sASK(t)=Ac∑k=−∞∞akp(t−kTs)cos(ωct)sPSK(t)=Ac∑k=−∞∞p(t−kTs)cos(ωct+φk)=Ac∑k=−∞∞p(t−kTs)cos(ωct)cos(φk)⏟Ik−Ac∑k=−∞∞p(t−kTs)sin(ωct)sin(φk)⏟QksQAM(t)=Ac∑k=−∞∞akp(t−kTs)cos(ωct+φk)=Ac∑k=−∞∞akcos(φk)⏟Ikp(t−kTs)cos(ωct)−Ac∑k=−∞∞aksin(φk)⏟Qkp(t−kTs)sin(ωct)sFSK(t)=Ac∑k=−∞∞p(t−kTs)cos(bk⋅ωct)=xASK1(t)+xASK2(t)sMSK(t)=Ac∑k=−∞∞aIkcos(π2Tbt)p(t−kTs)cos(ωct)−Ac∑k=−∞∞aQksin(π2Tbt)p(t−kTs)sin(ωct){\displaystyle {\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)}\\&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}}}}\\&s_{QAM}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{k}p\left(t-kT_{s}\right)\cos \left(\omega _{c}t+\varphi _{k}\right)}=\\&A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{k}\underbrace {\cos \left(\varphi _{k}\right)} _{I_{k}}p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)-A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{k}\underbrace {\sin \left(\varphi _{k}\right)} _{Q_{k}}p\left(t-kT_{s}\right)\sin \left(\omega _{c}t\right)}}\\&s_{FSK}(t)=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)\\&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)}\\\end{aligned}}}
Sabiendo que:
xm(t)=xI(t)cos(ωct)−xQ(t)sin(ωct)RxIxQ(τ)=RxQxI(τ)Rx(τ)=RI(τ)cos(ωcτ)2+RQ(τ)cos(ωcτ)2→Gx(f)=GI(f−fc)+GI(f+fc)4+GQ(f−fc)+GQ(f+fc)4{\displaystyle {\begin{aligned}&x_{m}(t)=x_{I}(t)\cos(\omega _{c}t)-x_{Q}(t)\sin(\omega _{c}t)\\&R_{x_{I}x_{Q}}(\tau )=R_{x_{Q}x_{I}}(\tau )\\&R_{x}(\tau )=R_{I}\left(\tau \right){\frac {\cos \left(\omega _{c}\tau \right)}{2}}+R_{Q}\left(\tau \right){\frac {\cos \left(\omega _{c}\tau \right)}{2}}\to \\&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}}\\\end{aligned}}}
Las señales digitales suelen visualizarse mediante su constelación:
xm(t)=xI(t)cos(ωct)−xQ(t)sin(ωct)x~m(t)=xI(t)+j⋅xQ(t){\displaystyle {\begin{aligned}&x_{m}(t)=x_{I}(t)\cos(\omega _{c}t)-x_{Q}(t)\sin(\omega _{c}t)\\&{\tilde {x}}_{m}(t)=x_{I}(t)+j\cdot x_{Q}(t)\\\end{aligned}}}
Se representa en un plano: