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0.13 Ortogonalización gram-schmidt

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Método Gram-Schmidt para cálculo de bases ortogonales por medio de los símbolos de una señal. Se explica también el concepto de constelación en comunicaciones. Se incluyen dos programas en LabVIEW acerca de este tema, uno de los dos hecho por medio de MATLAB Script

El siguiente módulo está hecho en base al módulo ORTOGONALIZACIÓN GRAM-SCHMIDT Y TEORÍA BÁSICA DE LAS CONSTELACIONES , realizado por Venuska González y Mariangela Mezoa.

En matemáticas, el concepto de Ortogonalidad está referido al de Perpendicularidad . Se dice que dos vectores x → size 12{ { vec {x}}} {} y y → size 12{ { vec {y}}} {} pertenecientes a cierto espacio vectorial (V) son ortogonales si se cumple que el producto escalar entre ellos es igual a cero , es decir:

x → ⋅ y → = 0 size 12{ { vec {x}} cdot { vec {y}}=0} {}

A partir de un conjunto de vectores linealmente independientes se puede construir un nuevo conjunto de vectores ortonormales (Que cumplan con las condiciones de ortogonalidad y norma vectorial). Esto se conoce como el método de Ortogonalización Gram-Schmidt (G-S). Pero, ¿cómo aplicar este concepto para un sistema de comunicación digital?

Ortogonalización gram-schmidt

Supóngase una señal S i (t) que representa a un símbolo m i . Se estima que esta señal pase por el receptor que está encargado de obtener cada símbolo de la misma. Sin embargo, es evidente que al pasar por el canal, la señal se contaminará debido a la existencia de ruido en el sistema. En una condición ideal, el resultado sería el siguiente:

Sistema ideal de recepción (sin ruido). Cada símbolo mi de la señal se recibe sin interferencia.

Al introducir ruido blanco gaussiano en el sistema , quedaría como sigue:

Sistema de recepción con introducción de ruido

La segunda situación ocasiona que a la salida del receptor no se obtenga precisamente el símbolo m i , sino que se obtenga un estimado del símbolo original.

Es en este punto en donde entra el concepto de ortogonalización G-S: La señal S i (t) puede expresarse en función de un conjunto finito de bases (o vectores) ortonormales ( U ), de forma tal que cada forma de onda estaría relacionada con un coeficiente que será denominado s . Matemáticamente se tiene que:

S i ( t ) = ∑ j = 1 n s ij . U j ( t ) size 12{S rSub { size 8{i} } \( t \) = Sum cSub { size 8{j=1} } cSup { size 8{n} } {s rSub { size 8{ ital "ij"} } "." U rSub { size 8{j} } \( t \) } } {}

Es decir, a cada símbolo m i se le asocia una forma de onda s. desarrollando la fórmula anterior, para todos los símbolos posibles , se obtiene un sistema de ecuaciones como sigue:

S 1 ( t ) = s 11 . U 1 ( t ) + s 12 . U 2 ( t ) + s 13 . U 3 ( t ) + . . . + s 1n . U n ( t ) S 2 ( t ) = s 21 . U 1 ( t ) + s 22 . U 2 ( t ) + s 23 . U 3 ( t ) + . . . + s 2n . U n ( t ) S 3 ( t ) = s 31 . U 1 ( t ) + s 32 . U 2 ( t ) + s 33 . U 3 ( t ) + . . . + s 3n . U n ( t ) ⋮ S m ( t ) = s m1 . U 1 ( t ) + s m2 . U 2 ( t ) + s m3 . U 3 ( t ) + . . . + s mn . U n ( t ) alignl { stack { size 12{S rSub { size 8{1} } \( t \) =s rSub { size 8{"11"} } "." U rSub { size 8{1} } \( t \) +s rSub { size 8{"12"} } "." U rSub { size 8{2} } \( t \) +s rSub { size 8{"13"} } "." U rSub { size 8{3} } \( t \) + "." "." "." +s rSub { size 8{1n} } "." U rSub { size 8{n} } \( t \) } {} #S rSub { size 8{2} } \( t \) =s rSub { size 8{"21"} } "." U rSub { size 8{1} } \( t \) +s rSub { size 8{"22"} } "." U rSub { size 8{2} } \( t \) +s rSub { size 8{"23"} } "." U rSub { size 8{3} } \( t \) + "." "." "." +s rSub { size 8{2n} } "." U rSub { size 8{n} } \( t \) {} # S rSub { size 8{3} } \( t \) =s rSub { size 8{"31"} } "." U rSub { size 8{1} } \( t \) +s rSub { size 8{"32"} } "." U rSub { size 8{2} } \( t \) +s rSub { size 8{"33"} } "." U rSub { size 8{3} } \( t \) + "." "." "." +s rSub { size 8{3n} } "." U rSub { size 8{n} } \( t \) {} #dotsvert {} # S rSub { size 8{m} } \( t \) =s rSub { size 8{m1} } "." U rSub { size 8{1} } \( t \) +s rSub { size 8{m2} } "." U rSub { size 8{2} } \( t \) +s rSub { size 8{m3} } "." U rSub { size 8{3} } \( t \) + "." "." "." +s rSub { size 8{ ital "mn"} } "." U rSub { size 8{n} } \( t \) {}} } {}

El objetivo cuando se tiene un sistema como el mostrado en la figura 2 es el de obtener el estimado que más se aproxime al valor real. Esto se hace minimizando la energía de la señal de error entre el símbolo original y el estimado:

s j = ∫ 0 T S ( t ) . U j ( t ) dt j = 1,2,3, . . . , N alignl { stack { size 12{s rSub { size 8{j} } = Int cSub { size 8{0} } cSup { size 8{T} } {S \( t \) "." U rSub { size 8{j} } \( t \) ital "dt"} } {} #j=1,2,3, "." "." "." ,N {} } } {}

Visto desde la perspectiva vectorial, el procedimiento será entonces el de obtener una representación de la señal en función de dos vectores en el plano. El estimado del vector original sería entonces la proyección de éste sobre el plano:

Ejemplo aplicado a vectores. ŝ(t) es el estimado de cada forma de onda original S(t) y e(t) sería la introducción de ruido en el sistema.

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Source:  OpenStax, Señales y sistemas en matlab y labview. OpenStax CNX. Sep 23, 2011 Download for free at http://cnx.org/content/col11361/1.4
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