PSK (Phase Shift Keying) [ editar ]
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{\displaystyle {\begin{aligned}&s_{PSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)=}\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}-A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)\underbrace {\sin \left(\varphi _{k}\right)} _{Q_{k}}}}\\&\cos ^{2}x+\sin ^{2}x=1\to \\&I_{k}^{2}+Q_{k}^{2}=1\\\end{aligned}}}
Dependiendo del numero de niveles tenemos diferentes tipos de PSK
BPSK (Binary Phase Shift Keying) [ editar ]
Tambien llamada PRK (Phase Reversal Keying).
Constellation diagram for BPSK.
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{\displaystyle {\begin{aligned}&s_{BPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)=}\\&\varphi _{k}=\left\{0,\pi \right\}\to s_{BPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}}\\\end{aligned}}}
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{\displaystyle \varphi _{k}}
I
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{\displaystyle I_{k}}
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{\displaystyle 0}
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{\displaystyle +1}
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{\displaystyle \pi }
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{\displaystyle -1}
Por lo que vemos, para este caso particular de PSK, la señal puede ser modelada como una codificacion polar modulada por un coseno
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{\displaystyle {\begin{aligned}&x_{BPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&x_{I}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)},x_{Q}(t)=0\\&{\bar {G}}_{I}(f)=G_{polar}(f)=A^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\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}}\to \\&G_{x_{BPSK}}(f)={\frac {A^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)}{4}}\\\end{aligned}}}
BER de BPSK
QPSK (Quadrature Phase Shift Keying) [ editar ]
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{\displaystyle {\begin{aligned}&s_{PSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)=}\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}-A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)\underbrace {\sin \left(\varphi _{k}\right)} _{Q_{k}}}}\\\end{aligned}}}
Existe más de un tipo de QPSK, la que se utiliza en la práctica:
Constellation diagram for QPSK with Gray coding. Each adjacent symbol only differs by one bit.
φ
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{\displaystyle \varphi _{k}}
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{\displaystyle I_{k}}
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{\displaystyle Q_{k}}
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{\displaystyle +{}^{\pi }\!\!\diagup \!\!{}_{4}\;}
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{\displaystyle {\frac {1}{\sqrt {2}}}}
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{\displaystyle +{}^{3\pi }\!\!\diagup \!\!{}_{4}\;}
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{\displaystyle -{\frac {1}{\sqrt {2}}}}
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{\displaystyle -{}^{3\pi }\!\!\diagup \!\!{}_{4}\;}
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{\displaystyle -{\frac {1}{\sqrt {2}}}}
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{\displaystyle -{}^{\pi }\!\!\diagup \!\!{}_{4}\;}
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{\displaystyle {\frac {1}{\sqrt {2}}}}
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{\displaystyle {\begin{aligned}&s_{QPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)-A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}}\\&s_{I}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)}\\&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)}\\\end{aligned}}}
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{\displaystyle {\begin{aligned}&s_{I}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)}\to \\&I_{k}=\left\{{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},{\frac {1}{\sqrt {2}}}\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}}={\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}+{\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}=0\\&P_{I_{k}}=\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{I_{k}}^{2}=P_{I_{k}}-m_{I_{k}}^{2}={\frac {1}{2}}\\&{\bar {G}}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+\underbrace {m_{a_{k}}^{2}} _{0}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}=\underbrace {\sigma _{a_{k}}^{2}} _{\left({}^{1}\!\!\diagup \!\!{}_{2}\;\right)^{2}}\cdot R_{s}\cdot T_{s}^{2}\operatorname {sinc} ^{2}\left(T_{s}f\right)=\\&{\bar {G}}_{I}(f)=A_{c}^{2}\sigma _{a_{k}}^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)={\frac {A_{c}^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\\end{aligned}}}
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{\displaystyle {\begin{aligned}&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)}\to \\&Q_{k}=\left\{{\frac {1}{\sqrt {2}}},{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}}\right\}\to m_{a_{k}}=0,\sigma _{a_{k}}^{2}={\frac {1}{2}}\\&m_{Q_{k}}={\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}+{\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}=0\\&P_{Q_{k}}=\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{Q_{k}}^{2}=P_{Q_{k}}-m_{Q_{k}}^{2}={\frac {1}{2}}\\&{\bar {G}}_{Q}(f)=A_{c}^{2}\sigma _{a_{k}}^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)={\frac {A_{c}^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\\end{aligned}}}
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{\displaystyle {\begin{aligned}&m_{I_{k}}=m_{Q_{k}}=0\\&\sigma _{I_{k}}^{2}=\sigma _{Q_{k}}^{2}={\frac {1}{2}}\\&{\bar {G}}_{I}(f)={\bar {G}}_{Q}(f)=A_{c}^{2}\sigma _{a_{k}}^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)={\frac {A_{c}^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\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_{QPSK}(f)=2{\frac {G_{I/Q}(f-f_{c})+G_{I/Q}(f+f_{c})}{4}}={\frac {G_{I/Q}(f\pm f_{c})}{2}}={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)\\\end{aligned}}}
Para la probabilidad de error (BER):
BER de QPSK
Existe otro tipo de QPSK:
4PSK
φ
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{\displaystyle \varphi _{k}}
I
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{\displaystyle I_{k}}
Q
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{\displaystyle Q_{k}}
0
{\displaystyle 0}
+
1
{\displaystyle +1}
0
{\displaystyle 0}
π
╱
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{\displaystyle {}^{\pi }\!\!\diagup \!\!{}_{2}\;}
0
{\displaystyle 0}
+
1
{\displaystyle +1}
π
{\displaystyle \pi }
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{\displaystyle -1}
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{\displaystyle 0}
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{\displaystyle -{}^{\pi }\!\!\diagup \!\!{}_{2}\;}
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{\displaystyle 0}
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{\displaystyle -1}
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{\displaystyle {\begin{aligned}&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)}\to \\&I_{k}=\left\{+1,0,-1,0\right\}\\&m_{a_{k}}=+1\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}=0\\&P_{a_{k}}=+1^{2}\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)^{2}\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{a_{k}}^{2}=P_{a_{k}}-m_{a_{k}}^{2}={\frac {1}{2}}\\&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)}\to \\&Q_{k}=\left\{0,+1,0,-1\right\}\\&m_{a_{k}}=0\cdot {\frac {1}{4}}+1\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)\cdot {\frac {1}{4}}=0\\&P_{a_{k}}=0\cdot {\frac {1}{4}}+1^{2}{\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)^{2}\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{a_{k}}^{2}=P_{a_{k}}-m_{a_{k}}^{2}={\frac {1}{2}}\\\end{aligned}}}
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{\displaystyle G_{4PSK}(f)={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)}
BER de 4PSK
Como consecuencia final, vemos que la media y varianza de una señal PSK es constante:
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{\displaystyle {\begin{aligned}&m_{I_{k}}=m_{Q_{k}}=0\\&\sigma _{I_{k}}^{2}=\sigma _{Q_{k}}^{2}={\frac {1}{2}}\\\end{aligned}}}
8-PSK,16-PSK… [ editar ] Como se ha visto, la media y varianza de la señal no cambia, por lo que la densidad espectral de potencia será siempre igual independientemente del número de símbolos usados:
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{\displaystyle G_{PSK}(f)={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)}
Constellation diagram for 8-PSK with Gray coding.
Para la probabilidad de error (BER):
BER de M-PSK
Signal doesn't cross zero, because only one bit of the symbol is changed at a time
Dual constellation diagram for π/4-QPSK. This shows the two separate constellations with identical Gray coding but rotated by 45° with respect to each other.
