Fourier Transform of Unit Step Function

Signals and SystemsElectronics & ElectricalDigital Electronics

Fourier Transform

For a continuous-time function $x(t)$, the Fourier transform is defined as,

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

Fourier Transform of Unit Step Function

The unit step function is defined as,

$$\mathrm{u(t)=\begin{cases}1 & for\:t≥ 0\0 & for\:t< 0\end{cases}}$$

Because the unit step function is not absolutely integrable, thus its Fourier transform cannot be found directly.

In order to find the Fourier transform of the unit step function, express the unit step function in terms of signum function as

$$\mathrm{u(t)=\frac{1}{2}+\frac{1}{2}sgn(t)=\frac{1}{2}[1+sgn(t)]}$$

Given that

$$\mathrm{x(t)=u(t)=\frac{1}{2}[1+sgn(t)]}$$

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

$$\mathrm{F[u(t)]=X(\omega)=\int_{−\infty }^{\infty}x(t)e^{-j\omega t} dt=\int_{−\infty }^{\infty} u(t)e^{-j\omega t} dt}$$

$$\mathrm{\Rightarrow\:X(\omega)=\int_{−\infty }^{\infty} \frac{1}{2}[1+sgn(t)]e^{-j\omega t}dt}$$

$$\mathrm{\Rightarrow\:X(\omega)=\frac{1}{2}\left [ \int_{−\infty }^{\infty} 1 \cdot e^{-j\omega t} dt + \int_{−\infty }^{\infty} sgn(t) \cdot e^{-j\omega t} dt\right ]}$$

$$\mathrm{\Rightarrow\:X(\omega)=\frac{1}{2}\{ F[1]+ F[sgn(t)]\}}$$

The Fourier transform of the constant amplitude and the signum function is given by,

$$\mathrm{F[1]=2\pi\delta(\omega)\:\:and\:\:F[sgn(t)]=\frac{2}{j\omega}}$$

$$\mathrm{\therefore\:F[u(t)]=X(\omega)=\frac{1}{2}\left [2\pi\delta(\omega) + \frac{2}{j\omega}\right ] }$$

Therefore, the Fourier transform of the unit step function is,

$$\mathrm{F[u(t)]=\left (\pi\delta(\omega) + \frac{1}{j\omega}\right )}$$

Or, it can also be represented as,

$$\mathrm{u(t)\overset{FT}{\leftrightarrow}\left ( \pi\delta(\omega) +\frac{1}{j\omega}\right )}$$

Magnitude and phase representation of Fourier transform of the unit step function −

$$\mathrm{Magnitude,|X(\omega)|=\begin{cases}∞ & at \:\omega = 0\0 & at\:\omega= −\infty & \omega= \infty\end{cases}}$$

The graphical representation of the unit step function with its frequency spectrum is shown in the figure.

raja
Updated on 02-Dec-2021 11:58:46

Advertisements