Mathematical Expression of Sinc Function

Since sinc function is a ratio we can define it mathematically as

Sinc(x)= 1 at x=0

otherwise,

Sinc(x)=sin(πx)/(πx)

The value of function at x=0 is calculated by L’Hôpital’s rule and is equal to 1. It is also important to note that the integral of function from -∞ to +∞ is π.

∫ sinc(x)dx = π

Fourier Transform

Let us see how we can get a frequency domain function from a time domain function and a time-domain function from a frequency-domain signal.

X(ω) = ∫x(t)e^−jωt dt Forward transform

x(t) = 1/2π ∫ X(ω)e^jωt dω Inverse transform

If X(ω) is the Fourier transform of a signal x(t) meaning x(t) is the inverse transform of X(ω)

x(t) ←→ X(ω)

x(t)= FT ←→ X(ω)

This means we can easily analyse a signal in frequency and time domain of either form is given to us

Fourier Analysis of Sinc Function

The main thing that makes Sinc Function a milestone in communication is its Fourier Transform. The Fourier transform of sinc function is rectangular pulse and a rectangular shape in the frequency domain is the idealized “brick-wall” filter response. This makes sinc(x) as the impulse response of an ideal low-pass filter.

Sinc Function is an important tool in the electronic industry. They are ubiquitous in modern electronics and are almost used in every daily appliance for analysis of various circuits working. Sinc Function is used in numerous electronic devices and systems, contributing to their design, analysis, and performance optimization.

In this Article, We will be going through the Sinc Function, First, we will start our Article with the Definition of the Sinc Function, Then we will go through the Mathematical Expression of the Sinc Function, then we will see how to generate Sinc Function. At last, We will conclude our article with Advantages, Disadvantages, Applications, and Some FAQs.