The sinc function is the shape hiding inside every digital recording
Divide the sine of x by x and you get a wave that swells to a peak of 1 and then ripples away in both directions. That humble curve, the sinc function, is how engineers rebuild a smooth signal from evenly spaced samples, and it defines the perfect low-pass filter.
There are two versions. The older, unnormalised sinc divides sin x by x and is sometimes called the sampling function. The normalised form, standard in digital signal processing and information theory, divides the sine of pi times x by pi times x. The only difference is a stretch of the horizontal axis by a factor of pi. Both are undefined at zero on paper, but the limit there is exactly 1, so that value is filled in and the function becomes smooth everywhere, what mathematicians call entire.
The name is a clipped form of the Latin sinus cardinalis, cardinal sine. Philip Woodward and I. L. Davies introduced it in a 1952 paper on information theory in telecommunication, remarking that the function turned up so often in Fourier analysis that it deserved its own notation. Woodward used it again in his 1953 book on probability and radar. The function had first been written in this form much earlier by Lord Rayleigh, in his formula for a spherical Bessel function.
Normalisation brings tidy properties. The area under the normalised curve over the whole real line equals 1, against pi for the older version, and its zeros land exactly on the nonzero integers, whereas the unnormalised curve crosses zero at nonzero multiples of pi. Its peaks and troughs sit precisely where the curve meets the cosine. It can also be written as an infinite product and ties to the gamma function through Euler's reflection formula.
The engineering payoff comes from the Fourier transform. Transform the normalised sinc and you get the rectangular function, equal to 1 between minus a half and a half and zero elsewhere. That makes the sinc filter the ideal brick-wall low-pass filter, passing low frequencies untouched and cutting higher ones completely, and it underlies the reconstruction of a continuous band-limited signal from uniform samples.
Source: Sinc function
