# Resumen y tablas

${\begin{aligned}&{\hat {x}}(t)=x(t)*h_{hilbert}(t)=\int _{-\infty }^{\infty }{\frac {x(\tau )}{\pi (t-\tau )}}\partial \tau {\text{ }}\to {\text{ }}{\hat {X}}(f)=X(f)\left(-j\operatorname {sign} (f)\right)\\&R_{yx}(\tau )=R_{xy}^{*}(\tau ){\text{ }};{\text{ }}G_{\hat {x}}(f)=G_{x}(f){\text{ ; }}R_{{\hat {x}}x}(\tau )=-R_{x{\hat {x}}}(\tau )\\&x_{+}(t)=x(t)+j{\hat {x}}(t)\\&x_{-}(t)=x(t)-j{\hat {x}}(t)\\&X_{+}(f)=\left\{{\begin{aligned}&2X(f),f>0\\&0,f<0\\\end{aligned}}\right.=X(f)\left(1+\operatorname {sign} (f)\right)\\&X_{-}(f)=\left\{{\begin{aligned}&0,f>0\\&2X(f),f<0\\\end{aligned}}\right.=X(f)\left(1-\operatorname {sign} (f)\right)\\&G_{x_{+}}(f)=\left\{{\begin{aligned}&4G_{x}(f),f>0\\&0,f<0\\\end{aligned}}\right.=2G_{x}(f)\left(1+\operatorname {sign} (f)\right)\\&G_{x_{-}}(f)=\left\{{\begin{aligned}&0,f>0\\&4G_{x}(f),f<0\\\end{aligned}}\right.=2G_{x}(f)\left(1-\operatorname {sign} (f)\right)\\&S_{x_{+}}=S_{x_{-}}=2S_{x}\\&R_{x_{+}x_{-}}(\tau )=0\\&{\tilde {x}}(t)=x_{+}(t)e^{-j\omega _{c}t}{\text{ ; }}{\tilde {X}}(f)=X_{+}(f+f_{c})\\&{\tilde {Y}}\left(f\right)={\frac {{\tilde {X}}\left(f\right)\cdot {\tilde {H}}\left(f\right)}{2}}\\&{\tilde {x}}(t)=x_{I}(t)+jx_{Q}(t)=e(t)e^{j\varphi (t)}\\&x(t):{\text{ Se }}\!\!{\tilde {\mathrm {n} }}\!\!{\text{ al paso-banda}}\\&{\tilde {x}}(t),x_{I}(t),x_{Q}(t),e(t){\text{ : Se }}\!\!{\tilde {\mathrm {n} }}\!\!{\text{ ales paso-bajo}}\\&x_{I}(t)={\text{Componente en fase de }}x(t){\text{ }}(I:{\text{ In phase)}}\\&x_{Q}(t)={\text{Componente en cuadratura de }}x(t){\text{ }}(Q:{\text{ Quadrature)}}\\&e(t)={\text{ Envolvente (el modulo) de }}x(t)\\&\varphi (t)={\text{ Fase de }}x(t){\text{ }}\\&x(t)=x_{I}(t)\cos(\omega _{c}t)-x_{Q}(t)\sin(\omega _{c}t)=e(t)\cos(\omega _{c}t+\varphi (t))\\&e(t)=\left|{\tilde {x}}(t)\right|={\sqrt {x_{I}^{2}(t)+x_{Q}^{2}(t)}}={\sqrt {x^{2}(t)+{\hat {x}}^{2}(t)}}\\&\varphi (t)=\arctan \left({\frac {x_{Q}(t)}{x_{I}(t)}}\right)\\&x_{I}(t)=e(t)\cos \left(\varphi (t)\right){\text{ ; }}x_{Q}(t)=e(t)\sin \left(\varphi (t)\right)\\&G_{x_{I}}(f)=^{1}\!\!\diagup \!\!_{4}\;\left(G_{x_{+}}(f+f_{c})+G_{x_{-}}(f-f_{c})\right)\\&S_{x}=S_{I}=S_{Q}\\\end{aligned}}$
| | $s(t)$ | $S(f)$ |
| AM | $A_{c}\left(1+mx(t)\right)\cos \left(\omega _{c}t\right)$ | ${\frac {A_{c}}{2}}\left(\delta \left(f-f_{c}\right)+\delta (f+f_{c})\right)+{\frac {A_{c}}{2}}m\left(X\left(f-f_{c}\right)+X(f+f_{c})\right)$ |
| DSB | $A_{c}x(t)\cos(\omega _{c}t)$ | ${\frac {A_{c}^{2}}{4}}\left(G_{x}(f-f_{c})+G_{x}(f+f_{c})\right)$ |
| SSB | ${\frac {A_{c}}{2}}\left(x(t)\cos(\omega _{c}t)-{\hat {x}}(t)\sin(\omega _{c}t)\right)$ | ${\frac {A_{c}}{4}}\left[X(f-f_{c})\left(1+\operatorname {sign} (f-f_{c})\right)\right]+\left[X(f+f_{c})\left(1-\operatorname {sign} (f+f_{c})\right)\right]$ |
| PM | ${\begin{aligned}&A_{c}\cos \left(\omega _{c}t+\varphi _{\Delta }x(t)\right)\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}^{tone}(t)=s(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cos \left(\omega _{c}t+k\omega _{m}t\right)}\\\end{aligned}}$ | $S^{tone}(f)={\frac {A_{c}}{2}}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\left[\delta \left(f-\left(f_{c}+kf_{m}\right)\right)+\delta \left(f+\left(f_{c}+kf_{m}\right)\right)\right]}$ |
| FM | ${\begin{aligned}&A_{c}\cos \left(\omega _{c}t+2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}^{tone}(t)=s(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cos \left(\omega _{c}t+k\omega _{m}t\right)}\\\end{aligned}}$ | $S^{tone}(f)={\frac {A_{c}}{2}}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\left[\delta \left(f-\left(f_{c}+kf_{m}\right)\right)+\delta \left(f+\left(f_{c}+kf_{m}\right)\right)\right]}$ |
| | $G_{s}(f)$ | $P_{s}$ | $\beta _{T}$ | $\left(^{S}\!\!\diagup \!\!_{N}\;\right)_{D}$ |
| AM | ${\frac {A_{c}^{2}}{4}}\left(\delta (f-f_{c})+\delta (f+f_{c})\right)+{\frac {A_{c}^{2}}{4}}\cdot m^{2}\left(G_{x}(f-f_{c})+G_{x}(f+f_{c})\right)$ | ${\frac {A_{c}^{2}}{2}}\left(1+m^{2}S_{x}\right)$ | $2W$ | ${\frac {m^{2}S_{x}}{1+m^{2}S_{x}}}\gamma$ |
| DSB | ${\frac {A_{c}^{2}}{4}}\left(G_{x}(f-f_{c})+G_{x}(f+f_{c})\right)$ | ${\frac {A_{c}^{2}}{2}}S_{x}$ | $2W$ | $\gamma$ |
| SSB | ${\frac {A_{c}^{2}}{8}}\left[G_{x}(f-f_{c})\left(1+\operatorname {sign} (f-f_{c})\right)\right]+\left[G_{x}(f+f_{c})\left(1-\operatorname {sign} (f+f_{c})\right)\right]$ | ${\frac {A_{c}^{2}}{4}}S_{x}$ | $W$ | $\gamma$ |
| PM | ${\frac {A_{c}^{2}}{4}}\sum \limits _{k=-\infty }^{\infty }{\left|J_{k}\left(\beta \right)\right|^{2}\left[\delta \left(f-\left(f_{c}+kf_{m}\right)\right)+\delta \left(f+\left(f_{c}+kf_{m}\right)\right)\right]}$ | ${\frac {A_{c}^{2}}{2}}$ | ${\begin{aligned}&2\left(\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}+1\right)W=\\&2\left(\varphi _{\Delta }A_{m}+1\right)W\\\end{aligned}}$ | $\varphi _{\Delta }^{2}S_{x}\gamma$ |
| FM | ${\frac {A_{c}^{2}}{4}}\sum \limits _{k=-\infty }^{\infty }{\left|J_{k}\left(\beta \right)\right|^{2}\left[\delta \left(f-\left(f_{c}+kf_{m}\right)\right)+\delta \left(f+\left(f_{c}+kf_{m}\right)\right)\right]}$ | ${\frac {A_{c}^{2}}{2}}$ | ${\begin{aligned}&2\left(\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}+1\right)W=\\&2\left({\frac {A_{m}f_{\Delta }}{f_{m}}}+1\right)W\\\end{aligned}}$ | $3\left({\frac {f_{\Delta }}{W}}\right)^{2}S_{x}\gamma$ |