Time t is in seconds and s is in s^(-1)

Time \(t\) is in seconds and \(s\) is in \(\mathrm{s}^{-1}\). All systems in this problem are causal, with ROC the open half-plane to the right of the rightmost pole. Let \(u(t)=1\) for \(t>0\) and \(u(t)=0\) for \(t<0\). Define the three rational transfer functions \[ H_{\mathrm{mp}}(s)=\frac{s+5}{s+2},\qquad H_{\mathrm{nmp}}(s)=\frac{s-5}{s+2},\qquad H_{\mathrm{ap}}(s)=\frac{s-5}{s+5}. \] (a) Locate the finite poles and finite zeros of each of the three functions. Classify each system as minimum-phase or non-minimum-phase according to whether every finite zero lies in the open left half-plane. Determine whether each system is BIBO stable. (b) Form the product \(H_{\mathrm{mp}}(s)\,H_{\mathrm{ap}}(s)\) and compare the product with \(H_{\mathrm{nmp}}(s)\). For \(\omega\in\mathbb{R}\), compute \(|H_{\mathrm{ap}}(j\omega)|\) and determine the relation between \(|H_{\mathrm{nmp}}(j\omega)|\) and \(|H_{\mathrm{mp}}(j\omega)|\). Evaluate each of the three magnitudes at \(\omega=0\) and at \(\omega=5\,\mathrm{s}^{-1}\). (c) Invert \(H_{\mathrm{ap}}(s)\) to obtain its impulse response \(h_{\mathrm{ap}}(t)\). (Polynomial division is required.) Evaluate \(h_{\mathrm{ap}}(0^{+})\) from the ordinary part on \(t>0\) and identify any impulsive term at \(t=0\). Compute \(\arg H_{\mathrm{ap}}(j5)\) in radians in \((-\pi,\pi]\).

Minimum-phase means every finite zero (and, for a causal stable system, every pole) lies in the open left half-plane. An all-pass factor can move a zero from the left half-plane to its mirror image.