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sinc and Polynomial interpolation

October 25, 2010

Another short one. This one bugged me for quite a while until I realized what the answer was a few years ago. The Sampling Theorem states that (under the right conditions)

\displaystyle x(t) = \sum_{n=-\infty}^{\infty} x(nT) \;\textrm{sinc}\left(\frac{t - nT}{T}\right)

where

\displaystyle \textrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}

(the normalized sinc function). The problem is this: Where does the sinc function come from? (In a philosophical sense. It plops out of the proof sure enough, but that’s not what I mean). Fourier theory is full of (trigonometric) polynomials (i.e. sine/cosine waves when you’re dealing with real-valued signals), so where does the factor of x in the denominator suddenly come from?

The answer is a nice identity discovered by Euler:

\displaystyle \textrm{sinc}(x) = \prod_{n=1}^{\infty} \left(1 - \frac{x^2}{n^2}\right)

With some straightforward algebraic manipulations (ignoring convergence issues for now) you get:
\displaystyle \textrm{sinc}(x) = \prod_{n=1}^{\infty} \left(1 - \frac{x}{n}\right) \left(1 + \frac{x}{n}\right)

\displaystyle = \prod_{\substack{n \in \mathbb{Z} \\ n \ne 0}} \left(1 - \frac{x}{n}\right) = \prod_{\substack{n \in \mathbb{Z} \\ n \ne 0 }} \frac{n - x}{n}

Compare this with the formula for Lagrange basis polynomials:

\displaystyle L_{i;n}(x) = \prod_{\substack{0 \le j \le n \\ j \ne i}} \frac{x_j - x}{x_j - x_i}

in other words, the sinc function is the limiting case of Lagrange polynomials for an infinite number of equidistant control points. Which is pretty neat :)

From → Maths

One Comment
  1. John Doe permalink

    As you say, sinc(x) is an interpolation function for the integers, and it is easy to see that the translates sinc(x-k) form an orthonormal set in L^2(R). In this language, the essence of the sampling theorem is that those guys really span all bandlimited functions with frequency <= 1/2. But the Fourier transforms of those guys are just the "cut-off" Fourier basis functions
    rect(f) * e^{-2 pi i n x},
    and Shannon's theorem follows at once from the definition of a band-limited function.

    Put slightly differently, an intuitive way of understanding Shannon's theorem is the following: Band-limited functions can be periodically continued in Fourier space, hence *their Fourier Transform* can be represented by a Fourier series cut off with a rectangular function. After transforming back, the family of sinc translates arises.

    Reply

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