With the help of Fourier analysis, prove that rectangular pulse consists of infinite number of frequencies

Fourier Analysis of Rectangular Pulse

To prove that a Rectangular Pulse consists of an infinite number of frequencies, first consider a rectangular pulse shown in the below figure:

To do a Fourier Analysis of this rectangular pulse let's find the Fourier transform of this signal.

The Equation of Fourier Transform is as follows:

$$ X (jw) = \int_{-\infty}^\infty x(t)e^{-jwt} dt \ \ \ \ ................(1)$$

$$ x(t) = \frac {1}{2π} \int_{-\infty}^\infty X(jw)e^{jwt} dw \ \ \ \ ..................(2)$$

Where

• $X (jω)$ is called the Fourier transform of the time function $x(t)$.

Whereas

• $x(t)$ is the inverse Fourier transform of $X =(jw)$

• The integral on the right-hand side of equation (1) is called the Fourier integral.

Therefore,

the Fourier transform of this rectangular signal according to equation (1) is as follows:

$$ X (jw) = \int_{-\frac{a}{2}}^\frac{a}{2} A e^{-jwt} dt $$

$$ = - \frac{A}{jw} \left(e^{-jw \frac{a}{2}} - e^{jw \frac{a}{2}}\right)$$

$$ = \frac{2A}{w} {e^{jw \frac{a}{2}} - e^{-jw \frac{a}{2}}\over 2j}$$

$$ = \frac{2A}{w} sin \frac{wa}{2}$$

$$ = aA = {sin \frac 12 wa\over \frac 12 wa}$$

Hence, we get

$$ X(jw) = {sin \frac 12 wa\over \frac 12 wa} \ \ \ \ ................(3)$$

Apart from the constant aA equation (3) has the following form of the function:

$$ sinc \ x = \frac{sin x}{x}$$

Where

$$ x = \frac 12 wa$$

The plot of this above function represent as follows:

Therefore, the plot of $ X (jw)$ as a function of $w$ is as shown in the below figure:

This is the required plot of the Fourier transform of the Rectangular Pulse shown in the first figure.

This proves that Rectangular Pulse consists of an infinite number of frequencies.