10 problem 4 specify the frequency response of a

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Unformatted text preview: 1 , ⎧ ⎧ 2j ⎧ ⎧ �2, � ak = 1 ⎧ 2j , ⎧ ⎧ ⎧ ⎧−1, ⎧ ⎧ ⎧ � 0, k k k k k k = −2 = −1 =0 =1 =2 = 3, 4, 5 And finally, we find the Fourier series coefficients, bk , of the output y [n]: � bk = ak H (e = ak · jk�0 1 + 2 e−jk�04 1 + 2 e−jk 4 4 ) = ak · = ak · � 1 1 − 4 e−jk�0 1 − 1 e−jk 4 4 1 + 2 e−jk� � 1 − 1 e−jk 4 4 We have only few non-zero coefficients, so we can go ahead and evaluate them. In doing so, it is sometimes useful while computing the value of the complex exponential, to visualize the the complex vector ejkω going around the unit circle. As the integer k increases by one, the complex vector’s angle increases by an angle of β . � ⎩ 1 + 2 e−j (0)� 1+2 6 b0 = a0 H ej (0)�0 = (2) = (2) 1=3 1 −j (0) � 4 1− 4 1− 4e 4 =8 �� �� � j (1)�0 ⎩ 1 1 + 2(−1) 1 1 + 2 e−j (1)� = b1 = a 1 H e = 1 1 1 −j (1) � 4 2j 1 − 4 e 2j 1 − 1 ( �2 − j �2 ) 4 = 0.1247 + j 0.5806 �� �� � j...
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This document was uploaded on 02/09/2014.

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