Produce a discrete time series y(t_i) by super positioning 5 cosinusoidal components,

for your own choice of the amplitudes (a_j) and frequencies (f_j).

Add some Gaussian noise to each digitised value. Experiment with amplitudes (including that of the noise term), and frequencies, showing results graphically. Then smooth your noisy time series with at least two different filters, e.g., a simple moving average smoother and an order two binomial filter.

Discuss the relative performance of the smoothers. You should consider quantifiable parameters such as; variance, r.m.s. deviations from the noise-free time series, and signal attenuation. Comment on the validity of the expression below, for your chosen smoother.