Sinc functionIn mathematics, physics and engineering, the sinc function (), denoted by sinc(x), is defined as either sinc(x)=sinxx{\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}} or sinc(x)=sinπxπx,{\displaystyle \operatorname {sinc} (x)={\frac {\sin \pi x}{\pi x}},}the latter of which is sometimes referred to as the normalized sinc function.Heaviside step function
Window functionFunction used in signal processingA popular window function, the Hann window. Most popular window functions are similar bell-shaped curves.In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval. Typically, window functions are symmetric around the middle of the interval, approach a maximum in the middle, and taper away from the middle.
Fourier transformMathematical transform that expresses a function of time as a function of frequencyThe Fourier transform applied to the waveform of a C majorpianochord (with logarithmic horizontal (frequency) axis). The first three peaks on the left correspond to the fundamental frequencies of the chord (C, E, G).Whittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949.
Dirac combIn mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodicgeneralized function with the formula ШT(t):=∑k=−∞∞δ(t−kT){\displaystyle \operatorname {\text{Ш}} _{T}(t):=\sum _{k=-\infty }^{\infty }\delta (t-kT)} for some given period T{\displaystyle T}. Here t{\displaystyle t} is a real variable and the sum extends over all integersk{\displaystyle k}.
Boxcar functionIn mathematics, a boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant A{\displaystyle A}. The function is named after its graph's resemblance to a boxcar, a type of railroad car.
Periodic summationIn mathematics, any integrable functions(t){\displaystyle s(t)} can be made into a periodic functionsP(t){\displaystyle s_{P}(t)} with period P by summing the translations of the function s(t){\displaystyle s(t)} by integer multiples of P. This is called periodic summation:sP(t)=∑n=−∞∞s(t+nP){\displaystyle s_{P}(t)=\sum _{n=-\infty }^{\infty }s(t+nP)}The resulting periodic function may not be defined everywhere.
Philip WoodwardPhilip Mayne Woodward (6 September 1919 – 30 January 2018) was a British mathematician, radar engineer and horologist. He achieved notable success in all three fields. Before retiring, he was a deputy chief scientific officer at the Royal Signals and Radar Establishment of the British Ministry of Defence in Malvern, Worcestershire. Achievements in mathematics and engineeringWoodward's career in the Scientific Civil Service spanned four decades.
ConvolutionIntegral expressing the amount of overlap of one function as it is shifted over anotherVisual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function f{\displaystyle f}, and assuming the height of f{\displaystyle f} is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point.
Sampling (signal processing)In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space. The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a reconstruction filter.Pulse wave*As an Amazon Associate I earn from qualifying purchases.