Friday, February 6, 2015

The Sinc Function

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss many mathematical functions, from common ones like the sine function and the exponential function to less familiar ones like Bessel functions and the error function. A simple but important example is the sinc function.

Sinc(x) is defined as sin(x)/x. It is zero wherever sin(x) is zero (where x is a multiple of π), except at x = 0, where sinc is one. The shape of the sinc function is a central peak surrounded by oscillations with decaying amplitude.

The most important property of the sinc function is that it is the Fourier transform of a square pulse. In Chapter 18 about magnetic resonance imaging, a slice of a sample is selected by turning on a magnetic field gradient, so the Larmor frequencies of the hydrogen atoms depend on location. To select a uniform slice, you need to excite hydrogen atoms with a uniform range of Larmor frequencies. The radio-frequency pulse you must apply is specified by its Fourier transform. It is an oscillation at the central Larmor frequency, with an amplitude modulated by a sinc function.

When you integrate sinc(x), you get a new special function that Russ and I never discuss: the sine integral function, Si(x)

This function looks like a step function, but with oscillations. As x goes to infinity the sine integral approaches π/2. It is odd, so as x goes to minus infinity it approaches –π/2.

The sinc function and the sine integral function resemble the Dirac delta function and the Heaviside step function. In fact, sinc(x/a)/a gets taller and taller, and the side lobes fall off faster and faster, as a approaches zero; it becomes the delta function.Similarly, the sine integral function becomes--to within a constant term, π/2--the step function.

Special functions often have interesting and beautiful properties. As I noted earlier, if you integrate sinc(x) from zero to infinity you get π/2. However, if you integrate the square of sinc(x) from zero to infinity you get the same result: π/2. These two functions are different: sinc(x) oscillates between negative and positive values, so its integral oscillates from above π/2 to below π/2, as shown above; sinc2(x) is always positive, so its integral grows monotonically to its asymptotic value. But as you extend the integral to infinity, the area under these two curves is exactly the same! I’m not sure there is any physical significance to this property, but it is certainly a fun fact to know.

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