Sinc function approximation

The sinc function

sinc(x) = sin(x) /x

comes up continually in signal processing. Ifx is moderately small, the approximation

sinc(x) (2 + cos(x))/3

is remarkably good, with an error on the order of x⁴/180. This could be useful in situations where you're working with the sinc function and thex in the denominator is awkward to deal with and you'd rather have a pure trig function.

Here's a plot:

Of course the approximation is only good for small x. For large x the sinc function approaches zero while (2 + cos(x))/3 oscillates with constant amplitude forever.

When the approximation is good, it is very, very good, which reminds me of this nursery rhyme.

There was a little girl,
Who had a little curl,
Right in the middle of her forehead.
When she was good,
She was very, very good,
But when she was bad, she was horrid.

More sinc posts

• Sinc and Jinc Sums
• Peaks of Sinc
• Sinc Approximation to a Bessel Fuction

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