<=
![endif]>H <=
/i>p <=
/i>( z) <=
![endif]>H <=
/i>β ( z) <=
![endif]>H <=
/i>s <=
/i>( z) <=
![endif]>H <=
/i>d <=
/i>( z) <=
![endif]>H <=
/i>ro =
( z) <=
![endif]>Figura 3=
Diagrama de bloques de la
implementación del algoritmo EKS en Simulink
A continuación, se analizan los filtros utilizados en la
implementación.
Filtro de línea de retardo (Delay Line Filter )
Este filtro se utiliza para modelar el comportamiento vibratorio de=
una
cuerda, ya que una cuerda vibrante física tiene un retardo inherente=
que
determina su frecuencia de oscilación.
Donde:
N es el número de muestras de retardo.
representa un
retardo de una muestra en tiempo discreto.
indica que la
señal de entrada se retrasa N muestras antes de salir como señ=
;al
de salida.
El número N controla la longitud de la línea de retar=
do
y, por lo tanto, determina la frecuencia de la nota que produce la cuerda
sintetizada. Un valor grande de N corresponde a un tono más bajo (po=
rque
el retardo es más largo), mientras que un valor pequeño de N
produce un tono más alto (retardo más corto). En la Figura 4 =
se
muestra en bloques de Simulink el cálcul=
o del
número de muestras de retardo para simular el sonido de la primera
cuerda, que vibra a una frecuencia de 329 Hz., para una frecuencia de muest=
reo
de 44100 muestras por segundo.
Figura 4=
Implementación del cálculo del número de muest=
ras
de retardo N necesario para simular el sonido de una cuerda a 329 Hz=
o:p>
En una cuerda real, una perturbación viaja a lo largo de la
cuerda y luego se refleja en los extremos. El retardo digital simula este proceso, haciendo que la
señal se retrase y se retroalimenta a través del sistema de
síntesis para continuar la simulación de la vibración =
de
la cuerda.
Filtro pasa-bajos dirección de la
pica (Pick-direction lowpa=
ss
filter )
Es un tipo de filtro que se utiliza en la síntesis de cuerdas
para simular el efecto que tiene la dirección en la que se pulsa (o
"pica") una cuerda en el sonido resultante. Es particularmente
relevante en la síntesis física de instrumentos de cuerda, co=
mo
guitarras o bajos, donde la manera en que se pulsa la cuerda afecta las
características tonales y el contenido armónico. =
span>
En instrumentos de cuerda reales, cuando una cuerda es pulsada en
diferentes direcciones (por ejemplo, hacia arriba o hacia abajo con una
púa, o con los dedos), esto afecta el espectro de frecuencias que se
produce, y el contenido armónico resultante varía. Cuando una
cuerda se pulsa con fuerza, especialmente en una dirección particula=
r,
se generan armónicos más agudos y ricos. En cambio, una
pulsación más suave o en una dirección diferente puede
resultar en un sonido con menos armónicos altos, más amortigu=
ado
o "apagado". El filtro de paso bajo se introduce para modelar este
fenómeno. Esta fórmula describe un filtro de primer orden con=
un
comportamiento específico que depende del valor de p.
Figura 5
Salida del filtro
p
es un parámetro que controla el nivel de filtrado, y su valor debe
estar en el rango 0≤p <1
1−p
controla la ganancia aplicada a la señal actual.
1−pz −1 introduce una retroalimentación
retardada, lo que suaviza la señal y atenúa las frecuencias
altas.
p cercano a 0: Permite un sonido más brillante con más
armónicos altos.
p cercano a 1: Produce un sonido más suave con menos
armónicos altos.
Filtro de peine de posición de
pulsación (Pick-position comb filter )
Este filtro introduce un retardo controlado por el parámetro
β, creando un filtro en peine que afecta las frecuencias
específicas del espectro de la cuerda.
La sección
introduce un retardo de una cantidad entera de muestras, controlado=
por
la posición de la pulsación β. La diferencia entre 1 y e=
sta
versión retardada crea cancelaciones periódicas en el espectr=
o,
lo que elimina ciertos armónicos y deja pasar otros.
Por ejemplo, si β =3D1/2 se obtiene la simulación de una
pulsación en el medio de la cuerda, donde el filtro cancelará=
los
armónicos pares, creando un sonido más suave o "hueco&qu=
ot;
porque estos armónicos coinciden con los nodos de la vibració=
n de
la cuerda, mientras que N es el número de muestras por períod=
o.
Un valor de β cercano a 0 o 1 implica un pulso cerca de los extremos d=
e la
cuerda refuerza más armónicos y genera un sonido más
brillante.
Figura 6
Salida del filtro
Filtro de retardo pasa-todo por rigidez=
de
cuerda (String-Stiffness A=
llpass
Filter )
El filtro de retardo todo-paso por rigidez de cuerda simula el efec=
to
de la rigidez en cuerdas reales, introduciendo el fenómeno conocido =
como
dispersión en la propagación de las ondas a través de
cuerdas reales, las cuales no son perfectamente flexibles, como las cuerdas=
de
piano, guitarra o violín., lo que provoca que las ondas de frecuenci=
as
más altas viajen más rápido que las ondas de baja
frecuencia. Esto introduce una dispersión de fase, lo que significa =
que
los armónicos superiores no están perfectamente alineados con=
los
múltiplos enteros de la frecuencia fundamental, lo que da al sonido =
un
carácter diferente, más "metálico" o
"brillante".
En lugar de afectar la amplitud de la señal, como lo
haría un filtro paso bajo o paso alto, este filtro altera la fase de=
las
componentes frecuenciales sin cambiar su magnitud.
El filtro todo-paso tiene la siguiente forma general:
=
Donde A es un parámetro que controla la cantidad de
dispersión, relacionada con la rigidez de la cuerda y su longitud:
cuanto mayor es la rigidez, mayor es la dispersión, y este
parámetro se ajusta para controlar cuánto se altera la fase de
las frecuencias más altas. El parámetro A generalmente debe
cumplir con la condición:
0≤A <1
A=3D0: No hay dispersión de fase, y el filtro no tendr&aacut=
e;
ningún efecto sobre la señal. El sonido resultante será
como si no hubiera rigidez en la cuerda, y los armónicos estar&aacut=
e;n
espaciados uniformemente.
0<A<1: En este rango, el filtro introduce dispersión de
fase. A medida que AAA se acerca a 1, la cantid=
ad de
dispersión aumenta, y las frecuencias más altas se desplazan
más hacia arriba. Esto simula un mayor grado de rigidez en la cuerda.
Valores de 𝐴 cercanos a 0 generan un efecto de
dispersión muy leve, lo cual es adecuado para cuerdas que son casi
flexibles, como las de una guitarra o un violín. Valores de <=
span
lang=3DES-MX style=3D'font-size:10.0pt;line-height:115%;font-family:"Cambri=
a Math",serif;
mso-fareast-font-family:"Cambria Math";mso-bidi-font-family:"Cambria Math"'=
>𝐴 cercanos a 1
introducen más dispersión de fase y simulan cuerdas con mayor
rigidez, como las cuerdas gruesas de un piano o las cuerdas de bajo.=
o:p>
A≥1:Si A está fuera de este
rango (es decir, si es igual o mayor a 1), el filtro puede volverse inesta=
ble,
lo que provocaría comportamientos no deseados en la síntesis,
como distorsiones o incluso oscilaciones no controladas.
Filtro de amortiguamiento de cuerda (String-dampling filter ) <=
/span>
El filtro de amortiguamiento de cuerda es un filtro básico de
primer orden que, dentro del algoritmo EKS , mod=
ela el
amortiguamiento de las vibraciones de una cuerda. Este filtro es crucial pa=
ra
controlar la pérdida de energía en cada oscilación de =
la
cuerda, lo que afecta directamente el decaimiento y la suavidad del sonido
generado.
<=
![if !msEquation]>
El algoritmo EKS utiliza una lín=
ea de
retardo (o bucle de retroalimentación) para simular la vibraci&oacut=
e;n
de la cuerda, y el filtro de amortiguamiento se incorpora en este bucle para con=
trolar
el decaimiento de las oscilaciones de la cuerda. Este filtro atenúa =
las
frecuencias más altas y permite que las bajas frecuencias sigan
resonando durante más tiempo.
Cuando se pulsa una cuerda en un instrumento real, las frecuencias
más altas tienden a decaer más rápido que las bajas de=
bido
a la resistencia del material de la cuerda, el aire circundante y otros
factores físicos. El filtro simula este
comportamiento.
El filtro atenúa más las frecuencias altas conforme el
valor de S aumenta y se aproxima a 1, lo que genera un sonido más su=
ave
y menos brillante con un decaimiento más rápido de las
frecuencias agudas, imitando una cuerda que pierde energía
rápidamente debido al amortiguamiento. Las Frecuencias bajas tienden=
a
persistir durante más tiempo, dado que el filtro tiene menos efecto =
en
ellas. Si S=3D1 El filtro realiza un máximo suavizado, eliminando ca=
si todas
las frecuencias altas. El resultado es un sonido apagado, sin brillo, como =
si
la cuerda estuviera fuertemente amortiguada o sofocada. =
p>
Para controlar el decaimiento, es necesario incorporar una ganancia=
Filtro pasa-todo de afinación de
cuerda (First-order string=
-tuning
allpass filter )
Este filtro se utiliza en el contexto del algoritmo EKS
(Extended Karplus-Strong ) para ajustar de manera
precisa la afinación de la cuerda sin alterar la magnitud del espect=
ro
de la señal, sino solo su fase.
=
Dado que la longitud del ruido inicial es un número entero, =
se
descarta la parte fraccionaria al contar, provocando inexactitudes, es deci=
r,
una cuerda desafinada. Por ejemplo, si la frecuencia de muestreo es 44100 y=
la
longitud del ruido es 133 y 134, entonces las frecuencias de señal
correspondientes son 331,57 Hz y 329,10 Hz. Y la frecuencia de las notas E =
de
la primera octava (la primera cuerda al aire) es 329,63 Hz. Aquí la
diferencia está en décimas, pero, por ejemplo, para el traste=
15,
la diferencia puede ser ya de varios Hz. Este filtro reduce este error en l=
as
frecuencias que produce una guitarra estándar (Tabla 2). Sin embargo=
, no
es útil si la frecuencia de muestreo es alta (muy alta: varios cient=
os de
miles de Hz, o incluso más) o la frecuencia fundamental es baja, com=
o,
por ejemplo, para las cuerdas de bajo.
Tabla 2 =
Frecuencias de las cuerdas de guitarra
estándar
C=
uerda
Guitarra
<=
![endif]>Frecuencia (Hz)
6ta
82,4069
5ta
110
4ta
146,8324
3era
195,9977
2da
246,9417
1era
329,6276
El parámetro C influye directame=
nte
en la fase del filtro
Con C=3D0 el filtro se convierte en un pase directo (no introduce
ningún cambio), porque el numerador y denominador resultan en 1. Esto
significa que la señal no sufre ningún retraso adicional ni
alteración en la fase, simplemente aplicando un retardo de una muest=
ra.
Para valores positivos de C, el filtro introduce un retardo
fraccionario positivo. Esto significa que la señal tiene una fase
retardada, pero sin alterar su amplitud. A medida que C aumenta hacia 1, el
retardo fraccional es mayor, lo que puede afinar la frecuencia de
oscilación de la cuerda hacia valores más bajos. =
span>
Para C=3D1, el filtro introduce un retardo completo, equivalente a =
dos
muestras. Esto significa que no hay modificación en la señal
(como si fuera una señal de paso directo), pero introduce un retardo
máximo en fase sin afectar la magnitud.
Para valores negativos de C, el filtro introduce un adelanto de fase
(fase negativa), lo que equivale a un retardo fraccionario negativo. Esto p=
uede
ajustar la frecuencia de la cuerda ligeramente hacia valores más alt=
os.
Para
C=3D−1, el filtro genera un cambio en la fase que podría
interpretarse como una inversión de fase o una alteración
más significativa del retardo fraccional.
RESULTADOS Y DISCUSIÓN
Una vez realizado el análisis de los filtros EKS
se procede a parametrizar los valores de los coeficientes de los filtros a
usar, mediante la ejecución de una secuencia de comandos (script) de
MATLAB (Figura 7). La determinación de los parámetros utiliza=
dos
se realizó luego de pruebas de audición y variación de
éstos dentro de los rangos especificados en el análisis.=
Figura 7
Código de Matlab para inicializar
los parámetros de los filtros a ser utilizados en la simulació=
;n
Con estos parámetros, se obtiene el sonido claro y limpio de=
una
primera cuerda (nota mi), metálica, con la duración audible de
aproximadamente de 2,5 segundos, con una onda que va decayendo paulatinamen=
te
luego de que su amplitud máxima se alcanza (Figura 8). La frecuencia=
de
muestreo utilizada de 44100 Hz permite la generación del sonido con =
las
características descritas, frecuencia que es usada también po=
r (Matti Karjalainen et al.,=
1993)
sin embargo, si se disminuye esta frecuencia a valores de 22050 Hz y 11025 =
Hz o
menores el sonido va perdiendo realismo.
Figura 8=
Onda de salida del sistema
Se realizaron variaciones en los parámetros de los diferentes
filtros y usando diferentes frecuencias de cuerdas a simular, dentro de los
rangos descritos para cada uno de ellos, logrando diferentes variantes del
sonido resultante, desde sonidos más o menos metalizados o similares=
a
una cuerda de nylon, pudiendo parametrizarse además su duració=
;n
con el parámetro de ganancia a. En el trabajo de (Smith-III, 2013) se
programa los filtros usando el lenguaje FAUST , =
un
lenguaje de programación de alto nivel diseñado
específicamente para la creación de procesadores de audio,
sintetizadores, efectos de sonido y otros sistemas, basado en la
composición de funciones matemáticas para definir cómo=
se
procesan las señales de audio; en cambio, el programa Simulink permitió implementar de forma
gráfica los filtros necesarios para llevar a cabo la simulació=
;n,
cuyo resultado se guarda en un archivo WAV.
CONCLUSIÓN
El algoritmo de Karplus-Strong permite
simular digitalmente el sonido de una cuerda de guitarra, pudiendo ser
implementado en varias herramientas de programación. La
implementación en Simulink permite la
visualización en un entorno gráfico del diagrama de bloques d=
el
sistema, lo que permite su mejor comprensión y parametrizació=
n.
Existen un sinfín de filtros que permiten configurar las
características del sonido obtenido, cada uno con sus propios
parámetros y alcances.
Dado que Simulink tiene las herramientas
necesarias para interactuar con placas de hardware externo, como por ejempl=
o la
plataforma de software libre Arduino o Raspberry, basados en los resultados
obtenidos en la presente investigación, se podría implementar
físicamente el algoritmo KS en las placas mencionadas para la
creación de instrumentos virtuales y físicos con fines
experimentales o didácticos.
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Sociales y Humanidades , publicados en este sitio está disponible=
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bajo Licencia Creative Common=
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o:p>
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LATAM
Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción,
Paraguay.
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