Sinc Function

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.

What is Sinc Function?

Sinc function is often denoted as Sinc(x). This function is a non-periodic waveform with an interpolating graph. It is an even function with a unity area. It is popularly known as a sampling function and is widely used in signal processing and in the theory of Fourier Transforms.

Also known as sine cardinal, this function is commonly abbreviated as sinc and is defined as the ratio of sin(x) to x resulting in an oscillating graph. The value of the function at origin i.e x=0 is calculated using limits but overall the function is very useful for various analysis.

Sinc Function Graph

If we plot the graph representing the magnitude of the Sinc Function with time, the graph of the Sinc Function looks like this.

sinc(x)

As we observe, we can see that the graph Sinc Function oscillates very quickly . We also see that the is 0 for all integral values of time except at t=0, where it has a maximum value of π. This property is used largely to avoid intersymbol interference in Digital transmission systems.

If we carefully observe the shape of graph we see that the graph is symmetric about origin making it an Even function.

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.

Fourier Analysis of Sinc Function

How To Generate a Sinc Function ?

Anyone who knows tye representation of sinc Function can easily generate this function. These are the steps:

• Install the required software for representation of the signals. This may include softwares like MATLAB or programming language like python. You can either install it or work on the browser itself.
• Define the input range for which required sinc function should be generated. After this store the corresponding output values using the formula of sinc Function i.e sinc(x)=sin(x)/x.
• In order to see the output use plotting tools to plot the graph which will show the sinc function over the desired input range. The output will look like the graph given below.
  
  

  MATLAB output

Advantages of Sinc Function

The advantages of Sinc Function are stated below:

• Sinc Function form basis of signal analysis. They are crucial for understanding many concepts involved in signal processing thereby making it the base of electronics.
• Its even symmetric nature along with its oscillating property is very useful which is applied in various modulation schemes. As seen in the graph the function shows a maxima at origin and later shows local maxima’s and minima’s.
• Sinc Function are often used in applications where a Fourier analysis is required. For example they are used in study of Fourier analysis theory.
• It has theoretical application because of its ability to keep original signal intact. This is fulfilled when sampling frequency is at least twice the highest frequency in the signal i.e when Nyquist-Shannon sampling theorem holds true.
• Sinc Function is also used in DACs( Digital to Analog convertors). The reconstruction of original analog signals involves staircase waveforms. To obtain this in frequency domain the idealized spectrum is multiplied by the sinc function to produce the outputs.

Disadvantages of Sinc Function

The disadvantages of Sinc Function are stated below:

• The major disadvantage of sinc Function is that it that it oscillates between negative and positive infinity without dropping amplitude to 0. This can be a major problem when using it in computer applications due to infinite support.
• Due to limited features, these signals have limited applications in communication system. Other signals like sine wave are used more often due to the simplicity and features it offer.
• It has a very slow decay rate which can cause prolonged influence over time. Therefore it is not a practical choice for filters as it stops their ability to differentiate between relevant information.
• Another problem of sinc Function is the presence of side lobes. These lobes can interfere with the message signal in-case the sampling time instants are small resulting in complicated filtering tasks with noise.

Applications of Sinc Function

Sinc Function is used in various applications like:

• Sinc function is popularly used in signal analysis system as it is has certain frequency analysis properties which makes it a useful function. This include tasks like filtering and interpolation of signals in communication.
• This function is widely used to understand Fourier transform theory. Transmission signals employ the use of rectangular signals as their input signals. In order to perform and analyse this signal it is necessary to convert it from time domain to frequency domain. sinc function appears in the frequency domain representation of a rectangular pulse and helps to understand it.
• These signals are largely used in pulse shaping and interpolation . Due to the unique shape of sinc Function it helps to interpolation of signals. It reconstructs regular signals from sampled data which are further used in processing.
• It is employed in the study of optics in phenomenon’s like diffraction. The study of patterns from slit experiment throught rectangular apertures is done using sinc function.
• In general sinc function us widely used in communication systems during transmission and analysis because it helps in frequency domain representation of various time domain signals.

Conclusion

As we have seen Sinc Function play an important role in modern electronics. We have already discussed the unique graph of this signal along with other properties. As we discuss the uses of this signal we realize it is used in communication circuits, signal processing, Fourier Transforms, interpolation or for various other purposes. These applications signify the need to find appropriate methods to generate Sinc Function. There are different methods to generate this pulse and one method has been discussed with the readers. There are various other types of signals, each used for a different purpose.

FAQs on Sinc Function

What are Sinc Function?

The mathematical sinc function, also known as the cardinal sine function is the ratio of sin(x) to x.

What are MATLAB and why is it used?

MATLAB is a high programming online platform which integrates computation, visualization and programming to analyse and design systems different signals like sinc function.

What is Fourier Transform, what is the Fourier Transform of rectangular pulse?

Fourier Transform is a mathematical tool used for analysing the signals between two different domains, such as transforming signal from frequency domain and vice-versa. Using computation is can be found that Fourier transform of rectangular pulse is Sinc function.

What are some other signals?

Some other basic signals are rectangular pulse, square pulse, signum signal and sine signal.