Question

Problem 32: (20 points) Consider a periodic signal f(t), with fundamental period To, that has the exponential Fourier series representation f(t) = Σ Dnejuont . where wo 2T/To and 1. (2 points) When f(t) is a real-valued, show that DD This is known as the complex conjugate symmetry property or the Hermitian property of real signals. 2. (2 points) Show that when f(t) is an even function of time that Dn is an even function of n 3. (2 points) Show that when f(t) is an odd function of time that Dn is an odd function of n. 4. (2 points) Using the results from parts 1 and 2, show that if f (t) is real-valued and an even function of time then the coefficients Dn are real-valued and an even function of n 5. (2 points) Using the results from parts 1 and 3, show that if f(t) is real-valued and an odd function of time 6. (4 points) Show that the signal f(-t) has the Fourier series coefficients D-n. This is known as the time 7. (6 points) Consider three periodic signals whose Fourier series representations are then the coefficients Dn are imaginary and an odd function of n. reversal property 2) f3(t) Using the properties of the exponential Fourier series, determine if the signals are (a) (3 points) real or complex valued, (b) (3 points) an even or odd function of time, or neither

Answer 1

Solution <Te) fo

dt 70 d fbf4e etn@wjet Kt '((t) สู่(-t) in T 7 lt) Real

ni ' lme Re (-t) e dt nwom To

(nん Vri