find impulse response from step response

Question

if we have LTI system and we know unit step response of this system(we haven't original signal) how we can calculate impulse response?

Answer 1

You can find the impulse response

Let's take the case of a discrete system.

If $s[n]$ is the unit step response of the system, we can write

$$s[n]= u[n]\ast h[n]$$

where $h[n]$ is the impulse response of the system and $u[n]$ is the unit step function.

Now using commutative property you can write $$s[n]=h[n]\ast u[n]$$

Expanding convolution we get $$s[n] = \sum_{k=-\infty}^{\infty}h[k]u[n-k]$$

which can also be written as

$$s[n]=\sum_{k =-\infty}^{n} h[k]$$

So we can see that unit step response is like an accumulator of all value of impulse response from $-\infty$ to $n$. So now impulse response can be written as the first difference of step response.

$$h[n]=s[n] - s[n-1]$$

Answer 2

With an LTI system, the impulse response is the derivative of the step response. Because the impulse function is the derivative of the step function. Derivative in, derivative out.