formulas funcom (parte 1)
DESCRIPTION
formulas para señales y fundamentos de comunicacionTRANSCRIPT
Formulario
RESUMEN
Pares clsicos de la transformada de FourierPulso rectangular
con
Grfica 1. Par transformado seal rect.
con
Grfica 2. Par transformado seal sinc.
Grfica 3. Par transformado seal rect.
Grfica 4. Par transformado seal sinc.
(Esto es vlido solo para la rect)Pulso triangular
con
Grfica 5. Par transformado seal triangular.
con
Grfica 6. Par transformado seal sinc2.
Sign
con
Grfica 7. Par transformado sign.
Funcin escaln unitario (Heaviside)
con
Grfica 8. Par transformado escaln unitario.
Funcin impulso unitario (Delta de Dirac)
con
Grfica 9. Par transformado impulso unitario.
Tren de Impulsos
con
Grfica 10. Par transformado tren de deltas.
Tonos puros
Grfica 11. Par transformado seal cosinusoidal.
Grfica 12. Par transformado seal sinusoidal.
Seal cuadrada
Serie de trigonomtrica:
Grfica 13. Transformada de Fourier seal cuadrada.Transformada de Hilbert
Grfica 14. Par transformado Transformada de Gilbert.
Potencia Promedio:
Teorema de Rayleigh:
Teorema de Parseval:
Densidad Espectral de Energa: Producto interno: (seales ortogonales)Transformada de Fourier de Seales Peridicas:
Densidad Espectral de Potencia de Seales Peridicas:
Densidad Espectral de Potencia y funcin de autocorrelacin:
Densidad Espectral de Potencia y LTIs:
Funcin de autocorrelacin:
Correlacin y Densidad Espectral Seales de Potencia:
Propiedades de la funcin de autocorrelacin: 1.
2.
3.
Propiedades de la Densidad Espectral de Energa:
1.
2.
3. Si es real
4. Si pasa por un SLIT la DSE de la salida es
Propiedades de la Densidad Espectral de Potencia:1.
2. Si es real
3.
4. Si pasa por un SLIT la DSP de la salida es
Transformada de Fourier: Transformada Inversa de Fourier:
Propiedad 1 Linealidad (superposicin):
Propiedad 2 Dilatacin o Escalamiento en tiempo:
Propiedad 3 Regla conjugacin o Complejo conjugado:
Propiedad 4 Dualidad:
Propiedad 5 Desplazamiento en el tiempo:
Propiedad 6 Desplazamiento en frecuencia:
Propiedad 7 rea Bajo g(t): Nivel DCPropiedad 8 rea Bajo G(f):
Propiedad 9 Diferenciacin en el dominio de tiempo:
Propiedad 10 Integracin en el Dominio del Tiempo: + mixtaPropiedad 11 Teorema de la Modulacin: Propiedad 12 Teorema de la Convolucin:
Propiedad 13 Teorema de la Correlacin:
Propiedad 14 Teorema de la Energa de Rayleigh:
Figura o cuadro mental seales
Probabilidad
Probabilidad condicional y eventos independientes:
Regla de Bayes:
Ensayos de BernoulliTeorema del Binomio de Newton: Nmero de formas de obtener k xitos en n intentos
Esto significa que la probabilidad de k xitos en n ensayos es
Donde denota la probabilidad de recibir y cuando x se transmiti.
Probabilidades marginales:
Probabilidad en subsistemas (si):
(f representa el fracaso del sistema) (Probabilidad de fallo de un subsistema es p)
(fu y fL representan el fracaso de la parte superior e inferior de los caminos)
La fiabilidad o confiabilidad es
Modelo de canal (AWGN)
Figura 1. Modelo de canal (AWGN).
Additive White Gaussian Noise
Grfica 15. Densidad Espectral de Potencia del ruido blanco gaussiano N(t).
Representacin del Ruido Rosado (Coloreado) en sus componentes en fase y en cuadratura
Grfica 16. Densidad Espectral de Potencia del ruido rosado n(t).
Envolvente compleja:
Representacin en magnitud y fase:
Aproximacin de Ingeniera:
Relacin Seal a Ruido SNR a la salida:
Relacin seal a ruido en el canal:
MODULACION DE ONDA CONTINUATcnicas de Modulacin en amplitud
Estndar AM
Figura 2. Modulador de AM.
Figura 3. Receptor no sincrnico o incoherente de AM.
Figura 4. Receptor sincrnico o coherente de AM.
Representacin temporal seal modulada:
Forma estndar:
ndice de modulacin:
Valor DC:
Amplitud moduladora:
Potencia Promedio:
Ancho de banda:
Eficiencia:
Espectro de la seal modulada:
Grfica 17. Espectro de la seal modulada en AM.Tcnicas de Modulacin exponencial (o de ngulo)Estndar FM
Figura 5. Modulador de FM.
Figura 6. Receptor Genrico sincrnico o coherente de FM
Figura 7. Demodulador de FM (Receptor Genrico PLL).Representacin temporal seal modulada: Representacin Fasorial de la seal modulada: ndice de modulacin: o
Fase instantnea:
Frecuencia angular instantnea:
Frecuencia instantnea:
f: Desviacin de frecuencia, es la distancia de separacin mxima entre la frecuencia instantnea fi y la frecuencia de la portadora fc.
Forma estndar:
Aproximaciones de Ancho de banda:
o Banda Angosta (NBFM) Caso AM particular:
o Banda Ancha (WBFM) Regla de Carlson: (Seales no sinusoidales)Representacin Series de Fourier de la seal modulada:
Representacin temporal seal modulada Coeficientes de Bessel:
Espectro de la seal modulada:
Grfica 18. Espectro de la seal modulada en FM.
De los valores de cada impulso de la grfica del espectro de la seal modulada y los de los coeficientes de Bessel, se tiene que:
Potencia Promedio:
Propiedades de las Funciones de Bessel
1. Impares signo contrario, Pares mismo signo (iguales)2. Si es pequeo en valor se cumple que:
3.
Estimacin Ancho de banda: donde
Donde c el nmero de coeficientes de Bessel representativos, de manera que el ancho de banda estimado de la seal en el rango de frecuencias que contiene determinado porcentaje (%) de la potencia de la seal modulada. La frmula anterior para el ancho de banda viene de lo siguiente:
Grfica 19. Espectro de la seal modulada en FM para el clculo del ancho debanda.
Filtros:
Moduladores de FM
Banda Angosta (NBFM)
Figura 8. Modulador FM de banda angosta.
Figura 9. Modulador FM de banda ancha.Anexos. Tabla de funciones de Bessel
Funcin de Bessel de Orden "n"
Beta01234567891011121314
0,10,9980,0500,001------------
0,20,9900,1000,005------------
0,30,9780,1480,0110,001-----------
0,40,9600,1960,0200,001-----------
0,50,9380,2420,0310,003-----------
0,60,9120,2870,0440,004-----------
0,70,8810,3290,0590,0070,001----------
0,80,8460,3690,0760,0100,001----------
0,90,8080,4060,0950,0140,002----------
1,00,7650,4400,1150,0200,002----------
1,50,5120,5580,2320,0610,0120,002---------
2,00,2240,5770,3530,1290,0340,0070,001--------
2,5-0,0480,4970,4460,2170,0740,0200,0040,001-------
3,0-0,2600,3390,4860,3090,1320,0430,0110,003-------
3,5-0,3800,1370,4590,3870,2040,0800,0250,0070,002------
4,0-0,397-0,0660,3640,4300,2810,1320,0490,0150,0040,001-----
4,5-0,321-0,2310,2180,4250,3480,1950,0840,0300,0090,0020,001----
5,0-0,178-0,3280,0470,3650,3910,2610,1310,0530,0180,0060,001----
5,5-0,007-0,341-0,1170,2560,3970,3210,1870,0870,0340,0110,0030,001---
6,00,151-0,277-0,2430,1150,3580,3620,2460,1300,0570,0210,0070,0020,001--
6,50,260-0,154-0,307-0,0350,2750,3740,3000,1800,0880,0370,0130,0040,0010,000-
7,00,300-0,005-0,301-0,1680,1580,3480,3390,2340,1280,0590,0240,0080,0030,001-
7,50,2660,135-0,230-0,2580,0240,2830,3540,2830,1740,0890,0390,0150,0050,002-
8,00,1720,235-0,113-0,291-0,1050,1860,3380,3210,2230,1260,0610,0260,0100,0030,001
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1
1
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Como se distribuye la energa de la seal en frecuencia
Vin(t)
VCO
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PA
LPF
BPF
m(t)
Sistema No lineal VCO
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Por qu se utiliza otra letra? Porque significa un retardo entre una seal y una misma seal. (No es en tiempo).
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Que tan parecida es la seal consigo misma en cuanto a energa (tiempo)
Como se comporta una seal en tiempo
Como se distribuye la seal en frecuencia
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S(t)=Acos[(t)]
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m(t)
BPF
LPF
Hcs(f)=1
PA
cos(2pifct)
DC
OL
S(t)=A(t)cos[2fc +0]
BPF
Sistema No lineal
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Capacitor discharge
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LNA
LPF
m(t)
BPF
Sincronizacin
LNA
cos(2pifc(ttd))
OL
LPF
m(t)
Sobre modulada
Modulada
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Antena
Sistema No lineal
Sistema No lineal
m(t)
LNA
HPF
Bloqueo de DC
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BPF
LNA
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Antena
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HPF
Bloqueo de DC
LPF
f(t)
m(t)
BPF
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PLL
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(t)
Discriminador
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h(t)=a(t-td)
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Tx
Ruido Blanco
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NBFM
DSB-SC
modulator
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/2
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WBFM
NBFM
MULTIPLICADOR EN FRECUENCIA
Frequency multiplier
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Contacto: [email protected] [email protected] Pgina 3
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