= Linear time-invariant system
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= LTI system
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A linear time-invariant system combines two properties. <Linearity> means that the <output> from a <linear combination> of inputs is the same <linear combination> of their outputs. Time invariance means that shifting an <input> by an integer delay shifts its <output> by exactly that delay. The operation has no preference for a particular time origin.
For discrete time, write the unit impulse as $\delta[n]=1$ at $n=0$ and zero otherwise, a <Kronecker delta>. Its response $h[n]=T[\delta][n]$ is the <impulse response>. A finite <input> can be decomposed into shifted impulses, giving
$$
x[n]=\sum_kx[k]\delta[n-k]
\quad\Longrightarrow\quad
y[n]=\sum_kx[k]h[n-k]=(x*h)[n].
$$
The <superposition principle> supplies the sum and time invariance supplies the shifted copies of $h$. Infinite sums need convergence assumptions; for a bounded input and an impulse response $h$ with <absolute summability>, the <discrete convolution> converges absolutely. For a physical system with stored initial <energy>, this construction describes the response with zero initial state; a separate free response accounts for nonzero initial conditions.
The gain is reuse. Measure or calculate one <impulse response>, and the <convolution> predicts the response to every allowed <signal> without solving a new dynamical problem each time. Alternatively, the <frequency response> predicts how each oscillatory component changes. https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/0400ba59406d52d94c079d99928156ee_MITRES_6_007S11_lec04.pdf[Oppenheim, lecture 4: convolution from shifted impulses]{ref}.
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