# MSK
## MSK (Minimum Shift Keying)

Es una señal PSK que tiene como moduladora la función coseno alzado, técnicamente, es una señal OQPSK que usa un señal senoidal en vez de la típica función rectangular. Tiene una eficiencia espectral alta, por el hecho de usar un filtro de coseno alzado, y de tener una envolvente constante, lo que facilita la detección. Además, como máximo puede haber un desfase de pi/2 (debido a que es OQPSK) , lo que hace que los lóbulos secundarios no sean tan grandes.
${\begin{aligned}&s_{MSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}\cos \left({\frac {\pi }{2T_{b}}}t\right)p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}-A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{Q_{k}}\sin \left({\frac {\pi }{2T_{b}}}t\right)p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}\\&s_{MSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+b_{k}{\frac {\pi }{2T_{b}}}t+\varphi _{k}\right)}\\&s_{MSK_{I}}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}\cos \left({\frac {\pi }{2T_{b}}}t\right)p\left(t-kT_{s}\right)}\\&s_{MSK_{Q}}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{Q_{k}}\sin \left({\frac {\pi }{2T_{b}}}t\right)p\left(t-kT_{s}\right)}\\\end{aligned}}$
${\begin{aligned}&b_{k}=\left\{+1,-1\right\}\\&\varphi _{k}=\left\{0,\pi \right\}\to a_{I_{k}}=\left\{+1,-1\right\}\\&a_{Q_{k}}=\left\{+1,-1\right\}\\&f_{i}(t)={\frac {1}{2\pi }}{\frac {\partial \phi _{i}(t)}{\partial t}}=f_{c}+b_{k}{\frac {1}{2\pi }}{\frac {\pi }{2T_{b}}}=f_{c}+b_{k}{\frac {1}{4T_{b}}}\\&f_{i\max }={\frac {1}{4T_{b}}}\\&m={\frac {{\frac {1}{4T_{b}}}-\left(-{\frac {1}{4T_{b}}}\right)}{R_{b}}}={\frac {1}{2}}\\\end{aligned}}$
${\begin{aligned}&s_{MSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+b_{k}{\frac {\pi }{2T_{b}}}t+\varphi _{k}\right)}\\&b_{k}=\left\{-1,+1\right\}\\&\varphi _{k}=\left\{0,\pi \right\}\\&s_{MSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(b_{k}{\frac {\pi }{2T_{b}}}t+\varphi _{k}\right)\cos \left(\omega _{c}t\right)-}\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\sin \left(b_{k}{\frac {\pi }{2T_{b}}}t+\varphi _{k}\right)\sin \left(\omega _{c}t\right)\to }\\&a_{I_{k}}=\cos \left(\varphi _{k}\right)\\&\cos \left(b_{k}{\frac {\pi }{2T_{b}}}t+\underbrace {\varphi _{k}} _{0,\pi }\right)=a_{I_{k}}\cos \left(\underbrace {b_{k}} _{\pm 1}{\frac {\pi }{2T_{b}}}t\right)=a_{I_{k}}\cos \left({\frac {\pi }{2T_{b}}}t\right)\\&\sin \left(b_{k}{\frac {\pi }{2T_{b}}}t+\varphi _{k}\right)=a_{I_{k}}\sin \left(b_{k}{\frac {\pi }{2T_{b}}}t\right)=a_{I_{k}}b_{k}\sin \left(b_{k}{\frac {\pi }{2T_{b}}}t\right)\\&a_{Q_{k}}=a_{I_{k}}b_{k}\\&s_{MSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}p\left(t-kT_{s}\right)\cos \left({\frac {\pi }{2T_{b}}}t\right)\cos \left(\omega _{c}t\right)-}\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{Q_{k}}p\left(t-kT_{s}\right)\sin \left({\frac {\pi }{2T_{b}}}t\right)\sin \left(\omega _{c}t\right)}\\&a_{I_{k}}=\left\{\pm 1\right\};a_{Q_{k}}=\left\{\pm 1\right\}\\\end{aligned}}$
Como se ve tenemos envolvente constante, lo que facilita la deteccion
Ahora, para sacar la densidad espectral de potencia:
${\begin{aligned}&{\frac {\pi }{4\beta }}\cos \left({\frac {\pi }{2\beta }}t\right)\prod {\left({\frac {t}{2\beta }}\right)}\leftrightarrow {\frac {\cos \left(2\beta \pi f\right)}{1-\left(4\beta f\right)^{2}}}\\&p(t)=s_{MSK_{I}}(t)=\left({\frac {\pi }{2T_{b}}}t\right)p\left(t\right)=\cos \left({\frac {\pi }{2T_{b}}}t\right)\prod {\left({\frac {t}{2T_{b}}}\right)}\\&P(f)={\frac {4T_{b}}{\pi }}{\frac {\cos \left(2T_{b}\pi f\right)}{1-\left(4T_{b}f\right)^{2}}}\\&\left|P(f)\right|^{2}=\left({\frac {4T_{b}}{\pi }}\right)^{2}\left[{\frac {\cos \left(2T_{b}\pi f\right)}{1-\left(4T_{b}f\right)^{2}}}\right]^{2}\\\end{aligned}}$
${\begin{aligned}&{\bar {G}}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\&{\bar {G}}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+\underbrace {m_{a_{k}}^{2}} _{0}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}=\\&{\bar {G}}_{I}(f)={\bar {G}}_{Q}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}=\sigma _{a_{k}}^{2}\cdot R_{s}\left({\frac {4T_{b}}{\pi }}\right)^{2}\left[{\frac {\cos \left(2T_{b}\pi f\right)}{1-\left(4T_{b}f\right)^{2}}}\right]^{2}=\\&\sigma _{a_{k}}^{2}\cdot T_{b}\left({\frac {4}{\pi }}\right)^{2}\left[{\frac {\cos \left(2T_{b}\pi f\right)}{1-\left(4T_{b}f\right)^{2}}}\right]^{2}\\&m_{I_{k}}=m_{Q_{k}}=0\\&\sigma _{I_{k}}^{2}=\sigma _{Q_{k}}^{2}=1\\&{\bar {G}}_{I}(f)={\bar {G}}_{Q}(f)\\&G_{x}(f)={\frac {G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}}+{\frac {G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}}\to \\&G_{MSK}(f)=2{\frac {G_{I/Q}(f-f_{c})+G_{I/Q}(f+f_{c})}{4}}={\frac {G_{I/Q}(f\pm f_{c})}{2}}=\\&A_{c}^{2}{\frac {\sigma _{a_{k}}^{2}}{2}}\cdot T_{b}\left({\frac {4}{\pi }}\right)^{2}\left[{\frac {\cos \left(2T_{b}\pi \left(f\pm f_{c}\right)\right)}{1-\left(4T_{b}\left(f\pm f_{c}\right)\right)^{2}}}\right]^{2}=A_{c}^{2}{\frac {1}{2}}\cdot T_{b}\left({\frac {4}{\pi }}\right)^{2}\left[{\frac {\cos \left(2T_{b}\pi \left(f\pm f_{c}\right)\right)}{1-\left(4T_{b}\left(f\pm f_{c}\right)\right)^{2}}}\right]^{2}\\\end{aligned}}$

## GMSK (Gaussian Minimum Shift Keying)

Para la probabilidad de error (BER):
BER de MSK
* Proyecto: Departamento de Teoría de la Señal y Comunicaciones
* Anterior: QAM — MSK — Siguiente: Detección de señales digitales con ruido