Fourier Transform of Two-Sided Real Exponential Functions

Fourier Transform

The Fourier transform of a continuous-time function $x(t)$ can be defined as,

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

Fourier Transform of Two-Sided Real Exponential Function

Let a two-sided real exponential function as,

$$\mathrm{x(t)\:=\:e^{-a|t|}}$$

The two-sided or double-sided real exponential function is defined as,

$$\mathrm{e^{-a|t|}\:=\:\begin{cases}e^{at} \:\: for\:t\: \leq\: 0\\\\e^{-at} \:\: for\:t \:\geq \:0 \end{cases} =e^{at}u(-t)+e^{-at}u(t) }$$

Where, the functions $u(t)$ and $u(-t)$ are the unit step function and time reversed unit step function, respectively.

Now, from the definition of Fourier transform, we have,

$$\mathrm{X(\omega)\:=\:\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\:=\:\int_{-\infty}^{\infty}e^{-a|t|}e^{-j\omega t}dt}$$

$$\mathrm{\Rightarrow\:X(\omega)\:=\:\int_{-\infty}^{\infty}[e^{at}u(-t)\:+\:e^{-at}u(t)]e^{-j\omega t}dt}$$

$$\mathrm{\Rightarrow\:X(\omega)\:=\:\int_{-\infty}^{0}e^{at}e^{-j\omega t}dt\:+\:\int_{0}^{\infty}e^{-at}e^{-j\omega t}dt}$$

$$\mathrm{\Rightarrow\:X(\omega)\:=\:\int_{-\infty}^{0}e^{(a\:-\:j\omega)t}dt\:+\:\int_{0}^{\infty}e^{-(a\:+\:j\omega)t}dt }$$

$$\mathrm{\Rightarrow\:X(\omega)\:=\:\int_{0}^{\infty}e^{-(a\:-\:j\omega)t}dt\:+\:\int_{0}^{\infty}e^{-(a\:+\:j\omega)t}dt }$$

$$\mathrm{\Rightarrow\:X(\omega) \:=\: \left[\frac{e^{-(a\:-\:j\omega)t}}{-(a\:-\:j\omega)}\right]_{0}^{\infty} \:+\: \left[\frac{e^{-(a\:+\:j\omega)t}}{-(a\:+\:j\omega)}\right]_{0}^{\infty}\:=\:\left[\frac{e^{-\infty}\:-\:e^{0}}{-(a\:-\:j\omega)} \right] \:+\:\left[\frac{e^{-\infty}\:-\:e^{0}}{-(a\:+\:j\omega)} \right]}$$

$$\mathrm{\Rightarrow\:X(\omega) \:=\:\frac{1}{a\:-\:j\omega} \:+\:\frac{1}{a\:+\:j\omega} \:=\:\frac{2a}{a^{2} \:+\: \omega^{2}}}$$

Therefore, the Fourier transform of a two-sided real exponential function is,

$$\mathrm{F[e^{-a|t|}]\:=\:X(\omega)\:=\:\frac{2a}{a^{2}\:+\:\omega^{2}}}$$

Or, it can also be represented as,

$$\mathrm{e^{-a|t|}\overset{FT}{\leftrightarrow}\frac{2a}{a^{2}\:+\:\omega^{2}}}$$

Magnitude and phase representation of Fourier transform of the two-sided real exponential function −

$$\mathrm{Magnitude,\:|X(\omega)|\:=\:\frac{2a}{a^{2}\:+\:\omega^{2}};\:\:for\:all\:\omega}$$

$$\mathrm{Phase,\:\angle X(\omega)\:=\:0;\:\:for\:all\:\omega}$$

The graphical representation of the two-sided real exponential function with its magnitude and phase spectrum is shown in the figure.