Sinc function

{{Infobox mathematical function | name = Sinc | image = Si sinc.svg | imagesize = 350px | imagealt = Part of the normalized and unnormalized sinc function shown on the same scale | caption = Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale | general_definition = \operatorname{sinc}x = \begin{cases} \dfrac{ \sin x } x, & x \ne 0 \\ 1, & x = 0\end{cases} | fields_of_application = Signal processing, spectroscopy | domain = \mathbb{R} | range = [-0.217234\ldots, 1] | parity = Even | zero = 1 | plusinf = 0 | minusinf = 0 | max = 1 at x = 0 | min = -0.21723\ldots at x = \pm 4.49341\ldots | root = \pi k, k \in \mathbb{Z}_{\neq 0} | reciprocal = \begin{cases} x \csc x, & x \ne 0 \\ 1, & x = 0 \end{cases} | derivative = \operatorname{sinc}'x = \begin{cases} \dfrac{\cos x - \operatorname{sinc} x}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases} | antiderivative = \int \operatorname{sinc} x\,dx = \operatorname{Si}(x) + C | taylor_series = \operatorname{sinc}x = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k + 1)!} }} In mathematics, physics and engineering, the sinc function ( ), denoted by , is defined as either \operatorname{sinc}(x) = \frac{\sin x}{x} or \operatorname{sinc}(x) = \frac{\sin \pi x}{\pi x}, the latter of which is sometimes referred to as the normalized sinc function.…

{{Infobox mathematical function | name = Sinc | image = Si sinc.svg | imagesize = 350px | imagealt = Part of the normalized and unnormalized sinc function shown on the same scale | caption = Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale | general_definition = \operatorname{sinc}x = \begin{cases} \dfrac{ \sin x } x, & x \ne 0 \\ 1, & x = 0\end{cases} | fields_of_application = Signal processing, spectroscopy | domain = \mathbb{R} | range = [-0.217234\ldots, 1] | parity = Even | zero = 1 | plusinf = 0 | minusinf = 0 | max = 1 at x = 0 | min = -0.21723\ldots at x = \pm 4.49341\ldots | root = \pi k, k \in \mathbb{Z}_{\neq 0} | reciprocal = \begin{cases} x \csc x, & x \ne 0 \\ 1, & x = 0 \end{cases} | derivative = \operatorname{sinc}'x = \begin{cases} \dfrac{\cos x - \operatorname{sinc} x}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases} | antiderivative = \int \operatorname{sinc} x\,dx = \operatorname{Si}(x) + C | taylor_series = \operatorname{sinc}x = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k + 1)!} }} In mathematics, physics and engineering, the sinc function ( ), denoted by , is defined as either \operatorname{sinc}(x) = \frac{\sin x}{x} or \operatorname{sinc}(x) = \frac{\sin \pi x}{\pi x}, the latter of which is sometimes referred to as the normalized sinc function. The only difference between the two definitions is in the scaling of the independent variable (the axis) by a factor of . 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 then analytic everywhere and hence an entire function. 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 filter is used in signal processing. The function itself was first mathematically derived in this form by Lord Rayleigh in his expression (Rayleigh's formula) for the zeroth-order spherical Bessel function of the first kind. The sinc function is also called the cardinal sine function. ==Definitions== The sinc function has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for by \operatorname{sinc}(x) = \frac{\sin x}{x}. Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x). In digital signal processing and information theory, the normalized sinc function is commonly defined for by \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}. In either case, the value at is defined to be the limiting value \operatorname{sinc}(0) := \lim_{x \to 0}\frac{\sin(a x)}{a x} = 1 for all real (the limit can be proven using the squeeze theorem). 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 ). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of . ==Etymology== The function has also been called the cardinal sine or sine cardinal function , since it returns 1 (a cardinal number) for x = 0 and 0 (another cardinal number) for every other integer. The term "sinc" is a contraction of the function's full Latin name, the and was introduced by Philip M. Woodward and I.L Davies in their 1952 article "Information theory and inverse probability in telecommunication", saying "This function occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own". It is also used in Woodward's 1953 book Probability and Information Theory, with Applications to Radar. \frac{d}{dx}\operatorname{sinc}(x) = \begin{cases} \dfrac{\cos(x) - \operatorname{sinc}(x)}{x}, & x \ne 0 \\0, & x = 0\end{cases}. The first few terms of the infinite series for the coordinate of the th extremum with positive coordinate are x_n = q - q^{-1} - \frac{2}{3} q^{-3} - \frac{13}{15} q^{-5} - \frac{146}{105} q^{-7} - \cdots, where q = \left(n + \frac{1}{2}\right) \pi, and where odd lead to a local minimum, and even to a local maximum. Because of symmetry around the axis, there exist extrema with coordinates . In addition, there is an absolute maximum at . 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 , as well as to Gauss' Pi function, through Euler's reflection formula: \frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1 + x)\Gamma(1 - x)} = \frac{1}{\Pi(x)\Pi(-x)}. Euler discovered that \frac{\sin(x)}{x} = \prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right), and because of the product-to-sum identity plot of ]] \prod_{n=1}^k \cos\left(\frac{x}{2^n}\right) = \frac{1}{2^{k-1}} \sum_{n=1}^{2^{k-1}} \cos\left(\frac{n - 1/2}{2^{k-1}} x \right),\quad \forall k \ge 1, Euler's product can be recast as a sum \frac{\sin(x)}{x} = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \cos\left(\frac{n - 1/2}{N} x\right). The continuous Fourier transform of the normalized sinc (to ordinary frequency) is : \int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f), where the rectangular function is 1 for argument between − and , 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 = \operatorname{rect}(0) = 1 is an improper integral (see Dirichlet 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., , and for nonzero integer . The functions ( integer) form an orthonormal basis for bandlimited functions in the function space , with highest angular frequency (that is, highest cycle frequency ). Other properties of the two sinc functions include: The unnormalized sinc is the zeroth-order spherical Bessel function of the first kind, . The normalized sinc is . where is the sine integral, \int_0^x \frac{\sin(\theta)}{\theta}\,d\theta = \operatorname{Si}(x). (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0. The other is , which is not bounded at , unlike its sinc function counterpart. Using normalized sinc, \int_{-\infty}^\infty \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \quad \Rightarrow \quad \int_{-\infty}^\infty \operatorname{sinc}^2(x)\,dx = 1, \int_{-\infty}^\infty \frac{\sin(\theta)}{\theta}\,d\theta = \int_{-\infty}^\infty \left( \frac{\sin(\theta)}{\theta} \right)^2 \,d\theta = \pi. \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}. The following improper integral involves the (not normalized) sinc function: \int_0^\infty \frac{dx}{x^n + 1} = 1 + 2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{(kn)^2 - 1} = \frac{1}{\operatorname{sinc}(\frac{\pi}{n})}. \lim_{a \to 0} \frac{\sin\left(\frac{\pi x}{a}\right)}{\pi x} = \lim_{a \to 0}\frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) = \delta(x). This is not an ordinary limit, since the left side does not converge. Rather, it means that \lim_{a \to 0}\int_{-\infty}^\infty \frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) \varphi(x) \,dx = \varphi(0) for every Schwartz function, as can be seen from the Fourier inversion theorem. In the above expression, as , the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of , regardless of the value of . This complicates the informal picture of as being zero for all except at the point , 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. We can also make an immediate connection with the standard Dirac representation of \delta(x) by writing b=1/a and \lim_{b \to \infty} \frac{\sin\left(b\pi x\right)}{\pi x} = \lim_{b \to \infty} \frac{1}{2\pi} \int_{-b\pi}^{b\pi} e^{ik x}dk= \frac{1}{2\pi} \int_{-\infty}^\infty e^{i k x} dk=\delta(x), which makes clear the recovery of the delta as an infinite bandwidth limit of the integral. \sum_{n=1}^\infty \operatorname{sinc}(n) = \operatorname{sinc}(1) + \operatorname{sinc}(2) + \operatorname{sinc}(3) + \operatorname{sinc}(4) +\cdots = \frac{\pi - 1}{2}. The sum of the squares also equals : \sum_{n=1}^\infty \operatorname{sinc}^2(n) = \operatorname{sinc}^2(1) + \operatorname{sinc}^2(2) + \operatorname{sinc}^2(3) + \operatorname{sinc}^2(4) + \cdots = \frac{\pi - 1}{2}. When the signs of the addends alternate and begin with +, the sum equals : \sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}(n) = \operatorname{sinc}(1) - \operatorname{sinc}(2) + \operatorname{sinc}(3) - \operatorname{sinc}(4) + \cdots = \frac{1}{2}. The alternating sums of the squares and cubes also equal : \sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}^2(n) = \operatorname{sinc}^2(1) - \operatorname{sinc}^2(2) + \operatorname{sinc}^2(3) - \operatorname{sinc}^2(4) + \cdots = \frac{1}{2}, \sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}^3(n) = \operatorname{sinc}^3(1) - \operatorname{sinc}^3(2) + \operatorname{sinc}^3(3) - \operatorname{sinc}^3(4) + \cdots = \frac{1}{2}. \frac{\sin x}{x} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n+1)!} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots The series converges for all . The normalized version follows easily: \frac{\sin \pi x}{\pi x} = 1 - \frac{\pi^2x^2}{3!} + \frac{\pi^4x^4}{5!} - \frac{\pi^6x^6}{7!} + \cdots Euler famously compared this series to the expansion of the infinite product form to solve the Basel problem. \mathbf{u}_1 = \begin{bmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{bmatrix} \quad \text{and} \quad \mathbf{u}_2 = \begin{bmatrix} \frac{1}{2} \\ -\frac{\sqrt{3}}{2} \end{bmatrix}. Denoting \boldsymbol{\xi}_1 = \tfrac{2}{3} \mathbf{u}_1, \quad \boldsymbol{\xi}_2 = \tfrac{2}{3} \mathbf{u}_2, \quad \boldsymbol{\xi}_3 = -\tfrac{2}{3} (\mathbf{u}_1 + \mathbf{u}_2), \quad \mathbf{x} = \begin{bmatrix} x \\ y\end{bmatrix}, one can derive the sinc function for this hexagonal lattice as \begin{align} \operatorname{sinc}_\text{H}(\mathbf{x}) = \tfrac{1}{3} \big( & \cos\left(\pi\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \\ & {} + \cos\left(\pi\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \\ & {} + \cos\left(\pi\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \big). \end{align} This construction can be used to design Lanczos window for general multidimensional lattices. \mathrm{sinhc}(x) = \begin{cases} {\displaystyle \frac{\sinh(x)}{x},} & \text{if }x \ne 0 \\ {\displaystyle 1,} & \text{if }x = 0 \end{cases} ==See also== (cartography)

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