Continuous-time systems act on real functions of t∈R with t in seconds, and…

Continuous-time systems act on real functions of \(t\in\mathbb{R}\) with \(t\) in seconds, and discrete-time systems act on real sequences indexed by \(n\in\mathbb{Z}\). The unit step satisfies \(u(t)=1\) for \(t\ge 0\) and \(u(t)=0\) for \(t<0\); \(\delta[n]\) is the unit pulse. Shifts are \((S_a x)(t)=x(t-a)\). BIBO stability means every bounded input produces a bounded output. Causality and memorylessness are the standard pointwise-in-time definitions. The four systems below are designed so that each fails exactly one of the four properties linearity, time invariance, causality, and BIBO stability, while satisfying the other three. For each system, identify the unique property that fails and the three that hold, and supply a complete witness for the failure (explicit signals, scalars, and/or a shift). Do not invoke frequency-domain tests. - \(P_1\): \(y(t)=x(t)+2.00\,\mathrm{V}\). - \(P_2\): \(y(t)=x(t)\cos(20\pi t)\). - \(P_3\): \(y(t)=x(t+1.50)\). - \(P_4\): \(y(t)=\int_{-\infty}^{t}x(\tau)\,d\tau\), defined on inputs that vanish on some half-line \((-\infty,t_1)\), and with the rest condition that \(y\) also vanishes on that half-line. Additionally, exhibit one system (possibly one of the four, or a fifth) that is linear and BIBO stable but neither causal nor time invariant, with an explicit formula.

An offset, a time-varying gain, an advance, and a running integral each spoil a different one of the four named properties.