Fourier Transform of the signal x(t) = t/(1+t²)² is ___ | Complex Plane Integration

 

Fourier Transform of \(x(t)=\dfrac{t}{(1+t^2)^2}\)

We want to determine the Fourier transform of

$$ x(t)=\frac{t}{(1+t^2)^2} $$

using the Fourier transform definition

$$ X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt $$

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Step 1 :

Let

$$ x_1(t)=\frac{1}{1+t^2} $$

Then

$$ X_1(\omega)=\pi e^{-|\omega|}. $$

Proof:

$$ \int_{-\infty}^{\infty}\frac{1}{1+t^2}e^{-j\omega t}\,dt $$

Use the Residue Theorem.

$$ \int_{-\infty}^{\infty}\frac{1}{1+z^2}e^{-j\omega z}\,dz $$

Or,

$$ \int_{-\infty}^{\infty}\frac{1}{(z+j)(z-j)}e^{-j\omega z}\,dz $$

Poles are at \(z=-j\) and \(z=j\).

For the pole at \(z=-j\):

$$ \operatorname*{Res}_{z=-j}\left(\frac{e^{-j\omega z}}{(z+j)(z-j)}\right) = \lim_{z\to -j}(z+j)\frac{e^{-j\omega z}}{(z+j)(z-j)} = \frac{e^{-\omega}}{-2j}. $$

Therefore,

$$ -2\pi j\left(\frac{e^{-\omega}}{-2j}\right) = \pi e^{-\omega}. $$

For the pole at \(z=j\):

$$ \operatorname*{Res}_{z=j}\left(\frac{e^{-j\omega z}}{(z+j)(z-j)}\right) = \lim_{z\to j}(z-j)\frac{e^{-j\omega z}}{(z+j)(z-j)} = \frac{e^{\omega}}{2j}. $$

Therefore,

$$ 2\pi j\left(\frac{e^{\omega}}{2j}\right) = \pi e^{\omega}. $$

Step 2 : Differentiate the Time-Domain Function

Differentiate $$ \frac{1}{1+t^2} $$ with respect to \(t\): $$ \frac{d}{dt} \left( \frac{1}{1+t^2} \right) = -\frac{2t}{(1+t^2)^2} $$ Rearranging, $$ \boxed{ \frac{t}{(1+t^2)^2} = -\frac12 \frac{d}{dt} \left( \frac{1}{1+t^2} \right) } $$ Thus, the required function is simply one-half of the derivative of the known function.

Step 3 : Apply the Differentiation Property

The differentiation property of the Fourier transform is $$ \boxed{ \mathcal F \left\{ \frac{dx(t)}{dt} \right\} = j\omega X(\omega) } $$ Applying it, $$ \mathcal F \left\{ \frac{d}{dt} \left( \frac{1}{1+t^2} \right) \right\} = j\omega \left( \pi e^{-|\omega|} \right) $$

Step 4 : Multiply by \(-\frac12\)

Since $$ \frac{t}{(1+t^2)^2} = -\frac12 \frac{d}{dt} \left( \frac1{1+t^2} \right), $$ its Fourier transform becomes $$ X(\omega) = -\frac12 \left[ j\omega \pi e^{-|\omega|} \right] $$

Step 5 : Simplify

Collect the constants: $$ X(\omega) = -\frac{j\pi\omega}{2} e^{-|\omega|} $$

Final Answer

$$ \boxed{ \mathcal F \left\{ \frac{t}{(1+t^2)^2} \right\} = -\frac{j\pi\omega}{2} e^{-|\omega|} } $$

Or,

$$ \boxed{ \mathcal F \left\{ \frac{t}{(1+t^2)^2} \right\} = \frac{\pi\omega}{2j} e^{-|\omega|} } $$

Option (A)

Complex Plane Integration using Residue Theorem in Fourier Transform

Evaluating Fourier Transform integrals directly is often difficult. Complex Analysis provides a powerful technique called the Residue Theorem, which converts real integrals into contour integrals in the complex plane. This method is frequently used in GATE Signals and Systems.

Why Use the Residue Theorem?

For integrals of the form

$$ \int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt, $$

direct integration may be impossible. By replacing the real variable with a complex variable \(z\), the integral becomes easier to evaluate using contour integration.

Residue Theorem

If \(f(z)\) has isolated poles inside a closed contour \(C\), then

$$ \oint_C f(z)\,dz = 2\pi j \sum \text{Residues inside }C. $$

GATE Tip: Select the contour carefully.

• \(\omega>0\) → Lower Half Plane
• \(\omega<0\) → Upper Half Plane

Fourier Transform Example

Find the Fourier Transform of

$$ x(t)=\frac{1}{1+t^2} $$

Fourier Transform:

$$ X(\omega) = \int_{-\infty}^{\infty} \frac{e^{-j\omega t}} {1+t^2} dt $$

The poles are located at

$$ z=\pm j. $$

Choose the contour depending on the sign of \(\omega\), compute the enclosed residue, and apply the Residue Theorem.

The final result is

$$ \boxed{ X(\omega)=\pi e^{-|\omega|} } $$

Exam Tips

• Always identify the poles first.
• Choose the contour based on the sign of \(\omega\).
• Only enclosed poles contribute.
• Use Jordan's Lemma for Fourier integrals.

Conclusion

The Residue Theorem is one of the fastest methods for evaluating difficult Fourier Transform integrals. Although contour selection requires practice, mastering this technique is valuable for GATE, ESE, and advanced Signals and Systems courses.

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