Fourier Transform of Single-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 One-Sided Real Exponential Function

A single-sided real exponential function is defined as,

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

Where, $u(t)$ is the unit step signal and is defined as,

$$\mathrm{u(t)\:=\:\begin{cases}1 \:\: for\:t \:\geq\: 0 \\\\0 \:\: for\:t\: \lt\: 0 \end{cases}}$$

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

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

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

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

$$\mathrm{\Rightarrow\:X(\omega)\:=\:\frac{1}{-(a\:+\:j\omega)}[e^{-\infty}\:-\:e^{0}] \:=\:\frac{0\:-\:1}{- (a\:+\:j\omega)} \:=\:\frac{1}{a\:+\:j\omega}}$$

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

$$\mathrm{F[e^{-at}u(t)]\:=\:\frac{1}{a\:+\:j\omega}}$$

Or, it can also be represented as,

$$\mathrm{e^{-at}u(t)\overset{FT}{\leftrightarrow}\frac{1}{a\:+\:j\omega}}$$

Magnitude and phase representation of the Fourier transform of a single-sided real exponential function 

The Fourier transform of the one sided real exponential function is given by,

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

Multiplying it by the rationalising factor, we get,

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

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

Therefore, the magnitude and phase of Fourier series of single sided exponential function is given by,

$$\mathrm{Magnitude,\: |X(\omega)|\:=\:\frac{1}{\sqrt{a^{2}\:+\:\omega^{2}}};\:\:for\:all\:\omega}$$

$$\mathrm{Phase,\:\angle X(\omega)\:=\:-tan^{-1}\left(\frac{\omega}{a}\right);\:\:for\:all\:\omega}$$

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