgobin]

Hello abcz4113,

At first thought, I would separate the cases where $t<0$, $t=0$ and $t>0$, assuming your impulse or unit step happens at $t=0$, and use the knowledge of $h(t)$ and $g(t)$.

I will have a closer look at this in a moment.

 

[rude man]

My first observation is that this system seems to be non-causal. Compare the first integral to the convolution integral relating input to output, then ask yourself what the second integral is doing there ...

 

[abcz4113]

Thank you for your hints. I know how to do it now.

 

[rezeew]

Your approach to finding the impulse response function is correct. The impulse response function is defined as the output of a system when an impulse input is applied, which is represented by the delta function δ(t). In this case, you have used the unit step function u(t) as the input. Both the unit step and delta functions are commonly used as test inputs in systems analysis.

Your resulting IRF h(t,τ) = 1 + e^(t-τ) * u(τ-t) is also correct. This function represents the output of the system at time t, given an input that occurred at time τ. To solve for the unit step response, you would need to integrate this function over all τ, from -∞ to +∞. This would give you the output of the system at time t for a step input applied at time 0.

Overall, your approach and answer seem correct. However, it would be helpful to provide more context and information about the system in question, as well as any assumptions made during the solution process. This would allow for a more thorough evaluation and potential suggestions for improvement.