Tent function, often used in signal processing
Exemplary triangular functionAtriangular function (also known as atriangle function ,hat function , ortent function ) is a function whose graph takes the shape of a triangle. Often this is anisosceles triangle of height 1 and base 2 in which case it is referred to asthe triangular function. Triangular functions are useful insignal processing andcommunication systems engineering as representations of idealized signals, and the triangular function specifically as anintegral transform kernel function from which more realistic signals can be derived, for example inkernel density estimation . It also has applications inpulse-code modulation as a pulse shape for transmittingdigital signals and as amatched filter for receiving the signals. It is also used to define thetriangular window sometimes called theBartlett window .
The most common definition is as a piecewise function:
tri ( x ) = Λ ( x ) = def max ( 1 − | x | , 0 ) = { 1 − | x | , | x | < 1 ; 0 otherwise . {\displaystyle {\begin{aligned}\operatorname {tri} (x)=\Lambda (x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-|x|,0{\big )}\\&={\begin{cases}1-|x|,&|x|<1;\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}} Equivalently, it may be defined as theconvolution of two identical unitrectangular functions :
tri ( x ) = rect ( x ) ∗ rect ( x ) = ∫ − ∞ ∞ rect ( x − τ ) ⋅ rect ( τ ) d τ . {\displaystyle {\begin{aligned}\operatorname {tri} (x)&=\operatorname {rect} (x)*\operatorname {rect} (x)\\&=\int _{-\infty }^{\infty }\operatorname {rect} (x-\tau )\cdot \operatorname {rect} (\tau )\,d\tau .\\\end{aligned}}} The triangular function can also be represented as the product of the rectangular andabsolute value functions:
tri ( x ) = rect ( x / 2 ) ( 1 − | x | ) . {\displaystyle \operatorname {tri} (x)=\operatorname {rect} (x/2){\big (}1-|x|{\big )}.} Alternative triangle function Note that some authors instead define the triangle function to have a base of width 1 instead of width 2:
tri ( 2 x ) = Λ ( 2 x ) = def max ( 1 − 2 | x | , 0 ) = { 1 − 2 | x | , | x | < 1 2 ; 0 otherwise . {\displaystyle {\begin{aligned}\operatorname {tri} (2x)=\Lambda (2x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-2|x|,0{\big )}\\&={\begin{cases}1-2|x|,&|x|<{\tfrac {1}{2}};\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}} In its most general form a triangular function is any linearB-spline :[ 1]
tri j ( x ) = { ( x − x j − 1 ) / ( x j − x j − 1 ) , x j − 1 ≤ x < x j ; ( x j + 1 − x ) / ( x j + 1 − x j ) , x j ≤ x < x j + 1 ; 0 otherwise . {\displaystyle \operatorname {tri} _{j}(x)={\begin{cases}(x-x_{j-1})/(x_{j}-x_{j-1}),&x_{j-1}\leq x<x_{j};\\(x_{j+1}-x)/(x_{j+1}-x_{j}),&x_{j}\leq x<x_{j+1};\\0&{\text{otherwise}}.\end{cases}}} Whereas the definition at the top is a special case
Λ ( x ) = tri j ( x ) , {\displaystyle \Lambda (x)=\operatorname {tri} _{j}(x),} wherex j − 1 = − 1 {\displaystyle x_{j-1}=-1} x j = 0 {\displaystyle x_{j}=0} x j + 1 = 1 {\displaystyle x_{j+1}=1}
A linear B-spline is the same as a continuouspiecewise linear function f ( x ) {\displaystyle f(x)} f ( x ) {\displaystyle f(x)}
f ( x ) = ∑ j y j ⋅ tri j ( x ) , {\displaystyle f(x)=\sum _{j}y_{j}\cdot \operatorname {tri} _{j}(x),} wherex j < x j + 1 {\displaystyle x_{j}<x_{j+1}} j {\displaystyle j} ordered pair ( x j , y j ) {\displaystyle (x_{j},y_{j})}
f ( x j ) = y j {\displaystyle f(x_{j})=y_{j}} For any parametera ≠ 0 {\displaystyle a\neq 0}
tri ( t a ) = ( 1 a ) rect ( t a ) ∗ ( 1 a ) rect ( t a ) = ∫ − ∞ ∞ 1 | a | rect ( τ a ) ⋅ rect ( t − τ a ) d τ = { 1 − | t / a | , | t | < | a | ; 0 otherwise . {\displaystyle {\begin{aligned}\operatorname {tri} \left({\tfrac {t}{a}}\right)&=\left({\tfrac {1}{\sqrt {a}}}\right)\operatorname {rect} \left({\tfrac {t}{a}}\right)*\left({\tfrac {1}{\sqrt {a}}}\right)\operatorname {rect} \left({\tfrac {t}{a}}\right)=\int _{-\infty }^{\infty }{\tfrac {1}{|a|}}\operatorname {rect} \left({\tfrac {\tau }{a}}\right)\cdot \operatorname {rect} \left({\tfrac {t-\tau }{a}}\right)\,d\tau \\&={\begin{cases}1-|t/a|,&|t|<|a|;\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}} The transform is easily determined using theconvolution property of Fourier transforms and theFourier transform of the rectangular function :
F { tri ( t ) } = F { rect ( t ) ∗ rect ( t ) } = F { rect ( t ) } ⋅ F { rect ( t ) } = F { rect ( t ) } 2 = s i n c 2 ( f ) , {\displaystyle {\begin{aligned}{\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{aligned}}} wheresinc ( x ) = sin ( π x ) / ( π x ) {\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)} normalized sinc function .
For the general form, we have:
F { tri ( t a ) } = F { 1 a rect ( t a ) ∗ 1 a rect ( t a ) } = 1 a F { rect ( t a ) } ⋅ F { rect ( t a ) } = 1 a F { rect ( t a ) } 2 = 1 a a 2 s i n c 2 ( a ⋅ f ) = a s i n c 2 ( a ⋅ f ) . {\displaystyle {\begin{aligned}{\mathcal {F}}\{\operatorname {tri} \left({\tfrac {t}{a}}\right)\}&={\mathcal {F}}\{{\tfrac {1}{\sqrt {a}}}\operatorname {rect} \left({\tfrac {t}{a}}\right)*{\tfrac {1}{\sqrt {a}}}\operatorname {rect} \left({\tfrac {t}{a}}\right)\}\\&={\tfrac {1}{a}}\ {\mathcal {F}}\{\operatorname {rect} \left({\tfrac {t}{a}}\right)\}\cdot {\mathcal {F}}\{\operatorname {rect} \left({\tfrac {t}{a}}\right)\}\\&={\tfrac {1}{a}}\ {\mathcal {F}}\{\operatorname {rect} \left({\tfrac {t}{a}}\right)\}^{2}\\&={\tfrac {1}{a}}\ {a}^{2}\ \mathrm {sinc} ^{2}(a\cdot f)={a}\ \mathrm {sinc} ^{2}(a\cdot f).\end{aligned}}}
