Answer Happy

Let T>0. For a given signal x(t),0≤t≤T, a filtered signal y(t) is obtained by finding a stationary path of the functional S[y]=∫0T​dt((y(t)−x(t))2+ky′(t)2),y(0)=x(0),y(T)=x(T). Here k>0 is a parameter that determines how much filtering occurs. (i) Obtain the Euler-Lagrange equation and explain how to solve the problem given the signal x(t). (ii) Let ω>0 be independent of t. For the case T=π/(2ω) and x(t)=t+21​sinωt, calculate y(t). (iii) Find the limits of the solution of (ii) in the limits k→0 and k→∞. Comment on your results.