Question

The Fourier transform of the signal x(t) = 1/πt is​

Answer 1

Answer:

What is the Fourier transform of 1πt ?

This function i.e H(t)=1πt is known as Hilbert function which provides a 90 degree of phase shift to the input signal.

And if we take Hilbert transform of the Dirac-delta function , δ(t) we will have H(t)=1πt.

Now to calculate it’s Fourier transform we will use Duality property , which is …

If ⟹f(t)⇔F(ω)

Then ⟹F(t)⇔2πf(−ω)

Now let take this signal x(t)=sgn(t) , it has a FT X(ω)=2jω

Now using duality..

sgn(t)⇔2jω

2jt⇔2π.sgn(−ω)

1πt⇔−j.sgn(ω)(1)

1πt⇔1j.sgn(

Step-by-step explanation:

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Answer 2

Given,

x(t)=1/πt

To Find,

The Fourier transform of the signal x(t)=1/πt.

Solution,

This function i.e. x(t)=1/πt is known as Hilbert function which provides a 90 degree of phase shift to the input signal.

Here if we take Hilbert transform of the Dirac-delta function, delta(t) we will have x(t)=1/πt

Now to calculate it's Fourler transform we will use Duallty property.

If, f(t)=F(ω)

Then, f(t)=2πf(-ω)

Now lets take this signal x(t)=sgn(t).

It has a FT x(ω)=2jω

Now using duallty property,

sgn(t)=2jω

2jt=2π.sgn(-ω)

1πt=j.sgn(-ω)(1)

1πt=1j.sgn(ω)

Hence, the fourier transform of the signal x(t)=1/πt is 1πt=1j.sgn(ω).