Timeline for Time invariance of a summation sequence

7 events

when toggle format	what		by	license	comment
Jul 19, 2020 at 11:34	comment	added	Granolaboy		Thank you very much!
Jul 19, 2020 at 11:33	vote	accept	Granolaboy
Jul 16, 2020 at 17:38	comment	added	Dilip Sarwate		If $n_0=-\infty$, then \begin{align}y[n]&=\sum_{k=-\infty}^nx[k]\\&=\sum_{k=-\infty}^nu[n-k]x[k]\\&=\sum_{k=-\infty}^\infty u[n-k]x[k]\\&=u\star x\big\vert_n\end{align} where we have used the fact that $u[n-k]=1$ whenever $k$ is such that $n-k\geq 0$, i.e. $k \leq n$ and $u[n-k]=0$ whenever $k$ is such that $n-k<0$, i.e. $k>n$. So, we have expressed the output as a convolution of $x$ with a fixed impulse response (the unit step function) and so we have a time-invariant system, in fact, a linear time-invariant system.
Jul 16, 2020 at 6:24	comment	added	Granolaboy		$- \infty$ on the left equation*
Jul 16, 2020 at 6:07	comment	added	Granolaboy		Would you please explain how the system becomes time-invariant when $n_0 = -\infty$? The left equation is $\sum_{k=\infty}^n x_2[k]$ and the right equation $\sum_{k=0}^{n-(-\infty)} x_1[k] = \sum_{k=0}^{\infty} x_1[k]$ ?
Jul 15, 2020 at 19:34	history	edited	Dilip Sarwate	CC BY-SA 4.0	added 75 characters in body
Jul 15, 2020 at 16:37	history	answered	Dilip Sarwate	CC BY-SA 4.0