(Solved): 1. The output of a causal discrete-time LTI system is governed by the equation \[ y[n]=\frac{1}{2} ...

1. The output of a causal discrete-time LTI system is governed by the equation \[ y[n]=\frac{1}{2} y[n-1]+\frac{1}{9} y[n-2]+2 x[n] \] a) Determine the frequency response \( H\left(e^{\mu n}\right) \) of the system. b) Determine the impulse response \( h[n] \) of the system. 2. A LTI system has the frequency response plotted below. a) If \( x[n]=3 \cos \left(\frac{\pi}{5} n+\frac{\pi}{2}\right)+2 \cos \left(\frac{\pi}{3} n+\pi\right) \) is input to the system, what will the output be? b) If \( y[n]=5 \cos \left(\frac{\pi}{4} n+\frac{\pi}{2}\right)+9 \sin \left(\frac{\pi}{2} n+\frac{\pi}{3}\right) \) is output by the system, what was the input? c) If an ideal lowpass filter having a cutoff frequency of \( \frac{\pi}{3} \mathrm{rad} / \mathrm{s} \) and unit phase delay is added to the system, what will the output be to the input \( x[n]=2 \cos \left(\frac{\pi}{2} n+\frac{\pi}{3}\right)+4 \sin \left(\frac{\pi}{4} n+\frac{\pi}{2}\right) \). 3. You wish to build a neurostimuation device to help people with incontinence void their bladder and use electronic circuit below to simulate how voltage \( \mathrm{V}_{i}(\mathrm{t}) \) applied a microelectrode (modeled by \( \mathrm{RI} \) and C1) would alter the voltage \( v_{0}(t) \) across the nerve membrane (modeled by \( R 2 \) and \( \mathrm{C} 2 \) ). a) Specify the nerve in the body that the electrode would be designed to stimulate. b) Determine the frequency response of the stimulation system. c) Sketch the amplitude and phase spectrum of the stimulation system. 4. Consider the anesthesia delivery system below for controlling mean arterial pressure (MAP) during surgery. Suppose that the rate of change of pump output is proportional to the input valve setting (i.e. \( \mathrm{dz}(\mathrm{t}) / \mathrm{dt}=\mathrm{v}(\mathrm{t})) \), the impulse response of the controller is \( \mathrm{c}(\mathrm{t})=\mathrm{A} \delta(\mathrm{t}) \), the impulse response of the body is \( b(t)=\mathrm{te}^{-p t} \mathrm{u}(t) \), and the impulse response of the sensor is \( s(t)=e^{-q t} u(t) \). Specify the frequency response of the anesthesia delivery system.