# Resumen y tablas para modulaciones digitales

${\begin{aligned}&{\bar {G}}_{x}(f)=\sigma _{x}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{x}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\end{aligned}}$
${\begin{aligned}&H(f)=H_{\beta }(f)*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}={\frac {\pi }{4\beta }}\cos \left({\frac {\pi }{2\beta }}f\right)\prod {\left({\frac {f}{2\beta }}\right)}*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}\\&h(t)=h_{\beta }(t)\cdot \operatorname {sinc} \left(Rt\right)={\frac {\cos \left(2\beta \pi t\right)}{1-\left(4\beta t\right)^{2}}}\cdot \operatorname {sinc} \left(Rt\right)\\\end{aligned}}$
Con filtro optimo: $h_{R}(t)=p^{*}\left(T_{s}-t\right)$
${\begin{aligned}&\int \limits _{V_{T}}^{\infty }{{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\partial t}=Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)\\&P_{e}=pr\left('0'sent\right)\cdot p\left(^{error}\!\!\diagup \!\!_{'0'sent}\;\right)+pr\left('1'sent\right)\cdot p\left(^{error}\!\!\diagup \!\!_{'1'sent}\;\right)=\\&pr\left('0'sent\right)\cdot Q\left({\frac {\left|V_{T}-m_{'0'}\right|}{\sqrt {\sigma ^{2}}}}\right)+pr\left('1'sent\right)\cdot Q\left({\frac {\left|V_{T}-m_{'1'}\right|}{\sqrt {\sigma ^{2}}}}\right)\\&s_{T}(t)=\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}\\&m_{y_{PR}(t)}=\int _{0}^{T_{s}}{s_{T}(t)kp^{*}\left(t\right)\partial t}\\&\sigma _{y_{PR}(t)}^{2}={\frac {\eta }{2}}\cdot E_{p(t)}={\frac {\eta }{2}}\int _{0}^{T_{s}}{p^{2}(t)\partial t}\\\end{aligned}}$
| | $s(t)$ | $a_{k}$ | ${\bar {G}}_{s}(f)$ | $Q(x)$ |
| Unipolar NRZ | $\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}$ | $a_{k}=\left\{0,+A\right\}$ | ${\frac {A^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)+{\frac {A^{2}}{4}}\delta \left(f-R_{s}\right)$ | $Q\left({\sqrt {\frac {E_{b}}{\eta }}}\right)$ |
| Unipolar RZ | $\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}$ | $a_{k}=\left\{0,+A\right\}$ | ${\frac {A^{2}}{16}}T_{s}\operatorname {sinc} ^{2}\left({\frac {T_{s}}{2}}f\right)+{\frac {A^{2}}{16}}\sum \limits _{k=-\infty }^{\infty }{\operatorname {sinc} ^{2}\left({\frac {k}{2}}\right)\delta \left(f-kR_{s}\right)}$ | $Q\left({\sqrt {\frac {E_{b}}{\eta }}}\right)$ |
| Polar | $\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}$ | $a_{k}=\left\{-A,+A\right\}$ | $A^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)$ | $Q\left({\sqrt {\frac {2E_{b}}{\eta }}}\right)$ |
| Bipolar | $\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}$ | $a_{k}=\left\{0,\pm A\right\}$ | ${\frac {A^{2}}{2}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)$ | $\simeq {\frac {3}{2}}Q\left({\sqrt {\frac {E_{b}}{\eta }}}\right)$ |
| Manchester | $\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}$ | $a_{k}=\left\{-A,+A\right\}$ | $A^{2}T_{s}\operatorname {sinc} ^{2}\left({\frac {T_{s}f}{2}}\right)\sin ^{2}\left(\pi f{\frac {T_{s}}{2}}\right)$ | $Q\left({\sqrt {\frac {2E_{b}}{\eta }}}\right)$ |
| ASK(OOK) | $A_{c}\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}\cos \left(\omega _{c}t\right)$ | $a_{k}=\left\{0,+A\right\}$ | ${\frac {A^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)+{\frac {A^{2}}{4^{2}}}\delta \left(f\pm f_{c}-R_{s}\right)$ | $Q\left({\sqrt {\frac {E_{b}}{\eta }}}\right)$ |
| BPSK(PRK) | $A_{c}\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}\cos \left(\omega _{c}t\right)$ | $a_{k}=\left\{-1,+1\right\}$ | ${\frac {A^{2}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)}{4}}$ | $Q\left({\sqrt {\frac {2E_{b}}{\eta }}}\right)$ |
| QPSK | ${\begin{aligned}&A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&-A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{Q_{k}}p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}\\\end{aligned}}$ | ${\begin{aligned}&a_{I_{k}}=\left\{-{\frac {1}{\sqrt {2}}},+{\frac {1}{\sqrt {2}}}\right\}\\&a_{Q_{k}}=\left\{-{\frac {1}{\sqrt {2}}},+{\frac {1}{\sqrt {2}}}\right\}\\\end{aligned}}$ | ${\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)$ | $Q\left({\sqrt {\frac {2E_{b}}{\eta }}}\right)$ |
| MSK | ${\begin{aligned}&A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}\cos \left({\frac {\pi }{2T_{b}}}t\right)p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&-A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{Q_{k}}\sin \left({\frac {\pi }{2T_{b}}}t\right)p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}\\\end{aligned}}$ | ${\begin{aligned}&a_{I_{k}}=\left\{-A,+A\right\}\\&a_{Q_{k}}=\left\{-A,+A\right\}\\\end{aligned}}$ | $A_{c}^{2}{\frac {\sigma _{a_{k}}^{2}}{2}}\cdot T_{b}\left({\frac {4}{\pi }}\right)^{2}\left[{\frac {\cos \left(2T_{b}\pi \left(f\pm f_{c}\right)\right)}{1-\left(4T_{b}\left(f\pm f_{c}\right)\right)^{2}}}\right]^{2}$ | $Q\left({\sqrt {\frac {2E_{b}}{\eta }}}\right)$ |
| M-PSK | $A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)}$ | ${\begin{aligned}&a_{I_{k}}=\cos \left({\frac {\pi }{M}}i\right);i=1,2,3...\\&a_{Q_{k}}=\sin \left({\frac {\pi }{M}}i\right);i=1,2,3...\\\end{aligned}}$ | ${\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)$ | $\simeq Q\left({\sqrt {\frac {2E_{s}}{\eta }}}\sin \left({\frac {\pi }{M}}\right)\right)$ |
| QAM | ${\begin{aligned}&A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&-A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{Q_{k}}p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}\\\end{aligned}}$ | ${\begin{aligned}&a_{I_{k}}=\left\{\pm {\frac {1}{\sqrt {2}}},\pm {\frac {3}{\sqrt {2}}},...\right\}\\&a_{Q_{k}}=\left\{\pm {\frac {1}{\sqrt {2}}},\pm {\frac {3}{\sqrt {2}}},...\right\}\\\end{aligned}}$ | ${\frac {A_{c}^{2}}{2}}\sigma _{a_{k}}^{2}\cdot T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)$ | $2\left(1-{\frac {1}{\sqrt {M}}}\right)\cdot Q\left({\sqrt {{\frac {3}{M-1}}{\frac {E_{s}}{\eta }}}}\right)$ |
| FSK | ${\begin{aligned}&A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(b_{k}\cdot \omega _{c}t\right)}=\\&x_{ASK_{1}}(t)+x_{ASK_{2}}(t)\\\end{aligned}}$ | $a_{k}=\left\{0,+A\right\}$ | ${\begin{aligned}&{\frac {A_{c}^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c1}\right)\right)+{\frac {A_{c}^{2}}{4^{2}}}\delta \left(f\pm f_{c1}-R_{s}\right)+\\&{\frac {A_{c}^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c2}\right)\right)+{\frac {A_{c}^{2}}{4^{2}}}\delta \left(f\pm f_{c2}-R_{s}\right)\\\end{aligned}}$ | $Q\left({\sqrt {\frac {E_{b}}{\eta }}}\right)$ |