Question

If Fourier transform of e-\x=21+p2, then find
the fourier transform of t2e-xl.
Select one:
A. -21+p2
B. -41+p2
C. 41+p2
D. 21+p2​

Answer 1

Answer:

$F(\omega)=e^{-\frac{\omega^2}{2}}$

Step-by-step explanation:

Step 1: A Fourier transform (FT) is a mathematical transformation that breaks down functions into frequency components. The result of the transform is shown as a function of frequency. Most frequently, functions of time or space are converted, and the result will depend on the frequency of time or space, respectively.

Step 2: Another name for the procedure is analysis. Decomposing the waveform of a musical chord into terms of the strength of its individual pitches is an example of application. The mathematical procedure that links the frequency domain representation to a function of space or time is referred to as the "Fourier transform," as well as the frequency domain representation itself.

Step 3:

$\begin{aligned}& \frac{d[f(x)]}{d x} \stackrel{\text { Fourier transform }}{\longleftrightarrow} \\& \longleftrightarrow f(x) \stackrel{\text { Fourier transform }}{\longleftrightarrow} \frac{j d[F(\omega)]}{d \omega}\end{aligned}$

Where $F(\omega)$ is Fourier transform of f(x)

We are given $f(x)=e^{-\frac{x^2}{2}}$

$\frac{d[f(x)]}{d x}=-x \cdot e^{-\frac{x^2}{2}}$........(1)

Take Fourier transform of above equation (1)

$\begin{aligned}& j \omega F(\omega)=\frac{-j d[F(\omega)]}{d \omega} \\& \Rightarrow \frac{1}{F(\omega)} \cdot d[F(\omega)]=-\omega d \omega\end{aligned}$.........(2)

By integrating the above equation, we get

$\begin{aligned}& \ln [F(\omega)]=-\frac{\omega^2}{2} \\& \Rightarrow F(\omega)=e^{-\frac{\omega^2}{2}}\end{aligned}$

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