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Question

Let x(t) be a periodic signal with time period T. Let y(t)=x(tt0)+x(t+t0) for some to.
The Fourier series coefficients of y(t) are denoted by bk. If bk=0 for all odd K. Thent0 can be equal to

A
T4
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B
T8
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C
2T
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D
T2
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Solution

The correct option is A T4
y(t)=x(tt0)+x(t+x0)

Since, x(t) is periodic with period T.
Therefore, x(tt0) and x(t+t0) will also be periodic with perriod T.

bk=akejkω0t0+akejkω0t0
(By property)
ak is Fourier series coefficient of signal x(t) therefore,

bk=ak[ejkω0t0+ejkω0t0]

bk=2akcoskω0t0

Since, bk=0 for odd k

i.e., kω0t0=kπ2,

where k=±1,±3,±5...

ω0t0=π2

t0=π2×T2π=T4

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