Difference between revisions of "HW4.1 Jungu (Joe) Choi ECE301Fall2008mboutin" - Rhea

Latest revision as of 09:57, 16 September 2013

Example of Computation of Fourier series of a CT SIGNAL

A practice problem on "Signals and Systems"

CT Signal

$ x(t) = 1 + sin(w_0 t) + 3cos(2w_0 t + {\pi \over 4}) $

This is a signal with period $ T = {2\pi \over w_0} $

$ x(t) = 1 + {1 \over 2j}[e^{j w_0 t} - e^{-j w_0 t}] + {3 \over 2}[e^{(j 2w_0 t + {\pi \over 4})}+e^{-(j 2w_0 t + {\pi \over 4})}] $

$ x(t) = 1 + {1 \over 2j}[e^{j w_0 t}] + ({-1 \over 2j})e^{-j w_0 t} + {3 \over 2} [e^{j 2w_0 t}e^ {j{\pi \over 4}}]+ {3 \over 2}[e^{-j 2w_0 t} e^{-j{\pi \over 4}}] $

$ e^{j {\pi \over 4}} = {1 \over \sqrt{2}} + j{1 \over \sqrt{2}} $

$ e^{-j{\pi \over 4}} = {1 \over \sqrt{2}} - j{1 \over \sqrt{2}} $

$ x(t) = 1 + {1 \over 2j}[e^{j w_0 t}] + ({-1 \over 2j})e^{-j w_0 t} + {3 \over 2} [e^{j 2w_0 t}e^ {j{\pi \over 4}}]+ {3 \over 2}[e^{-j 2w_0 t} e^{-j{\pi \over 4}}] $

$ x(t) = 1 + {1 \over 2j}[e^{j w_0 t}] + ({-1 \over 2j})e^{-j w_0 t} + {3 \over 2}({1 \over \sqrt{2}} + j{1 \over \sqrt{2}}) [e^{j 2w_0 t}]+ {3 \over 2}({1 \over \sqrt{2}} - j{1 \over \sqrt{2}}) [e^{-j 2w_0 t}] $

Thus, the Fourier series coefficients for this are:

$ a_0 = 1 $

$ a_1 = {1 \over 2j} $

$ a_{-1} = {-1 \over 2j} $

$ a_2 = {3 \over 2}({1 \over \sqrt{2}} + j{1 \over \sqrt{2}}) $

$ a_{-2} = {3 \over 2}({1 \over \sqrt{2}} - j{1 \over \sqrt{2}}) $

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