Question
Download Solution PDFThe Fourier transform π(π) of the signal π₯(π‘) is given by
π(π) = 1, for |π| < π0
= 0, for |π| > π0
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(X(Ο)= \begin{cases}1, & \text { for }|Ο|<Ο_0 \\ 0, & \text { for }|Ο|>Ο_0\end{cases}\)
By taking inverse Fourier transform,
\( x(t)=\frac{\sin Ο_0 t}{\pi t} \)
\(x\left(\frac{\pi}{2 Ο_0}\right)=\frac{2 Ο_0}{\pi \times \pi} \sin Ο_0 \times \frac{\pi}{2 Ο_0}\) \(=\frac{2 Ο_0}{\pi^2} \sin \frac{\pi}{2}=\frac{2 Ο_0}{\pi^2}\)
So, option (C) and (D) are wrong.
\(x(0)=\underset{t \rightarrow 0}{L t} \frac{\sin Ο_0 t}{\pi t}=\) \(\underset{t \rightarrow 0}{L t} \frac{Ο_0 \cos Ο_0 t}{\pi}=\frac{Ο_0}{\pi}\)
So, x(0) ∝ ω0 Option (B) is wrong.
When ω0β → ∞, X(ω) will be a D.C signal and inverse Fourier transform of a D.C signal will be impulse signal.
So, option (A) is correct.
Hence, the correct option is (A).
Last updated on Feb 19, 2024
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