What Is the Sinc Function and Why Is It Important in Electrical Engineering?

The Sinc Function and Its Normalized Variant

The mathematical formulation of the sinc function, also known as the cardinal sine function, is written as follows:

 

\[\text{sinc}(x)=\frac{\sin(x)}{x}\]

 

The function shown above is undefined for x = 0, and consequently we need to define sinc(0) based on the limit as x approaches 0, which is 1. Thus,

 

\[\text{sinc}(x)=\begin{cases}1 & \text{for } x = 0\\\frac{\sin(x)}{x} & \text{otherwise}\end{cases}\]

 

In the context of digital signal processing, we often use an alternative form in which the independent variable is multiplied by π:

 

\[\text{sinc}_\pi (x)\equiv\text{sinc}(\pi x)=\begin{cases}1 & \text{for } x = 0\\\frac{\sin(\pi x)}{\pi x} & \text{otherwise}\end{cases}\]

 

This second form is called the normalized sinc function, because the definite integral over the entire range of x is equal to 1:

 

\[\int_{-\infty}^{\infty}\text{sinc}_\pi(x)dx=1\]

 

I mentioned that the normalized version is common in digital signal processing. When we’re dealing with discrete data instead of a continuous-time variable, the normalization is expressed as follows:

 

\[\sum_{n=-\infty}^{\infty}\text{sinc}_\pi[n]=1 \]

 

The following plot shows the shape of the sinc function, and it also conveys the difference created when we multiply the independent variable by π.

 

The Sinc Function in Signal Processing

The Fourier transform of the sinc function is a rectangle centered on ω = 0. This gives sinc(x) a special place in the realm of signal processing, because a rectangular shape in the frequency domain is the idealized “brick-wall” filter response. In other words, sinc(x) is the impulse response of an ideal low-pass filter.

The use of the sinc function in filtering applications is more apparent in the digital domain. The following diagram illustrates the similarity between the impulse response of a FIR filter and a plot of sinc(x).

 

 

The Fourier transform of the sinc function is a rectangle, and the Fourier transform of a rectangular pulse is a sinc function. If we need to shorten a discrete-time signal for the purpose of spectral analysis, we can multiply it by a rectangular window, and this operation is equivalent to convolving the Fourier transform of the signal with a sinc function.

The sinc function also appears in analysis of digital-to-analog conversion. An idealized reconstruction of an analog signal is a sequence of impulses, but real-life DACs produce “staircase” waveforms by applying a zeroth-order hold to the output samples. In the frequency-domain, the zeroth-order hold results in an output spectrum that is equal to the idealized spectrum multiplied by the sinc function.