(3.1) Consider the motion of a damped, driven harmonic oscillator (such as a mass on a spring, a ball in a well, or a pendulum making small motions):
ODE Problem Set
(3.2) Explicitly solve (and try to simplify) the system of differential equations for two coupled harmonic oscillators [don't worry about the initial transient] and find the normal modes by matrix diagonalization.
(3.3) A common simple digital filter used for smoothing a signal is y(k) = ay(k - 1) + (1 - a)x(k) where "a" is a parameter that determines the response of the filter. Use z-transforms to solve for y(k) as a function of x(k) (assume y(k < 0) = 0). What is the amplitude of the frequency response?
