A periodic signal is given by the equation x (t) = 2 + 4 cos (40 ? t - ?/5) + 3

Question

A periodic signal is given by the equation x (t) = 2 + 4 cos (40 ? t - ?/5) + 3 sin (60 ? t) + 4 cos (120 ? t - ?/3)

A. Determine the fundamental frequency w0, the period t0, and the coefficients in the representation X(t) = x0 + Re{(sigma)k=1N Xkejkwot} for the above input. Remember, you can do this problem without evaluating any integrals.

C. Now consider a new signal y(t) = x(t) + 10 cos (50 ? t - ?/6). How is the spectrum changed? Is y(t) periodic? If so, what is the period?

note: the ? marks are supposed to be pi signs

Explanation / Answer

Please ask if you have any doubt.I will help you.

a) we have 3 frequency components 20 Hz, 30 Hz and 60Hz.

The highest common factor is 10 Hz which is f0.

0 = (2)10 = 20.

t0 = 1/f0 = 0.1 second

The signal is given as

x(t) = 2 + 4 cos (40 t - /5) + 3 sin (60 t) + 4 cos (120 t - /3)

Now we get the coefficients as

X0 = 2 , X2 = 2e-j(/5) ,X3 = 1.5ej(/2), X5 = 2e-j(/3) . Rest of the co efficients are 0.

c) y(t) = x(t) + 10 cos (50 ? t - ?/6) . Just an extra component is added in the spectrum.

yes the signal remains periodic. but period changes.

the frequency components are 20, 25, 30 and 60 Hz.

Thus the common factor is 5 Hz which gives

Tperiod = 1/5Hz = 0.2 seconds.

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