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Which of the following cannot be the Fourier series expansion of a periodic signal?
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- x(t) = 2 cos 6t + 3 cos 3t
- x(t) = 2 cos πt + 7 cos t
- x(t) = cos t + 0.5
- x(t) = 2 cos 1.5πt + sin 3.5πt
Correct Option: B
To solve such type of problem solve each and every options one by one.
(A) x(t) = 2 cos 6t + 3 cos 3t
Fundamental period of
| x1(t)= T1 = | = | |||
| 6 | 3 |
Fundamental period of
| x2(t)= T2 = | = | |||
| 3 | 3 |
For the signal x(t) to be periodic ratio
| = rational number = | = | |||
| T1 | 3 | 1 | ||
| T2 | 2 | |||
| 3 |
which is rational number
i.e., Fourier series expansion
(B) x(t) = 2 cos πt + 7 cos t
| T1 = | = | = 2 | ||
| ω | π |
| T1 = | = | = 2π | ||
| ω | 1 |
| = | = | = irrational numbers | ||||
| T2 | 2π | π |
i.e., Fourier series expansion is not possible.
Hence, alternative (B) is the correct choice, and no need to solve further other option.
