Question
Download Solution PDFFourier Transform of sgn(t) is _____, where sgn represents signum function.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Fourier Transform:
It is used for frequency analysis of any Bounded Input and Bounded Output (BIBO) signal.
Fourier Transform for any function x(t) is given by
\(X\left( w \right) = \mathop \smallint \limits_{ - \infty }^{ + \infty } x\left( t \right){e^{ - jwt}}dt\)
Calculation:
x(t) = sgn(t) = 1, t > 0
0, t = 0
-1, t < 0
also sgn(t) = u(t) - u(-t)
This signal is not absolutely integrable so we calculate Fourier Transform of sgn(t) as a limiting case of the sum of exponential e-atu(t) - eatu(t) as a → 0.
x(t) = sgn(t) = e-atu(t) - eatu(t)
Taking Fourier transform of the above equation:
\(X(\omega)=[\frac{1}{a \ + \ j\omega} \ - \ \frac{1}{a \ - \ j\omega}] \)
\(X(\omega)=[\frac{-2j\omega}{a^2 \ + \ \omega^2}]\)
Now as a → 0
\(X(\omega)=\frac{2}{j\omega}\)
Hence option (4) is the correct answer.
Important Points
Inverse Fourier Transform for any function X(w) is given by
\(x\left( t \right) = \frac{1}{{2\pi }}\mathop \smallint \limits_{ - \infty }^{ + \infty } X\left( w \right){e^{jwt}}dw\)
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