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The Fourier transform is a mathematical function that decomposes a waveform that is a function of time into the frequencies that make it up. The result produced by the Fourier transform is a complex valued function of frequency.

The Fourier transform is given by$\mathit{A}\left(k\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{π}}{\mathbf{∫}}_{\mathbf{-}\mathbf{∞}}^{\mathbf{∞}}\mathit{ψ}\left(x\right){\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{k}\mathbf{x}}\mathit{d}\mathit{x}$.

The Fourier transform of the function$D\left(t\right)$is given by:

$$\begin{gathered} A\left(\mathrm{Ó\neg }\right)=\frac{1}{2\mathrm{Ï€}}{∫}_{-∞}^{∞}D\left(t\right)e^{-i\mathrm{Ó\neg }t}dt \\ =\frac{1}{2\mathrm{Ï€}}{∫}_{-∞}^{∞}C{e}^{-i\mathrm{Ó\neg }t}dt \\ =\frac{C}{2\mathrm{Ï€}}{∫}_{-∞}^0{e}^{\frac{t}{t}}{e}^{-i\mathrm{Ó\neg }t}dt+\frac{C}{2\mathrm{Ï€}}{∫}_0^{∞}{e}^{\frac{t}{t}}{e}^{-i\mathrm{Ó\neg }t}dt \\ =\frac{C}{2\mathrm{Ï€}}{∫}_{-∞}^0{e}^{t\left(\frac{1}{t}-i\mathrm{Ó\neg }\right)}dt+\frac{C}{2\mathrm{Ï€}}{∫}_0^{∞}{e}^{t\left(\frac{1}{t}-i\mathrm{Ó\neg }\right)}dt \end{gathered}$$

Solve the above equation further:

$$\begin{gathered} A\left(\mathrm{Ó\neg }\right)==\frac{C}{2\mathrm{Ï€}}\left(\frac{1}{{\displaystyle \frac{1}{t}}-i\mathrm{Ó\neg }}+\frac{1}{{\displaystyle \frac{1}{t}}+i\mathrm{Ó\neg }}\right) \\ =\frac{C}{2\mathrm{Ï€}}\left(\frac{2}{1+{\left(t\mathrm{Ó\neg }\right)}^2}\right) \end{gathered}$$

Thus, the Fourier transform is $A\left(\mathrm{Ó\neg }\right)=\frac{Ct}{2\mathrm{Ï€}}\left(\frac{2}{1+{\left(t\mathrm{Ó\neg }\right)}^2}\right)$.

This is not an oscillatory function but an even function whose maximum occurs at$\mathrm{Ó\neg }=0$such that$A\left(\mathrm{Ó\neg }\right)=\frac{Ct}{\mathrm{Ï€}}$. As there is an inverse relation between$∆\mathrm{Ó\neg }$and time interval$∆t$. So, the value of$A\left(\mathrm{Ó\neg }\right)$approaches 0 as$t→Â\pm ∞$. So, the graph can be obtained as:

The width of the function $A\left(\mathrm{Ó\neg }\right)$is approximately equal to the distance between the points of half maximum. At $\mathrm{Ó\neg }t=Â\pm 1$, the half maximum occurs, which implies $\mathrm{Ó\neg }=Â\pm \frac{1}{t}$. So, the value of $A\left(w\right)$is directly proportional to $t$. The function $D\left(t\right)$is exponentially decaying function is direct proportionality to $t$.

Thus, $A\left(w\right)$is inversely proportional to $t$and $D\left(t\right)$is directly proportional to $t$.