Continuous-time time scaling x(t/2) is defined for every real t whenever x is defined on R

Continuous-time time scaling \(x(t/2)\) is defined for every real \(t\) whenever \(x\) is defined on \(\mathbb{R}\). Discrete-time time scaling is not automatic: if \(n/2\) is not an integer, \(x[n/2]\) is undefined unless a convention is supplied. Let \(x[n]\) be the finite-length sequence \[ x[n]=\begin{cases} n, & |n|\le 4,\\ 0, & |n|>4, \end{cases} \] and let \(x(t)\) be the continuous-time analog \[ x(t)=\begin{cases} t, & |t|\le 4,\\ 0, & |t|>4, \end{cases} \] with \(t\) dimensionless in this comparison. (a) List every sample of \(x[n]\) for \(-4\le n\le 4\). Define the decimated sequence \(d[n]=x[2n]\) and list every nonzero sample of \(d\). Define the zero-insertion expansion \[ e[n]=\begin{cases} x[n/2], & n\text{ even},\\ 0, & n\text{ odd}, \end{cases} \] and list every sample of \(e[n]\) for \(-8\le n\le 8\). (b) Determine the closed support of the continuous-time signal \(x(t/2)\), and give a piecewise formula for \(x(t/2)\). Contrast the support of \(x(t/2)\) with the support of \(e[n]\). (c) A real signal is even if it equals its time reverse and odd if it equals the negative of its time reverse. Determine whether \(x[n]\) is even, odd, or neither, and whether \(x(t)\) is even, odd, or neither. Compute the discrete-time energy \(\sum_{n=-\infty}^{\infty}|x[n]|^2\) and the continuous-time energy \(\int_{-\infty}^{\infty}|x(t)|^2\,dt\).

Decimation keeps even original indices; expansion places original samples on even new indices and forces zeros on odd indices; continuous-time scaling by \(1/2\) stretches the analog triangle to twice the width with no holes.