Suppose that E is a continuous-time system with impulse response g. Define E as the system...

   

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Suppose that E is a continuous-time system with impulse response g. Define E as the system with impulse response g, defined as g(1) := g(1) for all te R. Show that its Fourier Transform, let's call it G, is given by G(w) = G(-w) for all w ER, %3D where G is the Fourier Transform of g. Hint: you may want to use that for any function f we have (7)dr = T(7)dr. Question 2: Let y be a complex number written as y = a+bj, where a and b are real numbers. (a) (1 point) Express a in terms of y and y. (b) (1 point) Express b in terms of and y. Question 3: (3 points) Use your answers to Question 2 to show that the real-valued signal Re g has Fourier Transform G + G). Clearly indicate any properties of the Fourier Transform that you are using in your steps. Here the notation is as in Question 1. Question 4: (3 points) Express the Fourier Transform of the real-valued signal Im g in terms of G and G. Explain your answer, showing all steps. Here the notation is as in Question 1. Suppose that E is a continuous-time system with impulse response g. Define E as the system with impulse response g, defined as g(1) := g(1) for all te R. Show that its Fourier Transform, let's call it G, is given by G(w) = G(-w) for all w ER, %3D where G is the Fourier Transform of g. Hint: you may want to use that for any function f we have (7)dr = T(7)dr. Question 2: Let y be a complex number written as y = a+bj, where a and b are real numbers. (a) (1 point) Express a in terms of y and y. (b) (1 point) Express b in terms of and y. Question 3: (3 points) Use your answers to Question 2 to show that the real-valued signal Re g has Fourier Transform G + G). Clearly indicate any properties of the Fourier Transform that you are using in your steps. Here the notation is as in Question 1. Question 4: (3 points) Express the Fourier Transform of the real-valued signal Im g in terms of G and G. Explain your answer, showing all steps. Here the notation is as in Question 1.