Consider an LTI (linear time-invariant) system whose impulse response h[n] and input signal x[n] are given as:

h[n] = u[n-2]

x[n] = (1/2)^n u[n-4]

Find the output y[n] of the system for all values of n. Show every step of your work, making sure to label the axes, as well as any important points/values in your graphs.

An external consultant recommends that a plant installs a bank of capacitors for power factor correction. this will reduce the peak electrical demand charges by an average of 93 kw every month. the plant current pays $13 per kw in peak demand charges. the capacitor bank will include 223 kw of fixed capacitors, and 183 of variable capacitors. the fixed capacitors cost $59 per kw, and the variable capacitors will cost $65 per kw. the consultant charges 21% of the equipment costs to install the capacitors. because this project will reduce the demand for the electric utility, they are prepared to provide a one-time rebate of $42 per kw of reduced demand. what is the simple payback period for this project (in years)?