The purpose of this assignment is three-fold: (1) to give you practice manipulating the Fourier Transform of periodic function, (2) to give you practice manipulating the Matlab code that calculates the Fourier functions, (3) to teach you what happens to the Fourier Transform when you multiply two functions together.

Part 1: Pick two functions, x1(t) = cos(ω1t) and x2(t) = sin (ω2t) where ω1 and ω2 are different frequencies. To make life easy you should select these frequencies such that their ratio is a rational number. By hand, solve for the Fourier Transform of x(t) = x1(t).x2(t). to verify that your answer is correct, use Matlab to "rebuild" and plot x(t) using your newly derived Fourier coefficient An, and show that this plot is in fact identical to x1(t).x2(t). At what frequencies does x(t) have energy and how are those frequencies related to ω1 and ω2? In general, how do multiplying two signals appear to change their frequency content?

Part 2: In this part we will again be multiplying two functions and calculating the Fourier Transform of the product. The first function is y1(t) = cos(20πt). The second function is the square wave y2(t) shown below. By hand, calculate the Fourier Transform of the product y(t) = y1(t).y2(t). Create a plot of A vs ω. Using what you learned in Part 1 about multiplying signals, relate your hand calculation to the Fourier Transform plots of y1(t) and y2(t), explain this relationship clearly.