Calculate Fourier Transform of x(t): \frac{1}{(1+t^2)} & abs(t)

In summary, the Fourier transform of x(t) = \frac{1}{(1+t^2)} is X(f) = \int{x(t)*e^{-j2\pi{}ft} dt}. To solve the integral, you will need to use integration by parts. For the second signal, x(t) = abs(t) for -2\leq{}t\leq{}2, splitting it into two integrals is the correct approach for finding the Fourier transform.
  • #1
razorwind
1
0
Determine the Fourier transform of the following signals: [tex]x(t) = \frac{1}{(1+t^2)}[/tex]

I start off by doing [tex]X(f) = \int{x(t)*e^{-j2\pi{}ft} dt}[/tex]

So i plug in x(t) into that equation, but I'm lost as to how to integrate it. Am i going in the right direction?

Also: x(t)=abs(t) for [tex]-2\leq{}t\leq{}2[/tex]

I am splitting it to -t for [tex]-2\leq{}t\leq{}0[/tex] and t for [tex]0\leq{}t\leq{}2[/tex] then doing the transform with the equation above. Is it allowed to split it into 2 integrals? So it ends up being [tex]X(f) = \int{-t*e^{-j2\pi{}ft} dt}[/tex] + [tex]\int{t*e^{-j2\pi{}ft} dt}[/tex] and I simplify from there.
 
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  • #2
Is this the right approach?Yes, this is the correct approach. You will need to use integration by parts to solve the integral for the first signal. For the second signal, splitting it into two integrals is the way to go.
 
  • #3


Yes, you are on the right track. To integrate x(t) = 1/(1+t^2), you can use the substitution method. Let u = t^2 + 1, then du = 2t dt. So the integral becomes:

X(f) = \frac{1}{2}\int{\frac{1}{u}e^{-j2\pi{}fu} du}

Using the fact that \int{\frac{1}{u} e^{-j2\pi{}fu} du} = \frac{e^{-j2\pi{}fu}}{-j2\pi{}f}, we can write:

X(f) = \frac{1}{2}\cdot \frac{e^{-j2\pi{}fu}}{-j2\pi{}f} = -\frac{1}{2j\pi{}f}\cdot e^{-j2\pi{}fu}

To calculate the Fourier transform of x(t) = abs(t), you are correct in splitting it into two integrals. This is allowed because the Fourier transform is a linear operation. So your final answer would be:

X(f) = -\frac{1}{2j\pi{}f}\cdot e^{-j2\pi{}fu} + \frac{1}{2j\pi{}f}\cdot e^{j2\pi{}fu}

= \frac{1}{j\pi{}f}\cdot sin(2\pi{}fu)

= \frac{2}{j\pi{}f}\cdot sin(\pi{}fu) \cdot cos(\pi{}fu)

= \frac{2}{j\pi{}f}\cdot sin(\pi{}fu) \cdot \frac{1}{2}\cdot \left(e^{j\pi{}fu} + e^{-j\pi{}fu}\right)

= \frac{1}{j\pi{}f}\cdot sin(\pi{}fu) \cdot \left(e^{j\pi{}fu} - e^{-j\pi{}fu}\right)

= \frac{2}{j\pi{}f}\cdot sin(\pi{}fu) \cdot sin(\pi{}fu)

= \frac{4}{j\pi{}f}\cdot sin^2(\pi{}fu)

= \frac{4}{j\pi{}f}\cdot \frac{1}{2}\cdot \left(1 - cos(2\
 

Related to Calculate Fourier Transform of x(t): \frac{1}{(1+t^2)} & abs(t)

What is the Fourier Transform?

The Fourier Transform is a mathematical operation that decomposes a function or signal into its constituent frequencies. It is used in many fields, including mathematics, physics, engineering, and signal processing.

What is the formula for the Fourier Transform?

The formula for the Fourier Transform is F(k) = ∫f(x)e-2πikx dx, where f(x) is the original function, k is the frequency, and i is the imaginary unit.

How do you calculate the Fourier Transform?

The Fourier Transform can be calculated using the formula F(k) = ∫f(x)e-2πikx dx or by using a Fourier Transform calculator or software. The process involves breaking down the original function into its sine and cosine components at different frequencies.

What is the Fourier Transform of the given function?

The Fourier Transform of the given function, x(t) = 1/(1+t^2) & abs(t), is F(k) = √(π/2)e-|k|, which represents a decaying exponential function.

What are the applications of the Fourier Transform?

The Fourier Transform has many applications, including signal processing, image processing, data compression, and solving differential equations. It is also used in fields such as optics, acoustics, and quantum mechanics.

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