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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.
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
Mapping changes in continuous phase modulation
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{\displaystyle {\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
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7cb0aff71f3895a4cc5c5db424b2aad364736a)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a47e581b6dd995498b9416271b3db4a66b74765)