The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.

The sinc function: x ( t ) = sin ( a t ) t , a > 0 . As can be expected, its FCT behaves much like a step function, (2.20) X c ( ω ) = ( 2 π ) 1 / 2 ∫ 0 ∞ sin ( a t ) t cos ( ω t ) d t = ( π 2 ) 1 / 2 , if ω < a , = 1 2 ( π 2 ) 1 / 2 , if ω = a , = 0 , otherwise .

sinc(x) is the product of an oscillating signal sin(x) (of period 2π) and a monotonically decreasing function 1/x . Therefore, sinc(x) exhibits sinusoidal oscillations of period 2π, with amplitude decreasing contin- uously as 1/x. Its zero crossings are at all non-zero integer multiples of π.