### 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.

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