4.9 Suppose that we have a LTI system with frequency response H ( jw) = 1 for |W | 25 otherwise Using frequency-domain methods, find the output y(t) of the system if the input x(t) is given by x(1) =1+2cos 2t + 2cos4t + 2 cos 6t. Solution. We begin by finding the Fourier series representation of x(t). Using Euler's relation, we can rewrite x(t) as x(t) = 1+2cos2t + 2cos4t + 5 cos 6t = 1+2 [2(@12 + e-12) ] + 2 [2 (ejst + e-j#1) ] + ; [ (e161 + e-165)] = 1 te/2t te i2 +eith te itt + Leibt + he-jot. Thus, we have that the Fourier series representation of x(t) is given by x (1 ) = [ axeikcot where wo = 2 and for k = 0 for k = +1 ak = O Al- for k = 12 for k = 13 otherwise. Since the system is LTI, we know that the output y(t) has the form y (1 ) = [ breikcot, where bk = akH(jkwo). Using the results from above, we can calculate the by as follows: bo = aoH () [0] [2]) = 0, b1 = alH(j[1] [2]) = 0, b_1 = a_1H(j[-1][2]) = 0, b2 = a2H(j [2] [2]) = 0, b_2 = a_2H(j[-2] [2]) = 0, b3 = a3H(j[3][2]) = 4(1) = 4, and b_3 = a_3H(j[-3][2]) = 4 (1) = 4. Thus, we have O AI- for k = 13 otherwise