# Impulse response if initial conditions are given

In most books, impulse response of LTI systems are calculated by assuming initial rest condition, but how to calculate response of an impulse input if there are some initial conditions present ?

My approach - first calculate zero input response (only using initial conditions that are given) and then calculate zero state response (taking impulse as input) assuming zero initial condition and then add both of them and we got total response.

Problem But I think this method will give wrong answer because due to impulse input there must be some changes occur in initial conditions, so I wanted to know

1. Due to change in initial conditions, which equation will change, zero input or zero state or both?

2. What is best approach to get solution of problem like this?

3. Is by using Laplace transform this problem can be solved without any change in initial condition?

• "But I think thus method will give wrong answer because...." There is nothing wrong with your method but thinking makes it so as Shakespeare might have written in an early draft of Hamlet. Jul 18 '20 at 20:23
• I never doubted this method until yesterday when I'm reading' b.p lathi's linear system and signal ' where in an example of r-l-c circuit, initial conditions just before zero were given and zero input response was asked, so since only ZIR was asked and as usual solution given in that book was something that I was expected until this statement appears "we need initial conditions at exact zero for calculation of ZIR and these initial conditions are derived from given initial conditions(just before zero) and then it says that initial cannot change instantly in absence of impulsive input Jul 19 '20 at 5:23
• So that's why I asked this question because if it mentioned in book that initial conditions for ZIR will be same at just before zero and exact zero when no impulsive input are there Jul 19 '20 at 5:28

## 1 Answer

The reason you're having trouble with this is because the impulse response of a system tells you something considerably different from a system's behavior when it is starting with non-zero initial values -- so no one bothers to treat this in the literature.

Problem But I think this method will give wrong answer because due to impulse input there must be some changes occur in initial conditions, so I wanted to know

A way to reconcile the ambiguity you see (and, typically, something you need to do if you're getting down and dirty with the details around the Dirac delta functional) is to make a distinction between $$t = 0$$, $$t = 0^+$$, and $$t = 0^-$$, where $$0^+ = 0 + \epsilon$$ and $$0^- = 0 - \epsilon$$ where $$\epsilon$$ is that famous infinitesimal from calculus that's greater than zero but arbitrarily small.

Then recall that the Dirac delta functional is zero for $$t < 0^-$$, and zero for $$t > 0^+$$, and -- strictly speaking -- undefined for $$t = 0$$

• Due to change in initial conditions, which equation will change, zero input or zero state or both?
• What is best approach to get solution of problem like this?

With a bit of hedging about with what you mean by initial conditions, your original idea -- that you can just add the impulse response and the system behavior with initial conditions -- is correct.

You lead yourself to this by recalling that if $$\delta(t)$$ is undefined at $$t = 0$$, then so are its effects. So to be entirely picky and technical, the impulse response is only defined for $$t > 0^+$$, not before.

Then you hedge your initial conditions statement by specifying the initial conditions at $$t = 0^-$$. Without that $$\delta(t)$$ in there, the state of the system at $$t = 0^+$$ would equal its state at $$t = 0^-$$, so (to an engineering level of rigor) this works.

With these details specified, you can just add the two responses.

• Is by using Laplace transform this problem can be solved without any change in initial condition?

You can use the Laplace transform as an aid to solving this, but at least the way that I was taught to insert initial conditions into a Laplace-domain solution was to use Delta functionals and their derivatives. You are, essentially, doing the same thing I did before, it's just that instead of taking the initial conditions at $$t = 0^-$$, you're finding the correct weights for $$\delta(t)$$, $$\delta^2(t)$$, etc., so that the states of your system match your initial conditions at $$t = 0^+$$.

Then to add in the impulse response, you just add in an extra one times $$\delta(t)$$. The effect will be the same as your proposed solution.

• Thanks for answer, but isn't ZERO input response changes due to impulsive input? And how to calculate that change Jul 19 '20 at 5:31
• Your question isn't clear. Do you mean the response at time zero due to an impulse? If you're working in a system where the value of the impulse isn't defined at time $t=0$, then you cannot find the response precisely at time $t = 0$. Because, undefined. Jul 20 '20 at 17:02