PSK

PSK (Phase Shift Keying)

${\begin{aligned}&s_{PSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)=}\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}-A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)\underbrace {\sin \left(\varphi _{k}\right)} _{Q_{k}}}}\\&\cos ^{2}x+\sin ^{2}x=1\to \\&I_{k}^{2}+Q_{k}^{2}=1\\\end{aligned}}$

Dependiendo del numero de niveles tenemos diferentes tipos de PSK

BPSK (Binary Phase Shift Keying)

Tambien llamada PRK (Phase Reversal Keying).

Constellation diagram for BPSK.

${\begin{aligned}&s_{BPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)=}\\&\varphi _{k}=\left\{0,\pi \right\}\to s_{BPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}}\\\end{aligned}}$

$\varphi _{k}$	$I_{k}$
$0$	$+1$
$\pi$	$-1$

Por lo que vemos, para este caso particular de PSK, la señal puede ser modelada como una codificacion polar modulada por un coseno

${\begin{aligned}&x_{BPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&x_{I}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)},x_{Q}(t)=0\\&{\bar {G}}_{I}(f)=G_{polar}(f)=A^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\&G_{x}(f)={\frac {G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}}+{\frac {G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}}\to \\&G_{x}(f)={\frac {G_{I}(f\pm f_{c})}{4}}\to \\&G_{x_{BPSK}}(f)={\frac {A^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)}{4}}\\\end{aligned}}$

Para la probabilidad de error (BER):

BER de BPSK

QPSK (Quadrature Phase Shift Keying)

${\begin{aligned}&s_{PSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)=}\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}-A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)\underbrace {\sin \left(\varphi _{k}\right)} _{Q_{k}}}}\\\end{aligned}}$

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.

$\varphi _{k}$	$I_{k}$	$Q_{k}$
$+{}^{\pi }\!\!\diagup \!\!{}_{4}\;$	${\frac {1}{\sqrt {2}}}$	${\frac {1}{\sqrt {2}}}$
$+{}^{3\pi }\!\!\diagup \!\!{}_{4}\;$	$-{\frac {1}{\sqrt {2}}}$	${\frac {1}{\sqrt {2}}}$
$-{}^{3\pi }\!\!\diagup \!\!{}_{4}\;$	$-{\frac {1}{\sqrt {2}}}$	$-{\frac {1}{\sqrt {2}}}$
$-{}^{\pi }\!\!\diagup \!\!{}_{4}\;$	${\frac {1}{\sqrt {2}}}$	$-{\frac {1}{\sqrt {2}}}$

${\begin{aligned}&s_{QPSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)-A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}}\\&s_{I}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)}\\&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)}\\\end{aligned}}$

Ahora, para sacar la densidad espectral de potencia:

${\begin{aligned}&s_{I}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)}\to \\&I_{k}=\left\{{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},{\frac {1}{\sqrt {2}}}\right\}\\&{\bar {G}}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\&\left|P(f)\right|^{2}=T_{s}^{2}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\&m_{I_{k}}={\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}+{\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}=0\\&P_{I_{k}}=\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{I_{k}}^{2}=P_{I_{k}}-m_{I_{k}}^{2}={\frac {1}{2}}\\&{\bar {G}}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+\underbrace {m_{a_{k}}^{2}} _{0}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}=\underbrace {\sigma _{a_{k}}^{2}} _{\left({}^{1}\!\!\diagup \!\!{}_{2}\;\right)^{2}}\cdot R_{s}\cdot T_{s}^{2}\operatorname {sinc} ^{2}\left(T_{s}f\right)=\\&{\bar {G}}_{I}(f)=A_{c}^{2}\sigma _{a_{k}}^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)={\frac {A_{c}^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\\end{aligned}}$

${\begin{aligned}&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)}\to \\&Q_{k}=\left\{{\frac {1}{\sqrt {2}}},{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}}\right\}\to m_{a_{k}}=0,\sigma _{a_{k}}^{2}={\frac {1}{2}}\\&m_{Q_{k}}={\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}+{\frac {1}{\sqrt {2}}}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)\cdot {\frac {1}{4}}=0\\&P_{Q_{k}}=\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left({\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}+\left(-{\frac {1}{\sqrt {2}}}\right)^{2}\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{Q_{k}}^{2}=P_{Q_{k}}-m_{Q_{k}}^{2}={\frac {1}{2}}\\&{\bar {G}}_{Q}(f)=A_{c}^{2}\sigma _{a_{k}}^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)={\frac {A_{c}^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\\end{aligned}}$

${\begin{aligned}&m_{I_{k}}=m_{Q_{k}}=0\\&\sigma _{I_{k}}^{2}=\sigma _{Q_{k}}^{2}={\frac {1}{2}}\\&{\bar {G}}_{I}(f)={\bar {G}}_{Q}(f)=A_{c}^{2}\sigma _{a_{k}}^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)={\frac {A_{c}^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\&G_{x}(f)={\frac {G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}}+{\frac {G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}}\to \\&G_{QPSK}(f)=2{\frac {G_{I/Q}(f-f_{c})+G_{I/Q}(f+f_{c})}{4}}={\frac {G_{I/Q}(f\pm f_{c})}{2}}={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)\\\end{aligned}}$

Para la probabilidad de error (BER):

BER de QPSK

Existe otro tipo de QPSK:

4PSK

$\varphi _{k}$	$I_{k}$	$Q_{k}$
$0$	$+1$	$0$
${}^{\pi }\!\!\diagup \!\!{}_{2}\;$	$0$	$+1$
$\pi$	$-1$	$0$
$-{}^{\pi }\!\!\diagup \!\!{}_{2}\;$	$0$	$-1$

${\begin{aligned}&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{I_{k}\cdot p\left(t-kT_{s}\right)}\to \\&I_{k}=\left\{+1,0,-1,0\right\}\\&m_{a_{k}}=+1\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}=0\\&P_{a_{k}}=+1^{2}\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)^{2}\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{a_{k}}^{2}=P_{a_{k}}-m_{a_{k}}^{2}={\frac {1}{2}}\\&s_{Q}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{Q_{k}\cdot p\left(t-kT_{s}\right)}\to \\&Q_{k}=\left\{0,+1,0,-1\right\}\\&m_{a_{k}}=0\cdot {\frac {1}{4}}+1\cdot {\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)\cdot {\frac {1}{4}}=0\\&P_{a_{k}}=0\cdot {\frac {1}{4}}+1^{2}{\frac {1}{4}}+0\cdot {\frac {1}{4}}+\left(-1\right)^{2}\cdot {\frac {1}{4}}={\frac {1}{2}}\\&\sigma _{a_{k}}^{2}=P_{a_{k}}-m_{a_{k}}^{2}={\frac {1}{2}}\\\end{aligned}}$

$G_{4PSK}(f)={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)$

Para la probabilidad de error (BER):

BER de 4PSK

Como consecuencia final, vemos que la media y varianza de una señal PSK es constante:

${\begin{aligned}&m_{I_{k}}=m_{Q_{k}}=0\\&\sigma _{I_{k}}^{2}=\sigma _{Q_{k}}^{2}={\frac {1}{2}}\\\end{aligned}}$

8-PSK,16-PSK…

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:

$G_{PSK}(f)={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)$

Constellation diagram for 8-PSK with Gray coding.

Para la probabilidad de error (BER):

BER de M-PSK

OQPSK

Signal doesn't cross zero, because only one bit of the symbol is changed at a time

π/4–QPSK

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.

Proyecto: Departamento de Teoría de la Señal y Comunicaciones

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