Given continuous time waveform x1(t) and x2(t) are the cosine functions with a constant Xo. x(t)...

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Given continuous time waveform x1(t) and x2(t) are the cosine functions with a constant Xo. x(t) = x, + x,() + x,(t) 0. 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.4 time in second 0.1 0.2 0.3 0.5 0.8 0.8 0.9 1 The x(n) is a discrete time waveform generated by sampling x(t) at 32 samples/second. the Fourier series expansion for x(n): A, +EA cos(27 fan+x) %3D a) Write down the Fourier Series expansion for x(n) in cosine. Let-nS S b) Assume x(n) goes through a LTI system with amplitude response S(f) as following: 2. 2. 321 O S(f) system 0.5 2 4 6. 8. 10 12 14 16 k (f = normalized frequency in cycles/samples) Plot the amplitude spectrum for output y(n) (A versus k). label all the values. c) Write out the Fourier Series Expansion of output y(n). Amplitude Response Given continuous time waveform x1(t) and x2(t) are the cosine functions with a constant Xo. x(t) = x, + x,() + x,(t) 0. 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.4 time in second 0.1 0.2 0.3 0.5 0.8 0.8 0.9 1 The x(n) is a discrete time waveform generated by sampling x(t) at 32 samples/second. the Fourier series expansion for x(n): A, +EA cos(27 fan+x) %3D a) Write down the Fourier Series expansion for x(n) in cosine. Let-nS S b) Assume x(n) goes through a LTI system with amplitude response S(f) as following: 2. 2. 321 O S(f) system 0.5 2 4 6. 8. 10 12 14 16 k (f = normalized frequency in cycles/samples) Plot the amplitude spectrum for output y(n) (A versus k). label all the values. c) Write out the Fourier Series Expansion of output y(n). Amplitude Response

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