ASK

De Wikiversidad

[editar] ASK (Amplitude Shift Keying)

ASK, tambien llamado OOK(On-Off Keying).

Para calcular su DEP nos basamos en que es una señal unipolar NRZ modulada por un coseno:

ASKsignal.png


Constelacion ASK

\begin{align} & s_{ASK}(t)=A_{c}\sum\limits_{k=-\infty }^{\infty }{a_{k}p\left( t-kT_{s} \right)\cos \left( \omega _{c}t \right)} \\ & x_{I}(t)=\sum\limits_{k=-\infty }^{\infty }{a_{k}p\left( t-kT_{s} \right)},x_{Q}(t)=0 \\ & \bar{G}_{x_{I}}(f)=G_{NRZ}(f)=\frac{A^{2}}{4}T_{s}\operatorname{sinc}^{2}\left( T_{s}f \right)+\frac{A^{2}}{4}\delta \left( f-R_{s} \right) \\ & G_{x}(f)=\frac{G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}+\frac{G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}\to \\ & G_{x}(f)=\frac{G_{I}(f\pm f_{c})}{4} \\ & G_{x_{ASK}}(f)=\frac{A^{2}}{4^{2}}T_{s}\operatorname{sinc}^{2}\left( T_{s}\left( f\pm f_{c} \right) \right)+\frac{A^{2}}{4^{2}}\delta \left( f\pm f_{c}-R_{s} \right) \\ \end{align}

Para la probabilidad de error (BER):

BER de ASK

[editar] 4-PSK

4-ASK

\begin{align} & s_{4-ASK}(t)=A_{c}\sum\limits_{k=-\infty }^{\infty }{a_{k}p\left( t-kT_{s} \right)\cos \left( \omega _{c}t \right)} \\ & s_{4-ASK_{I}}(t)=\sum\limits_{k=-\infty }^{\infty }{a_{k}p\left( t-kT_{s} \right)} \\ & a_{k}=\left\{ 0,A,2A,3A \right\} \\ & \bar{G}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left| P(f) \right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum\limits_{k=-\infty }^{\infty }{\left| P(kR_{s}) \right|^{2}\delta \left( f-kR_{s} \right)} \\ & \left| P(f) \right|^{2}=T_{s}^{2}\operatorname{sinc}^{2}\left( T_{s}f \right) \\ & m_{I_{k}}=0\cdot \frac{1}{4}+A\cdot \frac{1}{4}+2A\cdot \frac{1}{4}+3A\cdot \frac{1}{4}=\frac{3A}{2} \\ & P_{I_{k}}=\left( 0 \right)^{2}\cdot \frac{1}{4}+\left( A \right)^{2}\cdot \frac{1}{4}+\left( 2A \right)^{2}\cdot \frac{1}{4}+\left( 3A \right)^{2}\cdot \frac{1}{4}=\frac{14A^{2}}{4} \\ & \sigma _{I_{k}}^{2}=P_{I_{k}}-m_{I_{k}}^{2}=\frac{14A^{2}}{4}-\left( \frac{3A}{2} \right)^{2}=\frac{14A^{2}}{4}-\frac{9A^{2}}{4}=\frac{5A^{2}}{4} \\ & \bar{G}_{x}(f)=\frac{5A^{2}}{4}\cdot R_{s}\left| P(f) \right|^{2}+\frac{9A^{2}}{4}\cdot R_{s}^{2}\sum\limits_{k=-\infty }^{\infty }{\left| P(kR_{s}) \right|^{2}\delta \left( f-kR_{s} \right)} \\ & \left| P(f) \right|^{2}=T_{s}^{2}\operatorname{sinc}^{2}\left( T_{s}f \right) \\ & \bar{G}_{I}(f)=\frac{5A^{2}}{4}T_{s}\operatorname{sinc}^{2}\left( T_{s}f \right)+\frac{9A^{2}}{4}\cdot \sum\limits_{k=-\infty }^{\infty }{\left| \operatorname{sinc}^{2}\left( k \right) \right|^{2}\delta \left( f-kR_{s} \right)}= \\ & \bar{G}_{I}(f)=\frac{5A^{2}}{4}T_{s}\operatorname{sinc}^{2}\left( T_{s}f \right)+\frac{9A^{2}}{4}\cdot \delta \left( f-R_{s} \right) \\ \end{align}

\begin{align} & G_{x}(f)=\frac{G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}+\frac{G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}\to G_{Q}(f)=0 \\ & G_{x}(f)=\frac{G_{I}\left( f\pm f_{c} \right)}{4}\to \\ & G_{4-ASK}(f)=\frac{5A^{2}}{4^{2}}T_{s}\operatorname{sinc}^{2}\left( T_{s}\left( f\pm f_{c} \right) \right)+\frac{9A^{2}}{4^{2}}\cdot \delta \left( \left( f\pm f_{c} \right)-R_{s} \right) \\ \end{align}

Para la probabilidad de error (BER):

BER de 4-ASK

[editar] M-ASK

\begin{align} & m_{I_{k}}=\sum\limits_{i=0}^{M-1}{\frac{1}{M}\cdot i}\Rightarrow \sum\limits_{i=1}^{n}{i}=\frac{n\left( n+1 \right)}{2}\Rightarrow \frac{1}{M}\cdot \left( \frac{\left( M-1 \right)M}{2} \right)=\frac{M-1}{2} \\ & P_{I_{k}}=\sum\limits_{i=0}^{M-1}{\frac{1}{M}\cdot i^{2}\Rightarrow }\sum\limits_{i=1}^{n}{i^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}\Rightarrow \frac{1}{M}\cdot \left( \frac{\left( M-1 \right)M\left( 2\left( M-1 \right)+1 \right)}{6} \right)= \\ & P_{I_{k}}=\frac{\left( M-1 \right)\left( 2M-1 \right)}{6} \\ & \sigma _{I_{k}}^{2}=P_{I_{k}}-m_{I_{k}}^{2}=\frac{\left( M-1 \right)\left( 2M-1 \right)}{6}-\left( \frac{M-1}{2} \right)^{2}=\frac{2M^{2}-M-2M+1}{6}-\frac{M^{2}-2M+1}{4}= \\ & \sigma _{I_{k}}^{2}=\frac{4\cdot \left( 2M^{2}-3M+1 \right)}{4\cdot 6}-\frac{6\cdot \left( M^{2}-2M+1 \right)}{4\cdot 6}=\frac{2M^{2}-2}{4\cdot 6}=\frac{M^{2}-1}{2\cdot 6} \\ & \bar{G}_{I}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left| P(f) \right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum\limits_{k=-\infty }^{\infty }{\left| P(kR_{s}) \right|^{2}\delta \left( f-kR_{s} \right)} \\ & G_{M-ASK}(f)=\frac{M^{2}-1}{12}\cdot \frac{1}{4}T_{s}\operatorname{sinc}^{2}\left( T_{s}\left( f\pm f_{c} \right) \right)+\left( \frac{M-1}{2} \right)^{2}\frac{1}{4}\cdot \delta \left( \left( f\pm f_{c} \right)-R_{s} \right) \\ \end{align}

Para la probabilidad de error (BER):

BER de M-ASK


Proyecto: Departamento de Teoría de la Señal y Comunicaciones
Anterior: Modulaciones digitales — ASK — Siguiente: PSK


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