The purpose of this problem is to show you the connection between Fourier series and linear algebra. For simplicity, we will consider a finite basis. All questions will be referred to the following basis:
B = {1, sin(400t), cos(400t), sin(800t), cos(800t)}.
(a) The output voltage, v(t), of an electronic device is given by
v(t) = 10 + 2 cos(400t) + 3 sin(800t) + 7 cos(800t) mV.
Express v(t) with respect to the basis B.
(b) Express
v(t) = 2 + 3 sin(400t-pi /4) + 8 cos(800t + pi/6) mV with respect to the basis B.
(c) Suppose that the voltage v(t) is then inputted to a differentiator circuit. Find the matrix representation of the differentiation operation with respect to the basis B.
(d) Use the matrix representation in part (c) to find the derivative (with respect to t) of the voltage v(t) in part (b). Verify your derivative using your knowledge from calculus.