ASK

ASK (Amplitude Shift Keying)

ASK, también llamado OOK(On-Off Keying).

Para calcular su DEP nos basamos en que es una señal unipolar NRZ modulada por un coseno:

Constelación ASK

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

Para la probabilidad de error (BER):

BER de ASK

4-ASK

4-ASK

${\begin{aligned}&s_{4-ASK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{k}p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&s_{4-ASK_{I}}(t)=\sum \limits _{k=-\infty }^{\infty }{a_{k}p\left(t-kT_{s}\right)}\\&a_{k}=\left\{0,A,2A,3A\right\}\\&{\bar {G}}_{x}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\&\left|P(f)\right|^{2}=T_{s}^{2}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\&m_{I_{k}}=0\cdot {\frac {1}{4}}+A\cdot {\frac {1}{4}}+2A\cdot {\frac {1}{4}}+3A\cdot {\frac {1}{4}}={\frac {3A}{2}}\\&P_{I_{k}}=\left(0\right)^{2}\cdot {\frac {1}{4}}+\left(A\right)^{2}\cdot {\frac {1}{4}}+\left(2A\right)^{2}\cdot {\frac {1}{4}}+\left(3A\right)^{2}\cdot {\frac {1}{4}}={\frac {14A^{2}}{4}}\\&\sigma _{I_{k}}^{2}=P_{I_{k}}-m_{I_{k}}^{2}={\frac {14A^{2}}{4}}-\left({\frac {3A}{2}}\right)^{2}={\frac {14A^{2}}{4}}-{\frac {9A^{2}}{4}}={\frac {5A^{2}}{4}}\\&{\bar {G}}_{x}(f)={\frac {5A^{2}}{4}}\cdot R_{s}\left|P(f)\right|^{2}+{\frac {9A^{2}}{4}}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\&\left|P(f)\right|^{2}=T_{s}^{2}\operatorname {sinc} ^{2}\left(T_{s}f\right)\\&{\bar {G}}_{I}(f)={\frac {5A^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)+{\frac {9A^{2}}{4}}\cdot \sum \limits _{k=-\infty }^{\infty }{\left|\operatorname {sinc} ^{2}\left(k\right)\right|^{2}\delta \left(f-kR_{s}\right)}=\\&{\bar {G}}_{I}(f)={\frac {5A^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)+{\frac {9A^{2}}{4}}\cdot \delta \left(f-R_{s}\right)\\\end{aligned}}$

${\begin{aligned}&G_{x}(f)={\frac {G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}}+{\frac {G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}}\to G_{Q}(f)=0\\&G_{x}(f)={\frac {G_{I}\left(f\pm f_{c}\right)}{4}}\to \\&G_{4-ASK}(f)={\frac {5A^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)+{\frac {9A^{2}}{4^{2}}}\cdot \delta \left(\left(f\pm f_{c}\right)-R_{s}\right)\\\end{aligned}}$

Para la probabilidad de error (BER):

BER de 4-ASK

M-ASK

${\begin{aligned}&m_{I_{k}}=\sum \limits _{i=0}^{M-1}{{\frac {1}{M}}\cdot i}\Rightarrow \sum \limits _{i=1}^{n}{i}={\frac {n\left(n+1\right)}{2}}\Rightarrow {\frac {1}{M}}\cdot \left({\frac {\left(M-1\right)M}{2}}\right)={\frac {M-1}{2}}\\&P_{I_{k}}=\sum \limits _{i=0}^{M-1}{{\frac {1}{M}}\cdot i^{2}\Rightarrow }\sum \limits _{i=1}^{n}{i^{2}}={\frac {n\left(n+1\right)\left(2n+1\right)}{6}}\Rightarrow {\frac {1}{M}}\cdot \left({\frac {\left(M-1\right)M\left(2\left(M-1\right)+1\right)}{6}}\right)=\\&P_{I_{k}}={\frac {\left(M-1\right)\left(2M-1\right)}{6}}\\&\sigma _{I_{k}}^{2}=P_{I_{k}}-m_{I_{k}}^{2}={\frac {\left(M-1\right)\left(2M-1\right)}{6}}-\left({\frac {M-1}{2}}\right)^{2}={\frac {2M^{2}-M-2M+1}{6}}-{\frac {M^{2}-2M+1}{4}}=\\&\sigma _{I_{k}}^{2}={\frac {4\cdot \left(2M^{2}-3M+1\right)}{4\cdot 6}}-{\frac {6\cdot \left(M^{2}-2M+1\right)}{4\cdot 6}}={\frac {2M^{2}-2}{4\cdot 6}}={\frac {M^{2}-1}{2\cdot 6}}\\&{\bar {G}}_{I}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\&G_{M-ASK}(f)={\frac {M^{2}-1}{12}}\cdot {\frac {1}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)+\left({\frac {M-1}{2}}\right)^{2}{\frac {1}{4}}\cdot \delta \left(\left(f\pm f_{c}\right)-R_{s}\right)\\\end{aligned}}$

Para la probabilidad de error (BER):

BER de M-ASK

Proyecto: Teoría de señales y comunicaciones

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