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Fourier Transform of Rectangular Function
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 Rectangular Function
Consider a rectangular function as shown in Figure-1.
It is defined as,
$$\mathrm{rect\left(\frac{t}{τ}\right)=\prod\left(\frac{t}{τ}\right)=\begin{cases}1 & for\:|t|≤ \left(\frac{τ}{2}\right)\0 & otherwise\end{cases}}$$
Given that
$$\mathrm{x(t)=\prod\left(\frac{t}{τ}\right)}$$
Hence, from the definition of Fourier transform, we have,
$$\mathrm{F\left[\prod\left(\frac{t}{τ}\right) \right]=X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}\:dt=\int_{−\infty}^{\infty}\prod\left(\frac{t}{τ}\right)e^{-j\omega t}\:dt}$$
$$\mathrm{\Rightarrow\:X(\omega)=\int_{−(τ/2)}^{(τ/2)}1\cdot e^{-j\omega t}\:dt=\left[\frac{e^{-j\omega t}}{-j\omega} \right]_{-τ/2}^{τ/2}}$$
$$\mathrm{\Rightarrow\:X(\omega)=\left[ \frac{e^{-j\omega (τ/2)}-e^{j\omega (τ/2)}}{-j\omega}\right]=\left[ \frac{e^{j\omega (τ/2)}-e^{-j\omega (τ/2)}}{j\omega }\right]}$$
$$\mathrm{\Rightarrow\:X(\omega)=\left[ \frac{2τ[e^{j\omega (τ/2)}-e^{-j\omega (τ/2)}]}{j\omega\cdot (2τ) }\right]=\frac{τ}{\omega(τ/2)}\left[\frac{e^{j\omega (τ/2)}-e^{-j\omega (τ/2)}}{2j} \right]}$$
$$\mathrm{\because \:\left[\frac{e^{j\omega (τ/2)}-e^{-j\omega (τ/2)}}{2j} \right]=sin\:\omega (τ/2)}$$
$$\mathrm{\therefore\:X(\omega)=\frac{τ}{\omega(τ/2)}\cdot sin \omega (τ/2)=τ \left[\frac{sin\omega (τ/2)}{\omega (τ/2)}\right]}$$
$$\mathrm{\because\:sinc \left(\frac{\omega τ}{2}\right)=\frac{sin\omega (τ/2)}{\omega (τ/2)}}$$
$$\mathrm{\therefore\:X(\omega)=τ\cdot sinc \left(\frac{\omega τ}{2}\right)}$$
Therefore, the Fourier transform of the rectangular function is
$$\mathrm{F\left[\prod\left(\frac{t}{τ}\right)\right]=τ\cdot sinc \left(\frac{\omega τ}{2}\right)}$$
Or, it can also be represented as,
$$\mathrm{\prod\left(\frac{t}{τ}\right) \overset{FT}{\leftrightarrow}τ\cdot sinc \left(\frac{\omega τ}{2}\right)}$$
Magnitude and phase spectrum of Fourier transform of the rectangular function
The magnitude spectrum of the rectangular function is obtained as −
At $\omega=0$:
$$\mathrm{sinc\left(\frac{\omega τ}{2}\right)=1;\:\:\Rightarrow|X(\omega)|=τ}$$
At $\left(\frac{\omega τ}{2}\right)=± n\pi$ i.e., at
$$\mathrm{\omega=±\frac{2n\pi}{τ},\:\:n=1,2,2,3,...}$$
$$\mathrm{sinc\left(\frac{\omega τ}{2}\right)=0}$$
The phase spectrum is obtained as −
$$\mathrm{\angle\:X(\omega)=\begin{cases}0 & if\:sinc\:\left(\frac{\omega τ}{2}\right)>0\±\pi & if\:sinc\:\left(\frac{\omega τ}{2}\right)<0 \end{cases}}$$
The frequency spectrum of the rectangular function is shown in Figure-2.
Note
The magnitude response between the first two zero crossings is known as the main lobe.
The portions of the magnitude response for $\omega\:< -\left( \frac{-2\pi}{τ}\right)$ and $\omega > \left( \frac{2\pi}{τ}\right)$are known as the side lobes.
From the magnitude spectrum, it is clear that the majority of the energy of the signal is contained in the main lobe.
The main lobe becomes narrower with the increase in the width of the rectangular pulse.
The phase spectrum of the rectangular function is an odd function of the frequency (ω).
When the magnitude spectrum is positive, then the phase is zero and if the magnitude spectrum is negative, then the phase is $(±\pi)$.
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