# How to determine the step response using convolution of the signal's impulse response?

The step response can be determined by recalling that the response of an LTI to any input signal is found by computing the convolution of that signal with the impulse response of the system. Therefore we can write

$s(t) = u(t)∗h(t) =\int_{−∞}^{∞} u(τ)h(t−τ) dτ$

The convolution is commutative, meaning that $u(t)∗h(t) = h(t)∗u(t)$

It is more convenient to write the step response in the following way:

$s(t) = h(t)∗u(t) =\int_{−∞}^{∞}h(τ)u(t−τ) dτ=\int_{−∞}^{t}h(τ) dτ$

How do you explain the last step? How did we get the following;

$\int_{−∞}^{∞}h(τ)u(t−τ) dτ=\int_{−∞}^{t}h(τ) dτ$

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## migrated from electronics.stackexchange.comFeb 24 '13 at 2:22

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What is the actual question? –  AndrejaKo Feb 23 '13 at 10:03
what is the author saying? –  Rajesh K Singh Feb 23 '13 at 10:05
how did we get \$\int_{−∞}^{∞}h(τ)u(t−τ) dτ=\int_{−∞}^{t}h(τ) dτ\$ –  Rajesh K Singh Feb 23 '13 at 10:06

@ apalopohapa:Do you mean both, t and \$\tau\$, are variable here. –  Rajesh K Singh Feb 23 '13 at 10:23
the step function under the integral is \$u(-(\tau - t))\$. –  Rajesh K Singh Feb 23 '13 at 10:31