[5].(20 points) The following figure show a Bode plot of a band-pass filter.\\
20dB\\
10dB\\
1\\
w\\
3 4 5 6 $log_{10}w$\\
1\\
3\\
6\\
$10^1$ $10^2$ $10^5$ $10^6$ w\\
2\\
$10^0$ $10^3$\\
(a). At w = w1, the gain in dB is 10 dB.\\
Find the corresponding H(jw) with identification of all zero and pole (angular)\\
frequencies. Hint: H(jw)= K (1 + j \omega/\omega_z) / [(1 + j \omega/\omega_{p1}) (1 + j \omega/\omega_{p2})] and w1 can be\\
found from H(jw)\\
Find H(s) by replacing jw by s and simplifying the terms such that\\
H(s)= M (N(S)/D(s)), N(s) and D(s) are polynomial functions of s.\\
Find output y(t) for x(t) = 10cos (100\sqrt{10} t + \theta_1) + 5cos (10000t + \theta_2).\\
Hint: y(t) = A1 cos (100\sqrt{10} t + \theta_{y1}) + A2 cos (10000t + \theta_{y2})\\
Find A1. A2 from the gain information in the Bode plot. For simplicity neglect \theta_{y1},\theta_{y2}.\\
Also, use an approximation as |1 + j K| = K for K > 3.