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Special mathematical function defined as sin(x)/x
engineering, the sinc function (/ˈsɪŋk/), denoted by sinc(x), is defined as either sinc ( x ) = sin x x {\displaystyle \operatorname {sinc} (x)={\frac
Sinc_function
Ideal low-pass filter or averaging filter
In signal processing, a sinc filter can refer to either a sinc-in-time filter whose impulse response is a sinc function and whose frequency response is
Sinc_filter
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
Applications to Radar" as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which
Rectangular_function
Form of error in digital signals; spurious signals near sharp transitions
time domain, the cause of this type of ringing is the ripples in the sinc function, which is the impulse response (time domain representation) of a perfect
Ringing_artifacts
Technique in signal processing
copy of the Lanczos kernel, which is a sinc function windowed by the central lobe of a second, longer, sinc function. The sum of these translated and scaled
Lanczos_resampling
Signal (re‑)construction algorithm
Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The
Whittaker–Shannon interpolation formula
Whittaker–Shannon_interpolation_formula
Oscillatory error in Fourier series
}{N}}\right)\right)} If we introduce the normalized sinc function sinc ( x ) = sin ( π x ) π x {\textstyle \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}} for
Gibbs_phenomenon
Sufficiency theorem for reconstructing signals from samples
sinc function scaled to the sample value, x ( n T ) . {\displaystyle x(nT).} Subsequently, the sinc functions are summed into a continuous function.
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Function used in signal processing
product of two sinc functions vs. an Airy function, respectively. Conventions: w 0 ( x ) {\displaystyle w_{0}(x)} is a zero-phase function (symmetrical
Window_function
Generalized function whose value is zero everywhere except at zero
origin. Other approximations to the identity of this kind include the sinc function (used widely in electronics and telecommunications) η ε ( x ) = 1 π
Dirac_delta_function
Tent function, often used in signal processing
{a}^{2}\ \mathrm {sinc} ^{2}(a\cdot f)={a}\ \mathrm {sinc} ^{2}(a\cdot f).\end{aligned}}} Källén function, also known as triangle function Tent map Triangular
Triangular_function
Functions of an angle
of periodic functions Polar sine – a generalization to vertex angles Sinc function Small-angle approximation Klein, Felix (1924) [1902], "Die goniometrischen
Trigonometric_functions
Undefined point on a holomorphic function which can be made regular
the resulting function is regular in a neighbourhood of that point. For instance, the (unnormalized) sinc function, as defined by sinc ( z ) = sin z
Removable_singularity
When a signal or function exceeds its target
step function by the integral up to a certain frequency, which yields the sine integral. This can be interpreted as convolution with the sinc function; in
Overshoot_(signal)
\right)\right],} in terms of the normalized sinc function: sinc x = sin π x π x . {\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.} a k
Sigma_approximation
Family of solutions to related differential equations
zeroth spherical Bessel function j0(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: j 0 ( x ) = sin
Bessel_function
Bessel–Clifford function Kelvin functions Legendre function: From the theory of spherical harmonics. Scorer's function Sinc function Hermite polynomials
List of mathematical functions
List_of_mathematical_functions
Inputs at which function values are highest
if and only if x − 5 = 0. {\displaystyle x-5=0.} "The Unnormalized Sinc Function Archived 2017-02-15 at the Wayback Machine", University of Sydney Rockafellar
Arg_max
Extension of the factorial function
(-z)={\frac {\pi z}{\sin(\pi z)}}={\frac {1}{\operatorname {sinc} (z)}}} using the normalized sinc function; while the multiplication theorem becomes: Π ( z m
Gamma_function
Type of signal filter
signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain. However
Low-pass_filter
Largest and smallest value taken by a function at a given point
Paul. "Minimization and maximization refresher". "The Unnormalized Sinc Function Archived 2017-02-15 at the Wayback Machine", University of Sydney For
Maximum_and_minimum
Integral of sin(x)/x from 0 to infinity
Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real number line. ∫ 0 ∞ sin x x d x = π 2 . {\displaystyle
Dirichlet_integral
Mathematical transform that expresses a function of time as a function of frequency
example, the Fourier transform of the rectangular function, which is integrable, is the sinc function, which is not Lebesgue integrable, because its improper
Fourier_transform
2-dimensional polar coordinate function
A sombrero function (sometimes called besinc function or jinc function) is the 2-dimensional polar coordinate analog of the sinc function, and is so-called
Sombrero_function
translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by C ( f , h ) ( x ) = ∑ k = − ∞ ∞ f ( k h ) sinc ( x h − k )
Sinc_numerical_methods
wavelet. First, we define the scaling function to be the sinc function. ϕ (Sha) ( t ) := sin π t π t = sinc ( t ) . {\displaystyle \phi ^{\text{(Sha)}}(t):={\frac
Shannon_wavelet
Mathematical function with no sudden changes
sufficient is the sinc function, which is defined by sinc ( 0 ) = 1 {\displaystyle \operatorname {sinc} (0)=1} and sinc ( x ) = sin x x {\displaystyle
Continuous_function
Mathematical function used in signal processing
\operatorname {sinc} (Lf)\triangleq {\frac {\sin(\pi Lf)}{\pi Lf}}.} Transforming each term: W 0 ( f ) = 1 2 sinc ( L f ) + 1 4 sinc ( L ( f − 1 /
Hann_function
Analytic function in mathematics
} are the polygamma function and Euler's constant, respectively, as well as ∑ n = 1 ∞ ζ ( 2 n ) − 1 n t 2 n = log ( 1 − t 2 sinc ( π t ) ) {\displaystyle
Riemann_zeta_function
Physics experiment
}}\right]~\mathrm {sinc} ^{2}\left[{\frac {\pi b\sin \theta }{\lambda }}\right]\end{aligned}}} where the sinc function is defined as sinc(x) = sin(x)/x for
Double-slit_experiment
Type of mathematical integrals
integrals involve products of sinc ( a x ) {\displaystyle \operatorname {sinc} (ax)} , where the sinc function is given by sinc ( x ) = sin ( x ) / x
Borwein_integral
Special function defined by an integral
the sinc function, and also the zeroth spherical Bessel function. Since sinc {\displaystyle \operatorname {sinc} } is an even entire function (holomorphic
Trigonometric_integral
Mathematical explanation of far field diffraction
Wx}{\lambda z}}}\\&=aW\operatorname {sinc} {\frac {\pi Wx}{\lambda z}}\end{aligned}}} where sinc(p) = sin(p)/p. The sinc function is sometimes defined as sin(πp)/πp
Fraunhofer diffraction equation
Fraunhofer_diffraction_equation
Foundational principle in quantum physics
positivity condition on a real star-square function. Given a Wigner function W(x,p) with star product ★ and a function f, the following is generally true: ⟨f∗⋆f⟩=∫(f∗⋆f)W(x
Uncertainty_principle
Technique for reducing low-resolution image distortion
filter or interpolation functions, as though the data were a two dimensional function of continuous x and y. The sinc function of the radius has too long
Spatial_anti-aliasing
Mathematical identity used to evaluate certain improper integrals
terms of distributions. One of those is the improper integral of the sinc function over the positive real line, ∫ 0 ∞ sin x x d x = ∫ 0 ∞ sin 2 x x
Lobachevsky_integral_formula
Fundamental trigonometric functions
generalization to vertex angles Proofs of trigonometric identities Sinc function Sine and cosine transforms Sine integral Sine quadrant Sine wave Sine–Gordon
Sine_and_cosine
Numerical computing environment and programming language
developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and
MATLAB
Mathematical function resembling a boxcar
boxcar function is selected as the impulse response of a filter, the result is a simple moving average filter, whose frequency response is a sinc-in-frequency
Boxcar_function
Number of prime factors of a natural number
the normalized sinc {\displaystyle \operatorname {sinc} } function sinc ( x ) = sin ( π x ) π x {\displaystyle \operatorname {sinc} (x)={\frac {\sin(\pi
Prime_omega_function
Pseudoazimuthal compromise map projection
\left(\cos \varphi \cos {\frac {\lambda }{2}}\right)} and sinc α is the unnormalized sinc function with the discontinuity removed. In all of these formulas
Aitoff_projection
Method of mathematical integration
the sinc function: sinc ( x ) = sin ( x ) x {\displaystyle \operatorname {sinc} (x)={\frac {\sin(x)}{x}}} over the entire real line. This function is
Lebesgue_integral
Type of filter in signal processing
the corresponding sinc function is zero at every other sample point (except the center one). The product with the window function does not alter the
Finite_impulse_response
Concept in mathematical analysis
signal processing, the Dirichlet kernel is often called the periodic sinc function: P ( ω ) = D n ( x ) | x = 2 π ω / ω 0 = sin ( π M ω / ω 0 ) sin
Dirichlet_kernel
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Integral transform and linear operator
a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform
Hilbert_transform
Topics referred to by the same term
Look up sinc in Wiktionary, the free dictionary. Sinc is a mathematical function. Sinc may also refer to: Sinc McEvenue, Canadian football coach Sisters
Sinc_(disambiguation)
Model in digital signal processing
{\displaystyle \mathrm {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}\ } is the normalized sinc function. The Laplace transform transfer function of FOH is found by
First-order_hold
Quick, temporary change in amplitude of electrical signals
transmission. An example of a pulse which meets this condition is the sinc function. The sinc pulse is of some significance in signal-processing theory but cannot
Pulse_(signal_processing)
Extension of cubic spline interpolation
the derivative of the sinc function at x = 1 {\displaystyle x=1} , but this has no significance other than for comparison with sinc interpolation. When
Bicubic_interpolation
Mathematical model in quantum mechanics
=Lπℏ(ℏnπℏnπ+pL)sinc(pL2ℏ−nπ2)e−ipxc/ℏei(n−1)π2e−iωnt,=i(n−1)Lπℏ(ℏnπℏnπ+pL)sin(pL2ℏ−nπ2)(pL2ℏ−nπ2)e−ipxc/ℏe−iωnt, where the sinc function, sinc(x)=sin(x)/x
Particle_in_a_box
∫0∞xnex−1dx=Γ(n+1)ζ(n+1) for n>0 (see also Riemann zeta function) ∫0∞sinxxdx=π2 (see sinc function and the Dirichlet integral) ∫0∞sin2xx2dx=π2
Lists_of_integrals
Pulse-shaping filter in digital modulation
auto-correlation function of raised cosine function is as follows: R ( τ ) = T [ sinc ( τ T ) cos ( β π τ T ) 1 − ( 2 β τ T ) 2 − β 4 sinc ( β τ T )
Raised-cosine_filter
Model of signal reconstruction in digital-to-analog (DAC) converters
fT}}=e^{-i\pi fT}\mathrm {sinc} (fT)} where s i n c ( x ) {\displaystyle \mathrm {sinc} (x)} is the (normalized) sinc function sin ( π x ) π x {\displaystyle
Zero-order_hold
Smooth nowhere-analytic function
transform is an infinite product of sinc functions, or equivalently a product of powers of cosines. The function also has an unusual arithmetic side:
Fabius_function
Periodic distribution ("function") of "point-mass" Dirac delta sampling
"multiplication with the rect function" is equivalent to "convolution with the sinc function". Hence, it restores the original function from its samples. This
Dirac_comb
Method for converting signals between digital and analog
Figures 7–9 of Lyons article). Windowed sinc-in-time (brick-wall in frequency) filters: Although the sinc function's infinite support prevents it from being
Delta-sigma_modulation
k\Delta )}{\pi k}}=\Delta \cdot \mathrm {sinc} (\pi k\Delta )\cdot \mathrm {e} ^{2\pi i\,kx}.} with the Sinc function. Petříček, V.; Van Der Lee, A.; Evain
Crenel_function
Sum of a function's values every _P_ offsets
implies that the periodic summation of any band-limited function, such as the sinc function, is a sum of a finite number of sine waves, or even just
Periodic_summation
Screening phenomenon in metals
the perturbation followed by an ongoing sinusoidal decay resembling sinc function. The phenomenon is named after Jacques Friedel, who predicted the effect
Friedel_oscillations
equation for the optical modulation transfer function associated with smear is the standard sinc function associated with an extended sample M T F s m
Smear_(optics)
Transition rate formula
an unstable subatomic particle into other particles Sinc function – Special mathematical function defined as sin(x)/x Time-dependent perturbation theory –
Fermi's_golden_rule
Periodic rectangular waveform
using the Sinc function, using the definition sincx=sinπxπx, as x(t)=AτT(1+2∑n=1∞(sinc(nτT)cos(2πnft))) or with d=τT as x(t)=Ad(1+2∑n=1∞(sinc(nd)cos(2πnft)))A
Pulse_wave
Signal processing effect
Aliasing. Brillouin zone Glossary of video terms Jaggies Kell factor Sinc filter Sinc function Spectral density Spectral leakage Stroboscopic effect Wagon-wheel
Aliasing
Distance over which a wave's shape repeats
position x via a squared sinc function: S ( u ) = s i n c 2 ( u ) = ( sin π u π u ) 2 ; {\displaystyle S(u)=\mathrm {sinc} ^{2}(u)=\left({\frac {\sin
Wavelength
Differential equation for the description of waves or standing wave
the Fourier transform is the sinc function G 1 ( t , x ) = 1 2 π ∫ R sin ( | ω | t ) | ω | e i ω x d ω = 1 2 π ∫ sinc ( ω ) e i ω x t d ω = sgn
Wave_equation
Sum of inverse squares of natural numbers
)^{2n}(-1)^{n+1}B_{2n}}{2\cdot (2n)!}}.} The normalized sinc function sinc ( x ) = sin ( π x ) π x {\displaystyle {\text{sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}} has
Basel_problem
Analog of the continuous Laplace operator
{r}}_{k})} with μ {\displaystyle \mu } being an appropriately dilated sinc function defined in n {\displaystyle n} -dimensions i.e. r ¯ = ( x 1 , x 2 .
Discrete_Laplace_operator
Signal processing sample technique
reconstructive interpolation function, or lowpass filter impulse response, is sinc ( t / T ) {\displaystyle \operatorname {sinc} \left(t/T\right)} with
Undersampling
Set of functions used to represent the electronic wave function
interpolation functions. Various methods have been proposed for constructing the solution in real space, including finite elements, basis splines, Lagrange sinc-functions
Basis_set_(chemistry)
Hungarian-American mathematician (1893–1974)
resampling is based on a windowed sinc function as a practical upsampling filter approximating the ideal sinc function, now widely used in video up-sampling
Cornelius_Lanczos
Interference phenomenon of waves
of energy, and sinc x = sin x x {\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}} , which is the unnormalized sinc function. This analysis
Diffraction
function. Shi – hyperbolic sine integral function. Si – sine integral function. sigmoid – sigmoid function. sin – sine function. sinc – sinc function
List of mathematical abbreviations
List_of_mathematical_abbreviations
Study of Galois symmetry groups of differential fields
elementary functions sinc function sin x x {\displaystyle {\tfrac {\sin x}{x}}} and x x {\displaystyle x^{x}} are also non-elementary functions. The concept
Differential_Galois_theory
Function of propagation delay and Doppler frequency
filter output is distorted into a sinc function. The greater the Doppler shift, the smaller the peak of the resulting sinc, and the more difficult it is to
Ambiguity_function
Characteristic of any structure that is periodic across a position in space
phase-variable spin-echo signals. Each of the spin-echo signal is a sinc function of time, which can be described by Spin-Echo = M 0 sin ω r t ω r t
Spatial_frequency
^{m}\left({\frac {t}{\mathrm {fb} ^{m}}}\right)e^{j2\pi f_{c}t}} where sinc function that appears in Shannon sampling theorem. m > 1 is the order of the
Fbsp_wavelet
Interpolation with trigonometric polynomials
{\mathrm {sinc} \,{\tfrac {1}{2}}N(x-x_{k})}{\mathrm {sinc} \,{\tfrac {1}{2}}(x-x_{k})}}.\end{aligned}}} Here, the sinc-function prevents any singularities
Trigonometric_interpolation
Mathematical model which is both linear and time-invariant
low-pass filter with impulse response equal to a sinc function is not BIBO stable, because the sinc function does not have a finite L1 norm. Thus, for some
Linear_time-invariant_system
Mathematical operation
the signal in the time domain is rectangular (as below), it becomes a sinc function in the frequency domain. But if one applies the fractional Fourier transform
Fractional_Fourier_transform
Function for integral Fourier-like transform
(t)=2\,\operatorname {sinc} (2t)-\,\operatorname {sinc} (t)={\frac {\sin(2\pi t)-\sin(\pi t)}{\pi t}}} with the (normalized) sinc function. That, Meyer's, and
Wavelet
Wave phenomenon
we have Ψ=aCsinkasinθ2kasinθ2=aC[sinc(kasinθ2)] where the (unnormalized) sinc function is defined by sinc(x) =def sin(x)x. Now, substituting in
Diffraction_from_slits
Type of uncertainty of meaning where several interpretations are possible
sometimes, the same name of the function is used, causing confusion. Examples of such under-established functions: Sinc function Elliptic integral of the third
Ambiguity
Online database of integer sequences
arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408. Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Inverse of a finite difference/the antidifference for various functions
F(x+1)-F(x)=f(x)} . Using the reflection formula for the digamma function, this simplifies to: ∑ x sinc x = sinc ( x − 1 ) ( 1 2 + ( x − 1 ) × ( ln ( 2 ) + ψ (
List_of_indefinite_sums
Representation of a signal as a rectangular wave with varying duty cycle
amplitudes of the harmonic groups are restricted by a sinx/x envelope (sinc function) and extend to infinity. The infinite bandwidth is caused by the nonlinear
Pulse-width_modulation
Algorithm to smooth data points
closely related to smoothing splines), and convolution with a windowed sinc function. sgolayfilt from Signal Processing Toolbox. Available since before version
Savitzky–Golay_filter
Mathematical concept
result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be
Infinite_product
Filter used to construct a smooth analog signal from a digital input
the sinc function, either by windowing or by giving a spline approximation, either by cubics or higher order splines. In the case of windowed sinc filters
Reconstruction_filter
Signal processing technique
of the autocorrelation function of s c ′ {\displaystyle s_{c'}} is reached at 0. Around 0, this function behaves as the sinc (or cardinal sine) term
Pulse_compression
INTERCAL, Colossal Cave Adventure Philip Woodward – ambiguity function, sinc function, comb operator, rep operator, ALGOL 68-R Steve Wozniak – Breakout
List_of_programmers
Fourier analysis technique applied to sequences
(\omega )} is the Dirac delta function sinc ( t ) {\displaystyle \operatorname {sinc} (t)} is the normalized sinc function rect [ n L ] ≜ { 1 | n | ≤
Discrete-time Fourier transform
Discrete-time_Fourier_transform
since a box filter has poor spectral properties. Conversely, the ideal sinc function is not practical, having infinite support with possibly negative values
Cone_tracing
data structures for representing level sets Sinc numerical methods — methods based on the sinc function, sinc(x) = sin(x) / x ABS methods Error analysis
List of numerical analysis topics
List_of_numerical_analysis_topics
Signal filtering technique
y-function values. Despite its ease of implementation, its practical use is limited as the real-space representation of a top-hat filter is the sinc function
Top-hat_filter
Process in electronics and telecommunications
bandwidth increases. When the spectrum of the signal is uniformly rectangular, a sinc shape results in the time domain. This happens if the bandwidth of the signal
Pulse_shaping
that sin ( x ) / x = sinc ( x ) {\displaystyle \ \sin(x)\,/\,x\ =\operatorname {sinc} (x)} , the sine cardinal function; and similarly for sinh as
Stumpff_function
Important functions in solving differential equations
cardinal sine function, sinc ( θ σ 1 ) = sinc ( θ ) I . {\displaystyle \operatorname {sinc} (\theta \sigma _{1})=\operatorname {sinc} (\theta )~I
Trigonometric functions of matrices
Trigonometric_functions_of_matrices
Mathematical description in crystallography
sinc function]. Thus the intensity, which is a delta function of position for the perfect crystal, becomes a sinc 2 {\textstyle \operatorname {sinc}
Structure_factor
Filter that has a linear response
boxcar function, then it would implement a low-pass filter with a low frequency gain of N+1 and a frequency response given by the sinc function. Superior
Linear_filter
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SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
SINC FUNCTION
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