Triangular function

Triangular function

The triangular function (also known as the triangle function, hat function, or tent function) is defined either as:

: $egin{align}operatorname{tri}(t) = and (t) quad &overset{underset{mathrm{def{{=} max(1 - |t|, 0) \&= egin{cases}1 - |t|, & |t| < 1 \0, & mbox{otherwise} end{cases}end{align}$

or, equivalently, as the convolution of two identical unit rectangular functions:

: $egin{align}operatorname{tri}(t) = operatorname{rect}(t) * operatorname{rect}(t) quad&overset{underset{mathrm{def{{=} int_{-infty}^infty mathrm{rect}( au) cdot mathrm{rect}(t- au) d au\&= int_{-infty}^infty mathrm{rect}( au) cdot mathrm{rect}( au-t) d au .end{align}$

The function is useful in signal processing and "communication systems engineering" as a representation of an idealized signal, and as a prototype or kernel from which more realistic signals can be derived. It also has applications in pulse code modulation as a pulse shape for transmitting digital signals and as a matched filter for receiving the signals. It is also equivalent to the triangular window sometimes called the Bartlett window.

Scaling

For any parameter, $a
e 0,$ :

: $egin{align}operatorname{tri}(t/a) &= int_{-infty}^infty mathrm{rect}( au) cdot mathrm{rect}( au - t/a) d au \&= egin{cases}1 - |t/a|, & |t| < |a| \0, & mbox{otherwise} .end{cases}end{align}$

Fourier transform

The transform is easily determined using the convolution property of Fourier transforms and the Fourier transform of the rectangular function:

: $egin{align}mathcal{F}{operatorname{tri}(t)} &= mathcal{F}{operatorname{rect}(t) * operatorname{rect}(t)}\&= mathcal{F}{operatorname{rect}(t)}cdot mathcal{F}{operatorname{rect}(t)}\&= mathcal{F}{operatorname{rect}(t)}^2\&= mathrm{sinc}^2(f) .end{align}$

ee also

*Tent map

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

Triangular distribution — Probability distribution name =Triangular type =density pdf cdf parameters =a: ain ( infty,infty)b: b>a,c: ale cle b, support =a le x le b ! pdf = left{ egin{matrix} frac{2(x a)}{(b a)(c a)} mathrm{for } a le x le c frac{2(b x)}{(b a)(b c)}… … Wikipedia
Rectangular function — The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized boxcar function) is defined as::mathrm{rect}(t) = sqcap(t) = egin{cases}0 mbox{if } |t| > frac{1}{2} [3pt] frac{1}{2} mbox{if } |t| =… … Wikipedia
Window function — For the term used in SQL statements, see Window function (SQL) In signal processing, a window function (also known as an apodization function or tapering function[1]) is a mathematical function that is zero valued outside of some chosen interval … Wikipedia
T-function — In cryptography, a T function is a bijective mapping that updates every bit of the state in a way that can be described as x i = x i + f(x 0, cdots, x {i 1}), or in simple words an update function in which each bit of the state is updated by a… … Wikipedia
Pfaffian function — Not to be confused with Pfaffian. In mathematics, the Pfaffian functions are a certain class of functions introduced by Askold Georgevich Khovanskiǐ in the 1970s. They are named after German mathematician Johann Pfaff. Contents 1 Basic definition … Wikipedia
Airy function — This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see Stress functions. In the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer George… … Wikipedia
Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… … Wikipedia
Arithmetic function — In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that expresses some arithmetical property of n. [1] An example of an arithmetic… … Wikipedia
Generating function — This article is about generating functions in mathematics. For generating functions in classical mechanics, see Generating function (physics). For signalling molecule, see Epidermal growth factor. In mathematics, a generating function is a formal … Wikipedia
Möbius function — This article is about the number theoretic Möbius function. For the combinatorial Möbius function, see incidence algebra. For the rational functions defined on the complex numbers, see Möbius transformation. The classical Möbius function μ(n) is… … Wikipedia

Mark and share
Search through all dictionaries
Translate…
Search Internet

Share the article and excerpts