Sinc function

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 pi|). 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. The term "sinc" is a contraction of the function's full Latin name, the It is also used in Woodward's 1953 book Probability and Information Theory, with Applications to Radar. == Properties ==

The zero crossings of the unnormalized sinc are at non-zero integer multiples of , while zero crossings of the normalized sinc occur at non-zero integers. The local extrema of the unnormalized sinc correspond to its intersections with the cosine function. That is, for all points where the derivative of is zero and thus a local extremum is reached. This follows from the derivative of the sinc function: \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})}. == Relationship to the Dirac delta distribution ==

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds: \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. == Summation ==

All sums in this section refer to the unnormalized sinc function. The sum of over integer from 1 to equals : \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}. == Series expansion ==

The Taylor series of the unnormalized function can be obtained from that of the sine (which also yields its value of 1 at ): \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. == Higher dimensions ==

The product of 1-D sinc functions readily provides a multivariate sinc function for the square Cartesian grid (lattice): , 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 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 \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. == Sinhc ==

Some authors, by analogy, define the hyperbolic sine cardinal function. :\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==