Question

1. Using the Fourier series analysis Equation 3 for the periodic function r(t) shown in Figure 2.1, determine both the DC coefficient ao and a general expression for the other Fourier series coefficients ak. Do this by hand, not in Matlab. Show all your work in your lab report. You can add these pages as hand-written pages, rather than typing them in to your lab report, if you prefer Hint 1: It will be easiest to integrate this function from -T2 to T/2, rather than 0 to T. Then you can combine the two exponential terms into a sin term. Your answer for ak should be of the form sin(k)/k, although k will be multiplied by some constant. Hint 2: Your ao term will be difficult to determine from the ak term, as both numerator and denominator will go to 0 for k-0. The easiest way to find ao is by recognizing that r(t)dt which is the area under r(t) divided by T (or the average area) An alternate way to find ag is to apply l'Hôpital's rule (also called l'Hospital's rule) to your equation for ak. Recall that l'Hopital's rule is used to find f(r)/g(x) when lim (lim g) 0, as f(r) lim g(r)-lima g,(z) where indicates the derivative. L'Hopital's rule may be applied successively if the derivatives of the numerator and denominator still go to 0 2. Write a Matlab script or function to generate the Fourier series coefficients from a-N to av, using the expressions you found for ak and ao in the previous step. Include the DC coefficient ao also. Your function should take as input N 3. Run your m-file for N 3. Print out your Fourier series coefficients 4-3 to as. Include your in-file in your lab report, as well as the Fourier series coefficients a-3 to ag found above. You may want to do the next lab exercise first to be certain that your m-file works properly 4. Generate a stem plot of the Fourier series coefficients a-3 through ag 1.2 0.8 0.8 0.2 0.05 0.1 0.15 0.2 Figure 1: Periodic signal r(t) with period T shown versus time where fo is the fundamental frequency of r(t) and T 1/fo is the fundamental period of r(t) If we are given a periodic function r(t), we can find the Fourier series coefficients according to the Fourier Analysis equation, as

Answer 1

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