Sinc function

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In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by sinc(x), has two slightly different definitions.[1]

In mathematics, the historical unnormalized sinc function is defined by

\mathrm{sinc}(x) = \frac{\sin(x)}{x}~.

In digital signal processing and information theory, the normalized sinc function is commonly defined by

The normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale.
\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}~.

In either case, the value at x = 0 is defined to be the limiting value sinc(0) = 1.

The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of π). As a further useful property, all of the zeros of the normalized sinc function are integer values of x. The normalized sinc function is the Fourier transform of the rectangular function with no scaling. This function is fundamental in the concept of reconstructing the original continuous bandlimited signal from uniformly spaced samples of that signal.

The only difference between the two definitions is in the scaling of the independent variable (the x-axis) by a factor of π. In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is analytic everywhere.

The term "sinc" /ˈsɪŋk/ is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine).[2] It was introduced by Phillip M. Woodward in his 1952 paper "Information theory and inverse probability in telecommunication" in which he said the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own"[3] and his 1953 book "Probability and Information Theory, with Applications to Radar".[2][4]

Properties[edit]

The local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue cosine function.

The zero crossings of the unnormalized sinc are at non-zero multiples of π, while zero crossings of the normalized sinc occur at non-zero integers.

The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(ξ)/ξ = cos(ξ) for all points ξ where the derivative of sin(x)/x is zero and thus a local extremum is reached.

A good approximation of the x-coordinate of the n-th extremum with positive x-coordinate is


x_n \approx (n+\tfrac12)\pi - \frac1{(n+\frac12)\pi}  ~,

where odd n lead to a local minimum and even n to a local maximum. Besides the extrema at xn, the curve has an absolute maximum at ξ0 = (0,1) and because of its symmetry to the y-axis extrema with x-coordinates −xn.

The normalized sinc function has a simple representation as the infinite product

\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)

and is related to the gamma function Γ(x) through Euler's reflection formula,

\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)}~.

Euler discovered[5] that

\frac{\sin(x)}{x} = \prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right)~.

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect( f ),

\int_{-\infty}^\infty \mathrm{sinc}(t) \, e^{i 2 \pi f t}\,dt = \mathrm{rect}(f)~,

where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the sinc filter is the ideal (brick-wall, meaning rectangular frequency response) low-pass filter.

This Fourier integral, including the special case

\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \mathrm{rect}(0) = 1\,\!

is an improper integral and not a convergent Lebesgue integral, as

\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right|\, dx = +\infty ~.

The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled bandlimited functions:

  • It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero integer k.
  • The functions xk(t) = sinc(t−k) (k integer) form an orthonormal basis for bandlimited functions in the function space L2(R), with highest angular frequency ωH = π (that is, highest cycle frequency ƒH = 1/2).

Other properties of the two sinc functions include:

  • The unnormalized sinc is the zeroth order spherical Bessel function of the first kind, j0(x). The normalized sinc is j0(πx).
  •  \int_0^x \frac{\sin(\theta)}{\theta}\,d\theta = \mathrm{Si}(x) \,\!
where Si(x) is the sine integral.
x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.\,\!
The other is cos(λ x)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
  •  \int_{-\infty}^\infty \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \,\! \rightarrow \int_{-\infty}^\infty \mathrm{sinc}^2(x)\,dx = 1~,
where the normalized sinc is meant.
  •  \int_{-\infty}^\infty \frac{\sin^3(\theta)}{\theta^3}\,d\theta = \frac{3\pi}{4} \,\!
  •  \int_{-\infty}^\infty \frac{\sin^4(\theta)}{\theta^4}\,d\theta = \frac{2\pi}{3} ~.

Relationship to the Dirac delta distribution[edit]

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds,

\lim_{a\rightarrow 0}\frac{\sin(\pi x/a)}{\pi x}=\lim_{a\rightarrow 0}\frac{1}{a}\textrm{sinc}(x/a)=\delta(x)~.

This is not an ordinary limit, since the left side does not converge. Rather, it means that

\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx
            = \varphi(0)~,

for any smooth function φ(x) with compact support.

In the above expression, as a → 0, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(πx), regardless of the value of a. This complicates the informal picture of δ(x) as being zero for all x except at the point x = 0, and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the Gibbs phenomenon.

Summation[edit]

Summation of the unnormalized sinc(x) over integer x from 1 to ∞ equals (π−1)/2,

\sum_{x=1}^\infty \mathrm{sinc}(x) = \mathrm{sinc}(1) + \mathrm{sinc}(2) + \mathrm{sinc}(3) + \mathrm{sinc}(4) +\cdots = \frac{\pi-1}{2}

When the signs of the addends alternate and begin with +, the sum equals 1/2.

\sum_{x=1}^\infty (-1)^{x+1}\,\mathrm{sinc}(x) = \mathrm{sinc}(1) - \mathrm{sinc}(2) + \mathrm{sinc}(3) - \mathrm{sinc}(4) +\cdots = \frac{1}{2}

Multidimensions[edit]

The tensor-product of 1-D sinc functions readily provides a multivariate sinc function for the square, Cartesian, grid (Lattice):  {\rm sinc}_{\rm C}(x, y) = {\rm sinc}(x) {\rm sinc}(y) whose Fourier transform is the indicator function of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian lattice (e.g., hexagonal lattice) is a function whose Fourier transform is the indicator function of the Brillouin zone of that lattice. For example, the sinc function for the hexagonal lattice is a function whose Fourier transform is the indicator function of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple tensor-product. However, the explicit formula for the sinc function for the hexagonal, body centered cubic, face centered cubic and other higher-dimensional lattices can be explicitly derived [6] using the geometric properties of Brillouin zones and their connection to zonotopes.

For example, a hexagonal lattice can be generated by the (integer) Linear span of the vectors u_1 = \left[\begin{array}{c}1/2\\ \sqrt{3}/2\end{array}\right] and u_2 = \left[\begin{array}{c}1/2\\ -\sqrt{3}/2\end{array}\right]. Denoting \xi_1 = 2/3 u_1, \xi_2 = 2/3 u_2, \xi_3 = -2/3(u_1 + u_2) and \mathbf{x} = \left[\begin{array}{c}x\\ y\end{array}\right], one can derive[6] the sinc function for this hexagonal lattice as:

\begin{align}
{\rm sinc}_{\rm H}(\mathbf{x}) = 1/3\big(
&\cos(\pi\xi_1\cdot\mathbf{x}){\rm sinc}(\xi_2\cdot\mathbf{x}){\rm sinc}(\xi_3\cdot\mathbf{x})+\\
&\cos(\pi\xi_2\cdot\mathbf{x}){\rm sinc}(\xi_3\cdot\mathbf{x}){\rm sinc}(\xi_1\cdot\mathbf{x})+\\
&\cos(\pi\xi_3\cdot\mathbf{x}){\rm sinc}(\xi_1\cdot\mathbf{x}){\rm sinc}(\xi_2\cdot\mathbf{x})\big)
\end{align}

This construction can be used to design Lanczos window for general multidimensional lattices.[6]

See also[edit]

References[edit]

  1. ^ Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Numerical methods", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 
  2. ^ a b Poynton, Charles A. (2003). Digital video and HDTV. Morgan Kaufmann Publishers. p. 147. ISBN 1-55860-792-7. 
  3. ^ Woodward, P. M.; Davies, I. L. (March 1952). "Information theory and inverse probability in telecommunication". Proceedings of the IEE - Part III: Radio and Communication Engineering 99 (58): 37–44. doi:10.1049/pi-3.1952.0011. 
  4. ^ Woodward, Phillip M. (1953). Probability and information theory, with applications to radar. London: Pergamon Press. p. 29. ISBN 0-89006-103-3. OCLC 488749777. 
  5. ^ Euler, Leonhard (1735). "On the sums of series of reciprocals". 
  6. ^ a b c Ye, W.; Entezari, A. (June 2012). "A Geometric Construction of Multivariate Sinc Functions". IEEE Transactions on Image Processing 21 (6): 2969–2979. doi:10.1109/TIP.2011.2162421. PMID 21775264. 

External links[edit]