Problem 3 Determine the Fourier series coefficients for the periodic signal xn depicted below. Plot the magnitude and phase of these coefficients Fundamental period N= 6 ∑le 6∑l-km_1 1 n Notice that the last two expressions will give the same result, but the latter would take advantage of the symmetry of some of the samples to combine them into sinusoids 4=6[071k3+(1-)()+(2)-1 -1)+(1) -ikwo(o) +(2)e-k()+(-1)e-1ka2) 6[-1=-c70+270+2710+1 6(2j)sin kwo2+2 (2)cos kwo+1 J sin kwo /1 1+4cC� � � �� � � � �� Problem 3 Determine the Fourier series coefficients for the periodic signal x[n] depicted below. Plot the magnitude and phase of these coefficients. x[n] 2 −12 −6 0 6 12 n 1 · · · · · · −1 Fundamental period N = 6 �0 = 2 6 � = . ⇒ 3 ⎨ 5 2 1 1 ⎨ 1 ⎨ ak = x[n]e−jk�0n = x[n]e−jk�0n = x[n]e−jk�0n N 6 6 n=<N> n=0 n=−3 Notice that the last two expressions will give the same result, but the latter would take advantage of the symmetry of some of the samples to combine them into sinusoids. 1 � ak = (0)e−jk�0(−3) + (1)e−jk�0(−2) + (2)e−jk�0(−1) + (1)e−jk�0(0)+ 6 +(2)e−jk�0(1) + (−1)e−jk�0(2)� = 1 e−jk�0(−2) − e−jk�0(2) + 2e−jk�0(−1) + 2e−jk�0(1) + 1 � 6 1 = [(2j)sin k�02 + 2(2) cos k�0 + 1] 6 1 2 j = + cos k�0 + sin k�02 6 3 3 1 2 1 2� � ak = + cos k + j sin k 6 3 3 3 3 ⎩ � �� 1 2� = 1 + 4 cos k + j2 sin k 6 3 3 11