A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
of height 1 and base 2 in which case it is referred to as ''the'' triangular function. Triangular functions are useful in
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
and ''communication systems engineering'' as representations of idealized signals, and the triangular function specifically as an
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
kernel function from which more realistic signals can be derived, for example in
kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on ''kernels'' as w ...
. It also has applications in
pulse-code modulation
Pulse-code modulation (PCM) is a method used to digitally represent sampled analog signals. It is the standard form of digital audio in computers, compact discs, digital telephony and other digital audio applications. In a PCM stream, the ...
as a pulse shape for transmitting
digital signal
A digital signal is a signal that represents data as a sequence of discrete values; at any given time it can only take on, at most, one of a finite number of values. This contrasts with an analog signal, which represents continuous values; at ...
s and as a
matched filter
In signal processing, a matched filter is obtained by correlating a known delayed signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal wi ...
for receiving the signals. It is also used to define the triangular window sometimes called the
Bartlett window.
Definitions
The most common definition is as a piecewise function:
:
Equivalently, it may be defined as the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of two identical unit
rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname(t) = \Pi(t) =
\left\{\begin{array}{r ...
s:
:
The triangular function can also be represented as the product of the rectangular and
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
functions:
:
![Triangle function](https://upload.wikimedia.org/wikipedia/commons/b/b6/Triangle_function.png)
Note that some authors instead define the triangle function to have a base of width 1 instead of width 2:
:
In its most general form a triangular function is any linear
B-spline:
:
Whereas the definition at the top is a special case
:
where
,
, and
.
A linear B-spline is the same as a continuous
piecewise linear function , and this general triangle function is useful to formally define
as
:
where
for all integer
.
The piecewise linear function passes through every point expressed as coordinates with
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
, that is,
:
.
Scaling
For any parameter
:
:
Fourier transform
The transform is easily determined using the
convolution property of Fourier transforms and the
Fourier transform of the rectangular function:
:
where
is the
normalized sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the ...
.
See also
*
Källén function, also known as triangle function
*
Tent map
A tent () is a shelter consisting of sheets of fabric or other material draped over, attached to a frame of poles or a supporting rope. While smaller tents may be free-standing or attached to the ground, large tents are usually anchored using g ...
*
Triangular distribution
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''.
...
*
Triangle wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Like a square wave, the triangle wave contains only odd harmonics. However, the ...
, a piecewise linear periodic function
*
Trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
References
{{Reflist
Special functions