A signal x(t) is composed by adding three sinusoids is given by the following eq

Question

A signal x(t) is composed by adding three sinusoids is given by the following equations: x_1(t) = 10 cos (800Pie t + Pie/4) x_2(t) = 7 cos (1200Pie t - Pie/3) x_3(t) = -3 cos (1600Pie t) The output from summer is connected to the oscilloscope for spectrum analysis. Do the following: Sketch the spectrum of the signal indicating the complex size of each frequency component. Is x(t) periodic If so, what is the period lfwe add another signal to the input : x_4(t) = 5 cos (800Pie t + Pie/2) how is the spectrum changed Is the composite periodic If so, what is the period

Explanation / Answer

High-frequency carrier wave
vc(t) = cos(2¼fct); fc = 200 Hz
Amplitude modulation (AM) of carrier (electromagnetic) wave:
x(t) = v(t) cos(2¼fct)

Frequency contents of transmitted signal
x(t) = v(t) cos(2¼fct)
= (5 + 2 cos(2¼f¢t)) cos(2¼fct)
= 5 cos(2¼fct) + 2 cos(2¼f¢t) cos(2¼fct)
Trigonometric identity:
cos ® cos ¯ =
1
2
cos(® ¡ ¯) +
1
2
cos(® + ¯)
)
2 cos(2¼f¢t) cos(2¼fct) = cos (2¼(fc ¡ f¢)t)+cos (2¼(fc + f¢)t)

x(t) = 5 cos(2¼fct) + cos (2¼(fc ¡ f¢)t) + cos (2¼(fc + f¢)t)

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