x ^ ( t ) = x ( t ) ∗ h h i l b e r t ( t ) = ∫ − ∞ ∞ x ( τ ) π ( t − τ ) ∂ τ → X ^ ( f ) = X ( f ) ( − j sign ( f ) ) R y x ( τ ) = R x y ∗ ( τ ) ; G x ^ ( f ) = G x ( f ) ; R x ^ x ( τ ) = − R x x ^ ( τ ) x + ( t ) = x ( t ) + j x ^ ( t ) x − ( t ) = x ( t ) − j x ^ ( t ) X + ( f ) = { 2 X ( f ) , f > 0 0 , f < 0 = X ( f ) ( 1 + sign ( f ) ) X − ( f ) = { 0 , f > 0 2 X ( f ) , f < 0 = X ( f ) ( 1 − sign ( f ) ) G x + ( f ) = { 4 G x ( f ) , f > 0 0 , f < 0 = 2 G x ( f ) ( 1 + sign ( f ) ) G x − ( f ) = { 0 , f > 0 4 G x ( f ) , f < 0 = 2 G x ( f ) ( 1 − sign ( f ) ) S x + = S x − = 2 S x R x + x − ( τ ) = 0 x ~ ( t ) = x + ( t ) e − j ω c t ; X ~ ( f ) = X + ( f + f c ) Y ~ ( f ) = X ~ ( f ) ⋅ H ~ ( f ) 2 x ~ ( t ) = x I ( t ) + j x Q ( t ) = e ( t ) e j φ ( t ) x ( t ) : Se n ~ al paso-banda x ~ ( t ) , x I ( t ) , x Q ( t ) , e ( t ) : Se n ~ ales paso-bajo x I ( t ) = Componente en fase de x ( t ) ( I : In phase) x Q ( t ) = Componente en cuadratura de x ( t ) ( Q : Quadrature) e ( t ) = Envolvente (el modulo) de x ( t ) φ ( t ) = Fase de x ( t ) x ( t ) = x I ( t ) cos ( ω c t ) − x Q ( t ) sin ( ω c t ) = e ( t ) cos ( ω c t + φ ( t ) ) e ( t ) = | x ~ ( t ) | = x I 2 ( t ) + x Q 2 ( t ) = x 2 ( t ) + x ^ 2 ( t ) φ ( t ) = arctan ( x Q ( t ) x I ( t ) ) x I ( t ) = e ( t ) cos ( φ ( t ) ) ; x Q ( t ) = e ( t ) sin ( φ ( t ) ) G x I ( f ) = 1 ╱ 4 ( G x + ( f + f c ) + G x − ( f − f c ) ) S x = S I = S Q {\displaystyle {\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}}}
A c ( 1 + m x ( t ) ) cos ( ω c t ) {\displaystyle A_{c}\left(1+mx(t)\right)\cos \left(\omega _{c}t\right)}
A c 2 ( δ ( f − f c ) + δ ( f + f c ) ) + A c 2 m ( X ( f − f c ) + X ( f + f c ) ) {\displaystyle {\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)}
A c x ( t ) cos ( ω c t ) {\displaystyle A_{c}x(t)\cos(\omega _{c}t)}
A c 2 4 ( G x ( f − f c ) + G x ( f + f c ) ) {\displaystyle {\frac {A_{c}^{2}}{4}}\left(G_{x}(f-f_{c})+G_{x}(f+f_{c})\right)}
A c 2 ( x ( t ) cos ( ω c t ) − x ^ ( t ) sin ( ω c t ) ) {\displaystyle {\frac {A_{c}}{2}}\left(x(t)\cos(\omega _{c}t)-{\hat {x}}(t)\sin(\omega _{c}t)\right)}
A c 4 [ X ( f − f c ) ( 1 + sign ( f − f c ) ) ] + [ X ( f + f c ) ( 1 − sign ( f + f c ) ) ] {\displaystyle {\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]}
A c cos ( ω c t + φ Δ x ( t ) ) x P M F M t o n e ( t ) = s ( t ) = A c ∑ k = − ∞ ∞ J k ( β ) cos ( ω c t + k ω m t ) {\displaystyle {\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 t o n e ( f ) = A c 2 ∑ k = − ∞ ∞ J k ( β ) [ δ ( f − ( f c + k f m ) ) + δ ( f + ( f c + k f m ) ) ] {\displaystyle 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]}}
A c cos ( ω c t + 2 π f Δ ∫ − ∞ t x ( λ ) ∂ λ ) x P M F M t o n e ( t ) = s ( t ) = A c ∑ k = − ∞ ∞ J k ( β ) cos ( ω c t + k ω m t ) {\displaystyle {\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}}}
A c 2 4 ( δ ( f − f c ) + δ ( f + f c ) ) + A c 2 4 ⋅ m 2 ( G x ( f − f c ) + G x ( f + f c ) ) {\displaystyle {\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)}
A c 2 2 ( 1 + m 2 S x ) {\displaystyle {\frac {A_{c}^{2}}{2}}\left(1+m^{2}S_{x}\right)}
2 W {\displaystyle 2W}
m 2 S x 1 + m 2 S x γ {\displaystyle {\frac {m^{2}S_{x}}{1+m^{2}S_{x}}}\gamma }
A c 2 2 S x {\displaystyle {\frac {A_{c}^{2}}{2}}S_{x}}
γ {\displaystyle \gamma }
A c 2 8 [ G x ( f − f c ) ( 1 + sign ( f − f c ) ) ] + [ G x ( f + f c ) ( 1 − sign ( f + f c ) ) ] {\displaystyle {\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]}
A c 2 4 S x {\displaystyle {\frac {A_{c}^{2}}{4}}S_{x}}
W {\displaystyle W}
A c 2 4 ∑ k = − ∞ ∞ | J k ( β ) | 2 [ δ ( f − ( f c + k f m ) ) + δ ( f + ( f c + k f m ) ) ] {\displaystyle {\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]}}
A c 2 2 {\displaystyle {\frac {A_{c}^{2}}{2}}}
2 ( β P M F M + 1 ) W = 2 ( φ Δ A m + 1 ) W {\displaystyle {\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}}}
φ Δ 2 S x γ {\displaystyle \varphi _{\Delta }^{2}S_{x}\gamma }
2 ( β P M F M + 1 ) W = 2 ( A m f Δ f m + 1 ) W {\displaystyle {\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 ( f Δ W ) 2 S x γ {\displaystyle 3\left({\frac {f_{\Delta }}{W}}\right)^{2}S_{x}\gamma }
![{\displaystyle {\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1759febec39a47ae621561d03c5655978d5419)
![{\displaystyle 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]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e518993f383998a8a8699f5e8479fac9d8ac6469)
![{\displaystyle {\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8ba27636b0101a17efd3f89bd7dda097f43e34)
![{\displaystyle {\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]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71243ba0a4fb616cc6406f73b9fb1dcb99695a86)
