Derivation of discrete-time steady-state and transient response
Given: the system is LTI and causal.
$$\Rightarrow y[n]=h[n]\ast x[n]=\sum_{k=-\infty}^{\infty}h[k]e^{j\omega (n-k)}u[n-k]\\~\\
\because \textrm{ causality implies }h[k]=0\ \forall k&lt;0\\
\because u[n-k]=0\ \forall k&gt;n\\~\\
\therefore \sum_{k=-\infty}^{\infty}h[k]e^{j\omega (n-k)}u[n-k]=e^{j\omega n}\sum_{k=0}^{n}h[k]e^{-j\omega k}
$$
Combining that with the fact that the output is causal-
$$
y[n] = \left\{\begin{aligned}
&amp;0 &amp;&amp;: n &lt; 0\\
&amp;e^{j\omega n}\sum_{k=0}^{n}h[k]e^{-j\omega k} &amp;&amp;: n\ge 0
\end{aligned}
\right.
$$