Consider I can find the step response of a system with the following method:

We can easily find the step input of a system from its transfer function. Given a system with input $x(t)$, output $y(t)$ and transfer function $H(s)$

$$H(s) = \frac{Y(s)}{X(s)}$$

the output with zero initial conditions (i.e., the zero state output) is simply given by

$$Y(s) = H(s)X(s)$$

so the unit step response, $Y_\gamma(s)$, is given by

$$ Y_\gamma(s) = \frac{1}{s} H(s) $$

If my input is a "differentiating system" then what will be the step response ?


If your system is an ideal differentiator with input-output relation


then its transfer function is


From the equation in your question you obtain for its step response


which in the time domain corresponds to a Dirac delta impulse:


This is also intuitively clear, because a step input has a derivative of zero everywhere except at $t=0$, where it is discontinuous.

  • $\begingroup$ Thanks. Just want to know, is this the general case for all the differentiating systems? $\endgroup$ – gocjack May 26 '17 at 6:45
  • $\begingroup$ @statisticalbeginner: As long as you have a linear time-invariant (LTI) system, the step response is always given by $H(s)/s$. $\endgroup$ – Matt L. May 26 '17 at 6:58

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