Impulse Response Function Problem

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Discussion Overview

The discussion revolves around the impulse response function (IRF) of a system, specifically focusing on the mathematical formulation and evaluation of integrals involving unit step functions. Participants explore the implications of their approaches and the characteristics of the system under consideration.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant describes their method of transforming integrals using unit step functions and expresses uncertainty about the correctness of their resulting impulse response function h(t,τ).
  • Another participant suggests separating cases based on the value of t (t<0, t=0, t>0) and indicates that they will further analyze the situation.
  • A third participant raises a concern about the system being non-causal, prompting a comparison between the integrals involved and the convolution integral.
  • A later reply indicates that the initial poster feels they have gained clarity on the problem after receiving hints.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are differing views on the nature of the system (causal vs. non-causal) and the methods for solving the problem. Uncertainties remain regarding the correctness of the proposed impulse response function.

Contextual Notes

Participants have not fully resolved the implications of their mathematical steps, particularly regarding the assumptions made about the system's causality and the handling of the integrals.

abcz4113
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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.
 
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Hello abcz4113,

At first thought, I would separate the cases where t&lt;0, t=0 and t&gt;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.
 
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 ...
 
Thank you for your hints. I know how to do it now.
 

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