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Question

# Consider the signal, x(t)=∞∑m=−∞δ(t−m3)+δ(t−2m3) Where Cn is the exponential Fourier series coefficient of signal x(t). Which of the following is/are correct?

A

C1=C3=C5=1.5

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B
x(t) is a periodic signal with time period 2.
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C

C2=C4=C6=0

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D
x(t) is a periodic signal with time period 23
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Solution

## The correct option is D x(t) is a periodic signal with time period 23(a,d) (xt)=∞∑m=−∞δ(t−m3)+δ(t−2m3) ∞∑m=−∞δ(t−m3) can be plotted as ∞∑m=−∞δ(t−2m3) can be plotted x(t)=∞∑m=−∞δ(t−m3)+δ(t−2m3) x(t) is periodic signal with, T=23 ω0=2π23=3π Cn=1TT∫0x(t)e−jnω0tdt Cn=322/3∫0[2δ(t)+δ(t−13)]e−jn(3π)tdt t2∫t1f(t)δ(t−t0)dt=f(t0)∀to∈(t1,t2) Cn=32[2+e−jnπ]=32[2+(−1)n] Cn=4.5forniseven Cn=1.5fornisodd x(t)=∞∑m=−∞δ(t−m3)+δ(t−2m3) T=23 ω0=2π23=3π Cn=1TT∫0x(t)e−jnω0tdt Cn=322/3∫0[2δ(t)+δ(t−13)]e−jn(3π)tdt Cn=32[2+e−jnπ]=32[2+(−1)n]

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